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Cancellation of arbitrary tone interference for all-digital high definition television transmitted over… Marsland, Ian D. 1994

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Cancellation of Arbitrary Tone Interferencefor All-Digital High Definition TelevisionTransmitted Over Coaxial Cable NetworksbyIan D. MarsiandB.Sc.Eng. (Honours), Queen’s University, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEInTHE FACULTY OF GRADUATE STUDIES(Department of Electrical Engineering)We accept this thesis as conformingto the requi ed standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© Ian D. Marsland, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.________________Department of iec4ccI Er;eerThe University of British ColumbiaVancouver, CanadaDate 2. rftIL L19DE-6 (2188)AbstractIn this thesis a novel canceller of a completely unknown tone which is interfering with adigital quadrature amplitude modulation (QAM) signal operating in an additive white Gaussiannoise (AWGN) environment is proposed, analysed and evaluated. This canceller can beapplied to protect all-digital high definition television (HDTV) signals from tone interference,which arises from intermodulation products, a common source of distortion in cable televisionnetworks.Expressions for the optimal weights for the linear minimum mean-square enor (MMSE)filter, consisting of L delay elements, for cancelling the tone interference are derived, underthe condition that the tone’s frequency and power are known to the canceller. It is shownthat the MMSE is directly proportional to the combined power of the QAM signal and theGaussian noise, and inversely proportional to L. Furthermore, as the characteristics of the toneare assumed to be completely unknown, novel fast Fourier transform (FFT) based methodsfor estimating the frequency and power of the tone are proposed and analysed. By using theseestimates in place of the true values for the optimal weights, a suboptimal filter is derived.Performance evaluation results have shown that the performance of the suboptimal cancelleris, for all practical purposes, identical to the optimal one.To improve the performance further, without increasing the number of the filter’s delayelements, a decision feedback mechanism is employed to reduce the power of the data signal.Through a combination of analytical and computer simulated performance evaluation it isfound that for all practical purposes the proposed decision feedback tone canceller removesthe tone interference completely.11Table of ContentsAbstract , . ,. iiTable of Contents , iiiList of Figures vNotation. viiAcknowledgments ixChapter 2:2.12.22.32.42.4.12.4.22.4.32.52.62.6.12.6.22.6.32.6.42.7Chapter 3:3.13.23.2,13.2.23.2,3Introduction . . 1Interference Encountered in Community Antenna Television (CATV)SystemsA Brief Review of High Definition Television (HDTV)Tone Cancellation Techniques 9Research Contributions of Thesis.Organization of ThesisIntroductionSource Coding . .Channel Coding.....ModulationSymbol Encoder .Transmit Filter . ,ModulationCommunication ChannelDemodulationDemodulationReceive FilterSignal Power Spectral DensitySymbol DetectionConclusionA Novel Tone InterferenceIntroductionTone Interference EstimationLinear MMSE Estimator, . .Sub-Optimal Linear Estimator. . . . 14141621222325• • . . 2828303033• . . . 374047484849515661Chapter 1:1.11.21.31.41.5.2.7System Model Description and1112AnalysisCancellerA Decision Feedback Tone Canceller .,.,.,... • . •••••.1113.3 Estimation of Frequency and Power Ratio 663.3.1 PSD Estimation , 673.3.2 Average Periodogram 763.3.3 Symbol Rate Sampling 783.3.4 Frequency Estimation 793.3.5 Power Ratio Estimation 843.3.6 Decision Feedback for Frequency and Power Ratio Estimation . . 863.4 Summary and Conclusion 87Chapter 4: Computer Simulation Results . 924.1 Introduction 924,2 Computer Simulation System 934.3 Cancellation Without Decision Feedback 934.4 Cancellation with Decision Feedback . . . . . 97Chapter 5: Conclusions and Future Research Topics 102References . 105Appendix A: MMSE Tone Estimator 107Appendix B: Derivation of Mean and Variance of AveragePeriodogram with One Sample per Symbol 112B.l Signal Samples . ....... . ....... . ....... 112B.2 Finite Fourier Transform 116B.3 The Periodogram . . . . . . 120B.4 The Average Periodogram . . . . 123Appendix C: Power Ratio Estimator . 125Appendix D: Source Code Listings . 132D.l Positions of CTB . . 132D.2 Probability of Symbol Error . . 135D.3 Tone Canceller Simulator . . . . . 137ivList of FiguresFigure 1.Figure 2.Figure 3.Figure 4.Figure 5.Figure 6.Figure 7.Figure 8.Figure 9.Figure 10.Figure 11.Figure 12.Figure 13.Figure 14.FigureFigureFigureFigureFigureFigureFigureFigureFigureFigureFigure15.16.17.18.19.20.21.22.23.24,25.Figure 26.Figure 27.Figure 28.Figure 29.FigureFigureFigureFigureFigure30.31.32.33.34.Figure 35.5101115171923242630=35dB.... 44=35dB.... 44=10dB.... 45=10dB.... 45Simplified Spectrum of Broadband CATV Signal.Adaptive Noise Canceller. ...,...........Adaptive FilterAll-digital HDTV Transmission SystemTypical Digital Video Source EncoderTypical Digital Video Source Decoder. .....Typical QAM Transmitter16—QAM Signal ConstellationFrequency Response of Root-Raised Cosine Filter.Typical QAM Receiver ..16—QAM State-space Diagram. SNR = 35 dB, SIR16—QAM State-space Diagram. SNR = 10 dB, SIR16—QAM State-space Diagram. SNR = 35 dB, SIR16—QAM State-space Diagram. SNR = 10 dB, SIRProbability of Symbol Detection Error 46SRRmax vs. SIR, L = 32 55Gmax VS. SIR, L = 32 56MSE vs. L. SNR = 15 dB, SIR = 10 dB, C = C, various . . . . 59MSE vs. L. SNR = 15 dB, SIR = 10 dB, C = C, various e. . . . 60MSE vs. L. SNR = 15 dB, SIR = 10 dB, C = C, = 0.0005. . . 61MSE vs. L. SNR = 15 dB, SIR = 10 dB, 0.0001, various C. . 62Number of samples vs. SRR, SNR = 15 dB, SIR = 5 dB. . . . 63Gax vs. SIR, L = 32, with decision feedback. . . . . . . . . . 65PSD of Received Signal. SNR = 15 dB, SIR = 15 dB, a = 0.2 67Periodogram of Received Signal. N = 16384, N = 2, SNR = 15dB, SIR = 15 dB..... . 74Periodogram of Received Signal. N = 2048, N = 2, SNR = 15 dB,SIR=l5dB 75Average Periodogram (AP) of Received Signal. D 8, N = 2048,N 2, SNR = 15 dB, SIR = 15 dB 78Plot of Zo(k) vs. k, for N = 2048 and fT8 = 0.05 81Plot of Zo(k) vs. k, for N = 2048 and fT8 = 0.05. For pointsaround fNT3 only 82Plot of GM(S) vs. Sj, for N = 2048 and for various values of M 85Tone Interference Canceller 88Interference Estimator 88Reduced Complexity Realization of Interference Estimator . . . . 90Theoretical Performance Gain vs. SIR, L = 32, N = 2048, D = 32,without decision feedback . . . . . , . . ..... . . , . . . , 94Simulated Performance Gain vs. SIR, L = 32, N = 2048, D = 32,without decision feedback . . •............... . 95VFigure 36. Simulated SRR vs. SIR, L = 32, N = 2048, D = 32, withoutdecision feedback 96Figure 37. Theoretical Performance Gain vs. SIR, L = 32, N = 2048, D = 32,with decision feedback 98Figure 38. Simulated Performance Gain vs. SIR, L = 32, N = 2048, D = 32,with decision feedback, . , , 99Figure 39. Theoretical Performance Gain vs. SIR, L = 1000, N = 2048, D =32 , . . 100Figure 40. Simulated Performance Gain vs. SIR, L = 1000, N = 2048, D =32. , . . . , . , , 100Figure 41. Simulated Performance Gain vs. SIR, L = 32, N = 2048, D = 128. . 101Figure 42. Simulated Performance Gain vs. SIR, L = 32, N = 32, D = 2048. . 101viNotationData Signalv(t) ÷-÷ Transmitted bandpass signalv(t) ÷—* Data component of received lowpass signal‘a Transmitted symbols—* Variance of transmitted symbolsAdditive White Gaussian Noisen(t) —* Received bandpass noisen(t) —* Noise component of received lowpass signalN0 <— Single-sided noise PSDTone Interferencez(t) —* Bandpass tonez(t) —* Tone component of received lowpass signalfi —* Bandpass frequency of toneBaseband frequency of toneMagnitude of bandpass tone—* Magnitude of baseband toneInitial phase of toneModulationT3 -* Symbol durationCarrier frequencyhT(t) —* Impulse response of transmit filterHT(f) -+ Frequency response of transmit filter—* Impulse response of receive filterHR(f) Frequency response of receive filterh(t) -* Combined impulse response of Tx and Rx filtersH(f) Combined frequency response of Tx and Rx filtersviiReceived Signalr(t) Received bandpass signalr(t) —* Received lowpass signalRa —* Sample of received lowpass signalNa —* Noise component of received sampleZa —* Tone component of received sampleTa Detected SymbolTone Interference CancellationL ÷—* Number of samples on which tone estimates are based—* Normalized tone frequency (in radians)C Power ratioN —* Periodogram block lengthD —* Number of blocks in average periodogramXd(k) —* Periodogram of block dXD(k) —* Average periodogram over D blocksZo(k) —* Shape of tone in periodogram—* Displacement of actual frequency from tone spikeGM (6) Fraction of tone power in tone spikeUi —* Estimate of signal plus noise power02 f—* Estimate of tone powervuAcknowledgmentsI would like to express my sincere gratitude to my supervisor, Dr. P. Takis Mathiopoulos,for all the support and encouragement he provided during the course of my M.A.Sc. thesisresearch. His excellent suggestions and continual optimism were instrumental in transforminga stack of equations into a coherent document.For providing financial support, I would like to thank the Science Council of B.C. andRoger’s Cable. Funding for this thesis was also provided through NSERC operating grant44312 and a B.C. Advanced Systems Institute (ASI) Fellowship.Thanks are also due to Dr. John Madden of the Canadian Cable Lab Fund and Dr.Alexander Futro of Cable Television Laboratories, Inc. for supporting and arranging my tripto Boulder, CO, to visit CableLabs facilities. I would also like to thank Tom Williams ofCableLabs for providing technical advice on CTB.I wish to thank my parents for convincing me that graduate studies are a good idea.Once again, they were right. Furthermore, I am deeply indebted to them for encouragingme to think independently, beginning in my early childhood, I would also like to thank myfather and my brother for taking the time to proofread this thesis and providing much-neededconstructive criticism.Finally, I wish to thank Kelly, my wife and dearest friend, who’s support and remarkablepatience made this entire endeavor possible.ixChapter OneIntroductionWith the recent advancements in all-digital high definition television (HDTV) [1—7], thecable industry must prepare itself for the distribution of digital signals over existing cablenetworks. In such cable networks, it is well known that several forms of interference exist.It is therefore important for reliable transmission that the effects of these distortions on thetransmission of all-digital HDTV signals be understood. Perhaps even more importantly,whenever appropriate, these distortions must be compensated for, or even eliminated.In this thesis distortion in the form of an additive deterministic but completely unknowninterfering tone is considered, along with additive white Gaussian noise (AWGN). In thefirst two sections of this introductory chapter the various distortions encountered in cable TVsystems are discussed, and recent developments of HDTV systems are reviewed. In Section1.3 a brief description of known techniques for tone interference cancellation is presented.In Section 1,4 the research contributions of the thesis are stated. Finally, in Section 1.5 theorganization of the thesis is presented.11.1 Interference Encountered in CommunityAntenna Television (CATV) SystemsSince the beginning of regular television broadcasting in the 1930’s, the popularity oftelevision as a communication medium has grown enormously [8]. In the 1950’s, communityantenna television (often referred to as cable TV) systems were first installed to allow viewersin remote areas access to television signals broadcast from distant cities [9]. In these earlysystems, war surplus coaxial cables were used to carry TV signals to viewer’s homes froma head end, usually an antenna situated on a nearby hill. Broadband amplifiers, constructedfrom vacuum tubes, were used to maintain signal power. Over the years, the popularity ofCATV systems grew to the point where most viewers in North America are connected to acable network [10]. This growth stemmed primarily from the wide range of programmingthat cable TV offers, and the superior image quality available. As a result of the increasedpopularity, the individual networks have expanded from serving only a few customers in asmall town to delivering programming to hundreds of thousands of viewers in large urbanareas. Accompanying this growth, many technological improvements to the cable systemshave been made. Instead of consisting of a single antenna, modem head ends combine signalsfrom a variety of sources, including local studios, satellites, and terrestrial broadcasts, andtransmit them together over the cable network.One important advantage cable TV has over terrestrial broadcasting is in signal quality.When signals are transmitted over the air, serious distortion may be introduced. If the receiveris a long way from the transmitter, or the transmission path is blocked by a large object suchas a hill or building, the signal strength is weakened, so external interference will appear likea snow storm on the screen. If the transmitted signal is reflected off surrounding objects, thesignal may be received more than once, at slightly different times. This appears as a “ghosting”effect, and is known as multipath reception. Depending on atmospheric conditions, co-channel2interference can occur if a signal from a distant broadcaster, using the same channel as thedesired signal, manages to reach the receiver [9].Because of the shielding provided by the coaxial cable, CATV networks are less susceptible to these types of distortion. In general, CATV provides a much cleaner transmissionenvironment than terrestrial broadcasting. Nonetheless, it is still far from perfect. Variousforms of other types of interference may be introduced from external sources through cracksin the cable or improperly fitted connectors, and transmission line reflections may arise inimproperly terminated cables [9]. The most serious problem with coaxial cable, however, isits relatively high signal attenuation [101. To balance the loss of signal power, broadbandamplifiers must be inserted into the cable network, generally at intervals of about 2000 feet,depending on cable thickness. The required use of amplifiers is the primary limitation onCATV system performance [lii.Thermal noise generated by components in the amplifiers reduce the signal carrier-to-noise (C/N) ratio. When the signal is passed through a cascade of amplifiers, the noise iscompounded. Doubling the number of amplifiers leads to a 3 dB degradation of the C/N ratio[11]. If the C/N ratio falls too low (below about 43 dB [121), “snow” begins to appear onthe picture tube. This impairment is common at receivers a long way from the head end, asthe signal must pass through a long cascade of amplifiers [11].In addition to introducing thermal noise, the amplifiers also distort the waveform of thetransmitted signal. Although amplifiers are intended to provide linear gain with respect toinput amplitude, in practice the response is non-linear. To study the non-linear distortion, itis convenient to model the relationship between the input voltage of the amplifier (T411) withits output (V011) by the non-linear equation(1.1)3where A1, A2, A3,A4,... are constants [9, 11, 13, 14]. The first term in the above equationrepresents the linear gain of the amplifier, the second term reflects the second-order distortion,the third term represents the third-order distortion, and so on. With the use of balanced push-pull amplifiers, non-linear distortion of even-order has been effectively eliminated in modemCATV networks [15]. In addition, most research into the effects of non-linear distortion inCATV networks ignores the higher-order distortion [9, 11, 13, 14]. High order distortion isnot considered in this thesis, either. As a result, the response of the non-linear amplifierscan be modelled byV011 = A14 + A3V. (1.2)In practical CATV systems, many video signals are multiplexed together by modulatingcarriers at different frequencies. As the broadband signal is passed through a non-linear device,the modulated carriers are mixed together in a variety of ways, resulting in the creation ofnumerous spurious signals. To illustrate the impact of the undesired signals, it is useful toconsider the simplified model of the broadband signal represented in Fig. 1, where each 6MHz channel consists of only the carrier and video signal. Suppose a total of N differentvideo signals, denoted by .s(t) for all n E [1, NJ, modulate the amplitude of N differentcarriers at frequencies given by f, for n E [1, NJ. The combined broadband signal, v(t),can be expressed asv(t)=Z s(t) cos (2ft). (1.3)The spectrum of the broadband signal is shown in Fig. 1.The third-order non-linear distortion is related to the cube of the broadband signal, whichis, using this model,Nv3(t)=s(t) cos (2fflt)]446MHzChannel 1 Channel 2 Channel 3 Channel 4 Channel NFrequencyFigure 1. Simplified Spectrum of Broadband CATV Signal.NNN= Sa(t)Sb(t)Sc(t) COS (2Kfat) COS (2lrfbt) cos (2irft)a=1 b=1 c=1NNN= Sa(t)Sb(1)Sc(t) cos [2(fa + fb + f)t1. (1,4)a=1 b=1 c=1Third-order distortion products arise in relationship to the set of frequencies{fa + fb + f V a, b, c E [1, N]}. The different ways that the frequencies can be combined produce different forms of interference. This point is best illustrated by rearranging thesummations in Eq. (1.4) to group together the terms where the indices are the same. Aftersome straightforward but cumbersome algebraic manipulation, Eq. (1.4) can be rewritten asv3(t) = (t) COS (2K fat) + s(t)S(t) COS (2Kfat)a=1 a=1 &1ba+ s(t) COS [2K(3fa)tl + s(t)s(t) COS [2K(2fa ± fb)t]a=1 a=1 ‘N N+ > > > Sa(t).Sb(t)Sc(t) cos [2t(fa + fb + f)t] . (1.5)a=1 b1 c114a caThe presence of the first term in Eq. (1.5) implies that each carrier will carry the modulationof the cube of desired signal in addition to the desired signal. This effect is known as self-modulation [14]. From the second term it is clear that each carrier also carries the modulationof the other signals, resulting in an effect known as cross-modulation. Visually, this results5in other signals appearing in the background of the desired one. The remaining three termsgenerate products that are spread throughout the frequency spectrum. They are significantbecause there are so many of them. In fact, there are N third-order harmonics, 2N(N — 1)beats of the form 2fa + fb, and N(N — l)(N — 2) distinct beats of the form fa + fb + f.For a typical fully loaded 35 channel cable system, a total of 28,595 third-order (or thple)beats occur. Since NTSC systems use a large carrier amplitude modulation (LC-AM) system[16], the carriers are the dominant part of the transmitted signals. As such, each of theintermodulation products discussed above can be considered to be a single sinusoidal tone[9, 11],In summary, only these two types of interference are considered in this thesis. Becauseof the random nature and relatively white spectrum associated with thennal noise, it canaccurately be modelled as an additive white Gaussian random process. For the purposes ofthis thesis, only a single intermodulation product is assumed. It is modelled as a single tonethat is characterized by its magnitude, its bandpass frequency, fj, and its phase, j. It isassumed that none of these parameters vary with time, in that they each maintain a constantalbeit completely unknown value. As such, this distortion is not random, but deterministic,and mathematically it can be represented asz(t) Kjcos(2irft + qj), (1.6)with 0 < I < cc, fcD > 0, and 0 qj < 2ir. This model provides a good base fromwhich analysis of the effect of the distortion can begin. Further investigation can be carriedout on a more elaborate model, such as multiple tones and nonstationary tones for example,although this will not be reported in this thesis.61.2 A Brief Review of High Definition Television (HDTV)In North America, colour television signals are transmitted using a system recommendedby the National Television Systems Committee (NTSC) of the American Federal Communications Commission (FCC) in 1953 [16]. Although the NTSC system was very innovativeat that time, it possesses some limitations. It does not provide enough spatial resolution foradequate display on the large screens that are presently available, and it displays images toonarrow for the wide aspect ratio of many motion pictures. Another drawback of current television standards is that there is no single worldwide standard. In fact, according to the CCIR(Comité Consultatif International Radio), an international body which attempts to coordinatebroadcasting standards, there are eleven different television standards in common use [8]. Inthe “global village” of the 1990’s, this barrier to international communication is unacceptable.In 1987, the FCC decided to consider the development of a new standard for terrestrialtelevision broadcasting. It established the FCC Advisory Committee on Advanced TelevisionService to investigate issues regarding the new standard [1]. Referred to as advanced television(ATV), or high definition television (HDTV), this standard was expected to incorporate greaterspatial resolution and a wider aspect ratio. Furthermore, it was hoped that it could be acceptedinternationally [2]. The Advisory Committee began studying twenty-three proposed systemsin its quest for a practical HDTV standard. Some proposals were deemed infeasible and otherwere discounted as they possessed serious disadvantages. In 1990 it was decided that onlyfive warranted further investigation. It is interesting to note that at the time of this decision,only one of these five proposals was all-digital.Beginning in 1991, the