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Analysis of techniques to enhance the performance of direct sequence spread spectrum signaling for wireless… Wong, Aries Y. C. 1995

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ANALYSIS OF TECHNIQUES TO ENHANCE T H E PERFORMANCE OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALING FOR WIRELESS DATA COMMUNICATIONS by  ARIES Y. C. WONG B.A.Sc.(EE), University of British Columbia, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF APPLIED SCIENCE  in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA © Aries Y. C. Wong, 1995  In presenting  this thesis  in partial fulfilment of  the  requirements  for an  advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. copying  I further agree that permission for  of this thesis for scholarly purposes  department  or  by  his  or  her  representatives.  may be granted It  is  by the head of my  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  extensive  copying  or  my written  Abstract The processing gain of a spread-spectrum (SS) system is generally defined as the ratio between the transmission bandwidth and data bandwidth, and is a measure of its interference rejection capability. Due to limited spectrum, most licence-free commercial ISM-band direct sequence (DS) SS modems suffer from insufficient processing gain if they are to support a reasonably high data rate as required for wireless local area network applications. This thesis presents two DS-SS signaling systems, Code-Phase Multiplexing (CPM) and Code-Phase-Shift Keying (CPSK), both designed to increase the processing gain without reducing the aggregate data rate. In C P M several data bits are modulated with different phase shifts of the same pseudonoise (PN) code sequence and multiplexed for parallel transmissions, whereas CPSK is an M-ary DS-SS modulation method which employs a signaling set consisting of different phase shifts of a single PN code sequence. The performances of both systems against a single tone jammer, over multipath fading channel and against a single tone jammer in the presence of multipath fading are analyzed. Theoretical and simulation results show that the proposed C P M scheme is able to increase immunity against single tone jamming by increasing the number of parallel signaling streams. Simulation results also show that the performance of C P M DS-SS signaling in the mobile fading channel is very similar to that of conventional DS-SS signaling, and could be improved by the use of a R A K E receiver. However, in the presence of tone jamming in the fading channel, C P M DS-SS signaling gives better performance than conventional DS-SS signaling. While CPM, like conventional DS-SS modulation, gives no advantage over BPSK with respect to thermal noise, CPSK offers increasing thermal noise immunity as M increases. With ii  M=64, CPSK offers a 3.5 dB reduction in E / N over BPSK. Furthermore, the effect of b  0  single-tone interference at the carrier frequency is totally mitigated. It is also shown that CPSK maintains good performance in a multipath fading channel with a dominant direct path, a situation common to wireless local communications. A R A K E receiver could also be used to improve performance in a multipath channel without a dominant direct path. At the expense of increased receiver complexity, CPSK is effective in increasing both the power and bandwidth efficiency of DS-SS signaling.  iii  Table of Contents Abstract  ii  List of Figures  vii  Acknowledgment  x  1 Introduction  1  1.1  Motivations and Objectives  1  1.2  Background  1  1.3  1.2.1  Spread Spectrum Transmissions  1.2.2  Direct Sequence Systems  .  1 2  Organization of the Thesis  6  2 System Models  7  2.1  Code-Phase Multiplexed Direct Sequence Spread Spectrum System  2.2  Code Phase Shift Keying Spread Spectrum System  . . . . .  3 Performance Against Single Tone Jammer 3.1  7 10  15  Channel Models With Single Tone Jammer  15  3.1.1  Single Tone Jammer at the Data Carrier Frequency  15  3.1.2  Single Tone Jammer at a Frequency Different from the Data Carrier Frequency  3.2  16  Performance of C P M DS-SS Against Single Tone Jammer at the Data Carrier Frequency  19  3.2.1  Derivation of Average BER Using Characteristic Functions  20  3.2.2  Numerical and Simulation Results  22  iv  3.3  Performance of C P M DS-SS Against Single Tone Jammer at a Frequency Different from the Carrier Frequency  3.4  3.5  3.3.1  Derivation of Average Probability of Error  3.3.2  Numerical and Simulation Results . . . . /  28 '  .29  Performance of CPSK DS-SS Against Single Tone Jammer at the Data Carrier Frequency  32  3.4.1  Derivation of Average Probability of Error  32  3.4.2  Numerical and Simulation Results .  .35  Performance of CPSK DS-SS Against Single Tone Jammer at a Frequency Different from the Carrier Frequency  37  3.5.1  Derivation of Average Probability of Error  37  3.5.2  Simulation Results  40  4 Performance in the Presence of Multipath Fading  43  4.1  Model of Multipath Fading Channel  4.2  Performance of C P M DS-SS in the Presence of Multipath Fading  4.3  28  4.2.1  Analysis  4.2.2  Simulation Results  . .  43 46 . . . 46 49  Performance of CPSK in the Presence of Multipath Fading  . 54  4.3.1  Analysis  54  4.3.2  Simulation Results  55 v  5  6  Performance Against Single Tone Jammer in the Presence of Multipath Fading59 5.1  Performance of C P M  59  5.2  Performance of CPSK  62  System Performance Improvement with R A K E Receiver  63  6.1  Background of the R A K E Receiver  63  6.2  C P M DS-SS With R A K E Receiver  64  6.3  7  6.2.1  System Model and Implementation  6.2.2  Simulation Results  . .  64 66  CPSK DS-SS With R A K E Receiver  68  6.3.1  System Model and Analysis  68  6.3.2  Simulation Results  70  Conclusions and Future Work  74  7.1  Summary and Conclusions  74  7.2  Areas for Future Work  77  A A n Efficient Implementation of C P S K System  79  B Computer Simulations for Performance Analysis  81  B.l  Sample Listings of Simulation Programs  84  C Glossary of the Abbreviations, Acronyms, and Symbols  91  Bibliography  97  vi  List of Figures 1.1  Conventional DS/BPSK-SS System  . . .  3  1.2  Autocorrelation function of an m-sequence  5  2.1  Code-Phase Multiplexed Direct Sequence Spread Spectrum System . . . . . . .  8  2.2  The Transmitter of Code-Phase-Shift Keying DS-SS System  2.3  The Receiver of the Code-Phase-Shift Keying DS-SS System .  3.1  Graph of 7j vs. // for Various Phase Shift of the Same PN Sequence  3.2  Graph of 7{ as a Function of Phase Shift for Various Differential Frequencies . 19  3.3  Conditional BER of C P M vs. JSR for Various Jammer Phase @ SNR =10dB  .11 13 .18  b  G =511  23  4  3.4  B E R of C P M Conditioned on Jammer Phase @ SNR =10dB, JSRi=50dB and G!=127 24  3.5  B E R of C P M vs. SNR @ JSRi=50 dB, Gi=511 . .  25  3.6  BER of C P M vs. JSRi @ SNR =6 dB, Gi=255 . . . . . . . . . . . . . . . . . .  26  3.7  B E R of C P M vs. JSRi @ SNR =6 dB, Gi=511 . .  26  3.8  BER of C P M vs. JSRi @ SNR =8 dB, d=255  27  3.9  BER of C P M vs. JSRi @ SNR =8 dB, Gi=511  27  b  b  b  b  b  b  3.10 BER vs. SNR of C P M System with JSRi = 10 dB, Gi=31, Aw=0.2u;c . . . . 31 b  3.11 BER vs. JSRi of C P M System with SNR = 14 dB, Gi=31, Acu=0.2u;c . . . . 32 b  3.12 B E R vs. SNR for CPSK System with Single Tone Jammer at Carrier b  Frequency  . 36  3.13 Comparison of the Numerical and Simulated B E R of CPSK System with Single Tone Jammer at the Carrier Frequency  vii  38  3.14 B E R vs. SNR for CPSK against Single Tone Jamming with AU;=0.1CJ , b  C  JSR=10 dB  4  3.15 B E R vs. JSR for CPSK against Single Tone Jamming with SNR =4 dB 4.1  Au=0.1u , c  .  b  1  42  B E R vs. S N R of C P M operating over Rayleigh Channels with Various b  Maximum Delay Spread and Number of Unresolvable Paths  51  4.2  BER vs. S N R of C P M DS-SS over Rayleigh Fading Channel  52  4.3  BER vs. S N R of C P M DS-SS in various Multipath Channels . . ... . . . . . 53  4.4  B E R vs S N R of CPSK Operating Over Various Multipath Fading Channels . 57  4.5  Comparison of C P M and CPSK Operating Over Rayleigh Channel  5.1  BER vs. S N R for JSRi=60 dB, d = 3 1 , over Rician Channel with K-factor=10 dB 61  b  b  b  . 58  b  1  5.2  BER vs. JSRi for SNR =12 dB, Gi=31, over Rician Channel with «-factor=10 b  dB . . . . . . . . . . .  61  5.3  B E R vs. JSRi for SNR =30 dB, Gi=31, over Rayleigh Fading C h a n n e l . . . . 62  6.1  Channel Tap Weight Estimation with Binary Antipodal Signals  64  6.2  C P M DS-SS with R A K E receiver  65  6.3  Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rayleigh Channel . . 67  6.4  b  Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with K-factor 6 dB  6.5  :  67  Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with  6.6  ^-factor 10 dB  68  Performance Comparison between CPSK Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rayleigh Channel  71  viii  6.7  Performance Comparison between CPSK Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with K-factor 6 dB 71  6.8  Performance Comparison between CPSK Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with /c-factor lOdB  12  A.l  CPSK Transmitter with Table Lookup  79  A.2  CPSK Receiver with Concatenated Matched Filters.  80  ix  Acknowledgment I would like to thank my supervisor, Dr. Victor C. M . Leung for suggesting this topic and for his constant guidance, helpful advice and support throughout the production of this thesis. I would also like to thank my colleagues and staff in the department for their kindness and assistance during my studies at U B C . Finally, I would like to acknowledge the Science Council of British Columbia for their financial assistance.  Chapter 1 Introduction 1.1 Motivations and Objectives The processing gain of a Direct Sequence Spread Spectrum (DS-SS) System is. generally defined as the ratio between the transmission bandwidth and the data bandwidth, and is a measure of the system's interference rejection capability [1]. Due to limited spectrum and equipment constraints, most DS-SS modems suffer from insufficient processing gain if they are to support a reasonably high data rate. For applications such as high performance wireless local area networks (WLAN's), it is necessary to enhance the performance of existing DS-SS modem designs. The main objective of this research is to design and analyze DS-SS signaling methods with high processing gain and high data throughput.  1.2 Background 1.2.1 Spread Spectrum Transmissions Recent progress in communication devices has reduced the cost of spread spectrum (SS) systems, enabling increased usage of these systems for nonmilitary applications in addition to the traditional military applications. Possible applications of SS systems range from antijam systems, to guidance systems, to code division multiple access (CDMA) systems, to systems designed to combat multipath fading. A SS system can be simply defined as one in which the transmitted signal is spread over a wide frequency band, much wider than the minimum bandwidth necessary to transmit the information [1-3]. The band spread is achieved by means of a code which is independent of the data, and synchronized reception with the code at the receiver enables despreading and data recovery. 1  Chapter 1. Introduction  2  Proper spectrum spreading can simultaneously provide multiple benefits including: Antijamming •  Interference rejection Selective addressing capability  •  Code division multiplexing or multiple access Low-density power spectra for signal hiding  •  Message screening from eavesdroppers High resolution ranging. There are several basic spread spectrum techniques. One popular technique is Direct  Sequence (DS) or Pseudonoise (PN) in which a fast pseudorandom sequence modulates the phase of the carrier containing data. Another popular technique is Frequency Hopping where the carrier frequency is shifted in a pseudorandom way. Time Hopping is another spread spectrum technique which is similar to the familiar pulse modulation; that is, the code sequence is used to turn the transmitter on and off pseudorandomly. Although they share many common properties, each method has their own distinct advantages and disadvantages. Therefore, hybrid combinations of these techniques are frequently used.  1.2.2 Direct Sequence Systems DS systems, the most widely used SS systems, are well known for their relative simplicity and high immunity to interference. In a DS-SS system, the baseband signal is spread by multiplying it with a PN sequence clocked at a much higher rate (the chip rate) than the data sequence. This spreading technique continuously distributes the power of the signal over a given bandwidth. After the data is multiplied by the PN sequence, it is transmitted over the channel. The data can be transmitted by various  3  Chapter 1. Introduction  Data  -(S^>  PN Generator  Estimate  PN Generator  V_7  Oscillator.coc  Oscillator,coc  Figure 1.1 Conventional DS/BPSK-SS System modulation techniques used for narrow band communications [4-8]. Figure 1.1 shows a conventional DS/BPSK spread spectrum system; the receiver generates a synchronous version of the transmitter's PN sequence, which is multiplied with the received signal. Since a PN sequence is a sequence of pulses of 1 or -1, multiplication of two synchronized sequences yields the constant 1. Therefore, the signalis effectively despread. The carrier demodulation can be performed either before or after despreading. The signal is then passed through an integration and dump lowpass filter with a bandwidth equal to that of the information signal. The decision device decides whether the received data value is "1" or "0" when the output of the filter is greater than zero or less than zero, respectively. The despreading function at the receiver gives the system its jamming rejection advantage over conventional systems. When the receiver multiplies its local PN sequence with the received signal, the signal component modulated with the same synchronized sequence will be despread while any other signal will be spread to the transmission bandwidth. The low pass filter then attenuates the jammer power, multipath interference, and signal power from other co-channel users while preserving the desired signal. This interference rejection capability of a DS-SS system is measured by the processing gain which is expressed as a ratio of the transmission bandwidth of the SS system to the bandwidth of the information.  Process Gain  =  (1.1)  4  Chapter 1. Introduction  where B  ss  is the bandwidth of the spreading code, and  is the baseband bandwidth of  the information. Since bandwidths of both the spreading code and data are proportional to the chip rate and data rate respectively, the processing gain can also be expressed as the ratio of chip rate, R , to the bit rate of data, Rdc  Processing Gain  =  (1.2)  DS-SS signaling allows users with different PN sequences to transmit simultaneously and occupy the same RF bandwidth while rejecting the interference from each other [2]; thus, C D M A is being considered as an attractive alternative to frequency division multiple access (FDMA) and time division multiple access (TDMA) for cellular mobile telephony. Extensive research work has been done in this area to analyze the performance of C D M A systems operating over various channels [9-14]. Typical C D M A systems operate at a data rate of less than or equal to 9600 bps and require one demodulator per user at the cell sites. However, in wireless data networks which do not employ code division multiple access, a high data rate is generally needed to enable effective channel sharing among many wireless terminals using some random access protocols such as Aloha or carriersense multiple access (CSMA) [15]. However, due to limited spectrum, most commercial DS-SS modems for wireless local area networking, particularly those operating in the licence-free ISM (industrial, scientific and medical) bands, are susceptible to interference as their needs to support a reasonably high data rate result in insufficient processing gains [16]. In this thesis, two DS-SS signaling methods known as the Code-Phase Multiplexing (CPM) and Code-Phase-Shift Keying (CPSK) are presented to overcome the data rate vs. processing gain limitation. By definition, maximal codes, commonly used as PN  Chapter 1. Introduction  5  codes, are the longest codes that can be generated by a given shift register or a delay element of a given length. An interesting autocorrelation property of a maximal length sequence (m-sequence) [1] is shown in Figure 1.2. This autocorrelation property of PN codes allows the receiver to distinguish signals modulated by a common PN code with different phase shifts as if they were different PN codes. Both signaling methods utilize this property for multiplexing.or for M-ary modulation.  