{t))) (2.7) Depending on the envelope, a(t), of this single strong interferer, IM2 will generate spe-cific baseband components affecting the performance of a DCR. For a constant envelope interferer, a(t) = Ac, IM2 generates an undesirable D C component which can be treated by methods of DC offset reduction. For a non-constant envelope, a(t) = Ac(l + m(t)), where m(t) is a function of time for amplitude modulation, the baseband beat generated from IM2 will be composed of several undesirable spectral components given by Equation Chapter 2. Background 27 (2.8): a2(t) _[l + Ac(l + m(t))}2 , , ^ 2 2 2 - -^[l + 2m(t)+m2{t)}. (2.8) These components include a DC component, A^/2, which can be addressed by known mit-igation methods, and other more troublesome baseband spectral components, ^[2m(t) + m(t)2}, for which mitigation would be very difficult. Protection of D C R against these undesirable effects requires a high second-order intermodulation rejection ratio (IMR2) between the amplitude level of the interferer(s) and resulting IM2 component. Typically, a figure of merit is specified by IIP2, which specifies a fictitious input amplitude at which the desired signal becomes equal in amplitude to the spectral component generated from IM2 (see Figure 2.13). Thus, a high IIP2 down-conversion mixer is required to minimize the effect of IM2 within a D C R [19]. 2.3 DC offset Reduction Methods One obvious solution to eliminate DC offset is to employ A C coupling; that is, use a highpass filter (HPF) in the down-converted data path, as in Figure 2.14(a). The transfer function of the A C coupling is HAC(S) = T+fic-This method has some drawbacks: 1. Many modulation schemes, e.g., G M S K and QPSK or Q A M followed by (root) raised-cosine pulse shaping, have significant signal energy in the center of the spec-trum. The center of the spectrum lies exactly at D C after down-conversion to zero-IF. In addition to removing unwanted DC, H P F also removes the desired signal Chapter 2. Background 28 Input Level (dBm) IIP2 Figure 2.13: Second-order input intercept point (IIP2) concept. energy and this results in degradation of B E R . If the modulation has no significant D C component (i.e., if there is no information close to DC), high-pass filters can remove DC offsets without significant degradation of the signal quality. This is the case in pagers using 2-FSK. This may also be a viable option for modulation schemes that have relatively wider channel bandwidth, e.g., W C D M A 5 . 2. The H P F is required to have a very small corner frequency to make sure that it does not unnecessarily filter out the spectral contents of the signal. As a rule-of-thumb, it is estimated that the 3-dB cut-off frequency of a high-pass filter should be approximately 0.1% of the symbol rate in these systems if significant degradation 5Wideband code division multiple access (WCDMA) is a type of 3G cellular network. Chapter 2. Background 29 Figure 2.14: D C offset cancellation techniques using (a)capacitive coupling, (b) linear feedback; and (c) sampling in the signal quality is to be avoided [24]. Such a low corner frequency requires prohibitively large capacitors (of the order nF) and resistors, or equivalent capacitors if implemented using switched capacitors. 3. Another major problem wi th A C coupling is that the coupling capacitors can take a significant time to charge up which means that the receiver can take tens of milliseconds to settle. In this regard, pre-charging techniques are often required. Another approach is a D C feedback loop, which uses negative feedback to cancel the D C offset, as depicted in Figure 2.14(b). A D C feedback loop forms a negative feedback at frequencies close to D C , thus filtering out the D C offsets. B o t h the D C offset of the Chapter 2. Background 30 input signal and the DC offsets of the amplifier are canceled at the output. The transfer function of the block diagram of Figure 2.14(b) becomes HFB(s) = Y ^ ^ f j ^ c -A major advantage of this approach over that in Figure 2.14(a) is that it employs only grounded capacitors and can therefore utilize MOSFETs [25]. Since the capacitance density of MOSFETs is much higher than standard parallel plate structures, this approach has a major area advantage compared to that in Figure 2.14(a). However, the nonlinearity of MOS capacitors can limit the system performance. The high-gain amplifier needed