Advisory Committee supervised a number of tests of the remaining systems to determine their relative merits [3]. The Advanced Television Test Centre(ATTC) performed laboratory and field testing of the systems from a terrestrial broadcasting7perspective. At Cable Television Laboratories, Inc. (CableLabs), similar cable television related testing was performed. The Advanced Television Evaluation Laboratory (ATEL) wasestablished in Ottawa to provide subjective evaluation of the image quality of the proposals[3]. In February 1993, the Advisory Committed published its ATV system recommendations[3], Although no standard was in sight, significant decisions had been made. Perhaps themost import decisions were that HDTV signals will be allocated channels no wider that the 6MHz ones used by NTSC-compatible broadcasters, and digital compression and modulationschemes will be employed exclusively. Four of the five proposals were modified to incorporate a digital design, and the fifth dropped out of consideration. The proponents of all-digitalHDTV transmission are currently implementing modifications to their systems to rectify someof the minor problems identified during the testing procedure. The remaining systems currently being considered for HDTV transmission are: 1) the DigiCipher system proposed bythe American Television Alliance (ATVA), a collaboration of General Instrument Corp. andMassachusetts Institute of Technology; 2) the Digital Spectrum-Compatible HDTV (DSCHDTV) system proposed by Zenith Electronics Corp. and AT&T Bell Laboratories; 3) theAdvanced Digital Television (ADTV) system proposed by the Advanced Television ResearchConsortium (ATRC), consisting of Thomson Consumer Electronics, Inc., Phillips ConsumerElectronics Co., NBC, and the David Sarnoff Research Centre; and 4) the ATVA-ProgressiveSystem proposed by the American Television Alliance. All these systems are all-digital, andin general it appears that they have more similarities than differences. Although a detaileddescription of each of the systems is beyond the scope of this thesis, their main similaritieswill be described in Chapter 2.To conclude this very brief review of HDTV, it is worth mentioning that research intoall-digital HDTV transmission has resulted in some major technological breakthroughs. Forexample, advanced image compression algorithms have successfully been developed, allowing8the enormous quantity of information associated with the high resolution, wide aspect ratioHDTV signals to be reduced sufficiently to allow transmission in 6 MHz channels [3]. Dueto this success, other applications of digital television are being considered. Telephonecompanies are very much interested in using the new image compression technology to providehigh quality video telephone communication. Also, cable television networks plan to applydigital compression to NTSC signals to reduce bandwidth, greatly increasing the number ofchannels available to cable TV subscribers. Since the cable industry will be transmittingboth HDTV and compressed digital NTS C signals, it will certainly be a major user of digitaltelevision technology.1.3 Tone Cancellation TechniquesAs will be shown in Section 2.6 of this thesis, a single tone of sufficient magnitude canimpede the ability of CATV networks to distribute all-digital HDTV signals, to the pointwhere video transmission almost completely fails. To maintain acceptable image quality,some means of removing the tone is required. The issue of removing sinusoidal interferenceis a classical communication problem. A wide variety of solutions have been proposed, eachpossessing their own advantages and limitations. In general, these solutions fall into threebroad categories: I) fixed notch filters, iii) transform domain processing, and iii) adaptivenoise cancellation.Early attempts at tone removal employed fixed notch filters. If the frequency of thetone is known, these filters perform adequately. However, practical implementations ofthese filters have nulls with non-zero bandwidth, leading to some distortion of the desiredsignal. Furthermore, since the frequency of the tone must be known a priori, these filters areunacceptable for removing tones arising from CTB.9Transform domain processing techniques employ discrete Fourier transforms (DFTs) toconvert the signal to the transform domain, where the signal is processed. One technique usesconditional median filtering to reduce the power of the tone without significantly altering theDFT coefficients of the desired signal [171. Another applies a set of non-uniform weights tothe coefficients in an attempt to maximize the signal-to-noise ratio [18]. After processing,the signal is transformed back to the time domain with an inverse DFT. In general, thesetechniques work well only for sufficiently strong tones.The most robust and popular techniques for removing tone interference involve the useof adaptive noise cancellation, as described by Widrow et at. [19]. Fig. 2 shows a typicaladaptive noise canceller. The adaptive filter processes the delayed input to produce an estimateof the interfering tone. This estimate is subtracted from the primary input to effectively cancelthe tone. As shown in Fig. 3, the adaptive filter is implemented with a tapped delay line.The adaptive algorithm heuristically adjusts the weights in an effort to minimize the meansquare error between the actual tone and the filter output. Numerous different techniquesfor determining the weights have been proposed (see, for example, [19—21]), all providingdifferent trade-offs between simplicity and accuracy. In general, adaptive noise cancellationtechniques are the most versatile and efficient methods for removing tones, even when thetone is relatively weak. Furthermore, no a priori knowledge of the tone is required.Figure 2, Adaptive Noise Canceller.CorrectedSignalSignalInput10+ + FilterOutputErrorFigure 3. Adaptive Filter,1.4 Research Contributions of ThesisIn this thesis, a novel technique for removing a completely unknown tone which isinterfering with a digital QAM signal is proposed, analysed and evaluated in an AWGNenvironment. The optimal weights for the linear minimum mean-square error (MMSE)filter for cancelling the tone interference are determined, under the condition that the tone’sfrequency and power are known to the canceller. Novel methods for estimating the frequencyand power of the tone are developed, which are based on the finite-length discrete Fouriertransform (FFT). Using these estimates in place of the actual frequency and power yieldsa suboptimal filter that performs practically identically to the optimal one under normaldistortion conditions. In addition, a decision feedback mechanism is employed to furtherimprove the performance of the tone interference canceller by reducing the effect the datasignal has on the cancellation process. Through a combination of analytical and computersimulated performance evaluation it is found that for all practical purposes the proposeddecision feedback tone canceller removes the tone interference completely.Input111.5 Organization of ThesisIncluding this introductory chapter, this thesis consists of five chapters and four appendices, and is organized as follows.Chapter 2 contains an overview of all-digital HDTV transmission, with analysis of errorsarising from AWGN and tone interference, In Section 2.2 the HDTV compression algorithmsare described in general terms, and the implications of transmission errors on the decodingprocess are discussed. Section 2.3 contains a descriptions of the error protection that isprovided in the form of channel coding. Quadrature amplitude modulation (QAM) is discussedin Section 2.4, with emphasis on the symbol encoder, transmit filter, and carrier modulation.Mathematical models for the AWGN and tone interference are given in Section 2.5. In Section2.6 the demodulation process is described and the propagation of distortion from the channelthrough the demodulator and receive filter to the symbol detector is analysed. The effect ofthe distortion on the probability of transmission error is studied.In Chapter 3 a novel tone interference canceller is proposed. A method of estimatingtone interference is present in Section 3.2 that is based on the linear minimum mean-squareerror (MMSE) family of estimators. In Section 3.3 a method for estimating the frequencyand power of the tone as well as the Gaussian noise power is presented. These quantities arerequired by the proposed tone interference canceller. Section 3.4 contains a detailed summaryof the canceller, including block diagrams of potential implementations.The results of extensive testing of the proposed canceller by means of computer simulation is presented in Chapter 4. Performance is analysed over a wide range of distortionconditions and compared with theoretical bounds. Different implementations are comparedand contrasted to provide an understanding of the relevance of certain design parameters.Chapter 5 summarizes the contributions of this thesis and suggests future research topics12that could be carried out to extend the effective performance of the proposed tone interferencecanceller.Appendix A provides a proof that the filer weights for linear MMSE tone interferencecancellation are optimal. Appendices B and C contain derivations of the means and variancesof some of the other estimators used in Chapter 3 of this thesis. Finally, Appendix D provideslistings of all the computer source code written for the research of this thesis.13Chapter TwoSystem Model Descriptionand Analysis2,1 IntroductionThis chapter contains an overview of the components employed in all-digital HDTVtransmission. Since the FCC has yet to approve a standard for HDTV transmission, a genericmodel is presented here. Although this model differs slightly from the proposals, it embodiesall their primary ideas. To illustrate some specific points, emphasis will be given to theDigiCipher system, which was the first all-digital HDTV proposal. A simplified model foran all-digital HDTV transmission system that is satisfactory for the purpose of this thesis ispresented in Fig. 4.The four data sources supply digital data to be transmitted, and their formats differ slightlyamong the proposed systems, For the DigiCipher system, the digital video data is used totransmit frames consisting of 960 lines with 1408 pixels per line at a frame rate of 29.97 Hz14Digial4Audio-I-RECEIVERFigure 4. All-digital HDTV Transmission System [4].[3]. It is derived from sampling an analogue composite red-green-blue (RGB) signal whichhas 1050 lines per frame scanned in a 2:1 interleaved format. The two fields per frameare scanned with a field rate of 59.94 Hz, and have 16:9 width-to-height aspect ratio. TheDigiCipher digital audio signal is produced from four separate audio channels sampled at 48kHz with 16—bit precision in the A/D conversion. The ancillary/control data is reserved forpurposes such as the transmission of TeleText and subscriber access control [3]. Additionaldata may also be included for the purpose of system synchronization. To allow all this data15Digial IVideoDigialAudioSync.Digia1VideoAncillary/ q—Control DataSync.to be transmitted over a single channel band-limited to 6 MHz, source coding is used toimplement compression schemes in the source encoder, with corresponding decompression inthe source decoder. Channel coding is used to protect the compressed data from any errorsthat may arise during transmission. Also, bandwidth efficient modulation schemes are usedto represent the digital data in a format suitable for transmission over the communicationchannel, A HDTV receiver reverses the transmission process by demodulating the receivedsignal, correcting transmission errors, and decompressing the digital signals. It is expectedthat any transmission errors will be controlled successfully, allowing the reception of highquality video and audio, free from serious impairments.To study the performance of HDTV transmission, the processes of source coding, channelcoding, and modulation are discussed in this chapter. As this covers an extremely large field,only a very short description is presented. The organization of this chapter is as follows: inSection 2.2 an overview of the source coding process employed in compressing HDTV signalsis presented; in Section 2.3 the techniques used for channel coding to protect the transmittedHDTV signal from errors are discussed; in Section 2.4 a bandwidth efficient modulationscheme is presented that allows the HDTV signal to be transmitted over the coaxial cable TVnetwork; in Section 2.5 the modelling of the coaxial cable TV channel is discussed in detail,with a description of the distortion it introduces; in Section 2.6 the demodulation process isdiscussed, and the propagation of distortion arising in the communication channel throughthe demodulator is studied.2.2 Source CodingThe purpose of the source encoders is to reduce the number of bits required to representthe video and audio signals. This compression must be carried Out in a manner that allows16the source decoder to recreate reliable replicas of the original signals. This is accomplishedby two means: noiseless compression, which removes redundant information that can bereproduced exactly by the decoder, and noisy compression, which removes information thathas little visible impact on the human eye or audible impact on the human ear. Because ofthe different natures of video and audio signals, different source coding procedures are used.A typical video source encoder is shown in Fig. 5, Both temporal and spatial compressionare applied to provide superior compression rates. The individual components of the sourceencoder must be considered separately to gain an understanding of the coding process.The digital RGB video signal is first converted into its luminance and chrominancecomponents (the YUV signal space). Since the human eye is less sensitive to fine detail inchrominance than in luminance [22], the chrominance signals are subsampled horizontally bya factor of four, and alternate lines are discarded [5]. This is one source of image degradationintroduced in the compression process.Figure 5. Typical Digital Video Source Encoder [4].17Temporal compression is achieved through motion compensated predictive coding. Byanalysing the relative displacement of small segments of the image between the current frameand the previous frame, the motion estimator produces a set of motion vectors. The motioncompensated predictor applies the motion vectors to the previous frame to produce a predictionof the current frame. Since it is expected that the predicted frame will closely resemble thecurrent frame, the difference of these two images typically contains less information that theactual frame. The motion estimation is based only on the luminance component, and thesame motion vectors are applied to the chrominance components.Spatial compression is applied to the difference frame to further reduce the amount ofinformation to transmit. The difference frame is divided into blocks containing 8x8 pixelseach. A two-dimensional discrete cosine transform (DCT) is applied to each block, providingspatial frequency representations of the blocks. The adaptive quantizer converts the DCTcoefficients into binary data. Since the human eye is less sensitive to intensity variationsat high spatial frequencies [22], fewer bits are required to represent these coefficients thanthe low frequency ones. The quantization process is the second source of image degradationintroduced from compression. To achieve additional compression, redundancy is removedfrom the quantized DCT coefficients by run-length and Huffman entropy coding.Since the compression algorithms produce data at a variable rate depending on the amountof redundancy in the video signal, a buffer is used to store the data and emit it at a constantrate. If the buffer fills to near capacity, the adaptive quantizer is signalled to allocate fewer bitsto the DCT coefficients. This increases the compression rate at the expense of image quality.As the amount of data in the buffer decreases, the quantizer can return to allocating more bits.Fig. 6 shows a typical digital video source decoder. Essentially, it reverses the operation ofthe decoder. The encoded coefficients are decoded to produce the quantized DCT coefficients.These are identical to the ones produced by the adaptive quantizer in the transmitter. The18inverse quantizer provides the inverse function of the adaptive quantizer. Quantizationrounding errors cause a slight difference between the output of the inverse quantizer in thereceiver and the input of the adaptive quantizer in the transmitter. The inverse transformconverts the rounded DCT coefficients back into the spatial domain, creating the updateinformation. This information is the same as the transmitter’s difference frame, expect for theeffects of quantization rounding errors. The received motion vectors are applied to the storedprevious frame, producing the predicted frame. The update information is added, resultingin the received current frame, which differs from the transmitted frame only in quantizationerror, Finally, the frame is converted to the RGB domain for display on the receiver’s screen.Encoded j Entropy Inverse I Inverse UPDATE CURRENT DigitalCoefficients J Decoder Quantizer Transform INFORMATION -) FRAME Video+ RGBPREDICTEDFRAMEI MotionMotion I Framei CompensatedVectorsPredictorMemoryFigure 6. Typical Digital Video Source Decoder [41.To prevent the propagation of quantization error from frame to frame, some extra workmust be done by the transmitter. Since the previous frame used by the receiver differs fromthe true previous frame, any quantization errors it contains will be carried into the currentframe as well. To prevent this problem, the encoder does not use the true previous framefor motion estimation, but produces its own copy of the update information that the receiverwill use by applying inverse quantization and an inverse transform. This is added to theencoder’s predicted frame and stored for use as the previous frame for the basis of the nextframe transmitted. As a result, the transmitted motion vectors define a mapping from thereceiver’s previous frame to the current frame, and interframe quantization error propagationno longer occurs.19Although compression errors do not propagate from frame to frame, the loss of informationassociated with the chrominance sub-sampling and adaptive quantization does imply thatthe receiver is unable to exactly reproduce the original video signal. Nonetheless, theseimpairments should not significantly affect the image quality since they primarily affectinformation imperceptible to normal human eyesight [22]. Image distortion is greatestwhen there is little intraframe and interframe redundancy in the video signal, since coarserquantization must be employed to compensate for the additional information. This usuallyoccurs immediately following a scene change, and during scenes involving rapid motion.Although compression errors slightly distort the video signal, transmission errors canbe much more serious. However, the source coding process also involves some inherentprotection against these errors. Because of the compression algorithms employed, a single biterror can affect the video signal in a variety of ways. An error in the encoded coefficientswill affect the amplitude of only one DCT coefficient. This would cause a slight alterationof the all the amplitudes in one 8x8 pixel block. This effect will be more noticeable ifa low frequency DCT coefficient is in error [7]. Although having distortion in one blockmay not be too severe, this distortion also propagates to succeeding frames. A much moreserious problem occurs when the error causes the entropy decoder to lose synchronization,causing all the coefficients in the remainder of the block to be scrambled. Furthermore, thisloss of synchronization may be carried into subsequent blocks, destroying a large area of theimage. To help maintain synchronization, end-of-block markers are inserted into the encodedcoefficients. Errors in the motion vectors also have a serious effect on the received image.Not only do objects in the image appear distorted, they also appear in the wrong position.This distortion also propagates to succeeding frames.To limit the effects of error propagation, frames without motion compensation areperiodically transmitted. These frames are referred to as intra-frames, or I-frames. Although20the use of I-frames results in a significant reduction the image compression rate, it preventserrors from propagating indefinitely. The source coder also employs some error concealmenttechniques to reduce the visual impact transmission errors may incur.The source coding process described in this section is only a simplification of the actualprocess. A number of details, such as channel acquisition, have been omitted for the sakeof brevity. Source coding is also applied to the audio signals. Although the proposals usedifferent compression systems, they are all capable of compressing the audio data to about0.5 Mbitls [3]. The audio data is multiplexed with the video data along with ancillary andsynchronization data to produce a single binary data stream.Although the video compression rates vary slightly between the proposals, they all allowfor an enormous reduction in the amount of data that needs to be transmitted. For example,the DigiCipher system compresses the 121 million samples per second of its analog RGBsignal into a 12.59 Mbitjs binary data signal [5]. Usually there is a trade-off between imagequality and image compression. Although some care has been taken to minimize the impactof transmission errors on image quality, the compressed data remains quite sensitive to errors.A single bit can have a substantial impact on image quality for a noticeably long duration.To provide additional protection against errors