0-shift (magnitude = code length)  l  -1 chip  _^ shift (chip) -1  +1 chip  Figure 1.2 Autocorrelation function of an m-sequence  The C P M DS-SS signaling approach overcomes the data rate and/or processing gain limitation by modulating successive data bits with different phase shifts of the same PN code sequence, and multiplexing several bits for parallel transmissions. The commutative signaling scheme [17] also staggers successive data bits for parallel transmissions, but they are modulated by different short PN code sequences to combat fading. In contrast, the C P M DS-SS system is designed to increase the processing gain for each data stream without reducing the overall system data rate. This approach is similar to that disclosed in several recent patents [18-20]; however, no specific type of spreading code is specified and performance analysis is lacking. In fact, such DS-SS signaling methods have received little exposure in open literature.  6  Chapter 1. Introduction  The other signaling method, the CPSK DS-SS, overcomes the data rate and/or processing gain limitation by means of M-ary modulation. Although several M-ary DS  T  SS systems have previously been proposed or implemented, they all employ signaling alphabets consisting of M distinct code sequences that are either orthogonal to each other [21], [22], or descendants of a common "parent" code [23]. CPSK, on the other hand, employs an M-ary signaling alphabet in which each of the M = 2 , W  where W is the  word/symbol length, signaling waveforms is obtained from a different phase shift (by an integer number of chips) of a single PN maximal length code sequence. As a result, there are performance advantages over conventional BPSK DS-SS systems in terms of increased power and bandwidth efficiency for a given spreading gain. CPSK also offers implementation advantages over conventional M-ary DS-SS systems as only one code generator is needed in the transmitter and correlation receiver, and only one tapped delay line of twice the code length is needed in the matched filter receiver.  1.3 Organization of the Thesis Chapter 2 describes the system models for the proposed C P M and CPSK signaling methods. The bit-error rate (BER) performance of these two systems against a single tone jammer is examined in Chapter 3. Two cases are examined: the frequency of the tone is the same as, and different from, the carrier frequency. Chapter 4 analyzes the performance of the systems in the presence of multipath fading. Then, the performance of the systems against single tone jammer in the presence of multipath fading is analyzed in Chapter 5. A possible performance improvement utilizing a R A K E receiver is investigated in Chapter 6. Final conclusions and possible future works are discussed in Chapter 7.  Chapter 2 System Models In this chapter, the system models for the proposed C P M and CPSK signaling methods are described to illustrate the mechanisms of these methods. The received signals of these models are described mathematically by way of equations. These mathematical models will later be used for performance analysis by numerical computations if closed form solutions are obtainable, and by computer simulations in all cases.  2.1 Code-Phase Multiplexed Direct Sequence Spread Spectrum System The idea of the C P M DS-SS system is to transmit K data streams in parallel, as shown in Figure 2.1. Each stream is modulated by the same PN sequence, offset in time by some integral number of chips (PN code symbol intervals) relative to the PN sequence modulating the next stream. That is, given p\(t), the PN code used for modulating stream 1, stream k(k  = l,....,K)  is modulated by  Pk(t) = Pi(t-n),  (2.1)  where  T = (k-1)[G/K\T , k  (2.2)  C  [x\ is the integer part of x, G is the length and T is the chip duration of the PN code. c  To simplify the analysis, the outputs of the serial-to-parallel register are assumed to be latched so that the data bits of the K streams change state at the same time. Thus, CO  d (t) =  d rect {t-nT ) k  k  n  71=  Tb  b  — CO  7  k=.l,..,K  (2.3)  Chapter 2.  System Models  8  p,(t) 2cos(cot) Integrate _ \ &Dump|-g,Ub> p (t) 2cos(cot)  Data  Data out  o— n(0 j(t)  P L p^t) Acos(cot)  P (t) 2cos(cot) K  Transmitter  Receiver  Figure 2.1 Code-Phase Multiplexed Direct Sequence Spread Spectrum System  change state at the same time. In equation 2.3, d\ is the data of £-th stream at n-th bit duration, T& = GT is the data bit interval, and rec^ (-) is a rectangular waveform of C  6  7Vsecond duration. The output of the transmitter is:  s{t) = A^2d (t)p {t) k=i k  k  cos (ut)  (2.4)  The received signal at the input of the receiver is:  r(t) = s(t)+j(t)  + n(t)  (2.5)  where j(t) is the jamming and interfering signal and n(t) is the additive white Gaussian noise (AWGN). The receiver consists of an array of K correlators or matched filters, each  Chapter 2. .System Models  9  matched to one of the K phase shifts of the PN code. In particular, a simple correlator as shown in Figure 1.1 can be used. Because of the symmetry of the receiver, only the output of the first correlator, gi{T ), will be examined. b  gi{T ) = Y Jr(t)p!(t)2 b  cos (u;t)dt  "  (2.6)  o With normalization relative to the amplitude of the desired signal,  (T )  9l  = di(T ) + I(T ) + J(T ) + N(T )  b  b  b  b  (2.7)  b  where  ( >)  7  T  =  ^ ¥ k-2  J WtoWn®  b  d  t  ( 2  -  8 )  0  J(T ) = JjrJmPi(t)  cos (wt) dt  (2.9)  n(t) {t) cos (ut) dt  (2.10)  b  o and N{T ) = -^rJ b  Pl  o For easy implementation, maximal length sequences are assumed to be used. The autocorrelation function of a maximal length PN sequence with period G is R (T) = C  G  h E CkCk+T and is given by jfc=i  1, R (r) c  = \ _  r = 0,G,2G.... 1  I —G  r otherwise  (2.11)  Therefore,  n frj  r  6  Pi(t)pi(t)dt o  = -±-  •• . V z ^ l  (2.12)  Chapter 2.  System Models  10  The second term of the filter output, I{T ), due to self-interference can then be simplified b  as  I(T ) = Y,dl(Tb)(=±)  (2.13)  b  k-2  with Pr{di(T ) b  = 1} = Pr{di(T )  ^  '  = -1} = 1/2  h  The last term, N(Tb) represents the thermal noise and is an independent zero-mean Gaussian random variable with the variance _  2  * where E /N b  _  1  A*T ~ -2(E /N )  -  h  b  ^  0  is the bit-energy-to-noise-density ratio, SNRb.  0  /(Tft) is the term due to external jamming and interference signals and will be derived later. Assuming no prior knowledge of J(Tb), since both the self-interference, I(Tb), and the thermal noise, N(Tb), can modeled as random variables with zero mean, the receiver simply estimates the data sent to be "1" if the filter output is greater than, or equal to zero and to be "0" otherwise. The probability of incorrect detection of the r-th data bit may be expressed as  Pe,i = -Px{ {T ) l  9l  b  < 0\'d (T ) = 1} + -Px{ {T ) l  t  b  gi  b  > Q\di{T ) = -1} h  (2.15)  2.2 Code Phase Shift Keying Spread Spectrum System Based on the idea of the C P M DS-SS system, a modified alternative is proposed. One foreseen problem with the C P M spread spectrum approach is that an increasing number of parallel branches would increase the processing gain but also increase the self-interference simultaneously. Depending on the operating environment, this may result in little gain in the BER performance of the system BER. An alternate method is to encode W bits  Chapter 2.  System Models  11  of data into M, where M = 2 ,  different phase shifts of the original PN sequence. The  W  CPSK transmitter, shown in Figure 2.2, groups the input data into W-bit data symbols with symbol period T . The data symbol value is used to select a signaling waveform s  from an M-ary alphabet. Each member of the signaling alphabet consists of a unique phase shift (by an integer number of code chips) of a single (maximal length) PN code sequence, pit), of length G and chip interval T . The selected waveform is up-converted c  for transmission at carrier frequency u>. Suppose the value of the data symbol currently being transmitted is m, where 1 < m < M. The output of the transmitter is therefore s(t) = Ap (t) cos (ut),  (2.16)  m  where p (t) m  = p(t — m T ) is the m-th waveform in the signaling alphabet with phase c  c  shift (in number of chips) m  = m\_G/M\.  c  Here we assume that adjacent signaling  waveforms in the alphabet are separated by approximately the same number of chips in terms of the phase of the PN code.  m(t)  W  S/P  /  data 1 t o W in  Code Phase *" Selector M =2  P N Seq. Generator  s(t)  m  Acos(cot)  '  transmit signal  Code Chip Delays P(t)  Figure 2.2 The Transmitter of Code Phase-Shift Keying DS-SS System :  The received signal at the input of the receiver is  r(t) = s(t)+j(t)  + n{t),  (2.17)  where j(t) is the jamming signal and/or the multipath interference depending on the channel, and n(t) is A W G N with two-sided spectral density N /2. 0  For ease of analysis,  Chapter 2.  12  System Models  we assume that the receiver employs a bank of M correlators, as shown in Figure 2.3, or equivalently, a bank of M matched filters each matched to the PN code with a specific phase shift as given above. The decision device in the receiver determines the most likely phase shift of the data stream in each symbol period and hence decodes the received data symbol into W data bits. The correlators or matched filters require both chip and symbol interval synchronization for successful demodulation. This can be accomplished through the use of a suitable preamble in burst transmissions, or by multiplexing the modulated data signal with a pilot signal consisting of the same PN code with zero phase shift but opposite polarity to facilitate continuous synchronization. We assume carrier, chip, and data symbol synchronization at the receiver. After normalization relative to the amplitude of the desired signal, the output of the m-th correlator at the end of the symbol period, corresponding to the received data symbol, is  Z  m  = 9m(T ) = 1 + J (T ) s  m  s  +  N (T ). m  s  (2.18)  The outputs of the other correlators are  Zi = g (T ) = Ji(T ) + Ni(T ) - i l  s  s  s  V1< i ^ m < M .  (2.19)  In the above equations,  (2.20)  o  (2.21)  o  Chapter 2.  13  System Models  Decision (Select Max.)  W  Wto 1 P/S  and Mto W  m(t) data out  Decode  Code Chip Delays Figure 2.3 The Receiver of the Code-Phase-Shift Keying DS-SS System  The decision device chooses the most positive output as the branch corresponding to the received data symbol, and decodes the W data bits corresponding to the m value. Output data is then obtained through a parallel to serial converter. The receiver makes a correct decision when Z  M  is greater than all the other decision variables, Z„ i ^ m. The  probability of correct detection of the data symbol may be expressed as  oo P = J  P{Z  c  > Zi,Z >  m  Z  m  2  , Z  > Z  m  M  IZ ) m  p{Z )  dZ ,  m  m  (2.22)  —oo where P(Z  m  > Z\,Z  m  > Z  2  , Z  m  > ZM \ Z ) denotes the joint conditional probabilm  ity that given Z , Z\, Z , • • • , ZM are all less than Z , and p(Z ) M  2  density function of Z . M  m  m  is the probability  Assuming all data symbol values are equally likely, the average  probability of a (W-bit) symbol error is PM  =  1 - Pc  (2.23)  Chapter 2.  14  System Models  For performance comparisons with other systems, it is of interest to obtain the biterror probability (or bit-error rate, BER). Assuming all symbol errors are equiprobable and occur with probability P /(M M  - 1) = P /(2  - l ) , the average number of bit  W  M  errors per W-bit symbol is W P M ^  W  / ( 2  W  — l)  and the average bit error probability  is therefore {w-i) — P  2  Ph =  W  i  M  .  (2.24)  Chapter 3 Performance Against Single Tone Jammer 3.1 Channel Models With Single Tone Jammer Various interference sources in the 902-928 MHz licence-free ISM band.are listed in [24]. In particular, ISM equipment which is premitted unlimited radiated energy could generate single tone with much higher power than the transmitting signal. To show the improvement of the performance of these two systems, both cases of single tone jammer with frequency the same as and different from the carrier frequency is considered. In general, a single tone jamming signal at frequency u>j can be written as:  j(t) = acos(ujt + 6)  (3.1)  where 6 is the phase difference between the data signal carrier and the jamming signal.  3.1.1 Single Tone Jammer at the Data Carrier Frequency In a simpler case where the single tone jammer is at the carrier frequency, the normalized correlator output term due to the jamming signal is: T  J(T) = ^  J a (t) cos (ut + 0) cos (ut)dt Pl  (3.2)  o where T can be either 7& or T depending on the signaling scheme being used. Since the s  signal at frequency 2u> will be filtered out, the jamming signal term becomes T  J{T) = ^  J {t)cos9dt  .(3.3)  Pl  o Let the spreading sequence be represented by  G-l (t)  Pl  =  a rect (t n  Tc  n=0 15  - nT ) c  (3.4)  Chapter 3.  where a  n  16  Performance Against Single Tone Jammer  are the chips of the spreading code and rectT (t) c  is a rectangular pulse of  duration of T . c  G-1  T  J{T) =  1  f—  -  /  nrect (t  a  Tc  - nTc) cos 6 dt  o »=° G-l n=0  It is a well known properties of binary maximal-length sequences that, in one period, the number of '—1' is exactly one more than the number of '1' [1], [25]. That is, G-l = -1  J > „ n=0  (3.6)  .  Therefore, the output of due to the jamming signal can be simplified as  where a/A is the square root of the Jamming-to-Signal Ratio, JSR, at the correlator input.  3.1.2 Single Tone Jammer at a Frequency Different from the Data Carrier Frequency In the case where the single tone jammer is at a frequency different from the data carrier frequency, the /-th correlator output term due to the jamming signal is:  T i \dt o  -£=[Ci{Au)  cos6 - Si(Au) sind]  (3.8)  where d(Au)  = Jpi(t)  cos (Aut)  dt  (3.9)  Chapter 3.  Performance Against Single Tone Jammer  17  T  Si{Au) = J (t)  sin (Aoot)dt  Pl  (3.10)  o and Au; = uij — LO  (3.11)  is the difference between the frequency of the jammer and the frequency of the carrier of the system. As shown by Nazari and Ziemer [26], Ji(T) can be written as  MT)  = ^rvi  .  (3.12)  where u>; is a sinusoidal function of uniformly distributed random phase in (0, 2TT) and amplitude IY Therefore, the probability density function (p.d.f.) ,fi(x), of /,(!) is given by i  ,  , ,'. /"n/r,-  '(5)(W^-((i5t5y) fi(x) = 0  where  (  A  -^T )  In equation 3.14, sinc(x) =  sm  Jj \ z  Y  c  . "  Y  otherwise  (3.13)  Q{1)cos(AuTJ)  (3.14)  and Ci(l) is the aperiodic autocorrelation of the  spreading code [11] of the /-th correlator.  