in this approach can also reduce the overall linearity of the system. A third approach uses the idle time intervals in digital wireless standards to carry out offset cancellation as shown in Figure 2.14(c). During the idle time intervals, the switch is closed and the offset is measured and stored on the capacitor. However, thermal noise of the switch mandates large values for the capacitor. Analog as well as mixed-mode solutions have been previously proposed to address the DC offset problem. Most solutions rely on good design and layout to reduce leakage and imbalances that cause these offsets. A n on-chip LO reduces DC offsets due to LO leakage, but this alone is not enough [16]. Some representative analog solutions are presented in references [2, 26-28]. A l l of these employ some form of a negative feedback loop to reduce DC offset. Figure 2.15 shows the architecture of the offset canceler proposed by Wang et al. in [2]. The summing amplifier has four inputs: in+ and in- are differential signals with DC offset component, Vref is a reference voltage and ctrl signal is a D C offset cancellation signal from the loop filter. The charge pump circuit will charge the loop filter if the comparator output is Chapter 2. Background 31 high and it will discharge the loop filter if the comparator output is low. Effectively, the output of comparator will have a 50% duty cycle in the absence of any DC offset in the input signal. If there is some non-zero D C offset, this will be reflected in the duty cycle of comparator output. When the loop converges, Ctrl will be adjusted to cancel the D C offset. The operation is similar to a Phase-Locked Loop (PLL). The response time and stability of this loop are determined by the charge pump and parameters of the loop filter. In summary, this solution has problems similar to an H P F and it is not sufficient by itself to remove DC offsets. As reported by Wang et al. in [2], after correction, the DC level is reduced to 5 mV and still needs further correction. Summing F r o m Loop Charge Filter Pump Figure 2.15: DC offset cancellation using analog techniques [2] Mixed-mode offset cancellation schemes are presented in references [2-4, 26-31]. While Lindquist and Isberg [26] and Shoval et al. [31] describe only one-step offset cancellation, Nezami [29] and Yoshida et al. [3] describe a two-step cancellation. Here, the solution presented by Yoshida et al. in [3], which is depicted in Figure 2.16, will be briefly described. Chapter 2. Background 32 In this scheme, time-invariant or static offsets are removed in the analog, or feedback, ADC LPF LPF A D C DAC Memory Averaging Circuit Averaging Circuit DAC Memory Figure 2.16: Mixed-mode DC offset cancellation [3] mode. The feedback loop includes A D C , averaging circuit, memory, D A C , summing circuit and L P F . Static D C offset is estimated by disconnecting the L N A from the antenna before the start of each burst. Time-varying or dynamic offsets are canceled in the digital feed-forward mode. The same averaging circuit in Figure 2.16 is used to estimate the D C level present at the A D C output and this estimate is then subtracted from A D C output to obtain a D C free signal. The presented approach is applicable to burst-based systems and the DC estimate made on one burst is applied to the data on the following burst. This will provide acceptable results only if data in contiguous bursts have more or less the same D C value; this may not be true for practical T D M A systems, especially in the presence of Chapter 2. Background 33 large nearby interferers. Some solutions utilize a known training data sequence to correct for DC offsets. A n example of this method is described in [30] where 5 training symbols in the preamble are used for offset correction. These schemes are not applicable to systems that do not have preamble and it also requires a very close collaboration with the digital baseband detector to achieve offset cancellation. In references [4, 32], an even harmonic mixer is proposed that, by design, eliminates LO leakage offsets if the transconductance (Gm) mismatch is considered negligibly small. Figure 2.17 shows a single-balanced version of the CMOS even-harmonic mixer as proposed by Fang et al. in [4]. The basic principle of an even-harmonic mixer is that the input R F signal is effectively mixed with the second harmonic of the LO generated