occurring in the channel, channel coding isapplied.2.3 Channel CodingTo protect the compressed binary data stream from transmission errors, a HDTV system must use channel coding. Proposed techniques include error detection, forward errorcorrection, and data interleaving [6]. A combination of some or all of these techniques areincorporated in all the HDTV proposals, ensuring reliable data communication, Through the21use of cyclic redundancy check codes, transmission errors can usually be detected. When anerror is detected, the source decoder is informed of the corrupt data so that error concealmenttechniques can be initiated. Although most errors can be detected, the error concealmenttechniques can not perfectly hide the resulting image defects, so they are used only as a lastline of defence. Most error protection comes from the use of forward error correction (FEC).Reed-Solomon codes are used to correct most transmission errors. These block codes areknown for their superior ability to correct multiple bursts of errors [231. For example, a (127,117) Reed-Solomon code transmits 117 seven-bit symbols in blocks containing 127 symbols,and can correct up to five symbol errors per block, or one error burst of up to 35 bits per889 bit block [7]. Additional protection against error bursts is achieved in some proposalswith data interleaving.Clearly, channel coding is capable of providing very adequate protection of the compressed digital data. Occasional random errors and even short error bursts are controlledsuccessfully, causing absolutely no degradation of image quality. Errors that occur more frequently may defeat the error correction, but their impact is minimized by error concealment,As the error frequency increases, however, all proposals exhibit rapid degradation from thepoint where errors are noticeable but tolerable, to the point of unusability, where the imageis far too severely impaired to be viewed [31.To study the likelihood of transmission errors, the modulation and demodulation of thedata signal must be analysed, including the effect of the channel distortion.2.4 ModulationTo transmit HDTV signals over a coaxial cable television network, a modulation schememust be employed. Modulation allows many different TV signals to be transmitted simul22taneously over a single cable. The two modulation schemes being considered for FIDTVare quadrature amplitude modulation (QAM), and vestigial sideband modulation (VSB). OnlyQAM is considered in this thesis, and results will be different if a different modulation schemeis used. A block diagram of a typical QAM transmitter is shown in Fig, 7. A symbol encoder combines bits from the data source (the output of the channel encoder) into multi-bitsymbols. By transmitting several bits at once in a symbol, a higher data transmission ratecan be attained. The digital symbol stream is passed through a pulse shaping transmit filterwhich limits the bandwidth of the transmitted signal. The filtered signal is then used to modulate the amplitude of a carrier wave. A more detailed description of the QAM transmissionprocess follows.BinaryData ChannelSourceReal-valued signalComplex-valued signalFigure 7. Typical QAM Transmitter [24].2.4.1 Symbol EncoderTo increase the data transmission rate without increasing the required bandwidth, a symbolencoder combines groups of binary digits from the channel encoder into multi-amplitude,complex-valued symbols. During each symbol interval, an M-QAM symbol encoder collectslog2 M bits from the data source, and based on the value of the bits, selects one of M possiblesymbols from a finite set of allowable symbols known as the signal constellation, which shallbe denoted by S. The points in the signal constellation describe the in-phase and quadraturephase amplitudes to be transmitted for each symbol. These amplitudes take on values fromthe alphabet {+A,+3A,. . .,+(J_ 1)A}.23For all the examples presented in this thesis, a 16—QAM system is assumed. This systemgroups bits into 4—bit symbols for transmission. The FCC standard for HDTV may requirethe use of a higher order modulation scheme, such as 32—QAM or 64—QAM, The theoreticalanalysis in this thesis can readily be extended to these higher order schemes. The symbolconstellation for a 16—QAM scheme is depicted in Fig. 8. For example, the group of bits1001 is mapped to the symbol S9 = (—3A, A) = —3A + jA.QS S S1O 3A- 2 , 3,So .SiI I I I-3A A 3AS;3 -A-3A- 57Figure 8. 16—QAM Signal Constellation.The selected symbols are emitted at a rate of 1/T3 symbols per second, where T3is the duration of the symbol interval. The resulting symbol stream will be denoted by{Ia 11a e S, — <a < oo}, where Ia is the symbol emitted at time t = aT8. To developa theoretical analysis of the performance of the transmission system it is convenient tomodel the transmitted symbol stream by a discrete random process. Under this model, eachtransmitted symbol is randomly selected from S, with each point in $ being selected withequal probability. In addition, it is assumed that each symbol is selected independently of the24others in the stream. As a result, the mean (denoted by E[.}) of the random process isE[Ia] = 0 , (2.1)and the autocorrelation isI2ff bE[IIb] = oS(b — a) = i a —. (2.2)0 otherwisewhere * denotes complex conjugate, 6(.) is the Kronecker delta function, and u is the averageenergy per symbol. For 16—QAM, this is a = bA2.2.4.2 Transmit FilterFilters are used in the transmitter and receiver to limit the bandwidth of the transmittedsignal and minimize the effect of noise introduced in the communication channel. In additionto serving these two purposes, the filters are customarily designed so that they introduceno intersymbol interference (1ST) [24]. Let HT(f) and hT(t) denote the frequency andimpulse response of the transmit filter, and let HR(f) and hR(t) denote the frequency andimpulse response of the receive filter. The combined frequency response of the two filters isH(f) = HT(f)HR(f), and the combined impulse response is h(t) = hT(t) hR(t), where® denotes the convolution operation.To maintain compatibility with existing television spectral usage, it is necessary to limitthe bandwidth required to transmit HDTV signals to 6 MHz. This is achieved by requiringthat HT(f) 0 for all If 3 MHz. To maximize the received signal-to-noise ratio, thereceive filter must be matched to the transmit filter [24], so that H(f) = HT(f). To preventthe introduction of ISI, it is necessary that the combined impulse response has the propertyh(cT8) = 0 for all integers c 0.One filter meeting these three requirements is the root-raised cosine filter, Because ofits simplicity of implementation, it is commonly used in communication systems. This filter25has a frequency response ofHT(f)= vcos[If—(l—a)], f< , (2.3)10, Ifwhere a is referred to as the rolloff factor and denotes the excess bandwidth used as comparedto the minimum Nyquist bandwidth which is given by a = 0. A plot of the frequency responsefor various values of a is shown in Fig. 9. When a 0.2, the symbol rate must be chosenso that T5 > 0.2 x 10—6 seconds to limit the bandwidth to 6 MHz, The impulse response ofthe root-raised cosine filter can be derived as11’) 1+cos L7t-?Tr-1 I\ ‘S(ITLL) =4aNote that this filter is non-causal, so in practical hardware implementations a truncated versionof the impulse response must be used, A truncated version was also used in the softwaresimulations that were executed to produce the results in Chapter 4.///‘ ‘\ \—-=0,2—- c1=0,6•1” ,Ij \\/ / I/ I/ ,‘ I \S.,./ I I/ II I I \I, I_____________ _____________I-l/27 0 l/27Frequency (Hz)Figure 9. Frequency Response of Root-Raised Cosine Filter,1 This follows from the straightforward but tedious inverse Fourier transform of Eq. (2.3). This result is notusually included in textbooks.26The corresponding receive filter is identical to the transmit filter, and the combinedimpulse response isirt rtcsin cos7In ‘SU&)—71-t 4a2t7---TrNote that for any integer, c,sin7rc cosrcah(cT8) =irc 1 — 4cv2c= S(c) { if = 0 (2.6)0 otherwiseso no ISI is caused by the filters.In this thesis it is assumed that ideal root-raised cosine filters are used in the transmitterand receiver. Furthermore, it is assumed that the rolloff factor is strictly less that one. Forall examples, a rolloff factor of a = 0.2 is used to limit the bandwidth to 6 MHz,The output of the transmit filter, v(t), can be expressed in the time domain asv(t) = IaliT(t — aT5). (2.7)a= — coFor theoretical analysis, it is useful to model v(t) as a random process. It has a mean ofE[vj(t)] = E[Ia]h(t—aT5) = 0, (2.8)a=—ooand an autocorrelation function given bycv(t;r) E[4(t)v(t+T)]00 00= E[I’Ib]h(t— aT5)h (t + r — bT5) . (2.9)a=—oo b=—oo27Substituting Eq. (2.2) for E[11b] yields00 00qv(t; T) = — a)h(t— aT)hT(t + r — bT)b=—oo=— aT8)h (t + r — aT5). (2.10)a = —00Since this is a function of t as well as r, and q,(t + T3; T) = c(t; T), v(t) is a cyclostationary random process, with a period of T5.2.4.3 ModulationThe purpose of modulation is to shift the spectrum of the data signal from the basebandinto a higher frequency range. In a QAM system, the data signal modulates the amplitudeof a carrier wave. More specifically, the real component of the data signal is used tomodulate an in-phase carrier, while the imaginary component modulates a quadrature carrier.Mathematically, the modulated signal can be expressed asv(t) = Re[vj(/eJ2t] , (2.11)where Re[.] denotes the real part of [.] and f is the carrier frequency. Note that the averagetransmitted signal power is= . (2.12)The resulting signal can be multiplexed with other TV signals that modulate carriers atdifferent frequencies, allowing several different TV signals to be transmitted simultaneouslyover a single cable.2.5 Communication ChannelAs the modulated signal passes through the cable television network, distortion is introduced. As mentioned earlier, only two types of distortion are considered in this thesis, It28is assumed that the transmitted signal passes through the channel unaffected except for theaddition of noise and a single interfering tone. To gain a theoretical understanding of theimpact these distortions have, it is necessary to employ mathematical models describing them.The white noise, denoted by n(t), is modelled as a Gaussian random process that hasa mean of zero and a flat double-sided noise spectral density of No/2. This implies that itsautocorrelation function isE[n(L)n(L + r)] = (2.13)where 6(.) is the Dirac delta. Throughout this thesis this type of distortion is referred toas additive white Gaussian noise (AWGN), or merely as noise. It is useful to describe thestrength of the noise in terms of the signal-to-noise ratio (SNR). For a M-QAM system withlog2 M bits per symbol, the SNR per bit is defined asSNR. (2.14)N0 log2 MAs discussed in the introduction, a single additive sinusoidal tone interferer is characterized by its magnitude, I, its bandpass frequency, fj, and its phase, q!j. It is assumedthat none of these parameters vary with time, in that they each maintain a constant albeitunknown value. This distortion is not random, but deterministic. Mathematically, the valueof the tone at any time is given byz(t) Kcos(27rft + i) , (2.15)where 0 < I( <, fc > 0, and 0 çtj < 2K. The average power of this tone is= A practical measure of the tone’s strength is the signal-to-interference ratio(SIR) per bit, which is defined asSIR 1 —____1— 2o (2 16—P log2M— iç log2M —T8Klog2M29When the transmitted signal, v(t), passes through the cable television network, the twotypes of distortion are added to it. The signal observed at the received, r(t), is the sum ofthe three independent signals, sor(t) = v(t) + n(t) + z(t), (2.17)From r(t), the QAM receiver must reproduce the transmitted symbol sequence.2.6 DemodulationA block diagram of a typical QAM receiver is shown in Fig. 10. It is composed ofthree main units: a demodulator, a filter, and a symbol detector. The demodulator convertsthe received bandpass signal into a baseband one, the filter removes out-of-band noise andfurther filters the signal, and the symbol detector attempts to determine the transmitted symbolsequence from samples of the filtered signal.ChannelReal-valued signalFigure 10. Typical QAM Receiver [24].2.6.1 DemodulationThe demodulation process shifts the frequency spectrum of the received signal fromaround the carrier frequency into the baseband. Mathematically, this is accomplished by theoperationr0(t) r(t) . (2.18)Complex-valued signal30where r(t) is the received bandpass signal and r0(t) is the resulting baseband signal. Thecarrier frequency is f, and, as signal synchronization is beyond the scope of this thesis, it isassumed that f is known at the receiver2.This is a requirement for a coherent demodulator.Since the demodulation operation is linear and the received signal consists of threeindependent components (the data signal, the noise, and the tone), the demodulated signalalso has three independent components. By definingv0(t) v(t) . ,,./e_i27rfct , (2,19)n0(t) n(t) ‘ ‘,/e_j27rf , (2.20)andz0(t) z(t) . (2.21)the demodulated signal can be expressed asr0(t) r(t) .= [v(t) + n(t) + z(t)]= v0(t) + n0(t) +z0(t), (2.22)where Eq. (2.17) has been used to substitute for r(t). Further analysis of the demodulatedcomponents is useful.Eq. (2.19), the expression for the demodulated data signal component, can be simplifiedby substituting Eq. (2.11) for v(t), yieldingv0(t) v(t) .= /Re [vj(t)/ei2fct]= v(t) + vW)e_J4t. (2.23)2 There are several suitable methods available for achieving synchronization for the AWGN channel. See,for example, [24, 25].31The effect of demodulation on the noise component of the received signal is best analysedin terms of the mean and the autocorrelation function ofn0(t). The mean is given byE[n0(t)] = E[n(t)] vej2t = , (2.24)since n(t) is a zero-mean random process, and the autocorrelation function isqn0(r) = E[n(t)n0( + r)]= E [n(t)ei2 n(t + r)e_22(T)]= 2E[n(t)n(t + r)]e_32T=2q5(r)e_3T, (2.25)Substitution Eq. (2.13) for q(r) yields7n0(T) = 2(r)e1T= N08(r). (2.26)Since n0(t) is merely a scalar multiple of n(t), it is also a Gaussian random process [261.By substitution Eq. (2.15) for z(t), the demodulated tone component can be expressed asz0(t) z(t)= ‘./i(j cos (2irfjt += 4KC + e_i(2it)]= 4K + 4K= 4 + 4Kej2(2 c+fi)t (2,27)where f fj — f is the relative frequency of the tone in relationship to the carrier.322.6.2 Receive FilterAfter the received signal has been demodulated, it is passed through the receive filter,which has an impulse response of hR(t). The purpose of this filter is to remove signals fallingoutside the bandwidth of the transmitted signal, to remove the high frequency componentsof v0(t) and z0(t), and to reduce the power of the noise signal, without introducing 1ST. Theoutput of the filter is defined asr(t) r0(t) hR(1)= [v0(t) + n0(t) +z0(t)] hR(t)= v0(t) ® hR(t) + n0(t) € hR(t) + z0(t) ®= v(t) + n(t) + z(t) (2.28)Clearly, the lowpass received signal consists of three components: v(t) v0(t) ® hR(t), thefiltered data signal; n(t) n0(t) hR(t), the filtered noise signal; and z(t) z0(t) * hR(t),the filtered tone. Analysis of these three signal components is straightforward.The filtered data signal can be expressed in terms of the transmitted symbol sequence bysubstituting Eq. (2.23) for v0(t), so thatv(t) = v0(t) hR(t)= [v(t) + v(t)e41] hR(t)= vj(t) ® hR(t) + [v(t)e_f4t] h(t). (2.29)Converting this to the frequency domain with the use of a Fourier transform yieldsV(f) = Vj(f)HR(f) + V*(_f— 2f)HR(f). (2.30)where Hft(f) is the frequency response of the receive filter, and (f) is given by= 1: vj(t)e32tdt3300= f —aT3)e2td—00 a=—Oo00 00= Ia fa=—oo= IaHT(f)e’2 (231)—00Since one of the requirements of the transmit filter is that it is band limited, HT(f) 0for f l/T3 (given a < 1). Therefore V(f) 0 for f > l/T8. Furthermore, since thereceive filter is matched to the transmit filter, HR(f) is also equal to zero for f lIT8.Since 2f >> 1/T8, the range off for which V*(_f— f) is non-zero does not coincide withthe range of f for which HR(f) is non-zero. As a result, the second term in Eq. (2.30) canbe discarded, leavingV(f) = V(f)Hft(f)= 1a11T(f)HR(f)e2. (2.32)a =— 00By substituting H(f) = HT(f)HR(f) as the combined frequency response of the two filters,this becomesV(f)=IaH(f)e2, (2.33)and the inverse Fourier transform isv(t)=V(f)e2dt00 00= laf H(f)e32tTdfa=—oo=Iah(t — aT3), (2.34)where h(t) is the combined impulse response of the two filters.34Since the demodulated noise signal is Gaussian and the operation of the receiver filter islinear, the filtered noise signal is also Gaussian. Its mean isE[n(t)j = E[ri0(t)] h(t)= 0 , (2.35)and its autocorrelation function isN(T) E[n*(t)n(t + r)]= E[{n(t) ® h(t)} {n0(t + r) hR( + r)}]00 00= E [f n(a)h(t-)d. f n()h(t + r_)d]100 100= / / E[n(a)n(/9)]h(t—a)hR(t+T—I3)d/3daJ—oo J—oor°° 100= I I qj3 — )h(t—cY)hR(t + r—/3)d/3dc. (2.36)J—oo J—cxSubstituting Eq. (2.26) for 40(r) yieldsp00 p00c/.N(T)= J J No8tB — a)h(t—a)hR(t + r—/3)dda-00 -00= N0 f h(t—a)hR(t + r—a)d= N0 f h(-)hR(r-)d= N0= Noh(r) . (2.37)In summary, n(t) is a complex-valued Gaussian random process, with a mean of zero andan autocorrelation of Noh(r).The final term to analyse is for the filtered tone interference, By following a proceduresimilar to the one used in the analysis of the data signal, a simple expression for z(t) can35be found. Clearly, from Eq. (2.27),z(t) z0(t) ® hR(t)= [Kcie32e + 4Kcje_322fc+fie_] h(t). (2.38)Its Fourier transform isZ(f) = [#Kci6U_fi)e + 4KciS(f+2fc+fi)e] HR(f)= 4KcjHR(fj)S(f—fj)e3 + (2.39)Note that(240)Because fj is defined to be positive, f + fj >> l/T3, so HR(—2f— f) = 0 since thereceive filter is band limited. As a result, the second term in Eq. (2.39) is equal to zero. Theinverse Fourier transform yieldsz(t)= f 4KHR(f)s(f —= KcjHR(fj)e32uie3. (2.41)It is convenient to define K = 4KjHR(fj) as the magnitude of the filtered tone, so thatz(t) = . (2.42)The signal-to-interference ratio, expressed in terms of K, is (from Eq. 2.16)2aSIR=T8Klog2 M2o—T32K/H(f) log2 M— aH(f) (243)— KT8log2M36Eq. (2.42) is the mathematical model that shall be used to represent the tone interferencethroughout the remainder of this thesis. When the tone signal falls within the bandwidth ofthe data signal (that is, fI < (1 + a)/2T5), it distorts the transmitted data signal, possiblyintroducing errors when an attempt is made to determine the transmitted symbol sequence.2.6.3 Signal Power Spectral DensityA useful means of describing the received signal is its power spectral density (PSD). Thiscan be found by considering the components of r(t) separately, and then combining the results.For the data component, it is useful to model v(t) by a random process. It has a mean ofE[v(t)] = E[I]h(t— aT) = 0, (2.44)a= — ocand an autocorrelation function given byg5v(t; r) E[v*(t)v(t + r)]00 00= E[II]h(t— aT)h(t + T — bT8). (2.45)a—oc b—ooSubstituting Eq. (2.2) for E[11b} yields00 00v(t; T) = — a)h*(t — aT5)h(t + T — bT3)a=—oo b=—oc=g h*(t_aTs)h(t+r_aTs)a= —0000 00= f H*()e32(t)dafa=—oo—00 —0000 00 00= 100 100 H*()H()e32 tej2(t+T)aoce2dda00 00 00 7= u / / H*(a)H()e22 _)te32T — —J—00 J—OO Sa=—oc00= Ta=-oo-cH* (a)H(cr + j27rj2r(a+*)rd . (2.46)37Since this function is periodic with respect to t, v(t) is a cyclostationary random process, witha period of T3. To remove the time dependence, it is convenient to compute the time-averageautocorrelation function by integrating over a single period. ThusT— Islj T cv(1; r)dt= 00t: H*(a)H( + ej2dtej2+)TdQaooH*(a)H( + siflaj2+)Td100 00aoo /00H*(a)H += 1-00 H*(a)H()e32Td. (2.47)The average power spectral density of the data component can be found by taking the Fouriertransform of v(r). Therefore,v(f) 1TiT00 00= f H*()H(a) f= uf H*()H(a)S( - f)da1S —00= H(f)2. (2.48)The PSD of the noise component, determined directly from the autocorrelation functiongiven by Eq. (2.37), isN(f) f N(r)e32Tdr= /00Noh(r)e_32Tdr00 fOG= N0 / J H()eJ2Tde_3drJ—OO—0000 00= N0 / H() / ej2)TdTdJ—00 J—OO38= N0 f H()S( - f)d= NoHR(f)2. (2.49)Since the interfering tone is deterministic and periodic, its PSD must be found from itsFourier series representation. Clearlyz(f) = IqS(f— ft). (2.50)Since the tone is a deterministic signal it doesn’t have an autocorrelation function, Nonetheless, it is useful to define the functionz(t)z(t + r)= Kej27T i i< (t+r)= Iqe32iT (2.51)Note that the Fourier transform of this functions is= j z(r)e_32Tdr= j Iej1Te_jTdr=K8(f-f)= Fz(f). (2.52)Clearly g5 (r) can be used in a manner similar to the autocorrelation functions of the dataand noise signals.Since v(t) and n(t) have zero mean and are statistically independent, the autocorrelationfunction for the received signal can be expressed asbR(t; T) E[r*(t)r(t + r)]39E[{v*(t) + n*(t) + z*(t)} {v(t + r) + n(t + r) + z(t + r)}]= E[v*(t)v(t + r)] + E[v*(t)]E[n(t + T)] + E[v*(t)]z(t + r)+ E[n*(t)]E[v(t + r)j + E[n*(t)n(t + r)] + E[n*(t)]z(t + r)+ z*(t)E[v(t + r)] + z*(t)E[n(t + r)} + z*(t)z(t + r)= E[v*(t)v(t + r)] + E[n*(t)n(t + r)] + z*(t)z(t + T)= cbv(t; r) + N(T) + z(T). (2.53)Since this is periodic, the average autocorrelation must be found.lf-TR(t;T)dt= V(r) + N(T) + z(T). (2.54)The average PSD can then be computed asR(f) f_00R(T)e32TdT00 00 00= / v(r)e2TdT + / N(T)e32TdT + / z(T)e_22TdTJ—co ‘00= v(f) + N(f) + z(f)= + NOIHR(f)12+ KS(f-f). (2.55)As expected, the PSD of the received signal is characterized by an impulse occurring at thefrequency of the interfering tone. This property is exploited in the next chapter, where amethod for removing the tone from the received signal is presented.2.6.4 Symbol DetectionOnce the received signal has been filtered, the symbol detector attempts to determinethe transmitted data symbols. This is accomplished with the use of samples of the receivedsignal. As with the carrier synchronization in Section 2.6.1, it is assumed in this thesis that40perfect symbol synchronization is achieved, so that samples can be collected at the symbolrate, 1 /T8, and at the precise symbol sampling instants. Let Ra denote the value of theequivalent lowpass received signal at time t = aT5 so thatRa r(aT3)= v(aT3)+ z(aT5)+ n(aT3) . (2.56)Therefore Ra is the value of sample a, and is a result of the addition of three independentterms.The first term, v(aT3), is due to the transmitted signal and can be evaluated with Eq.