Since i ^ depends on both the spreading  sequence of the correlator and the difference between frequencies of the carrier and the tone jammer, the effects due to the tone jammer would be different for different correlators.  In the case that jammer frequency is the same as the carrier frequency,  Chapter 3.  Performance Against Single Tone Jammer  18  Ti = — T , as shown before. In fact, this is very similar to the case where the tone c  frequency is the same as the carrier frequency except that the output due to the jamming signal at each correlator has a different gain of ( I V T ) . In order to be able compute c  probability of error with normalized quantities, we define  Ti  , and  7,- = —  where f  c  AuT  fi =  c  2TT  Af  (3.15)  fc  is the chip frequency in Hz. Figure 3.1 shows the dependence of 7,- on u- for  three m-sequences which are generated by the same frequency with different amount of phase shift (in number of chips). The graphs show that 7; fluctuates randomly, but in general, decreases with a- when u- is greater than 0.5. The relation between 7,- and \i also depends on the sequence and its phase shift.  Figure 3.2 shows how 7,- changes with  the phase shift of the same sequence at /x = 0.1, 0.2 and 0.25. The results show that the  15  1 1 1 1 1 1 1 1 1 1 i 1 1 1 i i i 1 i 1 i 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 J 1 1 1 1 1 1 1 1 1 J 1 1 1 1 1 1 1 1 1 ,| 1 1 1 1 1 1 1 i 1 1 i 1 1 1 1 1 1 1 i | iI i 1 n mu 1 n1 1 1 r 1 i| p 1 1 r r r 1 \ \ 1  shift=6ir shift=31Tc shift=0r c  c  0  0.1  0.2  0.3  o -Q-- A -  0.4  0.5  0.6  0.7  0.8  0.9  Figure 3.1 Graph of 7,- vs. JJL for Various Phase Shift of the Same PN Sequence  Chapter 3.  19  Performance Against Single Tone Jammer  relation between 7; of a m-sequence and its phase shift changes with \x and cannot be generalized by a simple function. In other words, the effective gain of a tone jamming signal at a frequency different from the carrier frequency would greatly depend on the amount of frequency difference, the chip frequency, the m-sequence for spreading and its phase shift, and cannot be described by a simple function in general. It is important to note that this problem is not unique to the systems presented in the thesis but to DS-SS systems in general.  0  10  20  30  40  50  60  shift of the original sequence Figure 3.2 Graph of 7/ as a Function of Phase Shift for-Various Differential Frequencies  3.2 Performance of C P M DS-SS Against Single Tone Jammer at the Data Carrier Frequency In this case, the effects of the tone jammer is the same on all parallel correlators, and the self-interference is a random variable with the same distribution at all correlators.  Chapter 3.  20  Performance Against Single Tone Jammer  By symmetry, analyzing the statistics of one correlator output would be sufficient. The analysis follows the approach in [27] and employs the B E R approximation in [10].  3.2.1 Derivation of Average B E R Using Characteristic Functions Assuming that each bit of data is independent identically distributed with  Pr{d {T ) k  b  = 1} = Pr{d (T ) k  b  = -1} = 1/2  Vk = l,2,.,K  (3.16)  the self-interference term l(Jb) is a random variable with binomial distribution. The characteristic function of I(Tb) is,  $ (u) = cos^- )(^)  (3.17)  1  7  For a fixed phase angle, 6, the jamming term J(T ) = (^) (~'g ) becomes a constant. s6>  b  The thermal noise term N(Tb) is an independent zero-mean Gaussian random variable with the variance a given by equation 2.14. Then, the characteristic function of N(T) is 2  $AT(U) =  expl —  \  (3.18)  Therefore, the characteristic function of the sum of the self-interference and thermal noise, Y{T ) b  = I(T ) + N(T ), b  b  is  ^y(u) = c o s ^  Qexp(^^j  .  (3.19)  Fixing the phase angle of the jamming signal, the conditioned average probability of error of the C P M DS-SS system at the output of correlator 1 is:  P ,l\e = \Pr{gi(T ) e  b  < 0 | hM)  = 1} + \Pr{ (T ) gi  b  >0|  = -1}  Chapter 3.  21  Performance Against Single Tone Jammer  J—1  I; J  oo  fy{y)dy + \ J  -oo  fr{y)dy  where J = ' a \ ^( — cos 9 G  J+l  J+l J  Mv)dy  J-I  J+l  2  CO  J 2TT 7 J-l -oo J+l oo  2  —— —  1 f —  2  7T  1  1  f  2J-l J  YK }  $Y( ) u  dy  c  o  s  ( y) dudy u  fyis real h even  J -oo  oo$ y ( « ) J+l ^ - 2~ y y cos (m/)  o  OO 1 r  1  j-l  j , . Vsin(u7/) V +  1 7  0  oo  1 1 /" $y(lt) — —— / (sin (Ju + u) — sinUu 2 2ir J0 u 1  oo  1  — u)) du  i V c o s (Ju)sin(u) ,  0  oo 0  Since the phase angle of the jammer is usually not know and assumed to be a random variable uniformly distributed from 0 to 2tr, the average probability of bit error of the C P M DS-SS system can be simply expressed in terms of the conditional probability of error as:  BER = -L J P  eMe  dd  (3.21)  Chapter 3.  22  Performance Against Single Tone Jammer  3.2.2 Numerical and Simulation Results  The relations between bit-error probability (BER) and the bit-energy-to-noise-density ratio, SNR = (E /N ), b  — 20 log (a/A),  b  0  and the jamming-to-signal ratio, JSR = (a/A)  2  or JSR (dB)  at the receiver have been evaluated by simulations and numerical  integrations for different number of parallel data streams, K. When K= 1, the system is simply the conventional BPSK DS-SS. In this case, the aggregate data rate R\, and the chip rate C\ and the length G\ of the spreading code, are related by: processing gain = G\ = C-[/R\.  For K>1, we keep the transmit power and bandwidth approximately  constant so that the amplitude for each data stream is AJC = Af\fK, rate of each data stream to RK = Ri/K,  and reduce the data  so that for each data stream, the processing  gain, GK, is increased to K(G\ + 1) — 1 and SNR remains unchanged. Thus, for constant jamming signal power, the jamming to signal ratio at the correlator input of each data stream, JSRjf = (a/Ax)  = K x JSRi is increased by the factor K. Since a DS-SS  system with a processing GK attenuates the jamming signal amplitude by a factor of GK, the net effect of increasing K of a C P M system is to reduce the jamming signal power at each correlator by a factor of  \fK.  Figure 3.3 shows the conditional BER vs. JSRi for jammer phase fixed at 0, 30, 45, 60 and 90 degrees relative to the carrier of a C P M DS-SS system with four parallel data streams and PN sequence length, G4, of 511 operating at an SNR of 10 dB. The b  graph indicates that depending on the jamming signal power, the conditional BER could change very little or significantly as the phase of jammer varies.  The conditional BER of  the system with different fixed jammer phases relative to the signal carrier are computed using equation 3.20 and shown in Figure 3.4.  Chapter 3.  Performance Against Single Tone Jammer  l.OE+00  l.OE-01  23  ~i—i—i—i—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—1—1—1—I—I—I—I—I—I—I—I—I—I—r—i—i—i—IT  U  1.0E-02  phase =90 phase =60 phase =45 phase =30 phase = 0  0-s  '''  / /  K/ / / // w  PQ  /  1.0E-03  '  ///  1.0E-04  /  0  /  /  //  /  /  Jo  /  /  '  /  /  8  1.0E-05  1.0E-06  20  _l  I I I I I I I I I I I I I L.  30  I I I I I I I I I I I I I [_  50  40  60  JSR (dB)  Figure 3.3 Conditional B E R of C P M vs. JSR for Various Jammer Phase @ SNR =10dB G =511 b  4  Notice that the conditional B E R is an even and periodic function of 9. Figure 3.4 shows that increases in the number of parallel data streams, K, has no effect on the BER performance at 9 = ± 9 0 ° . This is because the jamming signal has no effect on BER performance when it is orthogonal,^ = ± 9 0 ° , to the signal carrier, and increasing processing gain has no effect on Gaussian white noise. In most cases, the tone jammer is not orthogonal to the signal.carrier, 9 / ± 9 0 ° , and jamming degrades BER performance. In these cases, increasing processing gain suppresses, partially or completely, the tone interference. Hence, the B E R performance improves as K increases. The average BERs of the systems are first computed numerically using equations 3.20 and 3.21 with different values of SNR, JSRi, original PN code sequence length, G i , and number of parallel data streams, K. In order to corroborate the results obtained  Chapter 3.  24  Performance Against Single Tone Jammer  l.OE+00  l.OE-01  1.0E-02  l.OE-03  1.0E-04  l.OE-05  1.0E-06 1 ,UJJ W _  0  20  40  60  80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Jammer Phase (deg)  Figure 3.4 B E R of C P M Conditioned on Jammer Phase @ SNRb=10dB, JSRi =50dB and Gi=127 numerically, the average BER of the systems are also obtained by simulations based on the model described by the equations 2.7, 2.13, 2.14 and 3.7 with different values of SNRb, JSRi, G i , K. The details of the simulation method is described in Appendix B. Both results obtained by numerical integration and simulations are shown in Figures 3.5, 3.6, 3.7, 3.8 and 3.9.  In particular, Figure 3.5 shows the performance of the system  with Gi=511 subject to a jammer with JSRi =50 dB. This figure illustrates clearly the BER improvements as K increases from 1 to 64. At BER = 10~ , increasing K from 1 to 2  2 results in a 3 dB reduction in SNRb, enabling a doubling of the data rate for the same transmit power. Further increases in K result in diminishing returns, as half as much dB reduction in SNRb is achieved for each doubling of K. Notice that with K = 64, the effects of the jammer is almost completely negated, resulting in an overall SNRb saving of about 5 dB compared to K - 1 at BER = IO . Greater savings in SNRb due to increasing K -2  Chapter 3.  Performance Against Single Tone Jammer  25  l.OE-01 1.0E-02  k  w PQ  2  4  6  8  10  12  SNR (dB)  Figure 3.5 B E R of C P M vs. SNR @ JSRi =50 dB, Gi=511 b  are obtained at lower BERs. At SNRb=10 dB, increasing K from 1 to 64 reduces BER from a barely acceptable (for most applications) 1 0 , to a fairly adequate 5 x 1 0 . —2  -5  Similar results are obtained in Figures 3.6, 3.7, 3.8 and 3.9. These figures show that the effectiveness of increasing K depends on the JSRi and G i . At low JSRi, increasing K does not improve the BER performance if G i is already large enough to suppress the jamming signal. For example, with the SNR =6 dB and Gi=511, increasing K does not b  lower the B E R if JSRi is less than 30 dB. On the other hand, with the SNR =6 dB b  and Gi=255, if JSRi is large, say 65 dB, K has to be increased from 1 (BER^0.45) to 32 ( B E R « 0 . 2 ) or even larger to observe significant BER improvements. Therefore, depending on the SNR , there is a finite range of JSRi over which the C P M DS-SS b  scheme is effective. For a given achievable BER objective at a fixed SNR , the range of b  JSRi values the system is capable of handling increases approximately 3 dB when K is  26  Chapter 3. Performance Against Single Tone Jammer  l.OE+00  —1—I  1—I—I—I—I—T~  1  I  I  I  I  I  I  I  I  I  I L  I  I  I  I  I  I  I  I  I  1 I  I  I  I  I  I  1 I  I  I  I  I  I  I  I  I T  Simulation K =32 Simulation K= 16 Simulation K = 8 Simulation K = 4 Simulation K = 2 Simulation K = 1 Numerical K =32 Numerical K =16 Numerical K = 8 Numerical K = 4 Numerical K = 2 Numerical K = 1  l.OE-01  1.0E-02  l.OE-03  _1  20  30  40 JSR  50  60  (dB)  Figure 3.6 B E R of C P M vs. JSRi @ SNR =6 dB, Gi=255 b  1.0E+00  r  1.0E-01  t-  1.0E-02 b-  1.0E-03 JSR  (dB)  Figure 3.7 B E R of C P M vs. JSRi @ SNR =6 dB, Gi=511 b  doubled. For example, when SNRb is fixed at 8 dB (in Figure 3.8 ), Gi=255 and targeted  Chapter 3.  l.OE+00  l.OE-01  W PQ  27  Performance Against Single Tone Jammer  i—i—i—i—i—I—i—i—i—i—i—i—r—i—i—|—i—i—i—i—i—i—i—i—i—I—i—i—i—i-  - | I I I V  t-  1.0E-02  Simulation K =32 Simulation K =16 Simulation K = 8 Simulation K = 4 Simulation K = 2 Simulation K = 1 Numerical K =32 Numerical K =16 Numerical K = Numerical K = 4 Numerical K = 2 Numerical K = 1  l.OE-03 . -A'  l.OE-04  I  20  I  I  I  I  L_J  30  1_J  I  I  I  I I I I I I 40 JSR (dB)  1 I  L_J  I I 50  I  I  I  I  I L  Figure 3.8 BER of C P M vs. JSRi @ SNR =8 dB, Gi=255 b  1.0E+00  1.0E-01  £1 PQ  1.0E-02  n—i—i—r  T  1  1  1  1  1  1  1  Simulation K =32 Simulation K=16 Simulation K = 8 Simulation K = 4 Simulation K = 2 Simulation K = 1 Numerical K =32 Numerical K =16 Numerical K = 8 Numerical K = 4 Numerical K = 2 • Numerical K = 1  1.0E-03  JSR (dB)  Figure 3.9 B E R of C P M vs. JSRi @ SNR =8 dB, Gi=511 b  60  Chapter 3.  28  Performance Against Single Tone Jammer  BER = 10~ , the system can handle JSRi approximately up to 36 dB, 39 dB, 42 dB, 45 3  dB, 48 dB and 51 dB for K = 1, 2, 4, 8, 16 and 32, respectively. Since the processing gain, GK, increases by 3 dB when K doubles, the loss due to the self-interference is indeed very small. The results show that the C P M DS-SS signaling scheme has the following desirable properties: 1.  Given a fixed transmit power, bandwidth, jamming power, and a specific B E R objective, the aggregate data rate can be increased by increasing K.  2.  Given a fixed transmit power, bandwidth, data rate, and a specific B E R objective, the jamming margin, i.e., the degree of interference tolerance, can be increased by increasing K. All the figures also show that the simulation results agree with the numerical results  within 5%. The accuracy of the simulation results can probably be improved by increasing the number of simulated samples in all simulations.  3.3 Performance of C P M DS-SS Against Single Tone Jammer at a Frequency Different from the Carrier Frequency 3.3.1 Derivation of Average Probability of Error The analysis for this case would be very similar to the case with tone jammer at the carrier frequency except for the following differences.  First, the jamming effects  of the same tone jammer would be different on each correlator, as discussed in section 3.1.2. However, the effects would be the same as those caused by a tone jammer at the carrier frequency but with different amplitude. The effective J S R K at the i-th correlator  29  Chapter 3. Performance Against Single Tone Jammer  input would be  (3.22)  where I\ is given by equation 3.14, instead of  (OC/AK),  as in the case where the tone  frequency is the same as the carrier frequency. Therefore, the average probability of error for the /-th correlator, P j, can be derived using the same method, i.e., characteristic e  functions, as before.  Thus, the average probability of error of the /-th correlator  conditioned on the phase angle of the jamming signal is given by equation 3.20, modified with the effective JSR:  oo F  -i» = 2"~, J u  e  0  '  c o s (  'KG)  C O S  [ A T — V <">*• Sln  ( 3  '  2 3 )  -  and the average probability of error of the /-th correlator would be Pes = ^  J P \e dd . e>l  (3.24)  0  Since the jamming effect is different for different correlator, the correlators' outputs are no longer symmetrical. Thus, the average probability of error of the system becomes 1  K  BER=-^P ,i e  (3.25)  3.3.2 Numerical and Simulation Results Since the gain (Ti/T ) c  depends on both on Aui and the aperiodic autocorrelation of  the spreading code of the correlator, choosing a proper sequence is very important for achieving good system performance. In fact, for a given Au, the BER performance of a system may be improved as K increases if a particular PN sequence is used, and degraded as K increases if another PN sequence is used instead. Since the previous results show  Chapter 3.  