frequency. Therefore, any L O leakage from L O to R F input does not get mixed down to D C . Rather, it is translated to the L O frequency and is easily filtered by receiver filters. Figure 2.17: Single-balanced CMOS even-harmonic mixer [4]. Chapter 2. Background 34 2.4 I/Q Mismatch Compensation As observed in Figure 2.11, to acheive an IRR of 30 dB or larger, phase and gain mis-matches of 3° and 3% (or less) are required. Such values are very hard to achieve without employing any mismatch correction techniques, either in the analog domain or in the digital domain. Both analog and digital correction techniques have been used to mitigate the effects of phase and gain mismatches. A l l of these techniques use some form of an adaptive algorithm to find the coefficients of a correction filter. A n example of a more recent mixed-signal technique is presented by Der and Razavi in [23]. A sign-sign (SS) least-mean squares (LMS) method is used to calibrate gain and phase errors of a Weaver image-reject receiver to achieve an IRR of 57 dB. A pure analog approach is presented by Behbahani et al. in [33]. A five-stage polyphase filter is used to achieve an IRR of 60 dB for wireless L A N applications. As explained in [33], the higher the number of polyphase filter stages, the higher the IRR. However, the area impact of this technique precludes its use in a low-cost, low-power design. Amongst the digital methods, the work of Y u and Snelgrove [34] is representative of signal separation techniques in the digital domain to improve receiver image rejection. A complex L MS algorithm is employed to separate the image from signal. Two other techniques are presented in references [35, 36]. Valkama and Renfors [36] proposed a method that requires separate complex digital mixers and filters for the image and signal paths to achieve a cleaner reference signal for L M S / R L S 6 adaptation. Having separate demodulation and filtering paths for signal and image is an unnecessary compli-6The LMS (least means square) and RLS (recursive least squares) algorithms used for determining the coefficients of an adaptive filter. Chapter 2. Background 35 cation. Tubbax et al. in [35] assumed that the training sequence is available as part of the preamble of a T D M burst. This is not true for some of the wireless standards such as G S M [21], where the training sequences are in the midamble. 2.5 Summary In this chapter, different receiver architectures have been reviewed, and since D C R has the lowest cost and power, this work focuses on it. D C R design issues have been addressed in detail, and the existing methods to address these problems have been reported. As mentioned before, DC offset, flicker noise and I/Q mismatch issues should be re-solved carefully in D C R design. The existing methods to address these problems are divided in two categories, analog and digital. The trend is to perform as much as pos-sible in the digital domain and to minimize the R F front-end (analog). Also, most of these methods focused on the problem for a specific case (a standard or a system) rather than providing a general solution. In the next chapters, we introduce a novel system-level approach which addresses DC offset, flicker noise and I/Q mismatch problems more generally, and can be implemented in digital or analog. Chapter 3. Complex Quantized Feedback 36 Chapter 3 Complex Quantized Feedback As emphasized in previous sections, there is a need to address the important issues of D C R design including DC-offset, baseline wander effect, I/Q mismatch and low-frequency noise. In this chapter, we address these issues with a general approach applicable to an arbitrary D C R receiver. The basic idea is to use the information available in the I and Q paths to minimize the effect of low-frequency noise, phase error, and mismatch in both paths. Communication systems using quadrature demodulation whose baseband signal spectrum have considerable energy near D C can benefit from this approach. 