(2.34) asv(aT3) = Ich(aT — cT5). (2.57)c=— DoSince the root-raised cosine transmit and receive filters are designed to cause no intersymbolinterference (see Eq. (2.6)), h(nT5) = 6(n), As a result,v(aT5) = IcS(a — c) = Ia (2.58)c= — Dothe value of the symbol transmitted at t = aT5.The second term in Eq. (2.56) is caused by the additive noise introduced in the communication channel. By definingNa n(aT5) (2.59)and noting thatE[Na] = E[n(aTs)] = 0, (2.60)andE[N:Nbl = E[n*(aTs)n(bTs)] = &r(bTs — aT5)= N06(b— a) , (2.61)41it is possible to describe this term as a complex-valued discrete-time Gaussian random processwith a mean of zero and a variance of N0. Note that different samples are uncorrelated.The third term in Eq. (2.56) is results from the presence of the tone interference addedin the channel, and can be expressed asZa z(aT3)= I<iei2Ts f. (2.62)This is dependent on three constant but unimown parameters, K, f, and j, which completelydescribe the tone interference.Each received sample can be expressed as Ra = ‘a + Na + Za , where ‘a is the transmittedsymbol, Na reflects distortion due to additive white Gaussian noise, and Za is the distortiondue to the interfering tone. The set of samples {Ra} is all that is available to the decisiondevice. From these decision variables the decision device attempts to determine the transmittedsymbol sequence, {Ia}. For each sample, the detected symbol, ‘a, is the point in the QAMsignal constellation, 8, that is closest to Ra. That is, the decision device selects L1 = Sif and only ifSi — Ral < S — RaI (2.63)for all S, S3 E S. This selection is the symbol most likely to have been transmitted att = aT3, given Ra.If the noise and interference terms are sufficiently small, ‘a = ‘a and no detection erroroccurs. On the other hand, if either of these terms is large, an incorrect decision may be made,resulting in a detection error. The probability of a detection error depends on the separationbetween adjacent symbols in the signal constellation, the magnitude of the tone interference,and the average power of the noise samples. Since the additive noise is random, a high noise42power implies a high likelihood that the noise term will be large for each sample, resultingin a correspondingly high probability of detection error. This differs slightly from the effectthe tone has. A strong tone implies that the tone term is large for all samples, so its effecton the probability of error is much more drastic.Fig. 11 shows a state-space diagram for the received signal when a 16—QAM system isused. Each point represents the received in-phase and quadrature values for a single sample.A total of 8192 samples were used to generate this plot, when noise with a SNR of 35 dBand a tone with a SIR of 35 dB corrupt the transmitted symbols. By comparing this diagramwith the signal constellation shown in Fig. 8, it is evident that the received samples tend tofall in clusters around the symbol points in the signal constellation. Because both the noiseand tone are weak, transmission errors are unlikely to occur. When the noise is stronger,however, the points in the state-space diagram become more dispersed. This is shown inFig. 12 when the SNR is 10 dB. Although the samples still occur in clusters, the clusters aremuch wider, resulting in a greater likelihood of detection error. The effect of a large tone issignificantly different, as indicated in Fig. 13, when the SIR = 10 dB and the SNR = 35 dB.Note that the samples fall in rings around the symbol points. When the noise is weak, as inthis example, it is still possible to determine the transmitted symbols with a high likelihoodof success. However, if the noise is also large, detection errors become very probable. Fig.14 shows the state-space diagram when SNR = 10 dB and SIR = 10 dB. The samples nolonger appear in clusters or rings, but are spread throughout the state-space.Although a convenient closed form expression for the probability of a symbol detectionerror in terms of the noise power and tone parameters is not readily available, the probabilityof error can be calculated numerically by using the computer program listed in Appendix D.Fig. 15 contains a plot of the probability of symbol detection error (FE) vs. SNR for various43-3.0 -1.0 1.0 3.0I-ChannelFigure 11. 16—QAM State-space Diagram. SNR = 35 dB, SIR = 35 dB.a)U3.01.0-1.0-3.0I I I I I, I 4 I I* • •- - I— -4:----1 - - - -- 0 • I I 43.01.0-1.0-3.0-3.0 -1.0 1.0 3.0I-ChannelFigure 12. 16—QAM State-space Diagram. SNR = 10 dB, SIR = 35 dB.SIRs when 16—QAM is used. In general, increasing the SNR decreases P3. The exceptionto this occurs when the SIR is very small. In this case the SNR has no effect, and almostNote that these results are only exact for the cases in which the frequency of the tone is an irrationalnumber. When the frequency is rational, the results will vary slightly, but not signficantly for the purposes ofthe graph.44-3.0 -1.0 1.0 3.0I-ChannelFigure 13. 16—QAM State-space Diagram. SNRall symbols are detected incorrectly. The serious effect theerror is obvious.Referring back to Fig. 10, after selecting‘a as the detected symbol, the M-QAM receiverconverts it into the log2 M bit pattern associated with the symbol. These bits are emitted in3.01.0-1,0-3.0OooooIoIoIo.9I9I99.= 35 dB, SIR = 10 dB.3.01.0-1.0-3.0-3.0 -1.0 1.0I-ChannelFigure 14. 16—QAM State-space Diagram. SNR3.010 dB, SIR = 10 dB.tone has on the probability of45100l0’C -210Cio_510630SNRIbit (dB)Figure 15. Probability of Symbol Detection Error.sequence with the bits from the other detected symbols, producing a binary data stream. Thisdata stream should be identical to the one produced by the channel encoder in the transmitter.Because errors can occur during transmission, some of the bits may be in error, so the datastream is passed to the channel decoder. If the error rate is low, the error correcting abilityof the channel decoder will prevent visible distortion. Furthermore, the effect of uncorrectederrors will be concealed by the source coder.When there is no interfering tone it is known from the results of field testing performedby the FCC Advisory Committee that the DigiCipher system cannot operate reliably whenthe SNR is less than about 6.5 dB [3]. Performance for the other proposals is similar. Sincemaintaining signal levels above this point is not difficult in CATV networks, reliable digitalimage transmission is possible. However, if a strong interfering tone distorts the signal,image degradation may occur. For example, if a tone with a SIR of 10 dB is present, theSNR must be greater than about 9 dB for impairment-free transmission, If the SIR is 5 dBthe SNR must exceed 18 dB to prevent excessive transmission errors, Clearly the increased5 10 15 20 2546probability of error associated with a strong tone adversely affects performance to the pointwhere preventative measures need to be taken.2,7 ConclusionIn this chapter the all-digital transmission of HDTV signals was investigated. Thepowerful digital compression algorithms of the proposed systems are capable of reducingthe large amount of information associated with the high resolution, wide aspect ratio, HDTVsignals sufficiently to allow transmission over the 6 MHz channels currently used by traditionalNTSC signals. To protect the fragile compressed data from the ravages of transmission errors,strong error correcting codes are employed. Provided that the transmitted signal power issufficiently high, errors introduced from AWGN and an interfering tone arising from thecommunication channel can be effectively eliminated. Transmission of HDTV signals overexisting cable TV networks is definitely feasible.Unfortunately, the presence of a strong interfering tone has a dramatic effect on transmission errors, and complete collapse of the error correcting ability of the channel coder ispossible. While increasing signal power will ensure the integrity of the transmitted HDTVsignals, it is not an ideal solution. In the next chapter a robust method for cancelling toneinterference is proposed, allowing reliable data transmission even in the presence of strongtone interference.47Chapter ThreeA Novel Tone InterferenceCanceller3.1 IntroductionAs noted in the previous chapter, distortion introduced in the communication channel maylead to transmission errors. If too many errors occur, the channel decoder will not be ableto correct them, leading to image and sound quality degradation. Under severe interferenceconditions, the receiver will be unable to present even a vague facsimile of the desiredtelevision signal. Distortion in the form of a strong tone falling within the band of the desiredsignal significantly increases the likelihood of transmission errors. In a traditional QAMreceiver, samples of the received signal are used directly as decision variables to determinethe transmitted symbols. Unfortunately, an interfering tone added in the communicationchannel will distort the samples, possibly preventing correct symbol detection. To alleviatethis problem it is desirable to use a decision variable that has the effect of the tone removedor at least significantly reduced.48In this chapter a novel method is proposed for removing such a tone from the receivedsignal. This proposed system is designed to operate in conjunction with QAM digital signals.In Section 3.2 of this chapter a method of estimating the effect of the tone on each of thereceived samples is developed. For successful operation, the frequency of the tone must bedetermined. A method for achieving this is proposed in Section 3.3. This chapter concludeswith a summary of the proposed interference canceller in Section 3.4.3.2 Tone Interference EstimationIdeally, the frequency, magnitude and phase of the interfering tone would be known,allowing the receiver to calculate the effect of the tone on each decision variable by usingEq. (2.62). That is:Za Ije2Ts t . (3.1)A new decision variable could be generated by subtracting Za from each received sample.The new decision variable, R’a, would equalRRaZa—1a+Na+ZaZa1a+Na, (3.2)and would allow the decision device to determine the transmitted symbol more accurately.The interfering tone would no longer have any effect.Unfortunately, the parameters of the interfering tone are generally not known at thereceiver, so Za can not be calculated explicitly. Instead, an estimate, Za, of the effect of thetone must be used. In this case the new decision variable isRRaZa49=Ia+Na+ZaZa1a+Na+Ea, (3.3)where Ea Za — Za is the residue remaining when an imprecise estimator is used. If theresidue is small, the decision device is more likely to correctly determine the transmittedsymbol than if no attempt is made to remove the interference. Ideally, an estimator thatminimizes the probability of error would be used, ensuring the greatest possible reliability ofthe data. Unfortunately, the non-linear nature of the probability of error make this type ofestimator impractical [27]. In practice, estimators that minimize the mean of the square ofthe error (MSE) between the estimate and the desired value are used instead. This type ofestimator, known as the minimum mean-square error (MMSE) estimator [27] is nonethelesscapable of effective tone cancellation.The MMSE estimator is the one that minimizes the average residual power. Clearly, theresidue is an important measure of the accuracy of the estimator, It is convenient to measurethe residue in a manner analogous to the SIR. The signal-to-residue ratio (SRR) is defined asthe ratio of the average signal power (per bit) to the average residual power, soSRR______, (3.4)log2 Mwhere7E[l6a2] (35)is the average residual power. To reduce the likelihood of a transmission error it is necessaryto find an estimator for Za that results in a high SRRTo successfully cancel the interfering tone, accurate estimates of the tone at each samplinginstant is required. These estimates must be generated only from observations of the receivedsignal. Although there are numerous methods for finding estimates, it is important to use onethat is both simple and accurate.503.2.1 Linear MMSE EstimatorThe problem with using true MMSE estimators is that they are generally infeasible.Even if the form of the MMSE estimator could be found mathematically, it would likely beextremely difficult to implement. In practice, some limitations on the estimator must be made.To maintain simplicity, a estimator should rely only on discrete samples of the received signaland not on the entire analogue continuous-time signal. In addition, it is convenient to useonly samples collected at the symbol sampling instants. Therefore, an estimator of Za shouldbe based only on the samples {Ra_nV n e I}. For reasons to be discussed in Subsection3.2.3, it is further required that n be greater than or equal to one. Another requirement ofsimplicity is that the samples be combined in a linear fashion to produce the estimates. Alinear estimator of Za takes the formZa = WnRa_n, (3.6)where L is the number of samples used and {w } are weight coefficients for the samples.Thus Za, as generated by Eq. (3.6), is essentially a weighted average of L samples of thereceived signal. By carefully selecting the weights it is possible to control the accuracy ofthe estimates.As noted earlier, it is desirable to find an estimator that maximizes the SRR. Given thatit is also necessary to stick to the simple linear estimates of the form given by Eq. (3.6), itis desirable to use the linear minimum mean-square error estimator. Thus, it is necessary tofind the weights {w } which minimize22‘y=E 6a =E ZaZa , (3.7)to find the linear MMSE estimator. Although actually finding the optimal weights is a tedioustask, it is proved in Appendix A that the mean-square error is minimized whenWfl= L1 (3.8)51where C = (o + No) /K is the ratio of the data plus noise power to tone power, and= 2rfT is the normalized frequency of the tone in radians. ThereforeL7— 1 \p 3wna_nen=1is the linear MMSE estimator of Za.At this point it is useful to consider how this method provides accurate estimates. Bysubstituting Ra_n = ‘a—n + Na_n + Za_n into Eq. (3.9) it is clear thatZa= L+CIa_ne + L+CNa_ne + L+ClZa_. (3.10)The three components in this equation are due to the transmitted symbols, the AWGN, andthe interfering tone. The first component in Eq. (3.10) is essentially a weighted average ofL transmitted symbols. Since these symbols are assumed to be random with zero mean,this average tends to zero for large L. This is an application of the central limit theorem ofprobability theory [281. The second component is caused by the AWGN signal, and also tendsto zero for the same reason. The third component is due to the interfering tone, which has adeterministic nature. Each sample of the tone has the same magnitude but a different phase.The phase changes by w radians with each successive sample, that is, Za+i Zae’ Toalign the phases of each sample, they are multiplied by a the appropriate power of e3’, sothe third term is actually the sum of a constant. Mathematically, this can be shown asL c Za_ne’ = L1 Ijeia_eieiTh== L+CZa= ZaLLC, (3.11)52by substituting Eq. (3.1) for Za_n and Za. Clearly the third component of Eq. (3.10) tendsto Za for large L. By combining the results for these three terms for large L, it is clear thatZa Za as L —* oo. When L is finite, however, the effect of the data symbols and noisecomes into play. If the noise is strong or the tone weak, the estimates are less reliable. Tocompensate, the sum in Eq. (3.9) is divided by L + (o + N0)/K instead of by L alone,as might be expected.It is worth noting that the MMSE, calculated by substituting Eq. (3.8) for w, in Eq.(A.9), is7mm K? [i — wne — + wwme_im_]n=1 n=1 n=1 m=1+ (u+No)Zwwn— K —L0e_ez]+ K \1 1__ejme_j(m_n)L+Cn=1 m=12 L=Ki[l_L+c_L+C+(LC)2] +(Js+N0(L+G)2= K? [(L + C)2 — 2L(L + C) + L2] + K? ( + No) L(L+C)2 (L+C)2 iq= K?2 [L2 + 2LC + C2 — 2L — 2LC + L2 + LC](L+C)I2= [Lc+C2](L+C)2K— 1L+C—312— L+(+No)/K?53Eq. (3.12) represents the optimal performance of any linear estimator that attempts to minimizethe MSE. Since a number of tone cancellers fall into this category it is worthwhile to explorethe MMSE further. This provides a yardstick for comparing different cancellers.To begin this analysis first consider the effect of the number of samples, L. If L = 0,the MMSE is I. This is intuitive because if no samples are used to estimate the tone, noestimate is available for cancellation purposes and the residual power is the same as the powerin the tone. When L is small, the random nature of the transmitted symbols and Gaussiannoise make the estimate of the interference unreliable, so the residual power remains large. Asmore samples are used, the effect of the transmitted symbols and noise diminish, allowing forincreasingly accurate estimates. Finally, as L —* c the MMSE tends to zero, By selecting Lsufficiently large it is possible to generate estimates with any desirable accuracy. To comparedifferent cancellers it is necessary to compare them for the same value of L.As can be seen from Eq. (3.12), the MMSE depends not only on L, but on the power ofthe noise and tone. To simplify analysis it is more useful to study the maximum SRR, namelySRRmax = (3.13)7min10g2Minstead of the MMSE. Fig. 16 contains a plot of SRRmax vs. SIR for different SNRs and afixed value of L = 32. It is worth noting that the maximum SRR is essentially independentof the noise power, except when the tone is strong (SIR < 10 dB), and even then the noisehas little effect. In addition, note that the linear MMSE estimator can achieve cancellation toa SRR of greater than about 8 dB for all tones in SNR> 0 dB when L = 32. Clearly strongtones are attenuated, allowing for more reliable symbol detection. Of course, attenuation canbe increased by using a larger value of L, if necessary.Although the SRR is a useful measure of transmission performance, particularly inrelationship to the probability of transmission error, it is not particularly useful for measuring5460‘- 50C40300 10 20 30 40 50 60Signal-to-Interference Ratio (dB)Figure 16. SRRmax vs. SIR, L = 32.the cancellation performance. A more practical measure is the ratio of the average residualpower to the power of the tone, -y/iq. If this quantity is less that one, some performancegain has been achieved (i.e., the residual power after cancellation is less that the tone powerbefore cancellation). Furthermore, it is useful to work with a logarithmic scale. Define theperformance gain (in dB) asGIIdB —l0log= SRR11dB — SIRIIdB . (3.14)A positive value of G indicates a gain, while a negative value indicates system degradation.In Fig. 17 the theoretical bound on the gain, given by7mmGmax = —lOlog-j(2 + No)/K—10 logL + ( +N0)/K’ (3,15)55is plotted as a function of the SIR when L = 32. Curves for several different noise powers areincluded. From this graph it is apparent that linear estimates perform better at tone cancellationwhen the tone is strong. This is reassuring since the need to cancel strong tones is greaterthan the need to cancel weak ones. Note also that cancellation is completely ineffective forSIR > 20 dB when L = 32. Increasing L will allow weaker tones to be cancelled, but inSection 3.2.3 a better solution is presented.10.0C,)0.010 20 30 40Signal-to-Interference Ratio (dB)Figure 17. Gmx vs. SIR, L = 32.50 603.2.2 Sub-Optimal Linear EstimatorAlthough the estimator given by Eq. (3.9) is optimal in the linear MMSE sense, it hasa drawback. It requires knowledge of the frequency (ft) and magnitude (Kg) of the tone,along with the noise PSD (N0). Since, in general, this information is not known at thereceiver, this method is no more feasible that the ideal one presented at the beginning of thissection. Fortunately, estimates for wj = 2irfjT3 and C=(o’ + No) /K can be used instead056of the actual values. By using these estimated parameters, sub-optimal estimates of Za withacceptable accuracy can be generated. As will be shown in the next chapter, performancenearly identical to the theoretical bound given in Fig. 17 can be achieved. In the next sectiona procedure for producing the intermediate estimates is presented. Before then, however, itis useful to analyse the MSE when these estimates are used.When 5j and C are used as estimates for w and C, the MSE can be found by substituting= (3.16)L+Cfor w in Eq. (A.9) yielding= K — we’ — + wwme_t(m_]n=1 n=1 n=1 m=1+ (+No)ww= K 1 — — 1+ K1 1_ejme_j(m_n)n=1 m=1+ (u + No) 1 11L+C L+C= K 1— 1— 1L+C1 L+C1+K( 1n=lm=1(3.17)Since the mean-square error does not depend on the actual frequency of the interference buton the accuracy of the frequency estimate, it is useful to define2=j— (3,18)57as the error in the frequency estimate, Using this definition along with the property= = e2 ej27rnn=1 n=1 n=O—(irL)— sin (7rE)— sin (irL)ej(L+1) (3 19)— Sin(7r)allows the MSE to be expressed as-y = K i— 1 sin (irL)e (L+i) — 1 sin (irL) e_j(L+1)L + C sin (ir€) L + C sin (irE)K____2sin2(L)+(2+N)( 1L+C) sin2( r) L+C)2.22 2 sin (TrEL) / 1 “\ sin (7rEL)=K 1— cos[ir€(L+1)]+ I - I 2L + C sin (irE) \L + ci sn (irE)+Iqc(__1\L+CJ= K 1—2 sin(irL)cos[ir(L+1)]+ (__1__2sin2(ircL) +L + C sin (irE) \L + ci sin (irE)(3.20)Although Eq. (3.20) seems complicated, it does provide some insight into how the accuracyof LDj and C affect the mean-square error.To illustrate the effect of error in Lj, Fig. 18 contains a plot of -y versus L when SNR =15 dB, SIR = 10 dB, and C = C, for various values of E. For the sake of comparison, thecurve for the minimum mean-square error, which is achieved if e = 0.0, is also included. Forsmall L, the MSE when e is non-zero is very close to the MMSE, but for larger L the MSE isnoticeable larger than the MMSE. This discrepancy is more pronounced for larger values ofE. Although the slight discrepancies shown in Fig. 18 are undesirable, error in 5j has a muchmore serious implication. For larger values of L, the MSE becomes excessive, as shown inFig. 