Performance Against Single Tone Jammer  30  K  Polynomial of Generator  1  x +x + 1  11110  7.5462  2  x +X + 1  0 0 10 0 0  10.282, 10.52  4  x  0 0 0 0 10 1  13.913, 13.437, 12.61, 3.751  8  z  0 0 0 0 10 10  14.968 (for all correlators)  5  2  6  7  8  -t-.z + X + 1  + x  e  + x  4  5  +  2  x +X + 1 3  | Initial State  Jammer Amplitude Gains  Table 1 An Example of Implementation of C P M Simulation  that the jamming margin is increased by approximately 3 dB when K is doubled, the performance of a system will very likely be improved if the effective jamming amplitude is increased by no more than 3 dB when K is doubled. However, not all correlators have to be improved to yield an improved overall system performance. That is the overall system performance may be increased when K is increased, even though the effective jamming power is increased by more than 3 dB for some correlators, as long as the average B E R is lower. For example, considering the case where Au> = 0.2u> , where LO is the chip frequency c  C  in radian, a C P M DS-SS system is implemented with the m-sequences listed in Table 1 for different values of K's.  The table lists the polynomials and initial states of the  shift registers used to generate the sequence and the equivalent jammer amplitude gains for the correlators. Like the case with the jammer at the carrier frequency, graphs of average B E R vs. SNRb and average B E R vs. JSRi are evaluated both by simulations and numerical integrations. The simulation program is modified to take the effects of jamming frequency into consideration such that the jamming signal term at the correlator outputs are generated using equation 3.8. The numerical results are computed using equations 3.23, 3.24 and 3.25 which yield, respectively, the B E R conditioned on the phase angle, the average B E R for the /-th filter and the average B E R for the overall system. The  Chapter 3.  Performance Against Single Tone Jammer  31  l.OE+00  l.OE-01  Pi  1.0E-02  -  l.OE-03  r  1.0E-04  r  •  W CQ  l.OE-05  r  1.0E-06  -  1.0E-07  Simulation,K = Simulation,K = Simulation,K = Simulation,K = Numerical,K= Numerical,K= Numerical,K= Numerical,K= 8  0 • o  8 4 2 1 8 4 2 1  0\.  IP  A  N  10  12  14  16  SNR (dB)  Figure 3.10 B E R vs. SNR of C P M System with JSRi = 10 dB, Gi=31, ALO=0.2UJ b  C  results are summarized and shown in Figures 3.10 and 3.11 for Gi=3.1, Au>=0.2u> and c  different values of JSRi and SNR . b  The results show that the C P M system subjected to single tone jamming with the tone frequency not the same as the carrier frequency behaves similarly to the one subjected to single tone jamming with the tone frequency at the carrier frequency, except that the interference are often effectively amplified for the same jamming power when the tone frequency is different from the carrier frequency. As a result, the benefit of increasing the number of parallel streams would be less for the case with tone frequency different from the carrier frequency. In this particular example, the effective gain of jamming signal for the case of K=2 is about 3 dB more than that for the case of K=l.  Consequently,  the jamming margin is increased by less than 0.5 dB and the B E R performance is only slightly improved when K is doubled from 1 to 2. Both the numerical and simulation  Chapter 3.  32  Performance Against Single Tone Jammer  results agree with each other very well except for the case of K=4. This is probably due to the fact that one of the correlators (data channels) has very low average B E R at this simulated conditions. Therefore, much more data samples may have to be simulated in order to obtain more accurate average statistics.  3.4 Performance of CPSK DS-SS Against Single Tone Jammer at the Data Carrier Frequency 3.4.1 Derivation of Average Probability of Error In equation 2.21, • Ni(T), i =  1,2, ...,M,  are identical independent zero-mean  Gaussian random variables with variance 1  TA S  l.OE+00  (3.26)  2(E /N )  2  S  0  F  1.0E-01 0  1.0E-02 c4 w pa 1.0E-03  Simulation,K = Simulation,K = Simulation,K = Simulation,K = NumericaLK = Numerical,K= NumericaLK = Numerical,K=  X  1.0E-04 V-  - /  /  -x 1.0E-05  10  12  14  8 4 2 1 8 4 2 1  x 0 o A  16  18  JSR/dB)  Figure 3.11 B E R vs. JSRi of C P M System with SNR = 14 dB, Gi=31, Aw=0.2w b  c  20  Chapter 3.  Performance Against Single Tone Jammer  where E /N s  0  33  is the symbol energy to noise density ratio. The probability function of the  correlator outputs, given the ra-th data symbol (word) is transmitted, are therefore  p(Zi) = - ^ e x p ( ~( Vlira \  Zi+  ~  V 0  ^  2  ) )  \<i±m<M.  (3.27)  Since all symbols are equally probable, without loss of generality, the case of m=l is considered. The probability of correct decoding when m=l may be expressed as  oo P = J P(Z  (3.28)  c  where P(Z\  1  > Z ,Z 2  X  > Z , Zi > Z , . . . , Zi > Z 2  3  I Zi) (Zx)  M  P  dZ  x  > Z%,..., Z\ > ZM \ Z\). denotes the joint probability that Z , 2  Z3, . . . , ZM are all less than Z\, conditioned on Z\.  Since the {Z\\ are statistically  independent, the joint probability may be expressed as a product of M-l marginal probabilities of the form z • 1  P(Zi > Z \Zi)= l  J p(Z )  dZ  l  I < i < M-  l  ^1  Z  e  2TTCT J VSirtr  -( -+T- e) J  ^  dZt  —00  = --L  y/2ir  /  J  e~  x2/2  dx.  (3.29)  —00  These probabilities are identical for other decision variables and, hence, the probability  Chapter 3.  34  Performance Against Single Tone Jammer  of correct decision becomes M-l  e~^~ dx — CO  p(Zi) dZi  —oo  M-l  a-  CO  2  (M-l)  v^/( ~^ 1  r/C  Z  - I - J  I  -e '2/KC  dx  oo  =  - (  I  e  2^  9  )  2  dZ\  (y-jff e  (v7!)  2  dy  (3.30)  where (3.31)  G and the probability of a (W-bit) word error is P  M  =  l-P  c  (M-l)  (y-A'y/2(E /N j) s  0  (3.32)  x exp  Note that for G > 1, equation 3.32 reduces to P  M  =  l-Pc (M-l)  V2 -(y-y/2(E /N ))' a  x exp  0  dy,  (3.33)  Chapter 3.  Performance Against Single Tone Jammer  35  the average symbol error probability for conventional orthogonal signaling [6]. Since all M symbols are equal likely, the above expression for PM is also the average probability of a symbol error. In order to compare the performance to other schemes, it is desirable to convert the probability of a symbol error into an equivalent probability of bit error and express the B E R in terms of the bit energy to noise density ratio, SNRb, instead of symbol energy to noise density ratio, SNR . Since each symbol conveys W bits of S  information, the SNRb is simply l/W of the SNR . For equiprobable W-bit symbols, S  equation 2.24 is applied to obtain the BER. The above equations show that the BER performance of CPSK is independent of tone interference at the carrier frequency. Although it is unlikely that an interfering tone is right on carrier frequency, the above result has the important implication that the carrier synchronizer at the receiver does not need to phase lock to the received carrier, but it only needs to frequency lock to the received carrier.  3.4.2 Numerical and Simulation Results Unlike the C P M scheme described above, the CPSK scheme can increase the processing gain and/or the data throughput without any penalty of increasing any selfinterference. On the contrary, the desired, signal component of the correlator output is increased by a small fraction 1/G, where G is the length of the PN codes, and the selfinterference is cancelled. The average bit error rate is computed using the equations 3.32 and 2.24, and plotted against SNR in Figure 3.12 for different symbol sizes W. This b  figure shows that the performance of CPSK is similar to M-FSK, in that power efficiency increases with M. Compared to conventional DS-SS signaling employing BPSK or QPSK, which can do no better than non-spread BPSK in thermal noise, CPSK requires a lower SNRb at any given BER for W > 3. For example, in order to achieve an average B E R  Chapter 3.  36  Performance Against Single Tone Jammer  w pa  SNR.(dB)  Figure 3.12 BER vs. SNR for CPSK System with Single Tone Jammer at Carrier Frequency b  of lO" , BPSK requires SNR » 9.5 dB, while CPSK requires SNR « 8.3 dB, 7.3 dB, 5  b  b  6.7dB and 6 dB with W = 3, 4, 5 and 6 respectively. Thus CPSK with W = 6 achieves a power saving of 3.5 dB over BPSK at the above BER. Whereas the bandwidth of M-FSK increases with M without added interference suppression capability, the bandwidth of CPSK is fixed once a processing gain is selected, e.g., to satisfy interference suppression requirements. Increasing M thus increases the bandwidth efficiency of CPSK signaling, to the extent that the number of different code-phases can be accommodated within the chosen code length. For example, to maintain a phase separation of two chips between two signals in the signaling alphabet, M < 127 or W < 6 for G = 255. The minimum phase separation between two resolvable signals is one chip. Moreover, if the symbol duration, T , remains unchanged while W increases, the aggregate data rate also increases s  Chapter 3.  Performance Against Single Tone Jammer  37  by the same amount. On the other hand, if T is increased, a longer PN code can be used s  accommodating larger number of different code-phases, M. The average BER is also obtained by simulations, described in Appendix B, based on the model described by the equations 2.18, 2.19, 2.20 and 2.21 with different values of SNRb, JSR, G and W. The results are summarized and shown in Figure 3.13. Figure 3.13 shows the average B E R of the CPSK system with different word lengths (symbol sizes) for different SNRb obtained from simulations with JSR=10 dB and 30 dB and from numerical computation using equations 3.32 and 2.24. The fact that these three sets of results agree with each other confirms the claim that the system performance of CPSK system is immune to single tone interference at the signal carrier frequency. Further more, the results also show that the CPSK system can easily out perform the parallel scheme described above under single tone interference. In fact, as mentioned above, the CPSK system with symbol size of three bits even perform better than the BPSK over A W G N channel with no interference signal, the best possible performance of the C P M scheme.  3.5 Performance of CPSK DS-SS Against Single Tone Jammer at a Frequency Different from the Carrier Frequency  3.5.1 Derivation of Average Probability of Error Again the analysis for this case is quite similar to the case with tone jammer at the carrier frequency. However, one very important difference is that the effects due to the same tone jammer are different on different correlators with the effective JSR given by equation 3.22. Denote the output of i-th correlator due to the tone jammer conditioned on a given phase angle by J;,?. Then the probability functions of the correlator outputs,  Chapter 3.  38  Performance Against Single Tone Jammer  l.OE+00  l.OE-01  f  1.0E-02  tA w pa  l.OE-03  1.0E-04  t-  l.OE-05  1.0E-06  2  4  6 SNR (dB) b  Figure 3.13 Comparison of the Numerical and Simulated B E R of CPSK System with Single Tone Jammer at the Carrier Frequency given the m-th data symbol is transmitted, are P( m)  =  Z  1 nr-  / -(Z  - 1-  m  E  X  J ,e)' m  P  VZira  ^  )  \<i^m<M  v ^  =  6  X  (3.34)  P  Similar to equation 3.28, the conditional probability of correct decoding in this case is simply given by CO  P \m= c  J  P(Z > m  ZiVij:me[l,M]\Z )p(Z )dZ m  m  m  (3.35)  — CO  However, the marginal probabilities would become  P(Z  m  > Zi\Z ) m  = -)= V 27T  f J  e~ l dx x2  2  \<i^m<M  , (3.36)  Chapter 3.  39  Performance Against Single Tone Jammer  Thus, the conditional probability of correct decision becomes  I  M  0 0  iw»=/ n  \  a  /  —oo i = 1  e  2  dx  i y£ m -M  - w J-oo n. Z =1 0 0  ' - ?  r  /  <  :  v5—  I-  \e~~dy  (3.37)  i^ m  where 1+G  A' When G >  1,  (3.38)  ~G~  A ' ~ 1.  The probability of VK-bit symbol error, given the m-th data symbol is transmitted, conditioned on the phase angle and frequency difference for large G now becomes  Pe\9,m,Au> — -  Pc\6,m,Aui  y +  -oo  I= 1  v  x  1+Jm,e — Ji,e  y/2  i/ m x  exp ( - y  \dy  (3.39)  where cr is given by equation 3.26. Thus, the average probability of error if the m-th word is transmitted for a fixed Aco, is simply 2;r elm, Au  2 t t /  Pe\6,m,Au d&  (3.40)  Chapter 3.  40  Performance Against Single Tone Jammer  The average probability of word error conditioned on Aw would then be 1  M  (3.41)  M  Once again, equation 2.24 can be used to convert the probability of symbol error into an equivalent probability of bit error and express the BER in terms of SNR , instead of b  SNR . Notice that the average probability of bit error still depends on the choices of the S  spread sequence, amount of shift in integral number of chips between the data symbols, and the ratio of frequency difference to the chip frequency.  3.5.2 Simulation Results Since the effects on the correlator outputs due to the same tone jammer would be different depending on the choices of the spreading PN sequence, the amount of shift, in number of chips, of the code sequence for different correlators and the amount of difference between the tone frequency and the carrier frequency, the average BER would also depend on these factors. An arbitrary maximal length sequence is choosen for the simulations.  The results do not necessarily reflect the optimal, average or the worst  performance. This is just an example illustrating the effects of CPSK operating under a single tone jammer with the tone frequency different from the carrier frequency. In particular, a system implemented with an m-sequence of length 2048, as specified in Appendix B, subjected to a single tone jammer with the tone frequency at 10% of the chip frequency away from the carrier is considered.  The mathematical model of the  system consists the output of the correlator corresponding to the transmitted word and the outputs of the rest of correlators which are generated by equations 2.18 and 2.19, respectively. The jamming signal term and thermal noise term of these equations are  Chapter 3.  41  Performance Against Single Tone Jammer  l.OE+01  l.OE+00  word length=8 word length=7 word length=6 wordlength=5 word length=4 wordlength=3 wordIength=2 word length=l  b  -• A1 0x e-o-A  -A--  ft PQ  [} -  l.OE-01  -B-  --Q--  <6-_.  A-  - i\  • -a -  -J]  ••e—X—  <S-  —Q~  -X-  -0--  •-0--  1.0E-02  -+• —•-  l.OE-03  10  12  14  SNR/bit (dB)  Figure 3.14 B E R vs. SNR for CPSK against b  Single Tone Jamming with ACJ=0.1O; , JSR=10 dB c  generated by equations 2.21 and 3.8. The results of the simulations are summarized in the Figures 3~14 and 3.15. Figure 3.14 shows that increasing the SNR gives litde improvements in the B E R b  performance if the jamming signal is increased by the same ratio, so that JSR remains unchanged. This implies that the effect of the tone jammer in this case is so dominant that the thermal noise becomes an insignificant source of error. Therefore, CPSK may not be the most efficient choice for system with high transmitting power. However, in the case of low SNR , CPSK still out performs the BPSK system operating over A G W N only. b  For example, if SNR is at 4 dB, the average BER for BPSK in A W G N only is about b  0.02 whereas the average BER of the simulated system is about 0.002, an improvement. by an order of magnitude. Figure 3.15 once again shows that lower average B E R may be achieved by increasing the word length. The figure also shows that when the JSR  Chapter 3.  42  Performance Against Single Tone Jammer  becomes too large, increasing the word length has little or no effects. However, this effective range could be changed depending on the actual implementation and operating conditions of a particular system.  