3.1 Overview of the New Approach First a simple Q F B system is used in both I and Q channels to reduce the effects of baseline wander due to AC-coupling as well as the effects of 1/f noise. Then, the use of complex (in the mathematical sense) cross-coupled Q F B is proposed and it is shown that the system can reduce the undesired effects of carrier phase error and I/Q mismatch. A cost-effective approach to remove such low-frequency disturbances is to use simple AC-coupling, i.e., a high-pass filter (HPF) in the down-converted signal path. This ap-proach has been successfully applied in pager systems to receive frequency-shift-keying Chapter 3. Complex Quantized Feedback 37 (FSK) modulated signals [8] [37]. Since the baseband spectrum of F S K signal has little energy around DC, AC-coupling allows low-frequency disturbance to be removed with minimal distortion to the signal spectrum. For more spectrally efficient modulation schemes, such as quadrature amplitude modu-lation, the baseband signal spectrum has significant energy at low frequencies. Employing AC-coupling H P F in the I and Q paths, as shown in Figure 3.1, can remove and/or mini-mize the unwanted effects of DC-offset and 1/f noise. However, this filtering also removes low-frequency portions of the desired signal. The filtering of the desired signal can cause significant performance loss and introduce severe intersymbol interference (ISI) [9]. V A C coupling A C coupling Baseband Processing Data \"Out Figure 3.1: A direct-conversion receiver with AC-coupling • This inadvertent filtering out of the information bearing the low-frequency part of the signal, typically referred to as baseline wander (in connection with one-dimensional modulation schemes such as pulse amplitude modulation), makes the detection of the signal difficult and causes reduction of noise margin [38] [39]. To remove the baseline wander effect introduced by AC-coupling, a quantized feed-Chapter 3. Complex Quantized Feedback 38 back (QFB) circuit can be used [40]. Q F B is a DC restoration technique in which the low-frequency components of the desired signal are restored by post-decision feedback. Employing Q F B is essential for operation with input patterns having large low-frequency content [41]. For example, in 100 B A S E - T Fast Ethernet systems, Q F B is used to re-duce the baseline wander effect [42]. A conceptual block diagram of Q F B (also known as baseline wander correction) is shown in Figure 3.2. The corresponding time-domain waveforms at different nodes of the system are also shown. The basic idea behind this scheme is as follows: the low-pass filter (LPF) in the feedback path restores the low-frequency components from the output signal (assuming correct decisions) and adds it to the high-pass filtered input signal to reconstruct the entire spectrum [40] [43] [44]. Ideally, the L P F in the feedback loop has the same order and cut-off frequency as the H P F in the feed-forward path. A cross-coupled (CC) Q F B , an extension of the simple Q F B system, is the one of the main contributions in this dissertation, and is described in more detail in the next section. Vl(t) data in high-pass circuit v2(t) decision circuit r u i v4(t) v3(t) low-pass circuit data out Figure 3.2: Conventional quantized feedback technique Chapter 3. Complex Quantized Feedback 39 Simple AC-coupling and quantized feedback can be combined to obtain a cost-effective approach for removal of DC-offset and 1// noise. This scheme is shown in Figure 3.3. The received signal is down-converted into its I and Q components with high-pass filters HACI(S) and HACQ{S) performing AC-coupling in I and Q branches, respectively, and low-pass filters, Hn(s) and HQQ(S), complete the Q F B system. r(t). i LO r • m Figure 3.3: Simple system including high-pass filter and quantized feedback To improve this basic system, we propose the use of cross-coupled feedback circuits [45] to compensate for the carrier phase error and/or I/Q mismatch and to eliminate the crosstalk between the in-phase and quadrature paths. Figure 3.4 shows such a system which can be considered as a complex Q F B system. Cross-coupled filters are added between the two channels for carrier phase error compensation and/or I/Q mismatch Chapter 3. Complex Quantized Feedback 40 correction. HIQ(S), for example, represents the feedback system from in-phase channel to quadrature channel. HACI{S) and HACQ(S) are the transfer functions of the AC-coupling blocks in the in-phase and quadrature channels. The purpose of the in-phase feedback blocks, Hn(s) and HQQ(S), is DC restoration. 