19. The MSE does not approach zero for large values of L like the MMSE, but instead58tends to IQ, as shown in Fig. 20. This situation occurs for all non-zero values of e. Nomatter how small is, there is a point beyond which increasing L does not reduce the mean-square error. This phenomenon can be explained by recalling how Za is calculated. Whenthere is a slight error in the frequency estimate the system is unable to correctly compensatefor the varying phase of the tone in each of the samples. When L is small this does notproduce a serious problem. However, for large L the system completely loses track of thephase information in Za, causing the third term in Eq. (3.10) to tend to zero, forcing Za itselfto zero, As a result, there is a limit to the number of samples that can be used to generateaccurate estimates of the interference.Figure 18. MSE vs.Since there is an upper bound on L, there is a corresponding lower bound on the mean-square error that can be achieved when an imprecise estimate for w is used. This impliesthat the interference cannot be reduced below a certain threshold. For the example in Fig. 19,00 50 100 150 200Number of Samples (L)L. SNR = 15 dB, SIR = 10 dB, C = C, various .59(i’21)500 1000 1500 2000Number of Samples (L)Figure 19. MSE vs. L. SNR 15 dB, SIR 10 dB, C = C, various &when = 0.0001 the MSE has a lower bound of about one-eighth of the interference power.In practice, the frequency estimator proposed in the next section produces more accurateestimates, so the MSE is generally lower.The error in the estimate of C has only a slight affect on the MSE, even when C issignificantly incorrect. This is shown in Fig. 21 when = 0.0001, SNR = 15 dB, SIR = 10dB. Curves for C = 0.5C, C = C, and C = 2C are shown. For most typical values of C andC, the effect of any error is slight, and although undesirable, it does not produce the samelimit on L that error in the frequency estimate does. As a result, most of the emphasis in thenext section will be on accurately estimating the frequency of the interfering tone.For most applications the proposed scheme will eliminate the interference sufficientlyto allow for reliable data communication. Nonetheless, it is possible to further reduce themean-square error through the use of decision feedback,0600 5000 10000 15000 20000Number of Samples (L)Figure 20. MSE vs. L, SNR = 15 dB, SIR = 10 dB, C = C, = 0.0005.3.2.3 A Decision Feedback Tone CancellerThe presence of the data and noise signals in the received samples are the primary reasonswhy the tone can not be measured precisely. Increasing the number of samples used by theestimator will decrease the significance of these signals, allowing better tone estimation, butunfortunately, imprecise frequency estimates limit the number of samples that can be used.Even if the frequency is known, the number of samples required may be prohibitively large.Consider a tone with SIR = 5 dB in noise with SNR = 15 dB. Fig. 22 contains a plot showingthe number of samples required to achieve a given SRR, found by solving Eq. (3.12), inconjunction with Eq. (3.13), for L. For example, to increase the SRR to above 15 dB atleast L = 115 samples are required. Since the number of samples is fairly small, any slightinaccuracy in the estimate of the frequency will have little effect, However, recalling Fig.15, at a 15 dB SRR, the tone still noticeably affects the probability of transmission error.61200Figure 21. MSE vs. L. SNR = 15 dB, SIR = 10 dB, = 0.0001, various C.For negligible distortion due to the tone the SRR must be increased to above 25 dB. Thisis confirmed by recalling Fig. (15), which shows that the probability of error when the SIRis 25 dB is not substantially different from when no tone is present. Therefore, at least L1262 samples are required to effectively cancel the tone. Even if such a long tapped delayline were economically feasible, the immense susceptibility to inaccuracy in the frequencyestimate would prevent the residue from reaching this goal.As this example shows, the proposed system is capable of providing some improvementin system performance, but is generally incapable of achieving the ideal goal of a SRR above25 dB. However, with a slight modification system performance can be drastically improved.To illustrate this modification, consider how the interference estimator would perform if thedata signal was absent from the received signal. In this case, by following a procedure similar0 50 100 150Number of Samples (L)621500-d1000rJVCD500ci)030SRR (dB)Figure 22. Number of samples vs. SRR, SNR = 15 dB, SIR = 5 dB.to the one in Appendix A, the MMSE can be shown to be— N0321— L + N0/Iqproviding j and C are known. By using this equation instead of Eq. (3.12), it is clear that forthe tone in the previous example, the SRR would be 15.4 dB if the tone is estimated from onlyone sample. Only 10 samples would be need to increase the SRR to 25 dB. Clearly, if the datasignal could be removed from the received samples before estimating the tone, the residuecould easily be reduced to a negligible value, even if the frequency is not exactly known.Even though the transmitted data signal is not known at the receiver, it is possible toreduce the effect of the data signal on the samples by employing decision feedback. Sincethe estimate of the interference in the current sample is based only on previously receivedsamples (i.e. n 1), it is possible to subtract the previously detected symbols,‘a fromthe samples, before estimating the interference. By using this method the effect ofthe data signal is greatly reduced. Unfortunately, the minimum mean-square error and theoptimum weights are very difficult to find in this case as they depend on the probability5 10 15 20 2563of detection error, which in turn depends on the MMSE and the optimal weights. Even anumerical solution has proved elusive. However, using the estimator1 LZ—____‘ 1R — I (322)aLC/,,,d\ a—n a—n)produces excellent tone cancellation, as will be confirmed experimentally in the next chapter.As noted, detailed mathematical analysis of the system performance when decisionfeedback is used is extremely difficult since the correctness of the detected symbols depend onmany factors. Nonetheless, the usefulness of decision feedback can be shown by consideringa few scenarios. If only a few transmission errors occur, ‘an usually equals ‘an, so thatRa_n — Ia_n usually does not depend on the data signal. As a result, the effect of the datasignal is almost completely removed from Za. This leads to a lower residue, better tonecancellation, and fewer transmission errors, Since the tone canceller works without feedback,it is very tolerant of transmission errors. In fact, the feedback canceller generally performsno worse than the basic canceller when feedback is not used, even when nearly half thesymbols are detected incorrectly. Clearly, under normal operating conditions, situations thissevere are unlikely to arise. However, when the receiver tunes to a new signal, a large tonemay be present, potentially causing almost all symbols to be in error. In this situation, thefeedback canceller will initially performs worse than the basic canceller, usually achieving aSRR that is about 3 dB lower. Even at this lower performance level, however, the tone isreduced sufficiently to allow for a significant reduction in transmission errors. As the numberof errors declines, more accurate interference estimates are produced, further decreasing thenumber of transmission errors, eventually leading to a much better SRR. On the other hand,if many transmission errors arise from noise with extremely strong power, or perhaps an errorburst, the feedback canceller does not work well. However, since the source of the errors isnot the tone, removing the tone will not alleviate the error problem. In the next chapter, acomparison of the performance of the feedback and basic cancellers will be presented.64Although the true MMSE of the feedback canceller is unknown, a crude bound can befound by assuming that no detection errors occur. In this case the data signal is completelyremoved from the received signal, and the MMSE is given by Eq. (3.21). An uppper boundon the performance gain is thenGax = —10 log L J2(3.23)which is plotted in Fig. 23 for L = 32, Clearly these theoretical results are vastly superiorto the ones in Fig. 17. Without the presence of the data signal, the noise power becomesthe limiting factor controlling the performance. Cancellation can be applied to substantiallyweaker tones, and cancellation is much more effective, regardless of noise and tone power.60502010060Figure 23. G’ax vs. SIR, L = 32, with decision feedback.For the feedback canceller to work properly, another slight detail must be considered.Since the data signal power in the samples has been reduced, the power ratio, C should0 10 20 30 40 50Signal-to-Interference Ratio (dB)65estimate the quantityE 1a’a +Na(3.24)instead of (o + N0)/I, as in Eq. (3.8). This change reflects the decreased power ofthe data signal relative to the power of the tone, Fortunately, producing this estimate isstraightforward, as will be shown in the next section.3.3 Estimation of Frequency and Power RatioThe linear estimator of Za described in the previous section requires knowledge of thefrequency of the tone and the ratio of signal and noise power to interference power. Thisinformation can be readily extracted from the power spectral density of the received signal. Inthis section a simple method for estimating the PSD will be presented, followed by a detaileddescription of how the frequency and power ratio can be estimated.When attempting to determine the frequency of the interfering tone it is natural to analysethe received signal in the frequency domain. Fig. 24 contains a plot of the PSD of the receivedsignal when quadrature amplitude modulation is used with root-raised cosine filters with aroll-off factor of = 0.2. The noise and tone have powers given by a SNR of 15 dB anda SIR of 15 dB. The frequency of the tone is f = 0.1/Ta. The PSD is characterized by(see Eq. 2.55)(f) = H(f)I2 + NoHR(f)2+ K6(f- ft). (3.25)All the power in the tone is concentrated around the frequency of the tone, while the powerin the data and noise signals is dispersed smoothly throughout the signal frequency band. Thefrequency of the tone can be found by finding the frequency where the impulse in the PSD66(K)fFrequency (Hz)Figure 24. PSD of Received Signal. SNR = 15 dB, SIR = 15 dB, o = 0.2.occurs. This simple observation is the motivation for the proposed method for estimatingthe tone’s frequency.3.3.1 PSD EstimationThe PSD of a random process is based on all possible realizations of the process overall time. Since only one realization is observed at the receiver, only an estimate of the PSDcan be generated. Furthermore, practical considerations limit the observation to a finite timeinterval. Methods for estimating the PSD fall in the realm of spectral analysis. Numeroustechniques have been proposed in this mature field [29], offering solutions covering a widerange of accuracy and complexity. One technique, the periodogram, has been selected foruse in this thesis because of its simplicity and computational efficiency [29]. As will beshown later, the estimate it produces of the PSD is sufficient to allow the frequency of thetone to be accurately determined. The periodogram is based on the finite-length discreteFourier transform (FFT), a transform that can be performed efficiently and has hardwareimplementations readily available [301.(+N0)7-1/7; -1/27; 0 1/27;67The periodogram is a set of values that describe the PSD of the signal at a set of discretefrequencies. It is calculated in a straightforward manner from a block of samples of thereceived signal. The signal is sampled at a rate of 1/T samples per second. To preventaliasing, the sampling rate must meet the Nyquist sampling criterion, so T must be less thanor equal to T/2, since the raised cosine filters limit the bandwidth of the signal to fl < l/T8[24]. To simplify analysis, the relationship between the symbol duration and the samplingperiod is required to be T = T8 /N, where N is an integer representing the number ofsamples per symbol. Clearly N 2 is required to prevent aliasing.As the received signal is sampled, the samples are collected in blocks containing Nsamples each. The sample stored in the position of the jth block shall be denoted byrd(n) and is defined asrd(n) r(riT + dNT) , (3.26)for n = 0, 1,. . . , N — 1, and any integer d. The FFT of the dth block is given byRd(k) rd(n)e_32, (3.27)for k = — N/2,.. . —1, 0, 1,. . . , N/2 — 1. This transform has the property that the pointsin {Rd(k)} are equal to the Fourier transform of a sampled and truncated version of thereceived signal, evaluated at k/NT Hz. Since efficient algorithms exist for computing thediscrete Fourier transform when the block length is a power of two, it is necessary to choose Nto be a power of two. To simplify analysis it is also desirable that N be made a multiple of N.Once the FFT of a block has been calculated, its periodogram can be determined in astraightforward manner. The periodogram of the dfIz block shall be denoted by Xd(k) andis defined asXd(k) lRd(k)l2. (3.28)68The set of values {Xd(k)} are estimates of the PSD of the received signal at the set offrequencies { k E I, — k 4- — 1To show the relationship between the periodogram and the PSD of the received signal,it would be beneficial to find the probability distribution function (PDF) of the points inthe periodogram. Ideally, a simple expression for the PDF of {Xd(k)} could be found.Unfortunately, this is not the case. The difficulty arises primarily from the cyclostationarynature of the transmitted signal. Even finding an expression for the mean of Xd(k) is a nontrivial matter. The discussion that follows is included only to illustrate some of the strengthsand weaknesses of the periodogram approach to spectral estimation. It is not intended to bea thorough or exact analysis, as it was felt that this task is beyond the scope of this thesis.To express the mean of each point in the periodogram in terms of the PSD of the receivedsignal it is useful to use Eq. (3.27) and Eq. (3.28), so thatE[Xd(k)] E[Rd(k)I2]= E r(n)e2 rd(m)e2]n=O m=ON-iN-i -= . E[r(ri)rd(m)Je_22N n) (3.29)n=O m=OSubstituting Eq. (3.26) yieldsN-i N-ik(m—n)E[Xd(k)] = E[r(nT + dNT)r(mT + dNT)]e2 Nn=O m=ON-i N-ik(m—n)= - bj(nT + dNT; mT — nT)e327r N (3.30)n=O m=OThis equation provides an expression for the mean of the periodogram in terms of the receivedsignal’s autocorrelation function. By substituting Eq. (2.53) for R(flT + dNT; mT — nT)it is possible to express the mean asN-i N-iE[Xd(k)] = (nT + dNT; mT —n=O m=O69N-i N-i• k(m—n)+ c1N(mT - Nm=O m=ON-i N-i• k(m—n)+ bz(mT — Nn=O m=O= Ev(d, k) + EN(d, k) + Ez(d, k) , (3.31)whereN-i N-iE(d, k) qv(nT + dNT; mT — nT)e_32 V (3.32)n=O m—ON-i N-iEN(d, k) > cN(mT — nT)e22mN , (3,33)n=O m=OandN-i N-iE(d, k) qz(mT — nT)e_32 (mNri) (334)n=O m=OClearly, from the above equations, the mean is the sum of three components, with onecomponent depending on the data signal, one depending on the noise, and the third dependingon the tone. By analysing the three components separately it is possible to express the meanin terms of the PSD.The first component is the most difficult to analyse due to the cyclostationary nature ofthe data signal. From Eq. (2.46),15V(T1T + dNT; mT — nT)00 cc>= 1 f H*(f)H(f + )e2 T:N)ej2 )(m_n)Tdf (3.35)a— —coHowever, since qv(t; r) is periodic with respect to t with a period of T,cv(nT + dNT; mT — nT) = cbv(nT; mT — nT) (3.36)since dNT = dNTS/NX is a multiple of T8.4 In addition, when a > 1, H(f) andH(f + a/T8) do not overlap since H(f) is band-limited to If < lIT3. ThereforeSince N is a multiple of Na,, N/N,, is an integer. The block number, d, is also an integer,70H*(f)H(f + a/T8) 0 when al > 1. As a result, only three terms in the summationwith respect to a are non-zero. Therefore,1 00 a(nT + dNT; mT — nT) =aiH*(f)H (f + )e_i2Tei2)mTdf.(3,37)Substituting Eq. (3.37) for qv(nT + dNT; mT — nT) in Eq. (3.32) yieldsN-iN-i 1 00E(d,k) = >2 >2 - >2n=O m=O a=—1 —00= 1q21 >2 jH*(f)H(f+a)a=—1 n=O m=O(3.38)To clarify this expression defineG(f) >2 ej2T— 1 — exp (j2rfNT)1 — exp (j2irfT)= Sifl KfNTej2f(N_1)T (339)sin 7rfTso that1E(d, k)= a=-1H*(f)H(f +- + - - (3.40)The magnitude of G(f) is characterized by spikes occurring at multiples of 1/T = N/T8 Hz,and is small elsewhere. As a result, there can not be a spike at both G(f) and G(f + a/T3)for the same value off, so G*(f)G(f + a/T5) is negligible when a = +1 (if N 2).Therefore, the integrals when a = +1 can be ignored. Only when a = 0 does the integralcontribute to Ev(d, k), soE(d, k) f H(f)121G(f - ) df. (3.41)S—oo71To express this in terms of the PSD, recall from Eq. (2.48) thatv(f) = gIH(f)I2, (3.42)T3so that‘00Ev(d, k) = / v(f)IG(f- ) 2df. (3.43)i-coThe noise component, given by Eq. (3.33), can be expressed in a similar format byN-i N-i•, k(m—n)EN(d, k) &vfrnT — nT)e1 Nn=O m=ON-iN-i• k(m—n)= >Z / N(f)e22 mn)TdfeJr Ni-con=O m=Oco N—i_j2ir(f k \ N—i i27r’f_T)mTrJ N(f)e -ivr)nT “n=O m=O= J N(f)jG(f — ) 2df. (3.44)-coThe tone component, given by Eq. (3.34) can be expressed in terms of the PSD of thetone byN-i N-iEz(d, k) bz(mT — nT)e_32”n=O m=ON1N1. k(m—n)= I z(f)e12mTdf _3 Nf-com=O m=ON—i_J27r(f_ k \ N—i i2r(f_T)mTdf/ z(f)e 7T)flTZn=O m=O‘ 00r / ‘Fz(f)jG(f—)I2df. (3.45)J—ooCombining the three components yieldsE[Xd(k)j = Ev(d, k) + EN(d, k) + Ez(d, k)‘00= J v(f)tG(f - r)I2df+ J N(f)(G(f - )(2df-00 -0072+ L z(f)jG(f - )I2df= 1: [v(f) + N(f) + z(f)] IG(f - ) Idf. (3.46)By using Eq. (2.55) for the PSD of the received signal, this can be expressed asE[Xd(k)j= foo (f)G(f-2df=R(f)WN(f - (3.47)whereWN(f) = G(f)2 = sin2 NT (3.48)Nsin TrfTis referred to as a window function. Examination of Eq. (3.47) reveals that the mean of theXd(k) is equal to the result of the convolution of the PSD of the received signal with thewindow function evaluated at f = . Since the mean of periodogram is not equal to the PSDof the signal, the periodogram is known as a biased estimator [311. The convolution operationcauses the PSD to appear slightly widened, with more power shifted into the sidelobes. Forlarge N, the effect of the bias is less noticeable. In fact, as N — co, WN(f) —* 8(fT), so thaturn E[Xd(k)] = urn f R(f)WN(f — )dfN—*oo N—*oo=R(f)8([f - ]T)df= :R(f)8(f -R() . (3.49)The primary implication of the bias is that for finite N, the impulse that results from thetone that occurs in the PSD is estimated as a finite-magnitude pulse in the periodogram. Theshape of this pulse is proportionate to WN (f).Fig. 25 shows a typical periodogram of a received signal with SNR = 15 dB and SIR= 15 dB, calculated from N = 16384 samples collected at iV = 2 samples per symbol. The73predominant spike is due to the tone with frequency f = O.l/T, and occurs atk fNT = 0.1NT/T8 = 0.1N/N 819. (3.50)By finding the position of the spike in the periodogram, it should be possible to estimatethe frequency of the tone.1500 I I1000IC5000’-8192 -4096 0 4096 8192Position (k)Figure 25, Periodogram of Received Signal. N = 16384, N = 2, SNR = 15 dB, SIR = 15 dB.Fig. 25 also illustrates a problem inherent with the use of the periodogram. Since the PSDis dependent on the autocorrelation function of the received signal, it is essentially an averageall possible realizations of the signal. The periodogram, on the other hand, is calculated fromonly a single realization. Compare Fig. 25 with the actual PSD of the signal as shown in Fig.24. Note that the smooth portions of the spectrum appear very jagged in the periodogram.This is because the periodogram only estimates the PSD. Any given estimate can differ wildlyfrom its desired value, even though it equals its desired value on average. This differenceis a measure of the variance of the estimator. The smaller the variance, the more like likelythe estimator will be close to its desired value, The variance of the periodogram is a veryimportant consideration.74In the example given above, N was chosen to be excessively large to show how theperiodogram can be useful. A value of N = 2048 is much more reasonable in terms ofimplementation complexity, and Fig. 26 shows why the periodogram by itself is generallyimpractical. The tone, marked by an arrow in the graph, is indistinguishable. Because of thelarger bias associated with using a smaller value of N, the spike due to the tone is concealedby the random nature of the periodogram. If the tone had a weaker power, this problemwould be even more pronounced.150100I500-1024 1024Position (k)Figure 26. Periodogram of Received Signal. N = 2048, N = 2, SNR = 15 dB, SIR = 15 dB.The number of samples used to produce the periodogram is an important parameter. WhenN samples are used, the periodogram estimates the PSD at N discrete frequencies in the range[—p, ]. By increasing N, more estimates are produced, which are more closely spacedin the frequency range. This reduces the need to use interpolation to estimate the PSD atfrequencies not provided by the periodogram. In addition, increasing N causes the bias to bereduced which increases the magnitude of the spike due to the tone, making it easier to find.Clearly there is much to be gained by selecting a large value of N. Unfortunately, there are75some limitations. FFT algorithms have an order of N log N, and therefore the amount of timerequired to perform the operation grows non-linearly with increasing N. Similarly, hardwareimplementations become increasingly more complex. Another drawback is that more time isrequired to collect the block of data if N is increased.3.3.2 