Figure 3.15 B E R vs. JSR for CPSK against Single Tone Jamming with Au>=0.lu> , SNRb=4 dB c  Chapter 4 Performance in the Presence of Multipath Fading 4.1 Model of Multipath Fading Channel Multipath fading is one of the major sources of interference for indoor wireless communications. A number of studies have been done to develop statistical models and channel impulse responses of indoor multipath channels [28-32]. However, no single model can be used for all situations as the channel characteristics change with the amount and geometric shape of open space, material of objects and surrounding walls and many other factors. In fact, not all research results agree with each other. However, a majority of previous works agree that in most indoor channels: (1) the maximum delay spread does not exceed 400 ns, and (2) the typical delay spread is about 100 ns. If the spread bandwidth of the spreading signal exceeds the coherence bandwidth of the channel which is inversely proportional to the channel maximum multipath delay spread, the multipath components can be resolved into a discrete number of faded paths [33]. Thus, we assume that the indoor channel for the desired transmitter and receiver can be represented by an L-path fading model [6], [4] with L discrete multipath links between the transmitter and receiver. The low-pass equivalent impulse response of the passband channel, h(t), can be represented by L h(t) = Y m t - t i ) e ^  (4.1)  1=1  where <*>(•) is the Dirac delta function and /?/, <j>i and // are, respectively, the path gain, the path phase, and the incremental path delay of the /-th path of the multipath components. Because of the motion of people and equipment in and around the building, these 43  Chapter 4.  44  Performance in the Presence of Multipath Fading  parameters are randomly time-varying functions. However, the rate of their variations is very slow compared to any useful signaling rates that are likely to be considered. Based on G. Turin's [34] and M . Kavehrad's [33] descriptions of a discrete multipath fading channel, the following assumptions are made to enable the multipath components to be resolved. In case of no direct path between the transmitter and receiver, /?/ are assumed to be Rayleigh random variables for all 1=1 to L with the average power gain decreasing linearly with the corresponding path delay. This is a simplifying assumption which allows the path gains to be evaluated relative to each other. Transmitted signals are actually attenuated according to the inverse square to inverse fourth power of the absolute path lengths [35]. Furthermore, 4>i is assumed to be uniformly distributed between 0 and 2TT; and ti is multiple of T and uniformly distributed between 0 and T , where T c  m  m  is the  maximum multipath delay spread. In order to avoid the intersymbol interference, T , the s  data symbol interval has to be greater than T . This implies that the baseband signaling of m  a conventional SS system is limited to a rate less than the channel coherence bandwidth. With the proposed schemes, the symbol period for the same raw data rate can be easily increased by K times, where K is the number of parallel streams of data or the symbol length. A l l the parameters of all paths are also assumed to be independent identically distributed. This channel model is the well known Rayleigh Channel which is commonly used to model communication channels of mobile and indoor wireless communications where the receiver is not in the line of sight (LOS) of the transmitter.  The impulse response given in equation 4.1 is characteristic of a discrete multipath channel and has the same functional form as that given by G . Turin [34] and M . Kavehrad [33].  However, the number of discrete paths L is not determined in terms  Chapter 4.  45  Performance in the Presence of Multipath Fading  of communication system parameters. The basic result on the time resolution of signals using DS-SS signals is given in Section 1.5.3 of the text book by Simon et al [36]. As one would expected, the maximum value of L depends on the spread bandwidth, B , ss  because two signals must be separated by at least one chip duration, T , in order to be c  resolved. Using the result of time resolution of DS-SS signals shown by Kavehard et al [37], the maximum number of resolved paths for a maximum multipath delay spread, T, m  is given by  L max  m —  + 1 = [T * B \ m  ss  (4.2)  + 1  where [^J is the largest integer that is less than or equal to x and B  ss  is the one-sided  bandwidth of the spread-spectrum signal. However, there is not necessary a signal path for each integer multiple of T up to L c  max  Depending on the specific operating environment,  L could be any value between 1 and Lmax. The effects of choosing different values of T  m  and L will first be examined later by simulations. Copies of the transmitted signal  that arrive at unresolvable time difference, less than T , are assumed to combine to give c  rise to the Rayleigh distributed path gain as mentioned above. If the receiver is in the line of sight of the transmitter, a direct path signal exists. In this case, the channel model is the well known Rician Channel. The parameters of all paths are the same as those for Rayleigh Channel model except for the first path, which now consists a direct signal component as well as unsolved reflected signals. Thus, the first path can be represented by two components, the direct component and reflected component, as:  (4.3)  Chapter 4.  Performance in the Presence of Multipath Fading  46  where p is a real constant representing the attenuation of the channel for the direct path and ( is a Rayleigh distributed random variable with mean square ( . 2  One important  parameter of the Rician channel model is the K-factor which is the ratio of the direct path signal energy to the average scattered signal energy, K, with  P  2  * =  (4.4) Co  '  4.2 Performance of C P M DS-SS in the Presence of Multipath Fading 4.2.1 Analysis Using the model of the proposed C P M DS-SS system, the output of the transmitter is; K  ,s(t) = Aj2dk(t)Pk{t) k=i  cos (ut)  (4.5)  In the absence of other interfering signals, the received signal at the input of the receiver in this case is:  •(*) = Re\  J  h(T)s(t - T)exp(jut)  dr \  +  (4.6)  n(t)  where s(t) is the complex envelope of s(t) and n(t) is the white Gaussian noise with double-sided spectral density  N /2. 0  Combining the equations 4.1, 4.5 and 4.6 we have ' L  r(t) = 1=1  where <p = <t>\k lk  K  Y^  ^  lhdh  k=l  ~ u)Pk{t ~ m) cos (tot + <p ) + n(t) T  lk  ur . lk  Since the parameters of a path should be the same for all data streams, we have  fti = ft = 2  =  =  (4.8)  (4.7)  Chapter 4.  Performance in the Presence of Multipath Fading  fal  ~ fal = • • • = fax  47  =  fa]  (-) 4  9  and m = m = ••• = TIK = n  (4.10)  In case the receiver is in the line of sight of the transmitter, the magnitude of resultant signal of the direct path,component and scattered path component is: Pi = y/(p + (c°* fa) + ((smfa) 2  (4.11)  2  with the phase:  4 - Arc to (  <- ) 4  12  Then the received signal can be rewritten as L  K  r(t) = AY J2/3 d (t-Ti)p (t-ri)cos(ut /  l  k  + ^)  k  + n(t)  l  /=l k=i  '  (4.13)  ,  where (4.14)  T/=.77-TI  and $, = W - ^  (4.15)  Because of the symmetry of the receiver, only the decision variable of the first correlator, g\(Tt,), will be examined. Assuming the receiver is in synchronization with the phase and delay of the first desired modulated signal  n gi(T ) = Y J r(t) (t)2cos b  Pl  o = Afcdiin)  (ut)dt  '  + I(T ) + M{T ) + N(T ) b  b  b  (4.16)  Chapter 4. Performance in the Presence of Multipath Fading  48  where K  I(T )  A^YjrJ  =  b  -  TB  k-2  k(t)Pk(t) (t)  dt  d  Pl  A f M(T ) = — J ^ 0 , cos ($/) ^ d (t-T ) {t-Ti) (t)dt l b 1=2 k=l0 1  K  (4.17)  Q .  T  6  k  t  Pk  Pl  . (4.18)  J  N(T )  = ^-J  b  n(t) (t)  cos (cot) dt  Pl  (4.19)  o If an m-sequence is used for spreading the data, the second term of the decision variable, I(Tb), can be easily simplified as  I(T ) = J2d (T )(^\ b  k  (4.20)  b  k=2  ^  '  Since the k-th data signal stream d^(t) is a sequence of rectangular pulses taking on values of ± 1 as given in equation 2.3, the third term of the decision variable can be expressed as A  M(T ) b  =  1  r  K  -YJ2ft I  b  l  =  2  k  =  c  o  ^  s  i  t^w+^iW  (4.21)  where Tl  Rk,i( l) T  =  jPk(t ~ ri)pi(t)  dt  (4.22)  o and  T,  Rk,i{ri) In equation 4.21, d^  =  JPk{t ~ ri) i(t) P  dt  (4.23)  and d\ are, respectively, the previous and current data of the k-th  data stream. If the incremental delay of the /-th path is at the n-th chip interval, i.e. 0 < r/ = nT < T , the partial correlation functions can be expressed as c  b  R ,l(ri) k  =  C (n-G)T ktl  c  Chapter 4.  49  Performance in the Presence of Multipath Fading  Rk,l( l)  =  T  (4.24)  k,l{n)T  G  c  where Q y(.) is the discrete aperiodic correlation functions given as (  ]=0  Ck,i{n) = {  ° ^ i=o +  x  .(*)  n  (i)  <  !  r  <n  (  4  2  5  )  | ft | > G  0  If the current data bit of data stream k is the same as the previous one, d _ = d§, and k  x  we have the term Rk,i(Ti) the sequences  + R ,i(Tl) k  which is simply the periodic cross-correlation of  p and p\. Since pk{t) is just a shifted sequence of p\(t), the periodic k  cross-correlation is the same as the periodic autocorrelation of the selected m-sequence given by equation 2.11. On the other hand, if the current data bit is different from the previous one, we have the term Rk,l(ri) where 0k,\{ ) n  Pic and p\.  - R ,l(ri) k  = e ,li ) P n  k  T  (- ) 4  26  = Cfc,i(n) — C ,\{n — N) is the odd cross-correlation of the sequences k  Unfortunately, the properties of the odd correlation functions are not as  well studied as the periodic correlation functions. In fact, the odd correlation function varies depending on the choices of the m-sequence.  Therefore, system performance  greatly depends on the choices of the m-sequence as well. Extensive computer search may be required to find the optimal m-sequence of a given length to maximize system performance.  4.2.2 Simulation Results To evaluate the performance of C P M scheme, simulation models of C P M DS-SS in the presence of multipath fading have been constructed using equations 4.16, 4.19, 4.20,  50  Chapter 4. Performance in the Presence of Multipath Fading  4.21, and 4.24, which are used for generating, respectively, the filter output, the noise, the interference due to other parallel data streams that arrived at the same time, the multipath delay interference and continuous partial correlations. Simulations are first done with various channel parameters to show the effects of variation of channel parameters on the system performance. The results in Figure 4.1 show that for a fixed value of maximum possible multipath delay spread, T , of 125 m  T , the system B E R performance degrades as the number of unresolvable paths, L , c  increases. On the other hand, for the same L, the system B E R performance improves as T  m  increases since the unresolvable paths, which are uniformly distributed over T , m  would be further apart for larger T . Furthermore, the results can also be explained from m  the point of frequency-selectiveness of a communication channel. A channel is said to be frequency-selective if the coherence bandwidth of the channel is small in comparison to the bandwidth of the transmitted signal; that is, T is large in comparison to T . When m  T  m  c  is increases, relatively to T , the effects of fading varies more along the spectrum c  of the transmitting signal.  Besides, the effects of increasing the number of parallel  data streams of C P M seem to be slightly more significant for larger T . m  values of L and T  m  The actual  would depends on the actual operating environment. However, the  general behavior of the C P M system performance would be similar for different channel parameters' values. Unless otherwise specified, the following simulations are simulated with arbitrary choosen values of T =\0 T and L=5. m  c  The relations between BER and the average received bit-energy-to-noise-density ratio, SNRjj = E{/3 }(Eb/N ), where E{ •} denotes the expected value, have been evaluated by 2  0  simulations for different values of K and G\ over Rayleigh channel and Rician channels with different K-factors. Different m-sequences with small maximum odd-autocorrelation  Chapter 4.  Performance in the Presence of Multipath Fading  51  l.OE+00  SNR (dB) b  Figure 4.1 BER vs.. SNR^ of C P M operating over Rayleigh Channels with Various Maximum Delay Spread and Number of Unresolvable Paths  values for different lengths are arbitrary choosen for the simulations. As a result, the odd-autocorrelation changes as K increases. This may not necessary simulate the best possible or average performance. However, it guarantees the existence of systems that can perform the same or better and shows the effects of increasing number of parallel streams on the system performance. The results for a C P M system operating over a Rayleigh fading channel are shown in Figure 4.2. observations can be made: 1.  From this figure, the following .  Increasing the number of parallel streams has little effect on the performance. It does not necessarily improve the performance. In fact, using two data streams sometimes gives slightly better performance than using four data streams. This is probably due to the increase of odd-autocorrelations as different m-sequences are used for different  Chapter 4.  52  Performance in the Presence of Multipath Fading  l.OE+00  n—I—I—i—i—I—I—|—I—i—I—I—I—r-  no. of streams=8 no. of streams=4 no. of stream s=2 no. of stream =1  l.OE-01  0- &©- • A- -  w  1.0E-02 b-  l.OE-03  -j  i  i  i  i  i  i  i  i  i  i  10  i  i  i  i  15  i  i  i  i  i  i  i  20 SNR (dB)  i i_  25  30  35  40  b  Figure 4.2 B E R vs. S N R of C P M DS-SS over Rayleigh Fading Channel 5  K. Therefore, choosing a proper m-sequence with optimal odd-autocorrelations and odd-crosscorrelations is quite important to achieve good performance for C P M DSSS operating over Rayleigh fading channels. Also, there are (LxK)-l  self interfering  signals instead of K— 1 in a single path system. 2.  Very high SNR^ is required to achieve acceptable performance, say, B E R in the order of 10" . 3  In fact, an "error-floor" of about 2xl0" exists. 3  Thus, very little  improvements resulted from further increases in SNR beyond the point of 30 dB. One possible way to improve the performance of the system is to arrange the antennas of transmitters and receivers so that at least one direct path exists between the transmitter and receiver. In this case, the channel can be modeled as a Rician channel with a nonzero K-factor. Figure 4.3 shows the BER vs. SNR^ curves of a C P M DS-SS system with  Chapter 4.  Performance in the Presence of Multipath Fading  1 .OE+00  53  E~i—i—i—i—|—i—r—T—i—i—i—i—i—i—i—i—i—i—I—[—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—1~3  SNR  b  (dB)  Figure 4.3 BER vs. S N R of CPM DS-SS in various Multipath Channels h  various number of parallel data streams operating over Rician channels with different fc-factors. The performance of the system dramatically improves when operating in a Rician channel instead of the Rayleigh channel. For example, to achieve a system BER of 5xl0" , the CPM DS-SS system simulated requires SNR^ of about 18 dB at the input 3  of the correlators when operating over the Rayleigh channel but requires only about 10 dB and 7 dB when operating over Rician channels with n-iactors of 6 dB and 10 dB respectively. For the same transmitting power, the average received signal power over channels without direct path is usually less than that over channels with direct path. That is, E{f3 } of Rayleigh channels is usually smaller than those of Rician channels. 2  Thus, at a system B E R of 5xl0" , in the existence of a strong direct path signal such 3  that /c-factors = 10 dB, a minimum saving of over 11 dB SNRb is achieved over the  Chapter 4.  54  Performance in the Presence of Multipath Fading  case without a direct path signal.  