4N x(t) e H |Q(s) HQI(s) y(t) I * a(t) b(t) Figure 3.4: Complex Q F B system In this complex Q F B system, assuming that the properties of the high-pass filters, HACI and HACQ, are known, it is possible to derive the expressions for other filters in the system (namely, Hu, HJQ, HQJ and HQQ) that achieve approximate carrier phase error compensation and/or I /Q mismatch correction. Suppose the received signal is r(t) = a(t)cos(uct) + b(t)sin(uct), where the carrier frequency is uc, and a(t) and b(t) are information bearing baseband signals. As is done in Chapter 3. Complex Quantized Feedback 41 [46] in the context of an equalizer, if we assume that there are no decision errors, one can replace the output signals a and b with a and b, respectively. Therefore, in the Laplace transform domain, we can write: X(s) = HACi(s)I{s) + HII(s)A{s) + HQI(s)B(s) (3.1) Y(s) = HAcQ(s)Q(s) + HQQ(s)B(s) + HJQ(s)A(s). (3.2) Also, under ideal conditions and in the absence of noise, the sheer's output is the same as its input: X(s) — A(s) and Y(s) = B(s). Using the above equations and assumptions, the Q F B filters' transfer function can be derived, as discussed in the following subsections. 3.2 Carrier Phase Error Compensation In this section, we assume that there is a carrier phase error 9 between the transmitter and receiver LO. That is, both in-phase and quadrature L O signals encounter the same carrier phase error, 6. It is shown that a cross-coupled Q F B system can be used for compensating this carrier phase error. Recall that the received signal is r(t) = a(t)cos(uct) +b(t)sin(uct). Now let us assume that the carrier recovery system in the receiver has successfully recovered the carrier frequency; however, there is still a residual carrier phase error 9. That is, the L O signals are XLo,i(t) = cos(uct + 9) and XLo,o.(t) — sin(uct + 6). Multiplying r(t) by these two L O signals and filtering out the higher frequency components, the following baseband signals are obtained: xBB,i(t) = a(t)cos9 — b(t)sin9 and XBB,o.{t) = b(t)cos9 + a(t)sin9. Note Chapter 3. Complex Quantized Feedback 42 that the I and Q signals are now correlated and, in the s-domain, we have: I(s) = aA(s)-pB(s), (3.3) Q(s) = aA(s)+pB(s), (3.4) where a = cosO and (5 = sinO. At the summing nodes of Figure 3.4 in the I and Q signal paths, we have: A(s) = HACI(s)I(s) + HII(s)A(s) + HQI(s)B(s) (3.5) B(s) = HACQ(s)Q(s) + HQQ(S)B(s) + HIQ(s)A(s) (3.6) Substituting I(s) and Q(s) from Equation (3.3) and (3.4) into (3.5) and (3.6), we obtain: 0 = {aHACI(s) + Hn(s)-l}A(s) + {-PHAci(s) + HQI(s)}B{s) 0 = {0HACQ(S) + HIQ(s)}A(s) + {aHACQ(s) + HQQ(s) - l}B(s) Therefore, the following relationships can be established between the six filters: Hu(s) = l-aHACI{s) (3.7) HQI(s) = PHACi(s) (3.8) HIQ(s) = -(3HACQ(s) (3.9) HQQ(s) = l-aHACQ(s) (3.10) Chapter 3. Complex Quantized Feedback 43 This shows that the four Q F B filters, H N , HJQ, HQI and HQQ, are functions of the A C -coupling filters and residual carrier phase error. Assuming that Equation (3.7)-(3.10) are satisfied, the crosstalk can be minimized. Note that, in practice, the assumption of no decision errors is only valid for small phase errors, as confirmed by simulation results in Sections 3.3 and 3.4. 3.2.1 I / Q M i s m a t c h Compensa t ion I/Q mismatch can be characterized by two parameters: the amplitude or gain imbalance between I and Q branches, e, and the phase orthogonality mismatch, A 0 . Therefore, the L O signal for I and Q branches can be written as: After multiplying the received signal by the two L O components and low-pass filtering the result, we obtain the following baseband signals: xLO,i(t) = 2cosuct XLO,Q(t) = 2(1 + e)sin{ujct + A 0 ) XBB,l(t) = a(t) XBB,Q.{t) (1 + e)a{t)sinA(f) + (1 + e)b{t)cosA(f) or equivalently: I(s) A(s) (3.11) Chapter 3. Complex Quantized Feedback 44 Q(s) = [(l + e)sin/\\(p}A(s) + {{l + e)cosA(p]B(s). (3.12) Substituting I(s) and Q(s) from the above in the Equations (3.5) and (3.6), we obtain: Hn's) = l-HACi(s) (3.13) HQI(s) = 0 (3.14) HIQ(s) = -{l + e)sinA * » • > * * * * * l x sample 1 1 w , F F I Figure A . l : Spectral overlap of subcarriers in OFDM The frequency response of the ideal channel has a flat magnitude and pseudo-linear (i.e., linear + constant) phase. In the time domain, this translates to pure delay and frequency-constant gain. In the real world, the channels are not ideal thus introducing ISI (as well as noise) in the signal. However, if the pass-band of the channel is divided into several small bands, each of those bands can be considered a nearly ideal narrowband sub-channel: the magnitude of the frequency