Average PeriodogramBecause of the large variance, the periodogram of a single block can not be used byitself to estimate the PSD, Instead, it is desirable to compute the periodograms from severaldifferent blocks of samples and average the results together. The resulting estimate will bereferred to as the average periodogram (AP). When D periodograms are averaged together,the resulting AP will be denoted by XD(k) and defined asXD(k) (3.51)where Xd(k) is the periodogram calculated from the dt1 block of samples using Eq. (3.28).Note that the same number of samples must be used for each periodogram, and no sampleshould be used for more than one periodogram. The sample blocks must not overlap.The mean of the average periodogram can be determined easily by applying Eq. (3.47),so thatE[XD(k)] = E[Xd(k)]D-1 --= j R(f)WN(f — df=1: R(f)WN(f - )df. (3.52)Similarly, the variance isVar[XD(k)] = E[X(k)]—76D-1D-1 D-1D-1= E[Xd(k)Xe(k)1 — E[Xd(k)IE[Xe(k)]dO e=O d=O eOD-1 D-1= COV[Xd(k),Xe(k)1 , (3.53)d=O e=Owhere Cov[.,.] denotes the covariance [281. A detailed analysis of the covariance has notbeen included in this thesis for the sake of brevity. However, different periodograms areessentially uncorrelated, and the variances of all the periodograms are identical, independentof block number. Using these properties the variance of the AP can be expressed asD-1 D-1Var[XD(k)] Var[Xd(k)]6(e — d)d=O c=O= Var[Xd(k)]= Var[Xd(k)] . (3.54)As can be seen by Eq. (3.54), the variance of the AP is a fraction of the variance of a singleperiodogram. By increasing D it is possible to reduce the variance to any arbitrary level. Fig.27 shows an AP calculated for the signal in the preceding example, calculated with D = 8and N = 2048. Note that the random perturbations have been significantly reduced, allowingthe tone to be detected easily. As more blocks of samples are processed and included in theAP, the variance decreases further. Unlike N, D is not a fixed parameter and can increasewithout bound. Given enough time, even the weakest tone can be detected in the AP.Aside from N and L, the other important system design parameter is N, the number ofsamples per symbol. To prevent aliasing, N must be greater than or equal to two. However,the practice of taking N > 2 is questionable. Although the periodogram is proportionateto N, the tone spike is no more or less prominent with different values of N. IncreasingN widens the frequency range over which the PSD is estimated, but all this additionalrange falls outside the signal frequency band, Since N remains unchanged, the points in the77150-1024Position (k)Figure 27. Average Periodogram (AP) of Received Signal.D = 8, N = 2048, N = 2, SNR = 15 dB, SIR = 15 dB,periodogram become more spread out so the frequency resolution is decreased, with fewerrelevant estimates. On the other hand, collecting samples at a faster rate requires less timeto collect a full block of N. However, in that shortened time span, a less valuable estimateof the PSD is produced. Therefore, it is concluded that two is the best value for N toestimate the PSD.3.3.3 Symbol Rate SamplingIf the primary goal was estimating the PSD of the signal, the optimal value of N wouldbe two. However, for the proposed tone canceller, estimating the frequency of the interferingtone is the main concern, accurate estimates of the true PSD are not really important. Bytaking N = 1 (i.e., sampling below the Nyquist sampling rate), aliasing is introduced. Theresulting average periodogram does not reflect the PSD of the received signal, but does containsufficient information to accurately estimate the frequency of the interference. There are threereasons for doing this: i) the implementation is considerably easier, ii) the tone frequencycan be estimated with greater accuracy, and iii) decision feedback (which will be discussed78-512 0 512 1024in Subsection 3.3.6) can be used to further improve the results. Only one sample is collectedper symbol, and it is collected at the symbol sampling instant. Throughout the remained ofthis chapter, only the case when N = 1 will be discussed.In Appendix B, expressions for the mean and variance of the resulting AP are derived,when only one sample per symbol is collected. They areE[XD(k)] = o + N0 + KZo(k) (3.55)andVar[XD(k)] j[(cT +No)2 +2(u +No)KZo(k)] , (3.56)whereZo(k)=WN(fi—Jr)—sr2 [w(f—— Nsin2— N)Ts]— sin2 7rfNTS— 2 (3.57)Nsrn — k]is the mathematical formula that characterizes the effect of the tone on the points in the averageperiodogram. This spike is largest for the value of k closest to fNT8, and is small for allother values of k. Also, note from Eq. (3.56) that the variance is inversely proportional to D.3.3.4 Frequency EstimationBy using the average periodogram it is possible to estimate the frequency of the interferingtone. The presence of a tone in the received signal is indicated by a large spike in the AP,with the position of the spike depending on the frequency of the tone. By finding the positionof the largest spike in the AP, the frequency of the tone can be determined. To illustrate thisprocedure, assume that enough blocks have processed so that the random nature of XD(k)is minimal, so XD(k) E[XD(k)].79As noted in Appendix B, the mean of the average periodogram isE[XD(k)] = + N0 + sinfNT3 (3.58)Nsin [fNT—kJwhich attains its largest value when k = k, where k is the integer closest to fNT8. To usethis property, let k be the position of the largest value in the average periodogram, so thatXD (kg) > XD(k) for all k E {—, — 1]. Then, one possible estimate for f is(3.59)Under normal operating conditions, the largest value of the AP always occurs at k, so thisestimate can be quite reliable. The only problem arises when the tone is relatively weakand only a few blocks have been processed. In this case the tone is hidden in the randomnature of the AP. However, as more blocks are processed, the variance decreases and the tonespike becomes prominent. As this problem only occurs for an extremely short time whilethe receiver tunes to a new channel, it can be disregarded. In general, it is safe to assumethat k = k, and that assumption will be made for the remainder of this section. Althoughthe estimate of Eq. (3.59) is very reliable, it is limited by the frequency resolution of theAP to an accuracy of(3,60)There is no error only if fNT8 happens to be an integer.By considering the shape of the tone pulse, it is possible to fine-tune the estimate of thefrequency to achieve significantly better performance. The shape of the spike is governed bysin2 (7rfNT8)Zo(k) = 3.61N sin2 (*[fNT3— k])which is plotted in Fig. 28 for N = 2048 and fT3 = 0.05, so fNT = 102.4. As expected, itappears as a large spike, which falls at k = 102. Fig. 29 shows the same function for points80around k only. The points marked with a circle are for integral k, and indicate the pointsfor which the average periodogram has been calculated. The dashed line shows the values ofZo(lc) when k is not an integer. In this case, Zo(k) reaches its peak at some point not includedin the AP, That is, Z0(k) is at a maximum at fNT8 = 102.4, while the largest value in themean of the AP is at k = 102. It is useful to define the difference between the two points as= fNT8— k. (3.62)Note that 6jj < 1/2. By finding 6 as an estimate for j, the frequency of the tone can beestimated as(3.63)10005000 I-1024 -512 0 512 1024Position (k)Figure 28. Plot of Zo(k) vs. k, for N = 2048 and fT = 0.05.To estimate 6j, a simple interpolation procedure is used. The second largest point inE[XD (k)] occurs at k + 1 if 6 > 0 or at k1 — 1 if S < 0. Consider the ratio of the largestpoint in E[XD(k)] to the second largest point. LetB E[XD(k)j — (cr + N0)= E[XD(k+l)]—(u+No)812000-!1500-1000 1500 Q -e../0 — _—G — i I I98 99 100 101 102 103 104 105 106Position (k)Figure 29. Plot of Zo(k) vs. k, for N = 2048 and fT = 0.05. For points around fNT only.Z0 (kg)— Zo(k±1)sin2 (7rf,NT)— N sin2 (-y[fNT_k])— sin2 (irfNT)Nsin2 (-,[f,NT—(k±1)])— sn2 ([fNT8— (k + 1)]) (364)— i2 ([fNT8— k2])Substituting, 8 = fNT5 — k into the above equation yieldsB =sin2 (*[i + ii) = sin2 (i. (3.65)sin ( ) sin ( )Since < -, both and .[+1— S] are small, so the approximation Sill X x can beused. Therefore[]2[±1 612 +1 2B 2 9 = — — 1 . (3.66)[-k] [Sj SjTaking the positive square root of both sides yields— 1 (3.67)82which implies+1=. (3,68)To take advantage of this relationship, note that XD(k) E[XD(k)] for large D, so usingB — XD(k) — ( + N) (369)XD(k+l)—(+No)in Eq. 3.68 produces a satisfactory estimate of 6j. Since N0 is unknown, this method cannot beused directly. However, since typically o >> N0, the effect of the noise can be disregarded,and o used in place of + N0. Alternatively, a method for estimating o + N0, such as theone discussed along with estimating the power ratio in Section 3.3.5, can be used instead.In summary, the frequency of the interfering tone can be estimated as follows. First, findthe largest value in the average periodogram, and let k denote its position. If the largestneighbour of XD(kj) is XD(k + 1), find the ratioB = XD(kj) 2’ (3.70)XD(k + 1) — a8and estimate the frequency of the tone by1:_k 1+V 371NT3On the other hand, if the largest neighbour is XD(k — 1), find the ratioB—— 2’ (. )XD(kj — 1)—and estimate the frequency of the tone byk i+-./’ 3732NT8 NT3 ‘833.3.5 Power Ratio EstimationFor proper cancellation of the interfering tone, the ratio C=(o + N0)/K must beestimated as well as the frequency of the tone. This estimate can also be extracted from theaverage periodogram.To produce this estimate, a couple of intermediate quantities must be generated. The firstone is the average of all points in the AP. LetN1k=-(3.74)The second quantity is the sum of only a few points in the AP, centered around the positionof the tone spike, k. Let M denote the number of points to sum, where M is odd, and letc = (M— 1)/2. This quantity is then given byk +cS2 = XD(k) . (3.75)k—k,—cOnce S1 and 52 have been calculated, estimators for u + N9 and K can be generated withNGM(S)Sl—52= NGM(Si) — M (3.76)andS2—MS102= NGM(S) — M ‘ (3.77)respectively, where the functionk, +cGM(6) Z(k) (3.78)k=k—crepresents the fraction of the power of the tone that is contained in the M points centeredaround k. This is a function of j, which was defined in Eq. (3.62) and measures howclose the frequency of the tone is to the spike in the AR Fig. 30 contains plots of GM(6j) forvarious values of M when N = 2048, In practice, GM (Sj) can be approximated by using linear841.00Figure 30. Plot of GM(Sj) vs. 6, for N = 2048 and for various values of M,interpolation, and since S is unknown at the receiver, the quantity S = fNT — k is usedin its place. As confirmed by the experimental results in the next chapter, this approximationhas little impact the performance of the interference canceller.In Appendix C, it is shown that 01 and 02 are unbiased estimators for o + N0 and K,respectively. It is also shown that their variances are approximatelyVar[0i]— (N-M)D + N0)2 + ( — 2(u)2) N _M]andVar[02] + No)2NM + 2( + No)K] . (3.80)Note that Var[01] and Var{02] are roughly proportional to 1/ND. This implies that theseestimates are very accurate for large N, and, in addition, the accuracy increases as more blocksare added to the AP. Because of these properties, the quantity(3.81)02can be used as an estimate for the power ratio C = (o +N0)/IQ. Note, however, that Cis not an unbiased estimator for C since division is not a linear operation. Nonetheless, it0.950.900.85-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0,1 0.2 0.3 0.4 0.585is generally sufficient for the purposes of the tone canceller, usually providing very reliableestimates of the power ratio. There is, unfortunately, a situation where this is not the case.When the tone is very weak, C can be a very large number. In fact, C —> cx as K? —* 0.The same can not be said for C, however. By inspecting Eq. (3.80), it is apparent that thevariance of 02 does not tend to zero as I — 0. Because of this non-zero variance, 02 canbe much larger than I when the tone is weak, As a result, C does not tend to infinity asK? — 0, but instead levels off at some maximum value. In implications of this problem areexplored more thoroughly in the next chapter.In summary, the estimate for the power ratio can be found by first calculating S and52 with Eqs. (3.74) and (3.75), and then using the results to calculate 0 and 02 with Eqs.(3.76) and (3.77). Then calculate(3.82)as an estimate for C = (u + No)/K.. The estimate 0 could also be used by the frequencyestimator instead of o + N0 in Eq. (3.69).3.3.6 Decision Feedback for Frequency and Power Ratio EstimationAs in the case of the interference estimator (see Subsection 3.2.3), the strong presenceof the data signal in the received samples is the primary factor limiting the accuracy of theparameter estimators. Here, too, the use of decision feedback significantly improves resultsand is very easy to implement. As each sample, rd(n), is received by the parameter estimator,the corresponding detected symbol, 1n+dN’ is subtracted from it. The periodogram for thedth block is then calculated asN—i 2Xd(k) = (rd(n)—e22. (3.83)The average periodogram resulting when decision feedback is used has essentially the sameproperties as the one described earlier in this section. Theoretical analysis is difficult, but86by replacing o with o E Ia— Ia2],the mean and variance are approximately (seeEqs. (3.55) and (3.56))E{XD(k)] + N0 + KZo(k) , (3.84)andVar[XD(k)] + No)2 + 2(? + No)KZ0(k)] . (3.85)By using the same procedure described earlier, an accurate estimate of f can be produced. Also, the quantity C estimates (o +N0)/I when feedback is used, instead of(o- +N0)/K. However, if feedback is also used by the interference estimator, C estimatesthe desired value.3.4 Summary and ConclusionFig. 31 contains a block diagram of the proposed tone interference canceller. The receivedsignal, r(t), after demodulation and filtering, is sampled at the symbol sampling instants,t = aT3. From each sample, Ra, an estimate of the interference present in the sample, Za, issubtracted. The resulting decision variable, R, is used by the symbol detector to produce thedetected symbol, Ia, which is expected to be the same as the transmitted symbol. Since thetone interference has been removed from the decision variable, this is likely to be the case.To estimate the interference present in each sample, a simple tapped delay line is used,as is shown in Fig. 32. The tone is estimated asZa=Wn (Ra_n — Ia_n), (3.86)where {R0_} are previously received samples, and {ian} are previously detected symbols.The weights, { }, are calculated by=exp (j2nTs), (3,87)L+C87Rawhere fj is an estimate of the frequency of the interfering tone, and C is an estimate of(u? + No)/K. The quantity equals E [ia—2],an unknown value that depends onthe power of the transmitted symbols and the probability of detection error. Both f and Care generated by the parameter estimator. The number of samples used, L, is a fixed designparameter. It should be selected to be as large as possible, but inaccuracies in f place atheoretical limit on how large it can be,r(t) DataSinkFigure 31. Tone Interference Canceller.++ZaFigure 32. Interference Estimator88In practice, the cost of a long tapped delay line will be the primary limitation on L.However, an alternate realization can be used that is considerably cheaper. To motivate thisrealization, letQa=RaIa, (3.88)so thata=WnQan— 1 j2irfnTane— 1 j27rfj(fl+1)— a a_(n+1)en=O= L ++ — Qa_(L÷1)e32+1]= [L aQa_i_ne22nT8] ej2T + W1Qa_i— W(L+1)Qa_(L+1)= a_ie2Ts + WiQa_i— W(L+1)Qa_(L+1). (3.89)This equation provides a means for generating the interference estimate at t = aT3 by updatingthe estimate from the previous sampling instant. Using this property, the interference can beestimated with the device shown in Fig. 33. Since this implementation requires fewer addersand multipliers, a much larger value of L can be used without significantly increasing cost.The frequency and power ratio are estimated from the AP computed from the receivedsignal. The detected symbol is subtracted from the received sample, and the result is storedin a buffer of length N. Let rd(n) denote the rh value stored in the buffer while processingthe dth block. Thenrd(n) = R+dN— ‘n+dN (3.90)89ZaRaFigure 33. Reduced Complexity Realization of Interference EstimatorWhen the buffer is full, a FFT is performed on the data. The transformed block, {Rd(k)}also contains N values. The dt periodogram is computed from the transformed block asXd(k) = . (3.91)This periodogram is included in a running average of all the previously computed penodograms. This average periodogram isXD(k) = (3.92)where D is the total number of blocks currently processed. The block length, N, is anotherfixed design parameter. It too should be selected to be as large as possible, however economicconsiderations limit its size.From the data in the average peniodogram, fj and C are calculated. To estimate thefrequency, the largest value in the AP is found, and its position is denoted by k. If XD(k + 1)is found to be greater than XD(kj — 1), the quantityB = XD(kj) — 01 (393)+I;.1I90is computed, where 0 is an estimate for o + N0. If no estimate is available, a value ofzero can be used instead. With B, the quantity Sj is computed by = 1 / (i + \/), andthe frequency is estimated as = (k + ) /(NT5). If, however, XD(k + 1) is found tobe less than XD(kj — 1), the quantityB — XD(kj) — 0 (3 94)XD(k—1)—01is computed instead, and, with = _ii(i + the frequency is estimated as =(k + )/(NT3.To estimate the power ratio, the quantitiesN1=, (3,95)andk +cS2 = XD(k) (3.96)are calculated from the average periodogram, where c = (M — 1)/2, with M being the numberof points to sum around the spike caused by the tone. With 51 and 52, the quantitiesNGM(Si)S1—52Oi = , (3.97)NGM(Si) -Mand02 = 52—MS1 (3.98)NGM(Si) -Mare computed, where GM (82) is the fraction of the tone spike that is contained within theM points centered around k2. Note that 01 is an unbiased estimator of o’ + N0, and 02 is anunbiased estimator of I. The estimate C is finally generated by C = 01/02.Using the procedure outlined above, extremely effective interference cancellation will beachieved. Theoretically, near-optimal performance in terms of the MSE is possible. In thenext chapter, experimental results are presented to support this claim.91Chapter FourComputer Simulation Results4.1 IntroductionThe theoretical analysis of the proposed interference canceller included in the previouschapter indicated that it should have near-optimal performance in the MMSE sense. Furthermore, using decision feedback and increasing L are both expected to yield better tonecancellation since these actions directly affect the theoretical bound on performance. On theother hand, the block length, N, and the number of blocks processed, D, do not affect thebound, but they do affect how close to the bound the canceller performs.To support these theoretical claims, computer simulations were employed. This chaptercontains a summary of the simulation results, confirming that effective cancellation of toneinterference is possible. After a description of the simulation system in Section 4.2, thesimulation results when decision feedback is not employed are discussed in Section 4,3, InSection 4.4 the results when decision feedback is employed are studied.924.2 Computer Simulation SystemAll the experimental results presented in this chapter were produced by computer simulation of the communication system and interference canceller. Appendix D contains acomplete listing of the source code, written entirely in C. All results in this chapter pertainto a 16—QAM system with root raised-cosine filters with a roll-off factor of a = 0.2, and theaverage transmitted signal power is P = 10/T8. The frequency of the interfering tone andits initial phase were arbitrarily selected as = 0.05/T3, and çLj = 0.0. Of course, thesevalues were not available for use by the tone canceller. The tests were performed over awide range of tone and noise powers.The interference canceller was implemented using the reduced-complexity model of Fig.33, and the frequency and power ratio estimates were generated from APs using symbol-ratesampling. Unless otherwise stated, a filter length of L = 32 was used, and a total of D = 32blocks length of N = 2048 were used to compute the AP. In addition, a value of M = 21 wasused in Eq. (3.75) and Eqs. (3.76)-(3.77) to generate the intermediate estimates 01 and 02.To accurately measure the average residual power, a very large number of trials mustbe executed to ensure statistical convergence of the results. Since a total of ND = 65536samples were processed for each trial, convergence was hastened by using the average overthe last ND/4 = 16384 samples as the estimate of the residue for that trial. Since the samplesin a single trial are not strictly independent, each trial was repeated 2000 times with differentrandom data, and the results were averaged together.4.3 Cancellation Without Decision FeedbackIn Section 3.2, the theoretical performance of the linear MMSE estimator was studied,and the maximum performance gain was plotted in Fig. 17. For the sake of convenience93in comparison, this figure is repeated in Fig. 34. As noted, this data represents a boundthat limits the performance gain of all tone interference cancellers that use linear estimateswithout employing decision feedback to reduce the effect of the data signal. In Fig. 35 thesimulated results for the proposed canceller are presented. By inspecting this graph it is clearthat if the SIR is less than about 20 dB, some cancellation of the tone is possible. Thisperformance is almost identical to the theoretical bound in Fig. 17. If the SIR is in the 20—40dB range, very little cancellation occurs. Again, this performance is in agreement with thebound. However, if the SIR is greater than about 40 dB a minor problem is apparent. Inthis region the canceller exhibits a negative gain, indicating that additional distortion is beingintroduced. While this situation is undesirable, it should be pointed out that this additionaldistortion is very insignificant, since it is proportionate to the strength of the tone, whichitself is very weak in this region.6050E2010050 60Figure 34, Theoretical Performance Gain vs. SIR, L= 32, N = 2048, D = 32, without decision feedback0 10 20 30 40Signal-to-Interference Ratio (dB)9410.05.