Even in the case where a medium strength direct  path signal exists /c-factor = 6 dB a saving of about 8 dB is obtained. Moreover, the error-floor phenomenon is eliminated. Therefore, in distributed indoor wireless network applications, it is desirable to place the radio bridges so that all the workstations would have at least one radio bridge within line-of-sight. As observed for Rayleigh fading channels, increasing the number of parallel streams is found to have very litde effect on the system performance in Rician fading channels as well. Since the performance degradations due to multipath interference greatly depend on the odd-autocorrelations of the PN sequence, a proper choice of the PN sequences may improve the effect of increasing the number of parallel streams on the system performance. Furthermore, depending on the actual operating environment, the effects of increasing the number of parallel streams can be more significant if the maximum multipath delay spread is longer or the actual number of unresolvable paths is less.  4.3 Performance of C P S K in the Presence of Multipath Fading 4.3.1 Analysis As given by equation 2.16, the signal sent by the transmitter of a CPSK system is:  s(t) = Ap (t) cos (ut).  (4.27)  m  Using the same multipath channel model given by equation 4.1, the received signal in the absence of other interference signals at the input of the receiver is: L  K*) = Y Z=l  PlAPmtt  ~ l) T  C O S  M+  fa)  + (t) n  (- ) 4  28  Chapter 4.  Performance in the Presence of Multipath Fading  55  Assume that every sequence is shifted by an amount larger than the maximum spread delay, so that  Pm{t ~ Tl) ^  1 < / < L, 1 < m < (M - 1).  p +i(t), m  (4.29)  Let mo be the current data symbol being transmitted and m_i be the previously transmitted data symbol. In this case, the output of the correlator corresponding to the current data symbol being transmitted is:  9m (T ) = jr J r(t)p {t)2 cos (ut)dt 0  s  mo  o  = / M ^ r jPmMPmo(t)  dt +  A_  N (T ) mo  +  s  (TI) + R m ,m (Tl)  ¥  1=2  /M  +  0  —X>cos($,) R  (Tl)  0  +  R  mo,mo  +N (T)(430)  (TJ)]  mo  1=2  where i ? _ m  1)mo  ( r ; ) and  Rm m ( l) T  0l  ^  0  continuous partial cross-correlations given by  equation 4.24. and the outputs of the remaining matched filters are  A  A  G  Rm-uk^l)  r  9k(T ) = s  - ; ( i i - T 1=2 -Y,^ 7  r  + N(T ), S  c  o  s  +  Rm ,k(Tl) 0  ^ VI < k ^  m <M 0  (4.31)  4.3.2 Simulation Results Since both equations 4.30 and 4.31 greatly depend on the continuous partial crosscorrelations of the m-sequence used, the spreading m-sequence has to be known tb  Chapter 4.  Performance in the Presence of Multipath Fading  56  compute the outputs of the filters. Thus, direct computation of the B E R for general cases is not possible. Instead, computer simulations for the above model are employed with a set of arbitrary choosen m-sequence. The results are not necessary the optimal, worst case nor average. However, it guarantees the existence of systems that can perform the same or better and shows the effects of increasing the symbol/word length on the system performance. Similar to previous simulations, at least millions bits, binary digits, are generated randomly. The system performance is evaluated by computer simulations as described in Appendix B. The mathematical system model in this case consists the output of the correlator corresponding to the transmitted symbol and the outputs of the remaining M— 1 correlators which are generated using equations 4.30 and 4.31, respectively.  Simulation results for L=5, T =10T , and Gi=255 are shown in Figure m  c  4.4 for multipath fading channels with ^-factors of 0 (Rayleigh fading), 6 dB and 10 dB (Rician fading), and SNR^ ranging from 2 dB to 38 dB. The results show that substantial transmit power is needed to overcome multipath fading when the direct path component is absent. Similar to C P M , existence of the direct path component can dramatically improve the system performance. For example, to achieve a system BER of 10 , CPSK with word -3  length of 8 requires about 24.5 dB SNR^, at the input of the receiver, when operating over multipath fading channel with no direct path component but only 15 dB and 7.5 dB when operating over multipath fading channels with K-factors of 6 dB and 10 dB respectively. This represents a minimum saving of over 17 dB in signal power due to the existence of a strong direct path component with a /c-factor of 10 dB. Even in the Rayleigh fading channel with no direct path component, CPSK DS-SS performs better than conventional BPSK DS-SS or C P M DS-SS at high SNR~ , as shown in Figure 4.5. In this figure, b  the BER of CPSK DS-SS decreases linearly as S N R down to lO' , while conventional 4  b  57  Chapter 4. Performance in the Presence of Multipath Fading  l.OE+00  i—i—i—I—i—r—r  word length=8 word length=4 word length=2 word length=l K=10dB(Rician) K = 6 dB (Rician) K = 0 (Rayleigh)  l.OE-01  1.0E-02  ai  y  l.OE-03  w  pa 1.0E-04  l.OE-05  i  1.0E-06  10  12  14  16  18  20  22  24  26  i  28  30  32  34  • i  36  i  38  SNR (dB) b  Figure 4.4 B E R vs S N R  b  of CPSK Operating  Over Various Multipath Fading Channels BPSK DS-SS and C P M DS-SS hit the error-floor above lCr . Also, when the direct path 3  component is weak or absent, increasing the data symbol length W or signaling alphabet size M does not bring about much saving in transmit power.  However, as the direct  path component (i.e., the K-factor) increases, increasing M brings increasing savings of signal power and energy. In many wireless local communications systems, line of sight exists between the transmitter-receiver pairs. In this case the performance of CPSK can be quite acceptable without additional signal processing at the receiver. Moreover, the data rate of a digital communication system operating over a multipath channel is limited by the multipath delay spread in order to avoid severe intersymbol interference (1ST). Increasing the symbol length of CPSK means enlargement of the transmission alphabet which is one of the techniques for overcoming this ISI limitation [17]. Thus, higher data rates can be achieved.  Chapter 4.  58  Performance in the Presence of Multipath Fading  l.OE+00  n  i  i  i  i  i  i  i  i  i  i  . . . .  I .  i  i  i  i  i  i  i  I  I  I  I  i  i  i  i  I  I  I  I  i  i  i  r  1  I  I  l_  l.OE-01  1.0E-02 Pi pq CQ  l.OE-03  k  1.0E-04  1.0E-05  CPSK,W = 8 CPSK, W = 4 CPSK, W = 2 CPSK, W = 1 CPM, K = 8 CPM, K = 4 CPM, K = 2 CPM, K = 1 .  0  . i  10  J  -G-  -o^r-  I  I  L  I  15  20 SNR (dB)  _l  I  25  I  30  I  35  b  Figure 4.5 Comparison of C P M and CPSK Operating Over Rayleigh Channel  40  Chapter 5 Performance Against Single Tone Jammer in the Presence of Multipath Fading We consider the pessimistic situation where the jamming signal arrives via truelineof-sight propagation as in [4].  Thus, multipath fading affects only the data signals  but not the jamming signal. Therefore, the system model can be obtained by simply superimposing the two models for single tone jamming and multipath fading channel considered in the two previous chapters. In this chapter, only the case where the frequency of the jammer is the same as that of the data carrier is considered.  5.1 Performance of C P M The normalized output of the first correlator, g\(T ), in Figure 2.1 becomes b  (T )  gi  b  = di(T ) + J(T ) + I(T ) + M(T ) + N(T ) b  b  b  b  b  (5,1)  where d\(Jb) is the desired information. /(T&) is the output due to the jammer. For single tone jamming at the carrier frequency, (5.2) where /?i is the path gain of the signal locked on by the receiver as given by the equation 4.11. Thus JSRi in this case becomes  (ir~i\  (5.3) represents the self-interference due to other parallel streams. The multipath interference term is given as  1=1  P l  £i  59  60  Chapter 5. Performance Against Single Tone Jammer in the Presence of Multipath Fading  N(Tb) is an independent zero-mean Gaussian random variable with the variance a  2  N  _  0  E{f3l}T {Af b  1 2E{f3{}(E /N ) b  _  0  2  =  1 SNR ' b  All simulations, described in Appendix B, for this case are performed using equations 5.1, 5.2, 5.3 and 5.4 to generate, respectively, the output of the first correlator, component due to single tone jamming, self-interference component and multipath interference component.  The B E R vs.  SNR  b  curves and B E R vs. JSRi curves for different A"s  are obtained by simulations for Rician fading channel with K-factor of 10 dB and shown in Figures 5.1 and 5.2. The results show that the performance of C P M against single tone jammer in the presence of multipath fading is quite similar to the performance of the case with no multipath fading in that the BER performance improves as K increases from 1 to 8. For example, at BER of 2 x l 0 , increasing K from 1 to 2 results in over 5 -2  dB reduction in S N R , enabling more than tripling of the data rate for the same transmit b  power. Further increases in K result in diminishing returns. Greater savings in S N R due to increasing K could be obtained at lower BERs. At S N R  b  b  =16 dB, increasing K  from 1 to 8 also reduces BER from a barely acceptable (for most applications) 1.5xl0~ , 2  to a fairly adequate 2x10^. Also, for a given achievable BER objective and fixed S N R , b  the range of JSR values the system is capable of handling, or jamming margin, increases approximately 7 dB when K is doubled. For example, S N R is fixed at 12 dB (Figure b  5.2), Gi=31 and targeted BER=10 , the system can handle up to approximately 40 dB, -3  47 dB, 54 dB and 61 dB as K increases from 1 to 2, 4, and 8, respectively.  The BER  vs. JSRi curves for different K operating over Rayleigh fading channel are also obtained by simulations and shown in Figure 5.3.  As mentioned in the previous chapter, very  high S N R is required to achieve a BER that is barely acceptable for most applications b  when multipath fading is present with no direct path. However, the C P M scheme seems  Chapter 5. Performance Against Single Tone Jammer in the Presence of Multipath Fading 61  l.OE-01  —r fe  1.0E-02  Pi ui PQ  l.OE-03  I  K K K K  -  = = = =  e- —  8 4 2 1  -Qe-A  1.0E-04  12  10  16  14  SNR^dB) Figure 5.1 B E R vs.  SNR  for JSRi =60 dB,  b  .  Gi=31, over Rician Channel with Ac-factor= 10 dB 1.0E+00  i  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  ] i  I  i  "T—I—I—I—I—I—I—I—T  I r~  -  1.0E-01 —$~  4-- — - - G - 2 - —e-1 -  K: K: ui  pa  1.0E-02  .-A l.OE-03  1.0E-04  i  20  i  i  i  i  i  i  30  i  i  i  i  ,B'  .4r  A  i  i  i  i  i  Ai 40  i  i  i  i  i  i  i  i  50 JSR (dB)  Figure 5.2 B E R vs. JSRi for SNR =12 dB, 5  Gi=31, over Rician Channel with K-factor=10 dB  i  i  i  60  i  i  i  i  t i  i  i i  70  Chapter 5. Performance Against Single Tone Jammer in the Presence of Multipath Fading 62  l.OE+00  l.OE-01  .. .. ,.. ._, _. _. .... .._. „., T  b-  1  I  1  |  |  )  r  • T l T f l l ' l  |  I "I 1 T 1 I -T'T  I "T"T-|"|—i-pl  1 i l I —1—T—I—r 1 1 I I I I I 1 I I  K=32K = 16K = 8K = 4K = 2K= 1  P3  .-Ar"  1.0E-02  l.OE-03  i  20  i i i i  30  '  40  ' ' i i ' i  50  60  i i  70  ' i i  80  JSR (dB)  Figure 5.3 B E R vs. JSRi for SNR =30 dB, Gi=31, over Rayleigh Fading Channel to retain its desirable properties against single tone jamming. For example, with S N R b  b  = 30 dB (Figure 5.3), Gi=31 and targeted BER=10 , the system can handle a JSR up to -2  approximately 28 dB, 36 dB, 44 dB, 52 dB, 60 dB, and 67 dB, as K increases from 1 to 2, 4, 8, 16, and 32, respectively. The results show that the C P M has the same desirable properties against single tone jamming regardless of whether the channel is subject to multipath fading or not. Moreover, increased gains in jamming margin are obtained over a fading channel as K is increased, compared to a channel without fading.  5.2 Performance of C P S K Since the effects of single tone jamming at the carrier frequency are, as shown in the previous chapter, eliminated by CPSK, the performance of CPSK against single tone jamming at the carrier frequency in the presence of multipath fading would be the same as that shown in section 4.3.  Chapter 6 System Performance Improvement with RAKE Receiver 6.1 Background of the R A K E Receiver The results for both the C P M scheme and CPSK scheme indicate that they both perform quite well over Rician fading channel with a strong direct path signal (/c-factor of 10 dB). However, their performance may not be acceptable for practical applications when operating over Rayleigh fading channels where no direct path between transmitter and receiver is possible. Some measures have to be taken to improve the performance to an acceptable level. Different types of diversities are normally used for improving the system performance over multipath channels. In particular, the R A K E receiver proposed by Price and Green [6] is one of the most popular receiver structures used for spread spectrum applications. A R A K E receiver employs a single delay line through which the received signal is passed. Each tap is spaced to resolve the time-spreaded signal as the original signal with its duplicates arrive over a finite number of separate paths. The signal at each tap is correlated with the conjugate of the estimate of the corresponding path parameters and summed by a combiner. In effect, the tapped delay line receiver attempts to collect the signal energy from all the received signal paths that fall within the span of the delay line and carry the same information. Depending on the modulation signals, different ways are used to estimate the path parameters. In particular, if the modulation signals are antipodal such as BPSK, one way to estimate the path parameters is simply remove the information-bearing signal from the output of the correlator [6]. In order to feed this estimated path parameters into the input of the low pass filter, a delay of one signaling 63  Chapter 6.  64  System Performance Improvement with RAKE Receiver  interval must be introduced, as illustrated in Figure 6.1. That is, the receiver must first decide whether the information in the received signal is +1 or -1 and, then, it uses the decision to remove the information from the correlator output prior to feeding it to the lowpass filter.  r(t) COS((0/)  M X  1  Delay T r  Data Bit of previous decision from the integrator  Low pass Fi ter  cj(t> To summer and Integrator Figure 6.1 Channel Tap Weight Estimation with Binary Antipodal Signals  6.2 C P M DS-SS With R A K E Receiver 6.2.1 System Model and Implementation A receiver for C P M DS-SS can easily be modified into one with the R A K E receiver structure by inserting a transversal filter after the carrier demodulation and a combiner before the integrate and dump filter for each parallel stream, as shown in Figure 6.2. The transversal filter incorporates a delay line of T  m  every code chip period, T , seconds. c  seconds long, which is tapped at least  Chapter 6. System Performance Improvement with RAKE Receiver  r(t)  65  —(x>  Figure 6.2 C P M DS-SS with R A K E receiver In this case, the output of the first filter would becomes >> L  T  If  gi(T ) = Y ]Y, b  r  h  0  x  =  ( * + n)PxPi{t)2cos  b  x  X  (6.1)  Since  =  + ^ )dt  1  dt  A  1  where r(t) is given by equation 4.13.  Y J o r(t + T )a p (t)2cos(ut where  (ut + $ )  x  dlAd (T )+hm 1  (6-2)  l  (6.3) k=2 K  A-l  M (T ) x  b  = ^ J 2 J 2 ^ T 1=1 k=\  c  o  M +  (*'.