response is almost constant and the phase can be considered almost pseudo-linear. The basic idea of OFDM is to modulate several carriers with the center frequencies of each sub-channel at a symbol rate K times less than the single-carrier system that uses the same channel. The sub-carriers can transmit different bits/symbol (e.g., use M-QAM with various Ms). This way, for example, sub-channels with smaller attenuations (i.e., higher SNRs) can carry more of the data. The major contribution to the OFDM complexity problem was the application of the FFT algorithm to the modulation and demodulation processes. At the same time, Appendix A. Application: OFDM signaling 129 DSP techniques were being introduced in modems. The technique involved assembling the input information into blocks of N complex numbers, one for each subchannel. A n inverse F F T is performed on each block, and the resultant is transmitted serially. At the receiver, the information is recovered by performing an F F T on the received block of signal samples. This form of O F D M is referred to as discrete multitone (DMT). The spectrum of the signal on the line is identical to that of N separate Q A M signals at N frequencies separated by the signaling rate. Each such Q A M signal carries one of the original input complex numbers. The spectrum of each Q A M signal is of the form sin(kf)/f, with nulls at the center of the other sub carriers, as shown in Figure A . l . Figure A.2 shows the O F D M spectrum. Figure A.2: Spectrum of O F D M signal Care must be taken to avoid the overlap of consecutive transmitted blocks. This is solved by the use of a cyclic prefix. The process of symbol transmission is straightforward if the signal is to be further modulated by a modulator with I and Q inputs. Otherwise, it is necessary to transmit real quantities. This can be accomplished by first appending the Appendix A. Application: OFDM signaling 130 complex conjugate to the original input block. A 2N point IFFT yields 2N real numbers to be transmitted per block, which is equivalent to N complex numbers. A . 1.1 O F D M Implementa t ion In the following, we describe an implementation of O F D M signaling that is used in our simulations. The block diagram of the transmitter is given in Figure A.3. The serial-to-parallel converter divides each B bits of the input data to K groups, with the ith group having 6, bits. Using M r Q A M for sub-channel i, the multi-carrier modulator selects one of M ; = 2bi symbols, depending on the data. This way, we will have K symbols for every B bits of the system's input. Inverse D F T is then applied to the sequence of K symbols. The reason for this operation will be seen later. The real (xn\\ n = 0, • • •, K — 1) part of this sequence is then prefixed by cyclic repetition to make xx-m, XK-m+i, • • •, XK-I,XQ, XI, • • •, XK-I, in which m is the length of the channel sam-pled impulse response. The imaginary part of the signal, yn, undergoes the same process. The resulted sequences are converted to analog signals and are mixed with the carrier sine and cosine carriers to make the band-pass signal to be sent over the channel. Data _ ^ (bit stream) Serial-to-parallel Multi-carrier Modulator & I D F T Re{} Im{} A d d cyclic prefix Parallel-to-serial D / A | : u o 90Y*t_J -*g)— T o the channel Figure A.3: O F D M transmission. Unlike the O F D M implementation of [55] that gives real baseband signals, we used quadrature signaling that is complex in the baseband, but gives real quadrature and Appendix A. Application: OFDM signaling 131 in-phase band-pass components. Serial-to-parallel Remove cyclic prefix i[\"] + M\"] Multi-carrier de-modulation (DFT) ISI removal A Re{} Im{} CQFB QAM detectors Figure A.4: O F D M reception. The input to the receiver depicted in Figure A.4 is supplied by the R F front end shown in Figure 4.3. In [55], it is shown that the channel ISI corrupts the cyclic prefix (which is easily removed) and affects signal in the following manner. Xi = dXi + Noise, , i = 0,1, • • •, K - 1, in which Xi is the Q A M symbol transmitted through the i th sub-channel, {C;} is the K-point D F T of the sampled impulse response of the channel {co,Ci , • • •, c m _i}(padded by K — m zeros), and {Xi} is what we have (polluted by noise). Note that the proce-dure automatically de-correlates the received symbols, thus removing ISI. To recover the transmitted