°0.0-5.00 10 20 30 40 50 60Signal-to-Interference Ratio (dB)Figure 35. Simulated Performance Gain vs. SIR, L= 32, N = 2048, D = 32, without decision feedbackAnother way to express the effectiveness of the interference cancellation is consider to theSRR, which is plotted in Fig. 36 for the simulated results. Clearly all tones with a SIR greaterthan 0 dB are reduced to residues with SRRs of greater that 8 dB. As such, the probabilityof transmission error will be lower that if no attempt was made to cancel a strong interferingtone. As noted, when the SIR is greater that about 40 dB, the SRR is less that the SIR, butthe SRR is nonetheless greater than 40 dB when the tone is this weak. Clearly this will haveno practical effect on the probability of transmission error. As the SIR increases withoutbound, the SRR, instead of also increasing, actually levels off to a residue floor. Resultsof simulation when no tone is present have indicated that this floor is at about 60 dB. Theproposed canceller can not reduce the residue below the floor, and interference that is weakerthan the floor will be increased to that level. This phenomenon is a result of inaccuracies inthe estimate of the power ratio, and in particular, the problem with 02 not tending to zero95as I —* 0 that was discussed in Subsection 3.3.5. Fortunately, since the variance of 02 isinversely proportional to D, the residue floor decreases as D is increased. As the amountof time that the receiver has been tuned to a single TV station increases, more blocks areincluded in the AP, thereby reducing the variance of 02, leading to a lower residue floor. Thisis confirmed through simulation in the next section.60500201000 10 20 30 40 50 60Signal-to-Interference Ratio (dB)Figure 36. Simulated SRR vs. SIR, L = 32, N = 2048, D = 32, without decision feedbackWith the exception of the introduction of distortion when the tone is weak, the proposedcanceller performs as well as can be expected. For practical purposes, the bound has beenachieved. However, without the use of decision feedback, the gain is limited to less than 8dB. Significantly better results can be achieved when the detected symbols are used to reducethe power of the transmitted data in the received signal by means of a feedback mechanism,964.4 Cancellation with Decision FeedbackAs noted in Section 3,2, using the detected symbols to reduce the power of the data signalbefore attempting to estimate the tone is expected to provide substantially better interferencecancellation. The theoretical bound in this case, presented originally in Fig. 23, is repeatedin Fig. 37. Comparison with the simulated results, which are shown in Fig. 38, confirms thatthe performance gain is better than when feedback is not employed, and that the results arevery close to the bound over a wide range of distortion conditions. Of course, when the toneis weak (i.e., SIR > 50 dB) additional distortion is still introduced due to the residue floor,but it is at a noticeably lower level, Also, when the noise is strong (i.e., SNR < 10 dB), theperformance gain is somewhat lower than predicted by the bound. This occurs because ofthe high number of transmission errors that occur at these noise levels. Since the data is notcompletely removed less accurate estimates of the tone are produced. Unfortunately, withoutknowing the true MMSE it is impossible to determine whether this is a problem with theproposed estimator using sub-optimal weights or actually a property of the optimal estimator.Nonetheless, performance is better at these noise levels than when decision feedback is notused. In addition, since reliable communication is not possible when the noise is this strong,the performance of the canceller is not particularly relevant.Aside from using decision feedback, the theoretical bound can only be improved byincreasing the number of samples on which the estimate of the tone is based. In the previousperformance results a value of L = 32 samples was used. In Fig. 39 the theoretical boundon performance when L = 1000 is shown. Note that by using the reduced complexity modelof Fig. 33, it is possible to implement the proposed canceller economically with such a largevalue of L, since the number of multipliers and adders does not increase with L, as in thecase of the transervsal filter. This is a distinct advantage over traditional linear estimators.Simulated performance evaluation results are presented in Fig. 40. Other than the residue97502010010 20 30 40 50 60Signal-to-Interference Ratio (dB)Figure 37. Theoretical Performance Gain vs. SIR, L= 32, N 2048, D = 32, with decision feedback.floor, which is substantially higher, the performance is slightly sub-optimal in the regionswhere the performance gain is small (i.e. G’ < 10 dB). Inaccuracy in the frequency estimatesare the likely cause. Nonetheless, performance is considerably better that when L = 32, withabout 15 dB better cancellation.As more blocks are included in the AP, it is expected that more reliable estimates of fand C will be produced. In particular, the reduced variance of °2 should lead to a reductionof the residue floor. This is confirmed by the simulated performance results depiction in Fig.41, which shows that performance gain after D = 128 blocks have been processed, as opposedto the results presented in Fig. 38 when D = 32. Note that the residue floor does not affectthe performance for any of the interference conditions depicted in this graph.Although performance improves over time as D increases, the parameter N is much moreimportant. The results of repeating the simulation with N = 32 and D 2048 are presented09850c1)20100Figure 38. Simulated Performance Gain vs. SIR, L= 32, N = 2048, D = 32, with decision feedback.in Fig. 42. Note that even though the same total number of samples have been processedas in the simulation of Fig. 38, the residue floor is much higher. Otherwise, the results arestill very acceptable.0 10 20 30 40 50 60Signal-to-Interference Ratio (dB)99807050(tQ3020100Figure 39. Theoretical Performance Gain vs. SIR, L = 1000, N = 2048, D = 32.807060I)C)I0-1050 60Figure 40. Simulated Performance Gain vs. SIR, L = 1000, N = 2048, D = 32.0 10 20 30 40 50 60Signal-to-Interference Ratio (dB)0 10 20 30 40Signal-to-Interference Ratio (dB)10050-d00 10 20 30 40Signal-to-Interference Ratio (dB)-d2010050 60Figure 41. Simulated Performance Gain vs. SIR, L = 32, N = 2048, D = 128.50403020100-10-2050 60Figure 42. Simulated Performance Gain vs. SIR, L = 32, N = 32, D = 2048.0 10 20 30 40Signal-to-Interference Ratio (dB)101Chapter FiveConclusions and FutureResearch TopicsThe various all-digital HDTV systems currently being considered by the FCC are capableof delivering high resolution, wide aspect ratio video signals to home viewers over existingcable TV networks. The impressive compression rates of the source encoders allow fortransmission without the use of additional bandwidth. Furthermore, the powerful errorprotection ensures high quality image reception under normal operating conditions. Thepresence of a strong interfering tone, however, may defeat the ability of the error correctingcodes, leading to severe image degradation. In this case the use of some form of tonecancellation may be required at the receiver.In this thesis an effective tone canceller has been proposed. This device is based ona linear estimator, and is a suboptimal direct implementation of the minimum mean-squareerror estimator, with performance that is very close to the theoretical optimum under most102circumstances. Simulated performance analysis confirmed that cancellation can be applied toreduce the tone to negligible levels, thereby allowing impairment-free image reception.Finally, the following is a list of topics which can be considered for further research.• MULTIPLE TONESIf multiple tones, closely spaced in frequency as in the case of CTB, interfere with theHDTV signal, some cancellation is expected from using the proposed device directly, butperformance may be far from optimal. Further theoretical and experimental analysisis required. One the other hand, for cancelling tones that are well separated somemodifications will definitely have to be made, However, these modifications shouldbe relatively minor as this problem was a secondary design consideration of the proposedsystem. Nonetheless, extensive analysis and testing is required to support this claim.• NON-STATIONARY TONESThe performance of the proposed canceller should be investigation if the interfering toneis non-stationary. Methods for tracking slow variations in the tone’s frequency andpower should be developed.• THRESHOLD CANCELLATIONTo prevent distortion from being introduced because of the residue floor, methods fordetermining if the tone is below a certain threshold, and hence too weak to cancel shouldbe developed. Under this condition the canceller would be disabled.• ALTERNATIVE FREQUENCY ESTIMATION METHODSAlthough the use of the periodogram provides an extremely reliable estimate of the tone’sfrequency, it may be unduly complex. Since the accuracy provided is excessive, a lesscomplicated method may be capable of providing adequate results, In addition, theperiodogram is ill-suited for tracking frequency variations and distinguishing between103closely spaced tones. In hindsight, the periodogram may not be the most versatile toolfor frequency estimation. Alternate methods should be investigated.104References[1] R. E. Wiley, “A U.S. HDTV Update,” in HDTV ‘90 Colloquiem Proceedings, vol. 1,(Ottawa), pp. 1.4.1—1.4,5, 1990.[21 D. Wood and J. 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Goodwin et at., “Sinusoidal Disturbance Rejection with Application to HelicopterFlight Data Estimation,” IEEE Transactions on Acoustics, Speech, and Signal Processing,vol. ASSP-34, pp. 479—484, June 1986.[22] W. K. Pratt, Digital image Processing. New York: John Wiley & Sons, 2nd ed., 1991.[23] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications.Englewood Cliffs, N.J.: Prentice-Hall, 1983.[24] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 2nd ed., 1989.[25] H. Meyr and G. Ascheid, Synchronization in Digital Communication, vol. 1. New York:John Wiley & Sons, 1990.[26] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York:McGraw-Hill, 1984,[27] M. L. Honig and D. G. Messerschmitt, Adpative Filters: Structures, Algorithms andApplications. Boston: Kluwer, 1984,[28] S. M. Ross, A First Course in Probability. New York: Macmillan, 2nd ed., 1984.[29] S. Kay, Modern Spectral Estimation: Theory and Application. Englewood Cliggs, N.J.:Prentice Hall, 1988.[30] J. G. Proakis and D. Manolakis, Digital Signal Processing. New York: MacMillan, 2nd ed.,1992.[31] N. Mohanty, Signal Processing: Signals, Filtering, and Detection. New York: VanNostrand Reinhold, 1987.106Appendix AMMSE Tone EstimatorIn this appendix the assertion that Za, given byLZa = wnRa_n , (A.l)is the linear minimum mean-squared estimator of Za when—____)Wfl (A2L+Cwhere C= ( + No)/K and cj = 2fT is proved. This is accomplished by showingthat the partial derivatives of the MSE with respect to the real and imaginary parts of eachw are equal to zero. To begin, note that the MSE is7 E[Za_a]= Z Za — Z E[Za]— Za E [:] + E[a]= ZZa — ZwnE[Ra_n]— ZawE{R_nj+ wwmE[R_nRa_m]. (A.3)n=1 m=1To use this expression, recall from Section 2.6 that since ‘a and Na are zero mean randomvariables,E[Ral = E[Ia + Na + Za]= Zn , (A.4)andE[RjRm1 = E[IIm1 + E[NNm1 + ZZm=—n) + No6(m—n) + ZZm. (A.5)107Substituting these expressions into Eq. (A.3) yields7 = ZZa — Za*WnZa_n — Za w:za*_n+ > WWm [(cr + No)S(m — n) + Z_mZa_m]. (A.6)n=1 m=1From the definition of Za given by Eq. (2.62) it is clear thatZa_n = z(aT8 — nT3)== K ei2h1f.aT e_j2fmnTs= Zae_j2T$Zae3Th, (A.7)so Eq. (A.6) becomes7 = ZZ — ZZa wne2— ZaZ: we3+ Z:Za wwme_2wt(m_n) + ( + N) (A.8)n=1 m=1 n=1By noting that ZZa = K?, the above equation can be expressed as7 = K — — + wwme_m_]n=1 n=1 m=1+ ( + N0) (A.9)To find the partial derivatives it is convenient to express each w in terms of its realand imaginary components. Let x = Re{w] and y = Im[w] so that w = x, + jy.Then Eq. (A.9) becomes= I? [i — (x + jy)e — (x — JYn)e]108L L+ K (x71 — jyn)(Xn + jyn)e1(m_n=1 7n=1Ln=1L L= K [i — x(e’ + e’)—j yn(e —n=1 n=1L L+ (x, — jyn)(xn + jyn)e_3m_n=1 m=1Ln= 1L L L= (+No)(x +y) + KIln=1 L n=1 n=1 jL L+ K [XnXrn + XnYm — nXrn + ynym1e_Jm_. (A.1O)n=1 m=1The derivative of ‘y with respect to x for c = 1,.. . , L is87= (u+No)2xc+I[—2cos(wc)]CL L+ K [xS(m—c) +n=1 m=1L L[S(fl)y — y?6(mc)]en=1 m=1= 2(o + N0)x — 2I cos (L.’c)L+ p12 [ _Jwj(c_n) _w(n_c)]xe +xen=1L+ jiQ — yne7]n=1= 2(u + No)x — 2K cos (wc)L+ 2I [x cos [j(n—c)] + y sin [w(ri—. (A.l1)n=1To show that this is equal to zero when the coefficients are given by Eq. (3.8), it is necessary109to substitute1Re[w1]= cos(wn) , (A.12)L+Cand1= Im[wn]= L +sin (cLn) (A,13)into Eq. (A.11). This leads to1= 2(u + No) cos (cc) — 2K? cos (wic)L+CL 1+ 2K?[cos (win) cos wj(n— c)] + L + sin (wjn) sin [w(n — c)]jL+Cnzr=11= 2(u + No) cos (wc) —2K?(L + CL+C L+C)c0s(L+2K1i L+CC0S1n=1= 2 cos (wc)L[(u + N0)— K?(L + C) + K?L]___+No’\=2cos(wic)LiC[(u +No) _I(L+ ) +K?L]= 2 cos (wc)L + C[0]= 0 (A.14)for all c = 1,.. .,L.Following a similar procedure for the derivatives with respect to Yc yields— 2 + No)y— 2K? sin (wjc)L+ 2K? [—x sin w(n — c)] + y cos [w(n — c)]] , (A.15)n=1and substituting for x and y yields1= 2( + N0) sin (wjc)— 2K? sin (wjc)8Yc‘9 L+C110+ 2K [L+1c cos (win) sin [w(n — c)] + L sin (win) cos [wj(n — c)]]= 2(a + N0) L sin (wc) — 21q() sin (wc)+ 2I Zsin[wj(n_ n+c)]= 2sin(wic)L1+ No)— K?(L + C) + KL]= 2 SIll (wc)L += 0 (A.16)for all c = 1,. . . , L. The results in Eq. (A.14) and (A.16) show that the partial derivativesof y are all equal to zero when= L__)Wfl (A.17)To complete the proof it should be shown that the point given by {w} relates to theglobal minimum of -y, and not a local minimum, or a maximum or saddle. However, it wasfelt that such completeness is beyond the scope of this thesis. Nonetheless, it can safely beassumed that the coefficients given by Eq. (3.8) do minimize y and thereforeZa= L c Ra_ne’ (A.18)is the linear MMSE estimator of Za.111Appendix BDerivation of Mean and Variance of AveragePeriodogram with One Sample per SymbolIn this appendix, expressions for the mean and variance of the points in the averageperiodogram are developed. This analysis is only valid if one sample per symbol is collected,at the symbol sampling instant. It is tailored specifically for QAM signals. The proof beginsby finding various moments of the components in the received samples (Section B. 1). Theseare used to find similar moments of the finite Fourier transformed blocks (Section B.2). Theseresults are in turn used to find the mean and covariance of the periodograms (Section B.3).Finally, the mean and variance of the average periodogram is determined (Section B.4).B.1 Signal SamplesUsing Eq. (2.28), the samples of the received signal, rd(n), can be expressed in termsof samples of the three component signals asrd(n) r(nT + dNT)= v(nT + dNT8)+ n(nT8 + dNT) + z(riT8 + dNT)Vd(fl) + fld(fl) + Zd(fl). (B.l)The three components will be treated separately.The first component is due to the transmitted data signal, and each sample can be expressedby using Eq. (2.34) asVd(fl) v(nT + dNT)= Iah(nTs+dNT38a= —00112= Z I(+dN_)h(cTs)C —00= ‘(n+dN) , (B.2)since h(cT5) = S(c) because of there is no intersymbol interference due to the Nyquistfiltering assumed (see Eq. (2.6)).For all typical QAM signal constellations with each symbol equally likely, the followingproperties hold: E[Ia} = 0, E[IaIa] = 0, and E[IIaIa] = 0. Let o = E[IIa] and= E[IIaIIa]. As it is assumed that no correlation exists between the symbols transmittedat different times, the following properties hold:E[I1b] = crS(b— a) , (B,3)andE[IIbI1d] = oS(b — a)(d — c)6(c — a)+ ()2b- a)6(d - c)[i - 8(c - a)]+ (a)2S(d - a)S(c - b)[l-- a)]=- 2(] 6(b - a)(d- c)6(c - a)+ (u)2b- a)S(d -+ (2)6(d— a)6(c — b). (B.4)From these properties the following moments can be found:E[vd(n)] = = 0 , (B.5)E[v(n)ve(m)] = E [n+dN)I(m+eN)]=u6(m— n + (e — d)N)= — n)S(e — d), (B,6)113E[vd(n)v(m)] = E[I(n+dN)I(m+eN)]=0, (B.7)E[v(ni )vd(ml )Ve(fl2)] = E [1ni+dN) I(mi+dN)I(n2+eN ]=0, (B,8)andE[v(n1)vd(ml )v (m2)vd(m2)] = E [Ifl +dN) I(ml+dN)I+eN)I(m2+e )]=— 2(u)]8(mi — ni)6(m2— Th2)6(fl2 — ni + (e — d)N)+ (u)28(mi — m1)S(m2— n2)22+ (u5) S(m2 — n1 + (e — d)N)S(n2— m1 + (e — d)N)= [ — 2(u)2]S(m1 — nl)8(m2 — n2)(n2 — ni)S(e — d)+(2)28(—ni)6(m2— n2)+ (u)2S(m— nl)6(n2 — mi)S(e — d). (B.9)The second component of Eq. (B.1), nd(nj, is due to the AWGN signal. The samplesnd(n) are statistically independent complex Gaussian random variables with zero mean anda variance of N0. Any zero mean complex Gaussian random variable, Y, with variance 4,can be treated as two real Gaussian random variables, F and G, such that Y = F + jG. Fand G are statistically independent with identical distributions, zero mean, and variances of4/2. Therefore they have the propertiesE[F] = E[G] = 0, (B.lO)E[F2] = E[G2] =, (B.1l)E[F3] = E[G3] = 0 , (B,12)114E[F4] ===32 (B.13)andE[FG] = E[F]E[G] =0. (BJ4)Using these properties, the following moments of Y can be found:E[Y] = E[F+jG] E[F]+jE[G] = 0, (B.15)E[Y*Y]=E[F2+G = =4, (B.16)E[YY]=E[F2H-j2FG—G]= —j2(0)—=0, (B.17)E[Y*YY]= E[F3 + jF2G + FG2 + jG3]2 2=0+j(0)+(0)+j(0)= 0 (B.18)andE[Y*YY*Y1= E{F4 + 2FG + G4]= 32)22(uY)(uY) 32)2= 2(4)2. (BJ9)For statistically independent samples of a Gaussian random process, the following properties exist:= — a), (B.20)andE[Y*YbY*Yd]= 2(4)6 b— a)(d — c)6(c— a)115+ (4)26(b - a)6(d - c)[i - 6(c - a)]+ (4)2d- c)(c - b)[1 - - a)]= (4)26(b - a)6(d- c) + (4)26(d - a)S(c - b). (B.21)Extending these properties to samples of the noise signal yieldsE[nd(n)] = 0, (B.22)E[n(n)rie(m)] = No8(m — n)5(e — d) , (B.23)E[nd(n)n€(m)] = 0, (B.24)E[n(n1)nd(m1)ne(n2)] = 0 , (B.25)andE[n(n1)nd(m1)n(n2)n€(m2)] = NS(mi — n1)(m2— fl2)+ — ni)S(n2 — mi)S(e — d). (B.26)Finally, the third component of Eq. (B.l), zd(n), is deterministic and is a result of theinterfering tone. From Eq. (2.42), it can be expressed asZd(Th) z(nT3 + dNT8)= ic . (B.27)B.2 Finite Fourier TransformThe FFT of the 1rh block of samples, as defined in Eq. (3.27), isRd(k) rd(n)e J2N (B,28)71 0116This can be expressed in terms of the three component signals by definingVd(k) vd(n)e_12, (B.29)Nd(k) nd(n)e2, (B.30)andZd(k) Zzd(n)e_12, (B.31)so thatRd(k) = Vd(k) + Nd(k) + Zd(k) (B.32)Once again, each of the components will be treated separately,Moments of the first component can be found by applying the moments given in Eqs.(B.5)-(B.9). Therefore,E[Vd(k)] = E[vd(n)]eJ2= 0 , (B.33)N-i N-iE[V(k)V(l)] =n=O m=ON-i N-i= Zu5(m_n)6(e_d)e_j2mNn=O m=O= e_i26(e— d)= oNS(l— — d) , (B.34)N-i N-iIm knE[Vd(k)Ve(l)1 = E[vd(n)ve(m)]e_22e_3n=O m=O= 0 , (B.35)117N-i N-i N-iE[v;(k)vd(k)ve(l)1 = E[v(ni)vd(mi)ve(n2)je_j2e_j2mjvn1=O mj=On2=O= 0 , (B.36)andE [Vd* (k) Vd (k) V’ (1) V (1)]N-i N-i N-i N-i•,k(m1—r)_______= E[v(ni)vd(mi)v(n2)ve(m2)]e N — Nn1Om1=O fl20 m2Or [2(2)2] 6(mj —n)6(m2_nS(nj)S(e—d)N—i N—i N—i N—i X Nej2m22). k(m1 —n1)__ ____= k(m1-‘ii)_eJ21(m2_2)na=O mi=O n2=O m2=O + (a)2S(mi_ni)S(m_)e37r N — Nk(mj —n1) t(m2—2)[ + (J)2S(m_ni) (nmi)6(e_d)e N e2 NN-i N-ik(n1—).________=— 2(cT)2] > S(n2-ni)S e—d)e N Nn1=O fl20N-i N-i+(2)2 ej27 1—1) ej2i22)n1O fl2ON-i N-i__(2)2:: >)i2?rk(flfll) j2hh12)Nni=O fl20= — 2(u)jNS(e— d) + ()2N2 +()2N26(l — k)S(e — d). (B.37)Using Eqs. (B.22)-(B.26), the moments of the second component of Eq. (B.32), Nd(k)can be found. They are:N-iE[Nd(k)] = E[nd(n)]e_)27n=O= 0 , (B.38)N-i N-iE[N(k)Ne(l)1 = E[n(n)ne(m)]e 2iej2nO mON-i N-ilm—kn= No6(m— n)5(e — d)e2 Nm=O m=O118N-iN0 e_j2= N (e—d)nr0= N0S(l— k)S(e — d) , (B.39)N-i N-iE[Nd(k)Ne(l)j=E[nd(n)ne(m)1e27e_J2n=0 m=O= 0, (B.40)N-i N-i N-iE[N (k)Nd(k)Ne(l)] = E [n(ni)nd(mi)n(n2)]ei2‘e_22m1-ni)Nn1=0 m1=0n20= 0 , (B,41)andE[N (k)Nd(k)N(l)N(l)jN-i N-i N-i N-ik(mi —n1) t(m2—)= ) > E[n(nl)nd(mi)n(n2) eje N e2 Nnj=0 m1=0n2=0 m2=0N-i N-i N-i N-ik(m1—n) .21(m—n)= N,6(mi— ni)6(m2— n2)e’ Nm=0 mi=0 fl20 m2=0N-i N-i N-i N-ik(m1—‘ii)_________+ N5(m— n1)(n2 — mi)S(e — d)e2 N ej2m2!2)n1=0 m1On2Om2=0N-i N-i= N ) ej2khlv_1) e_j22fl2)n1=0n2=0N-i N-i. k(n2—1) . i(n1—n2)+ N — d)e3’ N Nn1=0 fl20= N02N2 + NN2S(l — k)5(e — d) . (B.42)For the third component of Eq. (B.32), the FFT can be computed directly. Using Eq.(B.27), Zd(k) can be expressed asN-iZd(k) zd(n)eJ2NnO119N-i= ::n=ON-i= Kej2jdN ej2(ji — — )nTn=O= Ijei2dNTse [(f — )NT8]ej_)NTsSill—= Kje22 NTe3 sin [(fNT8— k)] eui_k) (B.43)sin [-.(fNT8— k)] eji(fiNT5_k)Note that= Ksin2 [7r(fNT5— k)]sin [-CINTS — k)]— K2sin2 [TrfjNT5} (B.44)—Sill {-kCINTS — k)]It is useful to define the functionii2 ‘irfNT8Zo(k) = (B.45)Nsin(fNT8—k)so that= KNZo(k). (BA6)B.3 The PeriodogramFrom the definition of the periodogram in Eq. (3.28),Xd(k) (B.47)it has a mean ofE[Xd(lc)] = E[Rd(k)2] , (B.48)120and a covariance ofCOV[Xd(k),Xe(l)1 = cov[Rd(k)2, Re(l)2] . (B.49)To evaluate these expressions, substitute Eq. (B.32) for Rd(k) so that= + V(k)Nd(k) + V(k)Zd(k)+ N(k)Vd(k) + + N(k)Zd(k)+ Z(k)Vd(k) + Z(k)Nd(k) + Zd(k)2. (B.50)Taking the expectation yieldsE [jRd(k)2] = E [Vd(k)2] + E[V(k)]E[Nd(k)] + E[V(k)]Zd(k)+ E[N2(k)]Vd(k) + E [INd(k)2] + E[N(k)]Zd(k)+ Z(k)VdE[Vd(k)} + Z(k)E[Nd(k)] + Zd(k)12,= E [Vd(k)l2] + E [Nd(k)2] + (B.51)The expression for E Ed2R(l)12] needed for the covariance, has eighty-one terms, butmost are equal to zero. All terms with exactly one or three data or noise components havezero mean, and therefore are omitted. Also, terms with E[Vd(k)Ve(l)j and E[Nd(k)Ne(l)] arealso equal to zero, and are omitted too. The remaining terms areE [Rd(k)2Re(l)2]= E [vd(k)2ve(l) 2] + E [Vd(k)l2Ne(l)2] + E EVd(k)I2] z(l)2+ E [INd(k)2Ve(l)2] + E [Nd(k)I2Ne(l)2] + E [Nd(k)2] Ze(l)12+ Zd(k)12E[Ve(l)2] + lZd(k)2E[Ne(l)I2] + Zd(k)I2e l+ E[V(k)V (l)]E[Nd(k)N(l)] + E[Vd(k)V*(l)]E[N(k)JV(l)]+ E[Vc?