*) *-i U  s  R  b  o kAM +  d  R  (6.4) b  l=\-lk=l  and N (T ) x  b  =  2TA T  b  I  n (t)pi(t) cos (cot — $ ) dt x  (6.5)  x  the decision variable can thus be expressed as: L (T )  9l  b  = Y A=l  L  PlAdiW + E  ( A(^) + 7  M\(T ) b  +  N (T )) x  b  (6.6)  66  Chapter 6. System Performance Improvement with RAKE Receiver  6.2.2 Simulation Results Simulations based on decision variables of C P M DS-SS System described by equation 6.6 for multipath channels with K-factors of 0 (Rayleigh fading) 6 dB, and 10 dB are done as described in Appendix B. In order to compare the performance improvement of the of the system employing R A K E receiver structure, the average B E R is plotted against the average received bit-energy to noise density ratio of the signal with the shortest path, SNR^. Figures 6.3, 6.4 and 6.5 show the comparison between the performance of systems with a single path receiver and with a R A K E structure combining all non-zero signal paths, the optimum R A K E receiver, operating over Rayleigh fading channel and Rician channels with /c-factors of 6 dB and 10 dB, respectively. The results show that it is possible to dramatically improve the B E R performance of a C P M DS-SS system over different frequency-selective, slowly fading channels by employing the R A K E receiver structure. The lower the desired BER, the more the saving in SNR^ can be realized by employing the R A K E receiver structure. improvement of over 10 dB is easily achieved at B E R of 10  -2  In particular,  when operating over a  Rayleigh fading channel. Moreover, the error-floor is eliminated with the R A K E receiver structure. In the presence of direct path, the saving in SNR^ obtained by employing the R A K E receiver structure is less. In fact, the stronger the direct path signal, the less benefit is realized.  Chapter 6.  System Performance Improvement with RAKE Receiver  l.OE+00  67  —i—i—i—i—I—i—i—i—i—I—i—i—i—i—I—i—i—i—i—I—i—i—!—i—i—i—i—i—i—I—i—i—i—i—I—i—i—i—r  l.OE-01  F^-fS  1.0E-02  W 03  l.OE-03  1.0E-04  1.0E-05  1.0E-06  0  K =8 K =4 K =2 K = 1 RAKE with 5 paths Single Path J  0  10  I  I  I  15  I 20  i  I  I  I  I  I  I  I  25  •  o  A  I  I  I  I  L I  30  I  L_J  I L  35  40  SNR (dB) b  Figure 6.3 Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rayleigh Channel 1.0E+00  ~i—i—i—i—I—i—r~  1.0E-01  i  I  i  i  i  i  I  r  i  i  i  1  i  r  K =8 K =4 K =2 K= 1 RAKE with 5 paths Single Path  1.0E-02  gj  ~i  i—i—i—i—i—i—i—i—r - • -  0 • o A  1.0E-03  PQ  1.0E-04  \r  1.0E-05  1.0E-06 SNR (dB) b  Figure 6.4 Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with ^-factor 6 dB  Chapter 6. System Performance Improvement with RAKE Receiver  68  l.OE+00 E—r  SNR (dB) b  Figure 6.5 Performance Comparison between C P M DS-SS Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with /c-factor 10 dB  6.3 CPSK DS-SS With R A K E Receiver 6.3.1 System Model and Analysis Similar to the C P M DS-SS, the CPSK DS-SS receiver can easily incorporate the R A K E receiver structure by inserting a transversal filter after carrier demodulation followed by a combiner. Assume that the data is transmitting continuously and m_\, mo, m\ are the previous, current and next data word respectively.  In this case, the output of the correlator  corresponding to the signal being currently transmitted is  If 9m (Ts) = 7f / E 0  r  ( * + A)0APmo(O C O S (cot + $ ) dt r  A  69  Chapter 6. System Performance Improvement with RAKE Receiver  = Y g ,x(T ) t  po  (6.7)  + N {T  a  0  a  A=l  where r{t) =  J2Pl  Acos (^t'+-^l)Pmo(t-Ti)  (6.8)  n(t)  +  Thus,  gmoMTs)  = Y J (t +. T\)P\Pmo {t) COS (cut + $ ) di r  A  0  •  = Y J S  + rx)Pm (t) cos (u>t + $/) cos (u>< + $ )dt  PlP* P™o(t- l A  T  A  0  = A/JJ + 4 £ A"* "(« • ' *) /  pmi ( i  _  Tu)pmo(4)lft  . . (6  9)  / =1 A  where cos  = cos ($, - $ ) A  (6.10)  . /,A = / ~ T A T  T  The correlator output can also be expressed in terms of the continuous partial crosscorrelation as given by equations 4.22 and 4.23: A-l  gm ,x(T ) 0  s  =  A0l + Y,  APiPx  COS  +  ($,, ) A  1=1  A/3 /3 cos($, ) i=x+i . /  A  jA  Rm- mo(Tl,\) U  +  Rm ,m (Tl x) 0  0  t  (6.11)  Chapter 6. System Performance Improvement with RAKE Receiver  70  Similarly, the outputs of the remaining correlators are:  9k(Ts) = Y 9k,x(Ts) + f  (6.12)  N (T ) k  s  A=l  where  Aft G  ~  X  Rm ,m {T +  l  0  + Rmi,m  s  0  (T  s  +  T~l,\)  +  1=1 Rm-umo  J2  0  WACOS($,, ) A  ( l,\) T  + Rm ,m 0  ( l,\) T  0  (6.13)  /=A+1  6.3.2 Simulation Results Equations 6.7, 6.11, 6.12 and 6.13 are used to model the simulated CPSK system with the R A K E receiver operating over different multipath channels. Figures 6.6, 6.7 and 6.8 show the comparison between the performance of CPSK systems with a single path receiver and with a R A K E receiver combining five paths operating oyer a Rayleigh fading channel and Rician channels with ^-factors of 6 dB and 10 dB, respectively.  Chapter 6.  System Performance Improvement with RAKE Receiver  71  1.0E+00  1.0E-01  1.0E-02 b -  1.0E-03  V  1.0E-04  \r  1.0E-05  k  1.0E-06  0 • o  W=8 W=4 W=2 W= 1 RAKE with 5 paths Single Path J  0  I  I  10  I  I  I  I  15  I  I  I  I  I  I  20 SNR (dB)  I  I  1  I  25  A  I  I  I  !  I  30  I  I  I  1  1  L_  35  40  b  Figure 6.6 Performance Comparison between CPSK Systems with Single Path Receiver and.that with R A K E Receiver Operating Over Rayleigh Channel 1.0E+00 FT  ~T  1  1  1  I  I  I  I  I  I  I  I  I  I  I  I  1  I  I  I  n  I  I  r~  1.0E-01  1.0E-02 W  w  l.OE-03  1.0E-04  \r  1.0E-05  t-  1.0E-06  W=8 W=4 W=2 W= 1 RAKE with 5 paths Single Path _l  0  I  I  I  I  I  I  I  10  I  L  15  20  25  30  SNR (dB) b  Figure 6.7 Performance Comparison between CPSK Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with /c-factor 6 dB  Chapter 6.  System Performance Improvement with RAKE Receiver  l.OE+00  E r  SNRJdB)  Figure 6.8 Performance Comparison between CPSK Systems with Single Path Receiver and that with R A K E Receiver Operating Over Rician Channel with /c-factor lOdB  In the absence of a direct path between the transmitter and receiver, such that the channel exhibits Rayleigh fading, the R A K E receiver structure can dramatically improve the performance of the CPSK system by reducing SNR^ by some 20 dB at BER=10^ with W=8. However, the performance improvement obtained from employing the R A K E receiver structure diminishes as the direct path signal increases in relative strength, i.e., with an increased ^-factor.  In fact, when affected by Rician fading with a strong  direct path signal, the. system with R A K E receiver performs slightly worse than the one without R A K E receiver at high SNR^. Thus, in wireless local communications systems in which the transmitter-receiver pairs are not within line of sight, the use of a R A K E receiver would significantly improve system performance, at a modest increase in receiver complexity.  In fact, the channel estimator could direct the receive signal  72  Chapter 6.  System Performance Improvement with RAKE Receiver  73  to bypass the R A K E diversity-combiner if it detects a significant direct path signal. Otherwise, the received signal would pass through the R A K E diversity-combiner for performance improvement.  Chapter 7 Conclusions and Future Work 7.1 Summary and Conclusions We have presented two DS-SS signaling techniques known as C P M and CPSK. The performance of both C P M and CPSK in an AWGN channel with single tone jamming has been analyzed by numerical computations and simulations.  Results show that by  increasing the number of parallel data streams, C P M could improve the performance of a conventional DS-SS system in terms of increasing the aggregate data rate under constant jamming power, or alternately increasing the immunity against jamming at a fixed data rate, while maintaining a specific BER objective and fixed transmit power and bandwidth. When the tone jamming frequency is different from the signal carrier frequency, while C P M still possesses the desirable property of increasing processing gain without reducing data rate, choosing a proper spreading sequence is very important. This is because the jamming signal would have different gain for different choices of spreading sequence and amount of difference between the jamming frequency and carrier frequency. Whereas C P M , like other conventional DS-SS modulations employing BPSK or QPSK, gives no advantages over BPSK with respect to thermal noise, we show that CPSK can be more power-efficient. Furthermore, CPSK can be more bandwidth efficient than conventional DS-SS systems when a high spreading gain is needed to combat interference.  In fact, it may be possible to further improve the bandwidth efficiency  by a factor of four by employing antipodal (i.e. BPSK) modulation on each unique phase shift, and transmitting two independent CPSK signals in quadrature. Employing CPSK, it is therefore possible to design DS-SS systems with increased spreading gain without 74  Chapter 7.  75  Conclusions and Future Work  reducing the data rate, of alternately, systems with higher data rate without reducing spreading gain. For example, a 1 Mbit/s wireless local communications system could be implemented using CPSK with M=64 and G=127, and with quadrature signaling. Such a system would have a chip rate of 10.6 Mchips/sec, a transmit bandwidth of 21.2 MHz which fits within the 900 M H z ISM band (902-928 MHz), and a spreading gain of 21 dB which is about an order of magnitude better than that available in current commercial systems.  As is the case with any digital signaling system, the performance of both C P M and CPSK degrades in the presence of multipath fading.  In fact, results show that  the performance of C P M DS-SS signaling in the mobile fading channel is very similar to that of conventional DS-SS signaling. However, in the presence of tone jamming, we show that C P M DS-SS signaling gives better performance than conventional DS-SS signaling. Similar to the case without multipath interference, increasing the number of data streams, K, of C P M enables reductions in SNR at a fixed JSR or allows a higher JSR to be tolerated at a fixed SNR, while keeping B E R constant. Alternately, at fixed SNR and JSR, increasing K brings about substantial reductions in BER. Thus, C P M DS-SS signaling enables increases in system data rate without sacrificing interference immunity, or vice versa. While the results of C P M DS-SS operating over a Rayleigh channel in Figure 4.2 show an error-floor at approximately 2 x l 0  -3  BER, a phenomenon  characteristic of PSK signaling in fading channels, the results of CPSK DS-SS operating under the same condition in Figure 4.4 show that the error-floor is removed. It is possible to combat multipath fading using various diversity reception schemes. However, even without diversity reception, we have show that both C P M and CPSK perform well in  Chapter 7.  Conclusions and Future Work  76  a Rician fading channel when a strong direct path signal component is present at the receiver, which is the case for most line of sight local communications systems. In particular, increasing M of CPSK shows significant improvements in this case. System performance in a Rayleigh fading channel can be dramatically improved by using a R A K E receiver which coherently combines a number of multipath signals. However, the R A K E receiver is ineffective in a Rician channel with a dominant direct path component. The primary contributions of this thesis research are as follows: Two techniques, C P M and CPSK, were designed to enhance the performance of Direct-Sequence Spread-Spectrum Signaling for Wireless Data Communications. C P M DS-SS signaling enables increases in system data rate or reductions in SNR at a fixed JSR of a single tone jamming signal, regardless of the presence of multipath fading, without sacrificing interference immunity or vice versa. In addition to overcoming the data rate-processing gain limitation like C P M , C P S K s DS-SS signaling can be more power-efficient. In particular, an example of a wireless data communications system implemented with CPSK is shown to have a possible improvement in processing gain of about an order better than that available in current commercial systems. •  The system performances of both signaling techniques against single tone jamming, multipath interferences, and single tone jamming in the presence of multipath interferences have been evaluated by computer simulations, and numerical computations if closed form solutions are obtainable. Possible performance improvement, of both signaling techniques in the presence of multipath by using a R A K E receiver has also been investigated.  Chapter 7.  Conclusions and Future Work  77  7.2 Areas for Future Work This thesis was primarily concerned with the architecture and the performance of the proposed signaling schemes operating in different environments. There are a few areas require further research: Synchronization — Since all the system models for performance evaluations are based on perfect synchronization. The techniques to achieve both carrier and symbol synchronizations are very important and would require further research. Experimental realization of presented systems — Since both C P M and CPSK make use of a single PN sequence per system, they could offer certain implementation advantages. Efficient implementations of the transmitter for CPSK using table lookup to initialize the PN code generator for each data symbol, and the matched filter receiver using a shared tapped delay line for sequential comparison of filter outputs are presented in Appendix A. Similar ideas that may also be used to shorten the acquisition time and improve the signal tracking capability are interesting for further research. Performance analysis for different multipath channel models and against different type of jamming — The multipath channel models used for this thesis research are rather pessimistic.  Since the efficiency of increasing K of C P M and increasing M  of CPSK could be affected by the actual multipath channel model for operation, similar analysis could be repeated for specific model depending on the operating environment and system layout. •  Computer search for optimal PN sequences — The actual performance of both C P M and CPSK greatly depends on the partial-autocorrelations of PN sequence imple-  Chapter 7.  78  Conclusions and Future Work  merited in the presence of multipath interference and/or tone jamming at frequency different from the carrier frequency. Thus, intensive computer search on the optimal PN sequences could be beneficial. Effects of F E C coding — Since increasing the processing of SS system has no advantage against thermal noise, trading some processing gain for coding gain could be a possible way to improve the system performance. Thus, optimizing the trade-off between coding gain and processing gain is a possible area for further research. •  Performance in a multiple access environment — For most applications, it is likely 1  that more than one transmitter and/or receiver would be using the same signaling technique.. Therefore, various multiple access controls and the system performance in such environment should require further research. •  Equalization technique for tone jamming mitigation in CPSK — If the jamming tone is constant or sufficiently slow in variation in time, the effect of the jamming tone can be estimated and mitigated. This is particularly desirable because CPSK can possibly reduce the SNR while achieving a fixed B E R by increasing M .  Appendix A An Efficient Implementation of CPSK System An efficient way of generating the CPSK signal for transmission is illustrated in Figure A . l . In this implementation example, an /V-stage shift register (where G=2 -1) N  with feedback taps and parallel load inputs is employed for generating the PN code sequence. Using a lookup table in the form of a R O M or code-converter, the W-bit data symbol value is converted to an iV-bit initialization vector which is fed to the parallel load inputs of the code generator at the start of symbol period to initialize the code generator at the appropriate code phase.  W S/P / data 1 to W in m(t)  W: N Table Lookup  N /  PN Seq. Generator it. with Parallel Load  p-w.  Acos(cot)  s(t) transmit signal  Figure A . l CPSK Transmitter with Table Lookup.  