symbols, {Xi}, we can measure {C;} by passing a training sequence through the channel. Alternatively, assuming that we have a good initial guess for {Cj} and that the channel characteristics is constant through time or varies slowly, we can use L M S to adaptively find {Q} . Since ISI is automatically handled in O F D M signaling as described above, C Q F B filters are simply gains in this case. The detection equation becomes the following: Appendix A. Application: OFDM signaling 132 Re{Xi} Im{Xi} \\ ( I/Q 0 V 0 I/Q j + H Qi 1 Re{Xx} ^ Im{Xi} , i = 0 , - - - , A \" - l ( A . l ) in which {h+jQi} = DFTk{i[n] +jq[n]}, n = KN, KN + K-l and Qt{.) is Mr Q A M detector used for demodulation of the i th sub-channel. H is the C Q F B gain matrix that is given by Equation (4.3), if the receiver non-ideality parameters are known. Note that we assumed the same H for all sub-channels. That may not be the case as the sub-carrier frequencies are different for different sub-channels. If the receiver's characteristics significantly vary for different sub-channels, we can consider different Hs. Equation (A.l) above should be solved for every received symbol, although the D F T operation is performed once for every batch of K symbols. For an adaptive solution, we assume H and {C — i) (or an initial guess of them) are known and use the method of Sec-tion 4.3.2 to detect the symbols. Once Equation A . l is solved, it can be considered as two linear filtering equations: (a) with input Qi{(Re{X — i} , Im{Xi})T} (the decided sym-bol), output (Re{Xi}, Im{Xi})T , filter H, and ideal output Qi{(Re{X — i}, Im{Xi})T}, and (b) with input (Ii,Qi)T , output (Re{Xi}, Im{Xi})T, filter diag{(l/Ci,l/C — i)}, and ideal output Qi{(Re{X — i}, Im{Xi})T}. Using L M S , the update equations for {d} and H become the following. For z = 0, • • • ,K — 1: 1 ^ \\ B i J ( Re{X{} ImiXi} \\ H = H + pei(Ai B^, Appendix A. Application: OFDM signaling 133 Ci = d + p Qi) ti in which Ci = 1/Cj and (Ai BA = Qi{(Re{Xi}, Im{Xi})T} denotes the decision for the i th sub-channel. Note that Q is assumed to be real or the update equations would be a bit different. A.2 Adaptive ISI and non-ideality compensation in O F D M signaling In this section, an experiment and its results are reported that confirm good performance of the method described in Chapter 4. 6000 bits of randomly generated bits are to be transmitted by O F D M signaling through A W G N channel with SNR of 15 dB. O F D M employs two sub-channels, the first one with 4 -QAM (2 bits/symbol) and the second one with 16-QAM (i.e., carrying 4 bits/symbol) modulation. Therefore, the 6000 bits are transmitted via 2000 symbols. The effects of the channel and the DC-offset removal filter are considered as a single 5-point discrete filter, with the impulse response given below: hISi[n] = {0.82 -0 .18 -0 .18 -0 .18 -0 .18}. Note that in O F D M signaling we cannot use a DC-removal filter with zero DC gain or use a channel that has a zero gain somewhere within in the allocated signaling bandwidth. That is because the C* corresponding to the sub-carrier that falls on the zero, becomes very Appendix A. Application: OFDM signaling 134 small, making the detection equation (Equation (A.l)) ill-conditioned. In other words, if the channel (i.e., channel + DC-removal filter) has very small gain in a certain frequency (e.g., a zero gain at D C if a DC-removal filter is used), that sub-channel is not suitable for communication and the transmission will suffer from a large probability of error. In practice, the channel does not have a zero in its pass-band but the DC-offset removal filter at the receiver can make negligible, and make Co the first sub-channel unusable. To alleviate this problem, we can simply put aside the first sub-channel and use the rest of them for transmission. Alternatively, we can use a filter with non-zero (but small) D C gain for DC-offset reduction. Such a filter reduces the DC-offset that is necessary for good performance of the receiver and, at the same time, allows for using of the first sub-channel perhaps at a lower bit-rate as compared to the rest of sub-channels with higher gains and thus higher SNRs. In this experiment, we used the second approach above (the frequency response of hisi[n] is shown in Figure A.5): the DC-offset reduction filter has a small D C gain (-20dB). The receiver non-ideality parameters are considered 9 = 0.2rad,