(1C)Ve(l)lZd(k)7(l) +121+ E[Iv;(k)1v:(l)lzd(k)z:(l) + E[Nd(k)N:(l)1z(k)ze(l) . (B.52)From Eqs. (B.51) and (B.52), it is clear thatCoy [Rd(k) 2 Re(l)2]E [Vd(k)I2Ve(l)2]- E [vd(k)I2] E [Ve(l)2]+ E [Nd(k) 2 N(l) 2] - E ENdk 2] E [N(l) 2]+ E[V1(k) Ve (l)IE[Nd(k)N: (1)] + E[Vd(k) V*(l)]E[N (k)N: (1)]+ E[V(k)Ve(l)]Zci(k)Z’(l) + E[Vd(k)*(l)]Z(k)Zc(l)+ E[N(k)N:(l)jzd(k)z:(l) + E[Nd(k)N (l)]Z(k)Ze (1) . (B.53)Substituting Eqs. (B.34), (B.37), (B.39), and (B.42) into Eqs. (B.51) and (B.53) yieldsE [Rd(k)j2] = + N0+ KNZ0(k), (B.54)andCoy [Rd(k)I2, R(l)2]=- 2(u] N6(e - d) + (g)2N2+()2N26(l — — d) — ()2N2+ NN2 + J\/N2S(l — k)(e — d) — N12+ uNoN2S(l — k)6(e — d) + uNoN25(l — k)6(e — d)+ oNS(l —— d)zd(k)z:(l) + oPvTS(l—k)6(e— d)Z(k)Ze(l)+ N0S(l — k)S(e — d)Zd(k)Z(l) + N0S(l — k)6(e — d)Z(k)Z(l)=— 2()]N(e — d) + [(2)2 + 2No + N]N2S(l — — d)+ 2 [a + No] NZd(k)I26 l— k)S(e — d)= ( + No)228(l — — d) + 2( +N0)KN2Z(k)S(l — — d)122+ — 2(u)]N6(e — d). (B.55)As a result,E[Xd(k)} a + N0 + KZo(k) , (B.56)andCov[Xd(k), Xe(l)1= ( + N0)2S(l — k)S(e — d) + 2( +N0)KZo(k)8(l — k)S(e — d)(B.57)B.4 The Average PeriodogramFrom Eq. (3.51), the average periodogram is computed asXD(k) = Xd(k). (B.58)Using Eq. (B.56), the mean of the average periodogram isE[XD(k)] = E[Xd(k)]== u + N0 + KZo(k) . (B.59)The covariance between different points in the average periodogram isD-1 D-1Cov{XD(k),XD(l)] = COV[Xd(k),Xe(l)]d=0 e=OD-1 D-1[(u+No)2+2(u+No)KZo(k)]S(l_k)S(e_d)d=0 e=0D-1 D-1+ y — 2(o)2] 6(e — d)d=0 e=0123= + No)2 + 2( + No)KZo(k)] 6(1 — k) + [_2()2].(B,60)Note that for two different points in the periodogram, there is a slight correlation, as indicatedby the second term in Eq. (B.60), Note that the second term is much smaller that the first,since it is divided by N. Therefore for the variance the second term is negligible, soVar[XD(k)} = b[(u +iv0)2+2(o +No)KZo(k)] . (B,61)In summary, Eqs. (B.59) and (B.61) provide expressions for the mean and variance ofthe points in the average periodogram, when D blocks of N samples collected at the symbolsampling instants are used to calculate the average periodogram. These expressions are interms of the transmitted energy per symbol, o-, the one-sided noise spectral density, N0, thepower of the interfering tone, K, and the functionZo(k) = sin2 (fNT5). (B.62)N sin2 (--[fNT8— k])These results are used extensively throughout Section 3.3.124Appendix CPower Ratio EstimatorIn Section 3.3.5, two estimators, denoted by Oi and 02 were presented in conjunction witha method for estimating the power ratio, C= ( + No)/I. In this appendix it is provedthat 01 and 02 are unbiased estimators of o + N0 and I, respectively, and expressions fortheir variances are derived.Recall that the two estimators are calculated from the two intermediate quantities, Si and52, that are define by Eq. (3.74) and Eq. (3.75), respectively. To analyse 01 and 02 it is firstnecessary to find the means, variances, and covariance of Si and S2.In Eq. (3.74) the first quantity, S, is defined asN1k=-1(C.l)Its mean can be found by using Eq. (B.59) as an expression for E[Xj(k)], so thatE[S1j== [r + N0+ KZo(k)]= a + N0 + *K Z0(k). (C.2)To simplify this equation it is important to note thatZ0(k) =N—iN—i 1’ k \= (C.3)n=0 m=0125which follows from the definition of Zo (k) given in Appendix B, in particular Eq. (B.43).Using this identity it is easy to see thatN—i N—i / k ‘\Zo(k) =k=— n=0 m=0N-iN-i>n0 m=0 k=—N-i N-i= > e_2i(m)TS(m — m)n=0 m=0=ej27)=N. (C.4)Substituting this into Eq. (C.2) yieldsE[S1} = + N0 + K, (C.5)which is, of course, the total power in the received symbols.To find the variance of S1, Eq. (B,60) is used for the covariance of XD(k) and XD(l).This leads toN1 N1Var[Si]=- 1 ( [( + No)2 + 2(u + No)KZo(k)] 8(1 - k)2k=_J2Ll=_i \[(+N0)+2 +No)Kzo(k)] +N2[a_2()= + No)2 + 2( + N0)K+ ( - 2()2)]. (C.6)Recall that the second quantity, S2, defined in Eq. (3,75), is the sum of only a few pointsin the AP, centered around the position of the tone spike, k. Let M denote the number of126points to sum, where M is odd, and let c = (M — l)/2. This quantity is therefore defined ask +cS2 XD(k). (C.7)k=k—cIts mean can be found by using Eq. (B.59) for E[XD(k)], and isk +cE[S2]= E[XD(k)]k=k—ck +c= > [+No+Kzo(k)]k=k—ck+c= ( + N0)M + K Zo(k). (C.8)k=k—cThe summation of Zo(k) over k e [k — c, k + c] requires some attention. From Eq. (B.62),it is useful to note thatk+c k+c 2sin (irfNT8)k=k-ck= k=k-c N sin2 (*[fjNT — k])— C (fNT)— k=-c N sin2 ([fNT8— (k + k)])—-_, sin2 (ir[ + kg])— L.d 1\T Sifl \N1= k=-c N sin2 ([•- kfl’ (C.9)which depends on S and on the number of points used in the sum. The functionk +cZ0(k) (C.lO)k=k—crepresents the fraction of the total power of the tone that is contained within the M pointscentered around k. Since most of the power is contained in the spike, GM(6j) is usuallyvery close to unity. For the remainder of this Appendix, GM (Si) will be abbreviated to G tosimplify notation, Substituting Eq. (C.lO) into Eq. (C.8) yieldsE[S2]=(o + N0)M + KNG. (C.ll)127As in the case of S1, the variance of 32 can be found by using Eq. (B.60) as an expressionfor the covariance of XD(k) and XD(l). This leads tok+c k+cVar[S2j= Cov[XD(k),XD(l)]k=k1—c l=k—ck+c k+c [(u +No)2+2( +No)KZo(k)]6(l_ k)k=k—cl=k—c-f [o — 2(o) ]k, +c= > [( + No)2 + 2(cr + No)KZo(k)] + M2[ - 2()2]k=k—c= + N0)2M + 2(cr + No)KNG] + — 2(u)2], (C.12)To find expressions for the variances of 0 and 02, the covariance of S and 32 must beknown. This is easily shown to beN1_1 k,+cCov[Si,S2]= .Cov[XD(k),XD(l)]k=— l=k—c1 k+c + 2(u + No)KZo(k)] 8(1 — k)NZZ 2+[u_2(o) ]k +c= [(+N0)2+2(+NK?z(k)]+NM[_2()l=k—c= + N0)2M + 2( +N0)KNG + ( — 2())M] . (C.13)Armed with the joint moments of S and S2, analysis can proceed to 0 and 02. Recallfrom Eq. (3.76) that the estimator 0 is generated byNGS1— S201= NG—M(C.14)Using Eqs. (C.5) and (C.l 1) for E[S1] and E[S2] respectively, it is clear thatNGE[S1]— E[S2jNG-M128— NG{c+No+K?] — [(o+No)M+KNG]- NG-M( + No)[NG — M}= NG-M=+No. (C.15)Therefore 01 is an unbiased estimator of o + N0, Its variance can readily be found to beVar[Oij — 1— (NG_M)2/ar[NG9— S2]1= (NG-M)2[N2GVar[Sij — 2NGCov[Si, S2] + Var[S2}].1— (NG-M)2(C.16)By using Eqs. (C.6), (C.12), and (C.13), this variance can be expressed asN2’ 1 [(+N0+2(-+N)I (_2()2)]Var[0i] — 1— 2NG 1 I— (NG-M)2 ND I[ +(u-2()2)M JM2r2+[(u+No)M+2( +No)ING] +L4_2(u9)])(o+No 21N2G2 9iVMG iwl)[NDND+j2NGINGnNG 11 Ij [ +2(+No)K I= (NG-M)1N2G2 NMG+M21__-ND j)[(o+No)[NG2_2MG+M]+ 2(u + No)K?NG[G — 2G + 1]M211+ — 2()) {NG2 — 2MG+ j2r M(1—G)][ (o + No) LG + NG-M j= (NG_M)D +2(+No)KG1—G1____________________(C.17)t NG-M1\NG—M IN IThis expression, although exact, is a little too cumbersome for practical use. A reasonableapproximation can be made by assuming that G is exactly equal to one. This assumption129greatly simplifies Eq. (C. 17), yieldingVar[Oi] (N-M)D + No)2 + ( — 2()2) N —M] (C.18)as an approximate expression for the variance.The second estimator, 02, can also be analysed in a straightforward manner. Recall fromEq. (3,77) that.$2—MS102= NG—M (C.19)By using Eqs. (C.5) and (C.ll) its mean can be found to beE 0 1 — E[S2] — ME[S1][2j- NG-M—- NG-MK[NG — M]= NG-M= K. (C.20)Therefore 02 is an unbiased estimator of K. Its variance can readily be found to beVar[02]= (NGM)21[S2 — MS1]= (NG_M)2 [Var[82]— 2MCov[81,S2] +M2Var[8i]] . (C.21)By using Eqs. (C.6), (C.12), and (C.13), this variance can be expressed as[(u+No)M+2( No)KNG] +[u2(g)2](u + No)2M + 2( + No)KNGVar[02]= (NG—M)2 — 2M+(cry — 2(o)2)M+ M2[(at + N0)2 + 2( + N0)K+(cry - 2()2)]M 2M M2+JVO)1 c-/ 2 ir ‘ jz2 NG nMNG M2= (NG—M)2 + s + 1VO)11 D — L ND + ND/ 2\ 2 2 2— ) — ND V7Y130- 1((J+No)2MM— (NG-M)2 I+ 2(c + N0)K [G(N — M)— *(NG — M)]/( 2 N 2M (N—M)I s + 0) N (NG-M)1 22— (NG-M)D+2(+No)K?{G—*]As in the case of the variance of 01, this expression, is a little too cumbersome for practicaluse. By assuming G equals one Eq. (C.22) simplifies toVar[02J + No)2NM + 2(o + No)K?] (C.23)as an approximate expression for the variance.To summarize, the preceding proof showed that 0 and 02 are unbiased estimates of cr +N0and K, respectively. Furthermore, these estimators have variances that are approximatelygiven byVar[0i] (N-M)D + No)2 + (a — 2(u)2)N _M] (C.24)andVar[02] +No)2NM +2(u+No)K] (C.25)131Appendix DSource Code ListingsThis appendix provides complete listings of the source code for all the programs usedin generating results for this thesis. Section D.1 contains listings for the programs used todetermine the positions of the CTB, Section D.2 contains listings for the program used todetermine the probability of transmission error, and Section D.3 contains a complete listingthe tone canceller simulator.D.1 Positions of CTBTo determine the positions of the triple beats, three separate filters are used. The first, list,generates a list of all the distinct beats, The second, merge, determines the distinct positionsof the beats and the number of beats at each position. The final filter, channel, sums thenumber of beats falling within each channel. They are executed in order with the command:% list I sort I merge I channel132for)a=0;a<N;ato){for)bao1;b<N;b++)for)o=bal;C<N;ott)printf)”%6.2fE\n’,faba)freqa)a]+freqs]b]+freqs]o]));printf)”%6.2fE\n’,fabs)freqs[a]+frega[b)-freqa]oJb;prlntf)”%6.2fEl,n”,faba)freqa]a]—freqa[b]•freqs)o]U;printf)”%6.2fE\n’,fabs)freqs]a]—freqa[b]-freqs[o]H;for)a=0;a<N;aa+)primtf(‘86.20F\n,2’freqa]a]);for)a=0;a<N;a+÷)for)b=a.1;b<N;b+.)fprintf)’%6.2fd\n’,labs)freqs[a]+freqs[b]b;printf)’%6.2fC\n’,faba)freqs[a]-freqs]b]));8defineNV7main)) doublef,18;oharohr)2];mti,n)NV];for)il;i<NV;itt)o]i]=I;aoanf)”tlf%la,600,ohr);n)ohr)l]—while)aoanf)%1f%la”,&f,ohr)==2)if(I;=fI)printf)’%6.2f,fo;;for)i=i;0tNV;0’—)printf)’830”,a)i]);n[i]=0;printf)”\n”);fl=n[ohr[l]—‘A’]ta;prinrf)’%6.2e,fO)for)i=l;0<NV;itt)printf)”630”,n]i]);printf)‘\n”)I’hatthepositianaofallthebeats(Unsortedoutput)linolade<math.h>#defineN31doublefreqa]N]=54,60,66,76,82,174,182.186,182,188,204,210,123,126,132,138,144,150,156,162,168,226,222,228,234,240,246,252,258,264,270,276,282,288,294main)) jaila,b,0;jot6;for)a=l;a<N;a+÷)freqs]a)s=1.25;for)a=0;a<24;aat)priotf(‘66.21A\n,3freqs]a]);/t3rdHarmoniofor)a=0;a<N;att)printf)’%6.2f6\n”,freqa]a]);for)a=0;a<N;a)priatf“66.21C\n”,freqa]a]for)a=0;a<N;aao)forhal;beN;bat)if)b==a)oontloue;printf)’%6.2f0\n’,faba)2’freqa]a]afreqab)));printf)’%6.2f0\n’,faba)2tfreqa]a]-freqs)b];);#defineN359defineNV7doublefreqs[N]54,60,66,76,82,120,126,132,136,144,150,156,162,268,174,180,186,192,198,204,210,216,222,228,234,240,246,252,259,264,270,276,282,288,294incchan[NJ2,3,4,5,6,14,15,16,17,18,19,20,21,22,7,8.,9,10,11,12,13,23,24,25,26,27,28,29,30,31,32,33,34,35,36mainU doublef,fa;jOta,i,n[NVJ,nc[NV+1J;scanf)”%lf”,Sf);for)i0;iuNV;i*+)scanf)%d,&n[iJ);for)a=0;a<N;a,.)fa=freqsJaJwhile[1<fa)scanf)%1f,Sf)for)i=0;0<NV;0,.)scanf)%d,&n[iJ);for)i=0;i<NV*l;i**)nc)i[=0;while)f<fa+6)if)nJOJ>0I)n[3]>0IIn)4[>0)nc[NVI++;for)i=0;i<NV;ic.)nc[iJ+nJiJucanf)’%1f,Sf)for)i0;i<NV;i++)scanf)’%d”,&n[i[);printf)“%2d”,ohan[aJ)for)i=0;iu001,1;i*+)printf)”%34”,nc[i]);printf)“——%d\n”,nc[0]•nc[3]*no[4]D.2 Probability of Symbol ErrorTo determine the probability of a symbol detection error in the presence of AWGN aninterfering tone, the program error was used. Numerical integration is implemented withfive-point Gaussian quadrature interpolation.135return(Pale-1S(7la-lIP;tntSIR,INN;for(510=0;INN435;NNRo+(printf(”%d”,INN);for(SIR=0;SIR<30;SIRo0)printf(’%g’,theoryl(INN,SIR));priotf(”%g\n’,theoryl{SNR((;statiodoubletheoryl(intINS)doubleNo,P.a;No=11.1I4.0*explo(—SNRI1O.0(;P=elf(l.0Isqrt(NoH;a=575*p+0.20;P=1-a°a;return(Pala-il)7lu-tOF;stalicdouble0(5]=(—0.9061790459,—0.0384693101,0.0,0.5184693101,0.9061798459staticdoubleA(S)=(0.2369268851,0.4786286701,0.0688888888,0.4786286705,0.2369268851staticdoubletheorv2(intlION,mtSIR)doubleNo,Si;doubleF;mtn,5;No10.0/4.0*explo)-NNR/ll.0);(Ci=sqrt)lO.0/4.0expll(—SIR/l0.0(P=0.0;for(n=0;na20;n-*+)for(m=0;m<0;xo++(P+=Aim]*eval(Ni,No,(S]m]+2*n+l)*(M_P1180.0(M_PI/80.0;P=4.0*P/(2.0*IOPI(;P=1.0—P;/5ProgramtogenerateplotofFevs.INNforavatiousSIRsinrtude<stdio.h>*inolude<sac’n.h>1Localfunctionprototypes1stoticdoubletheoryl(intINN]stoticdoubletheory2(intINN,tntSIR)staticdoubleeval(doubleNi,doubleNo,doublephi(;staticdoubleP1(doubleNo,doubleeps(main)staticdoubleeval(doobleNi,doableNo,doublophilreturnF1(No,Nicos(phi(•FllNo,S*sin(p0iH;staticdoubteFt(doubleNo,dcubeego)doubleEl;El=erfHl+eps(/surt(No((+erlI(l—epu(/ogrl(No(return0.20+0.175*11;w C’D.3 Tone Canceller SimulatorIn the following pages the source code for the simulator is listed. The program is separatedinto two files: tx.c contains the main routines for the transmitter, channel, and receiver; andcanceLc contains the routines required by the interference canceller.137#inolude<stdio.h>(include<fcntl.h>(incLude<uys/utat.h>(include<math.h>(incLude<memcty.h>((defineTRUE1((defineFALSEtypedefstructdoubleEl,freq,phi;IMITYFE;typedefdrooldouble*R,*1;CMFLE;,Globalfuoctiooprototypes/doubledrand48(void);CMPLXcancel(CMFLXrobJGLobalparameters/mtN=2048;mtD=32;mtL=32;mtN=0;7Looalfunotionpfotstypes‘Iutatiodoubledoblock)intd(staticCMFLXtraosmit(001N)utatioCMFLXnoise)doubleNo,jotN);statioCMFLXimi)IMITYFEV,jotd,jotN)statiodoublepetFower)OMFLXrv,CMFLXrn,OMFLEro,CNFLX10);utatiomtread_params)intargo,ohararpv[J);doubleNo;IMITYFEV•main)intarpo,ohareargv[))doubleIto;mtd;/5Setupmainparameters5/if)Iread_parems)argc,argv)esit(1)for)d=O;d<24;d++)dablack)d)En=0.0;for)d24;daU;d++)Rn+=doblock)d);Rn1=(0—24);prjntf(“Seaidue=%g\n”,ES)statjcdoubledoblock)intd)staticmtdoinit=TRUE;staticOMFLXrc;OMFLXcv,rn,rc,cc;jot0;if(duinit)Irc.E=(double*(oalloo)N,aizeof(double));rc.I=(double*(calloc)N,sineof(double));doinjt=FALSE;/Generatedatauignal*/cv=lransmit(N(;/GenerateANGE=1rn=noise(Na,N)/GenerateIMI/rn=imi)V,d,N);/5Generatereceivedusnples5/far)n=0;naN;n*-*rc.E)nj=rv.E[n]+rn.E[n)+rn.Efn]rc.I(n]=rv.I)n)+rn.I[n)+rn.I[nj;/5canceltheinterferencera=cancel)rc(;returngetFawer)rv,rn,rc,ru)staticcMFLXtransmit)jntN)staticmtdoinjt=TRUE;staticOMFtEData;duublerptr,iptr;mtn;if(doinit)Uata.Edoublecallcc)N,sineof)dcublefl;fata.I=doublebcalloc)N,aiceof)duublefl;doinit=FALSE;rptr=DateR;iptr=Deta.I;far)n=0;naN;n++)Itrptr+.=))lrand4o))>>lO)&0s06)—3;5iptra+=)lrand4l))>mlO)&0x06)—3;returnData;staricDMFLXncise)dcubleNc,mtN)staticmtduinit=TRUE;staticOMFLXNoise;dauble5rptr,iptr;daublerl,r2;mtn;if(dainlt)INalse.E=doublebcalloc)N,sinecf(double));Ncise.f=(doublebcalloc)N,sizeof(double));dcinit=FALSE;/ApplynonoiseifSEEissetcorrectly/if(Na==0.0)returnNuise;rptr=Nsise.E;5— 1.300PNumberofsamplesparblock//Numberofblockstoprocess5/PNumberofsymbolstouse5/PWidthof52/iptr=Nuise.I;7’Thiscreatescosplecdasssiennoise—N)0,No)‘7far(0=0;flaN;n+o(ti=sgtt(-No’log)drand4lUH;r2=2.0*M_PI*drand4ofl;‘rptr++=rlcos(r2);‘iptr+c=rlsin(r2);returnNoise;ststicCMPLXilni)IMITYFE‘0,jotd,jotN)staticmtdsinit=TRUS;ststicCMFLX1141;dasble0rptr,‘iptr;doubletl,theta;jotn;jf)dainit)IMI.R=)dosble‘(calloc)N,sizeof(double));1141.1=)dosble‘(calloc(N,sizeof(double));dninit=FAtSE;7*ApplynotoneifSIRissetcorrectly‘7if)V—o’Ki==01)returnINS;rptr1141.R;iptr=1141.1;ti=2.0*N_Ft•V->freq/N;for)n=i;n<17;uc,)theta=tI•(double))n.d’N)+V->phi;trptt++=V->Ki*cos)theta)‘iptr++=V—aKiain)theta)return1141;statIcdoablegetPower)L’MPLXrv,CI1PtErn,O1PLXrc,CMFLXto)doableRz;doabledot,dci;rOt0;No=0.0;far(n’S;n<N;nea)dcr=rc.R)nJ-)rv.R)o)+rn.R)ofl;dci=rc.I[n)—)rv.I[n)+ro.I[nfl;RI+=)dztdztedzidzi);t’etotnRc/N;staticjotread_paranu)intargo,chat‘argv[Jcharjuni[80);doableSNRdb;langSeed=17t;jot0;FILEfp;Ip=fapen(paraas”,t”(;if)fp==(FILEHNUtL(fptintI(atdert,%a:unabletoopenparameterfole\n’,argo’(iiireturnFAtES;tacanf)fp,•%d%)”\cI’,RN,junk);Iacaof)fp,“%d%V\n)”,00,junk);Iscaol)fp,“%d*IHn)”,&t,junk);Isuaol(fp,%lf%[”\n(”,RSNRdb,junk);V=(IMITYFEHca110c)l,aiceof)IMITYFE));fsaanf(fp,%lf%if%lt%[’\nJ’,RV-aEi,&V—alteg,eV-aphi,junk);tcloae)fp(if(argo>=2)Seed=atol)argw)1))while(atgc>=4&kargv)2J10]==switch(atgv]2]11)case‘s’N=atoi(argv[3H;break;case‘0’0=atoi(argv[3](;break;case‘t’t=atoi(atgv)3)(;break;case‘14’N=atoi(argv(2)(;break;case‘N’INRdb=atof(arg’,’)Ofl;break;caae‘K’V-uKi=ataf(atgv(3]);break;case‘I’V->freq=atof)argv[3]);break;default:fprintf(atderr,“usage’%uheed-#wal))\n”,atgv)0)(;returnFALSE;argo-=2;arc”+2;if(ENRdbatOO)Nc=0.0;elseNu=10.0/4.0*expll(—SNRdb/l0.0);if)V-aRia100)V-aRi=0.0;elseV—aM=sgrt(l0.0/4.0*espll)—V-aKi/lb.0(I;7’InitializeSystemstand48(SeedSeed:tine(Ot)returnTRUE;L.a8include.cstdio.h>8inuludeufcntl.h,inolude‘csyslacat.h>includeanath.h>#defineTRUE1#defineFALSE0typedefatructdouble4R,*1;CMPLXCMFLXcancel)CMFLXrc);staticvoidFFTrnMPLXdata,i.otN);staticvoidIFrP)049LXdata,mtN);staticvoidFnls)cMFLXdata,mtN)staticvoidIFFT’NS)O1PLXdata,mtN);staticvoidswapOWLSdata,totN);staticvoidfftshift)OIPLXdata,incN);#defir.eDNTECT1)X)HR<—2)0—3;))X<0)71HR<2)713)))edefineONTECT)Ras))CMPINX)(ONTECT1)Raa.R),DETECT1)Ras.I)mtMo)]=(3,3,3,1,3,3,1,9,11,21,41,84,hO,337,673);doubleou[)=I0.146447,0.216621,0.280929,0.286841,i.288309,0.288i76,0.288767,1.166199,4.0740840,0.0386030,0.0197762,0.00976602,0.00479896,0.00240006,0.001204311;PGlobalparameters4/externjotN,I,N;typedefatructdoubleR,I;CMPLEX;typedetatruntdoublefreq,C;TONS;#detinePTR,OET)x,n)#definePTR_PUT)x,n,y)/4Localfunctionprototypes/staticvoidgetTone)CMFLXN,TONStone)CMPLXcancel)CMFLXrc)staticmtdoinit=TRUE;staticTONEtone;staticCMFLXro,fb,Na;staticCRPLEXSe;staticdoubleCl,01,CL,EL;staticdoubleofraq;staticmtF;CMPLEXIa,Ra,Rap,Qa;doubletr,ti;mta;if)doinit)Iro.R=double5)oalloc)N,aioeof)double));ro.I=double5)caltoc)N,aioeof)duuble));fb.R=double)calloc)N,sizeof)double));fb.I=double5)oalloc)N,aizeof)duuble));Ea.R=double5)calloc)t,sizeof)duuble)I;Ea.I=double5)calloo)L,aizeof)duuble));Za=)O1PLEX)10.0,0.01;tone=TONE)10.0,0.01;ofreq0.0;00;doinit=FALSE;Cl=coa)tOne.freq)01=ain)tone.freq)CL=coa)tone.freqt);OL=ain)tone.freqt);for)a=0;a<N;are)Ra=FTR_GNT)rc,a);Rap.R=Ra.N-Za.R/)L+tone.C);Rap.I=Ra.I-Za.t/)L+tune.C);PTR_PUT)rc,a,Rap);Ia=OETECT)Rap);PTR_PUT)rn,a,Ia);Qa.R=RaN-OaR;Qa.i=Na.t-Ia.I;PTR_PUT)Fb,a,Qa);/5SubtractoldsasWte5/ifa<I)Itr=coa)tone.treqa.ofreq)L-afl;ti=ain)tone.freqa+ofreq)L-aH;Za.R-=Oa.R[P)tr-Ea.I)Pti;Za.I-=Ea.R)P)ti+Ea.I[P)5tr;else2a.R-=Ea.RFJ5CL-Na.I)PISL;Za.I-=Ea.R)P)4SL+Ea.I[P]CL;/Addnewsample0/Za.R÷=QaR;Sal÷=Qa.I;/Rotatephase/tr=la.RC1-la.10l;ti=ta.R0l4la.t5Cl;ta=)CMFLEX)Itr,tiI;/5SavenewsampleejEaR)?)=Qa.R;Nat)?)Qa.I;F=0+1)8t;staticvoidgetTone)CMPLNN,TONE0tone)staticmtdoinit=TRUE;staticdouble0,OC;ataticdouble5Xataticmt0;double01,N2,Tl,T2;doublehigh,low,deltai;mtN,Ni,ialeft;-5 0I’DetectedsynOd5//‘Correcteddecisionvar‘//‘Detectedsymbol5//5Samplew;odata5/)CRPLEX)I)x).R)n),5).SIn)II0).R[n)=)y).R;)x).0In)=)y).0;ofreq=tone.freq;getTone)fb,Stone);returnro;if(doinit){mtn;x=(double(oalloo(N,sizeol(double));0=0;n=(ml)(1002((double)N)+0.0)-2;M=(N==0(?Mo(n]:PC=do[n;0=1.0;doinit=FALSE;JFindtonespikesodcalculateEl°/Ni=0;El=0.0;for(k0;k<N;k++({01+=(fiN];if(X]k]>X(ki((Ni=N;SiJN°p(/Calculate02fi02=0;for(kki—(N—i)12;Nuits(14—1(12;k+s02+02/‘,0;isleft=((([Ni—i)>X(ki+i((low=(isleftX[ki—lJX(ki+l((I0;high=X[ki)I0;/Csloolste‘NETAl‘ITi=(NdEl-02)I(N’G-14(;Ti=(Ti<0)00.0:TI:IEobtrsotdataandnoisepowerfromAF*1high-=Ti;low-=Ti;1InterpolatefreqsencfestImatefideitsi=(low>0.0)01.1/(1.0+sqrt(high!low((;0.0;tone->freq=2.0M_?I’i(ki-N/2)÷(isleft?-deirsi:deltas))/N;C=1.0-GCPCeltsi;/CalculateTiandT2/Ti=(N0t0i-02)I(oth-N(;T2=(02-N0i(I(0N-N(;Ti=(Ti<0(00.0Tl;T2=(T2<0(?0.0:T2;/CalculateC0/tone—>C=(T2>0)0(Ti/T2(:999999;tonalfunctionprotntypes0/stalinmtflnitFi.ip(intN(;stetinvoidFill(intflip,mtvsl,mtn(voidFFT(CMFLXdata,mtN(FFINE(dsta,N);swsp(dsts,N);fftshift(data,N)voidIFFT(cNFtXdata,mtN)ftshift(data,N)swsp(dsts,N);IFFTNO(dsta,N)voidFP’I’NE(CNFLXdeta,mlN)mti,m,max,step;dnsbledir,dii,0djr,°dji;doubletemp,tempr,tenpi,thete;doubleWir,Nh,W2r,PCi;for(max=N/2;max>=1;Sax>>=1)step=2°mmx;theta=N_FlI(double)max;tempsin(0.fltheta(;Wlr=-2.0*tesp*temp;Nh=sin(thete(;W2r=1.0;W2i=0.0;for(m=0;mumex;m÷+(dir=&datm.R[m(;dii=&datm.I)m(;djr=&date.R]m+msx]dji=&dats.I(m+msx(for(i=m;0<N;i+=step)tempr=dir-djr;tempi=+dii—dji;tdiro=djrtdtiodji;tdjr=W2r*dji=W2r*dir+=step;dii+=step;djr+=step;djio=step;tempr=W2r;tempiW2i;W2r+=tsmpr*Wir-tempi*Nh;W2i+=tempr*Nfl5tempi•Wir;voidIFF’I150)CMFLXdata,mtN)mti,N,mex,step;doabledir,dii,°djr,=dji;doubletemp,tempr,tempi,theta;doubleWlr,Nh,W2r,0(21;doablemv;for(max1;maxuN;max<u=1)1step=2msx;theta=N_FlI(double(max;tempsin(0.E*theta(;Wir=-2.0*temp*temp;Nh=sin(thetm(;W2r=1.0;FFT(R,N);fork0;N<N;k+÷(X)N(÷=(R.R)NE.R(N(sR.I[kflx.I)N)(/N;a- atempt+N2itempi;tempi—N2i*tempr;tempr=W2r;tempi=W2i;W2r+=tempr*Wlr-tempi*Wi!;W2i+=tempr*Wi!+tempi*Wir;returnflip;staticvcidFillmttflip,mtval,mtn(registermti;fcr(i=0;ian;iu+(ilip(i+n(=flip[i(+vat;W2i=0.0;icr(a=0;aacmx;m++(dir=&data.R(m(;dii=&date.I(m);djr=&data.R(rm+cax(;dji=&data.I(mucax(;fcr(i=m;0aN;i+=step(tempr=W2r+5djr-102!*dji;tempi=W2r*dji+W2i*djr;djr=*dir-tecpr;5dj!dfl-tecpi;*dir+=ternpr;+=tempi;dir+=elep;di!+=etep;djr.=etep;dji*=step;staticmtfnitFlip(intN(staticmtoN=0,5flip=(lot*(NUlL;mtvet,n;if(N==cN(returnflip;elee(free(flip(flip=(ict*(callcc(N,sizecf(int((oNN;flip(0(=0;c=1;vet=Ncc1;while(vel(IFill(flip,vel,n(veleu=1;n<e=1;mv=1.0/(dcuble(N;fcr(i0;iaN;i+a(Idets.Nfi)mv;dets.I(i(=mv;veidfitshift(C(4FIXdate,lotN(doublettemptemp=(double*(calloc(N12,sizecf(dcubleH;memcpy(tecp,&date.R(0(,N/2sizeof(doublefl;memcpy(&deta.RId],&data.R[N12(,N/2sizeoi(doublefl;memcpy(&dete.RIN/2(,leap,N/2*sizeof(doublefl;memcpy(temp.&dete.I(0(,N/2siceof(double)(;cemcpy(&dete.I[d(,&dete.I(0012(,N/2*sizeof(doublefl;memupy(&deta.f(N/2(,temp.N/2*sizsof(dceblefl;free(te*np(vcidswep(cNPtfdate,mtN(mti,j,tflip;doublereap;flip=InitFlip(N(;fcr(i=0;iaN;i**(Ij=flip(iJ;if(j>i)Itemp=datedete.R[i)=deta.R[j];data.R[jJ=temp;temp=dmte.I(!I;dele.I(i(=deta.I(j(;deta.IIj(=temp;

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