The M-parallel correlator receiver configuration shown in Figure 2.3 is generally applicable to any M-ary DS-SS signaling method. Alternately, the receiver for CPSK can be configured with M parallel matched filters. Consider digital matched filters employing tapped delay lines with tap weights given by the code values and summed over the length, G, of the code sequence [38]. With conventional M-ary DS-SS signaling employing, e.g., M orthogonal code, M different matched filters requiring a total of GM delay elements and multipliers and M summers are needed. However, in the case of CPSK, since the M code sequences are phase-shifted versions of the same PN code, the matched filters can be overlapped to share a common tapped delay line, as shown in Figure A.2, and 79  Appendix A. An Efficient Implementation of CPSK System  80  Baseband Signal  Each tap multiplied by appropriate PN code chip value e{-1,1}  Figure A.2 CPSK Receiver with Concatenated Matched Filters.  the filter outputs sampled at appropriate time delays are available for sequential pairwise comparisons. In this configuration, the CPSK receiver needs no more than.2G — x delay elements and multipliers, where x = [G/M\,  and M summers, giving substantial savings  in hardware complexity compared to other M-ary DS-SS signaling methods when M is large.  Appendix B. Computer Simulations for Performance Analysis  81  Appendix B Computer Simulations for Performance Analysis  Throughout the thesis, computer simulations are used to analyze the BER performance of both the C P M and CPSK signaling techniques at different channel conditions. A l l simulations are based on the mathematical models derived for the signaling schemes operating under different conditions.  The decision variables, g(T), of each model  are usually described by equations containing components of the desired signal, d(T), cochannel interferences, I(T), tone, J(T), and/or multipath, M(T), interferences and thermal noise, N(T). For example,  g(T)  = d(T) + I(T) + J{T) + M(T)  + N(T).  . (B.l)  All of the interferences and noise are independent random variables of different distributions described by the corresponding terms. In order to simulate and analyze the B E R performance, binary data bits are first randomly generated. For each transmission, which may transmit one or several bit(s) simultaneously, the interferences and noise are generated according to their own p.d.f.'s described by the equations, and the decision variables are computed. Depending on the signaling technique to be simulated, the received data bits are determined differently. For CPM, output of each correlator represents a single data bit. The data bit is received as a "1" and "0" if the output of that correlator is greater than or equal to 0 and less than 0, respectively. The received data bits are then compared with the original data bits, and marked as errors if they are not the same. On the other hand, all the outputs of the correlators of the CPSK system are compared with each other.  82  Appendix B. Computer Simulations for Performance Analysis  The data word corresponding to the correlator with largest output is determined as the received data word. The received data word would then be compared with the original data word bit-by-bit. Bits that are different from the corresponding bits of the original data word are marked as errors. Like most simulations, data generation stops when certain number of bits of errors are recorded. In particular, 5000 bits of errors are choosen as a terminate condition of the simulations. In addition, in order to speed up the simulations, the simulations also terminate when a million bits are simulated and more than 50 bits of errors are recorded or when 100 millions data bits are simulated. Unless the actual average BER is very low, the number of simulated data bits should be sufficient. In some cases, specific PN codes are required to analyze the B E R performance because the p.d.f.'s of the interferences depend on the choice of the PN codes. Unless otherwise specify, the m-sequences listed in table 2 are used to implement the system. Notice that these m-sequences are arbitrary choosen from the class of auto-optimal maximal-length sequences with least sidelobe energy [25], [39], [40].  This may not  simulate the best possible or average performance. However, it guarantees existence  Sequence Length I Polynomial of Generator  Initial State  1 .  31  x  63  x  + X  127  x  + x  3  + x  255  x  + x  6  +  511  x + x +1  1023  x  10  + x +1  0 1110  2047  x  11  + x +1  11000001000  5  6  7  s  9  +  + x  2  +  1  0 0 0 0 10 2  + X  x + x 4  4  3  2  110 0 1  3  +  1  + x  0 0 0 0 10 1 2  + X  +  1  10 1 1 0 10 0 10 0 0 10 0 0 0  Table 2 m-sequences for Simulations  110  10  Appendix B. Computer Simulations for Performance Analysis  of systems that can perfom the same or better and shows the behavior of the system performance as the system parameters and operating conditions change.  83  84  Appendix B. 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In r> <^ n n t»> n n n n n n M n  Ci f>  ° ^  N  In J In in in  89  Appendix B. Computer Simulations for Performance Analysis  82  — V —E 82  X X) c e  8  E  Si H E  •H II  N 1>  ii  o o -< c  8 &  X t> U o i v  5  E w U X o jj u B  0  °i  2  t» — o *o * e * fc. •Of IP fc. Jfc.5 0 IH I a ti a M in«. fN X —O  0  — t>  i-i  U — • o  1—1  TJ-H  s—  *1  n^i/i«) o o> o H r- © WN O IH N HHHHH HHHNtN o o o o o o Ol/l OLfl ©1/1•-<U*l«-Hl/l in in in in in m i/iui in i/i  *  ct a.  3 "S  P --O•  n  2 t>  -H I) U -l e xi X 9 °| £ X »M x> XI •S, .5  -ST I 0J • x > C J —Xx>Cr>C--'• — C xi Oil" xi c d tt 8 -! fc, o- c — xi -H t n M fc, i n o> law ti > llt ti0 O xi I B x> C 9 fct • — x H 0 IIfc>V uX .H tl 3 U ^ — a fc. n u o a o — X u i 1 ux II -H | x> , O• C II f > Cf) -H A -H ~* a x> a fc. O IX X Q k. Xt E fc> e o n ct- a. > ti -H XJ a a — 0 xi " xi X c£ n ItCn0 n« C n— fc. 1 3 B S .» XI XI Xl Xl B C -H 3 — : DiQ,— nfc.^ fc. .5.5.5? Cn >, — o > fc. B fc.fc. t> B a a a ©  B  •3 S a l> O  1 / 1 4 / 1 l/l l/l I T I l/l  2  g-  •ss  H fllfl ^ l / l yt  o  o J  X*  •  8 & N  XI  r» <o er. i I Nn m *£> r* intninu>« I S 5 5 5 S in r > f) rt n r in in in in >inninminmi/iininmt/i in ui ui in in i/i in m m i/i inin ui m in m 1  90  Appendix B. Computer Simulations for Performance Analysis  |o>  5  •3  c a  a n  II dx x£ V V fc, fc, 0 0  It % 41  *l * X> 4J ?•£•. — %»  V fc. «xi axx>v n  fcl fcl fc. fcl  E I J* co Q. U  E  2 TJ C fc.  fc. * _ —*  i *  sXI —.-8 n  n  ci •» EII IE I 10 I o u » u fct u i- 0 3 u a s t< 3 M o X «-<  co  <N »^ ^ i/> *o 60 *4> sp >*3 *0 vO *6 i/> i/i in i/i i/i u i ifl  t \ - a a x.fci •— >,wi Cn o — ~* H* o —* TJ -H i a X — IB V Cn 1 •*H aC •H |B fcl0*r» ^0>,- * Xu P>—— >X 0 Oie— o  u— -H a V —XUExi —  E E  V --TJ O —fclxi —oc •o "it i > B fci xi C C D xi 0 IM C -H fc, 0 0 -HX XI u XI x> X II II V O 0 0 — • en t H ^ C x> c fc,--. p II «fcl-H — d c ^- o » «M  a  *>. OTJ  -H  X °l,, tl M I  0 xi II II TJ 0 — II ti tl --TJ xi TJ _ •44fclTJ TJ I Ofc,fc,II £ 0 | 0  x. C  0  0  c  xi  s U —— II  «M  *> I « « TJ In f-<  X XI e tM J* 0  while for no  a a. £ s  I  —  0 •  > o xO . --4  x•» — ^a C o  0 i 0t H + — H  •&.--o i a „ -5°|V, «x o o  >,* x ojxi a a 0 IN xl ( 0 * *v  x> xi 0 3 3 C n d jjO Ofcl— -H ^ -o * n—< ^|o1 o, a - j x> axi Httoxio 0fc. 0 i c V TJ CS xi 0 * o a k i A Q tea | xi xt X »«"•*• < * c i-t — a— q x fciTJ >, 0 0 0 OO — c o 0 P < M X> - H I >* II- T3 *>. o OfcifixO — xiO r-l 0 - H 0 0 X i 0 II tl •>fc(TJ&U>MdHX> 0»x> — TJ — O xt x a "H 1-H C 0 —II II— II 0n — 1R1 —" • 8 " » —O XU —C x>CHil©1—O wo —xi MX — & 0 C tIi • H i-ITI sV'dX © >w 0 B-H tTx) — 0 d n ' " J ti QTJ o cna ^Bl - He n0 aac nII oft-H 0 xi ti* a-H atxx « IM -r> a X —C 1 C n xi 1 Jxi 0 0 X x . f c . - H 11 ox 0 X E O O 'ill.£11*39 -H XI XI "H O | | XI X* xi — u x> a a 1 H  -•Sg.S.' -B.C  SI  91  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  Appendix C Glossary of the Abbreviations, Acronyms, and Symbols  •  A — Amplitude of transmitting signal with AK as the amplitude of the signal of each data stream of CPSK with K data streams. A W G N -— Additive white Gaussian noise.  •  a — n-th chip of an PN sequence. n  Bt,b — Baseband bandwidth. B  ss  — Bandwidth of the spreading code.  BER — Bit-error rate. BPSK — Binary phase shift keying. bps — Bits per second, is usually used for data rate. •  • C\ — Chip rate of C P M system with K=l. C((/) — Aperiodic autocorrelation of the spreading code of the /-th correlator.  •  Cfc,i(0 — Discrete aperiodic correlation of the spreading code of the k-th correlator and that of the first correlator. C D M A — Code division multiple access.  •  C P M — Code phase multiplexing. CPSK — Code phase shift keying.  • • C S M A — Carrier-sense multiple access. •  DS — Direct Sequence.  •  dk(t) — Data bits of k-th stream with d\ as the n-th bit of d (t). k  Ej, — Bit energy.  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  •  Ei,/N  •  E — Symbol energy.  0  92  — Bit-energy-to-noise-density ratio.  s  E /N s  — Signal-energy-to-noise-density ratio.  0  erfc(x) — The complementary error function. F D M A — Frequency division multiple access. G — The length of a PN code with GK represent the length of the PN code of C P M system with K parallel data streams. 9k(T) — Output of the k-th correlator sampled at time T where T can be either Tb or T . s  h(t) — Low-pass equivalent impulse response of the passband channel. ISI — Intersymbol interference. •  ISM — Industrial, scientific and medical. I(T) — Signal component of the output of a correlator due to self-interference.  •  /• Q — The output of i-th correlator due to the tone jammer conditioned on a given phase angle, 0. JSR — Jamming to signal ratio with JSR/f as the JSR at the input of each correlator of a C P M system with K data streams.  •  J(T) — Signal component of the output of a correlator due to jamming signal.  •  j(t) — Jamming and/or interfering signal.  •  K — Number of parallel data streams of C P M . L — Number of discrete multipath links of a multipath fading model. M — Signaling alphabet size of CPSK, i.e. CPSK.  number of distinct sequence shift of  93  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  m — Data symbol of CPSK with m_i, mo, and mi as the previous, current, and next, respectively, symbol being transmitted, m is the same as mom — Phase shift (in number of chips) of the m-th waveform in the signaling alphabet c  of CPSK. N  0  — Noise spectral density of AWGN.  •  N(T) — Signal component due to thermal noise.  •  n(t) — Additive white Gaussian noise. PM — Average probability of a symbol error of CPSK with signaling alphabet size M. Pb — Average bit error probability of CPSK with signaling alphabet size M.  •  P — Probability of correct detection of a symbol in CPSK.  •  P \m — Conditional probability of correct symbol detection given data word m is  c  c  transmitted. ^c|0,m,Au>  — Conditional probability of correct decision  for a given value of 9, m,  and Aw. Pe,i\e — Conditioned average probability of bit error of the i-th correlator on 9. Pe\Au — Conditional probability of word error for a given value of Aw. Pe\e,m,Au — Conditional probability of word error for a given value of 9, m, and Aw. •  PN — Pseudonoise. P(Z  m  > Zi\Z ) m  — Marginal probabilities of the output of the m-th correlator larger  than that of /-th correlator for a given value of Z . M  •  p.d.f. — Probabiltiy density function. Pk{t) — The PN code for the k-th correlator. p(X) — Probability density function of a random variable X.  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  R  94  — Chip rate.  c  Rd — Bit rate of data. •  R\ — Data rate of C P M system with K=l. RK — Data rate of each data stream of a C P M system with K data streams.  •  R (T) — The autocorrelation function of a maximal length PN sequence with a C  sequence shifted by r. •  Rk,i( ) — The continuous partial cross-correlation function of the code sequences, T  Pic and pi, with a sequence shifted of r. r(t) — Signal at the input of the receiver. rectT(-)  — Rectangular pulse of ^-second duration.  SNRb — Bit-energy-to-noise-density ratio. •  SNR — Signal-energy-to-noise-density ratio.  •'  SNR^ — Average received bit energy to noise density ratio.  S  SS — Spread Spectrum. s(t) — Transmitting signal. •  s(t) — Complex envelope of s(t) Tb — Data bit interval.  •  T — Chip duration of the PN code. c  T  m  •  — The maximum multipath delay spread.  T — Symbol period. s  T D M A — Time division multiple access. •  ti — incremental path delay of the /-th path.  •  W — Length of word/symbol of CPSK.  95  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  Z, — Output of /-th correlator of GPSK system. Z  m  — Output of correlator corresponding to the current data symbol in CPSK.  T/ — The product of 7; and T . c  •  Acu — The difference between the frequency of the jamming tone and the frequency of the data carrier.  •  $x{u)  — Characteristic function of a random variable X.  '  a — Amplitude of single tone jamming signal. fli — path gain of the /-th path.  •  7; — The effective amplitude gain of the jamming signal to the /-th correlator of CPSK when the tone jamming is at a frequency different from the carrier frequency. £(•) — The Dirac delta function.  •  ( — The scatter component of the first path of a Rician channel. (  2  — the mean square of (.  •  9 — Phase difference between the data signal carrier and the jamming signal.  •  #fc,i( ) — is the odd cross-correlation of the sequence p  •  JJL—  •  p — A real constant representing the attenuation of the channel for the direct path  n  k  and p\.  The ratio of frequency difference to chip frequency, i.e. Au>/u> . c  signal component in case of Rician channel. •  K — The ratio of the direct path signal energy to the average scattered signal energy. This is also an important parameter of Rician channel. a  1  — Variance of the output due to thermal noise.  • • Tfc — The amount shift in chips between the &-th PN code and the 1-st P N code. 77 — The delay of the /-th path signal in chips.  Appendix C. Glossary of the Abbreviations, Acronyms, and Symbols  <f>i — path phase of the l-th path. u — Carrier frequency. •  u  c  — Chip frequency in radian.  UJ — Frequency of the single tone jamming signal in radian.  96  97  Bibliography  Bibliography  [1] Robert C. Dixon. Spread Spectrum Systems. John Wiely & Sons Inc., 1984. [2] Raymond L . Pickholz, Donald L . Schilling, and Laurence B. Milstein. Theory of spread-spectrum communications - a tutorial. IEEE Trans. Commun., C O M 30(5):855-884, May 1982. [3] Robert A. Scholtz. The spread spectrum concept. IEEE Trans. Commun., C O M 25(8):748-755, Aug. 1977. [4] Inn-Kiel Chang, Gordon L . Stuber, and Aubery M . Bush. Performance of diversity combining techniques for DS/DPSK signaling over a pulse jammed multipath-fading channel. IEEE Trans. Commun., COM-38(10): 1823-1834, Oct. 1990. [5] Gwo L . Chen and R. Schilling Adhami. The error probability for the BPSK/BPSK and OQPSK/BPSK spread spectrum communication systems in the presence of single tone jammer. System Theory, 1989 Southeastern Symposium, pages 584—588, 1989. [6] John G . Proakis. Digital Communications. McGraw-Hill Book Company, 1989. [7] Donald L . Schilling, Laurence B. Milstein, Raymond L . Pickholz, and Robert W. Brown. Optimization of the processing gain of an m-ary direct sequence spread spectrum communication system. IEEE Trans. Commun., COM-28(8): 1389-1398, Aug. 1980. [8] M . Tanda. Bit error rate of DQPSK signals in slow Nakagami fading. Electronics Letters, 29(5):431-432, Mar. 1993. [9] E. Geraniotis. Direct sequence spread-spectrum multiple-access communications over nonselective and frequency-selective Rician fading channels. 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