8 Carrier Recovery for ‘Sub-Coherent’ CDMA 8.1 Overview In this chapter we examine possible methods of carrier recovery for the SE- CDMA presented in Chapter 6. In particular, we propose, evaluate and compare two techniques; namely Symbol-Aided Demodulation (SAD) and the Pilot-Aided Demodulation (PAD). The performance analysis of each scheme (SAD and PAD) includes both Rician and Rayleigh multipath fading channels, and thus are also useful (in addition to the satellite) in terrestrial mobile applications. Both schemes are promising alternatives to differentially coherent demodulation for scenarios characterized with uncertainties in the carrier phase that make coherent demodulation unfeasible. The frequency selective fading (multipath), the Doppler phenomenon due to user mobility and/or to satellite drift motion, and the temperature variation and ventilation conditions at the sites of the various local oscillators that generate the transmitted signals cause the carrier phase uncertainty. Coherent demodulation requires the extraction of a reliable (perfect) phase reference from the received signal. A traditional alternative is the differentially coherent demodulation that uses the phase of the previous bit (symbol) as a reference, but requires almost 3dB (for M-ary PSK modulation in AWGN channels, it is less than that for BDPSK) of additional signal-to-noise (E b /N 0 ) in order to achieve the same bit error rate as coherent demodulation. This problem is more severe in DS/CDMA systems, which are limited by other-user interference: the additional cost in dBs of differentially coherent over coherent demodulation increases linearly with the number of users in the system, so as to render the fully-loaded multi-user system impractical [1]. Recently, SAD [2] and PAD [3] have been considered a form of ‘sub-coherent’ demodulation. In the proposed SAD and PAD schemes, estimates of the channel multipath phases and amplitudes are extracted by smoothing and interpolation of the transmitted known bits in the SAD scheme or the pilot in the PAD scheme. The SAD (or PAD) performance then consists of evaluating the additional Signal-to-Noise Ratio (SNR) needed by either scheme to achieve the same Bit Error Rate (BER) as the coherent demodulation. In this chapter we first present the system model and the design issues of the SAD scheme in Section 8.2, and its BER analysis (for the uncoded system) in Section 8.3. Then in Section 8.4 we present the system model, the design and the BER analysis CDMA: Access and Switching: For Terrestrial and Satellite Networks Diakoumis Gerakoulis, Evaggelos Geraniotis Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-49184-5 (Hardback); 0-470-84169-9 (Electronic) 188 CDMA: ACCESS AND SWITCHING (for the uncoded system) of the PAD scheme. The BER analyses of the coded systems are presented in Section 8.5. The coded system is based on a proposed new iterative decoding algorithm. The performance of the coded system of both schemes has been evaluated via simulations. The performance results are presented in Section 8.6. 8.2 Symbol-Aided Demodulation 8.2.1 System Model In symbol-aided demodulation, known symbols are multiplexed with data bearing symbols. The known symbols are multiplexed with the data symbols at a constant ratio, so that one known symbol is followed by J − 1 unknown data symbols. This ratio implies a loss in the throughput of 1/J. At the receiver the known symbols are used to estimate the channel for other sampling points. The system is as shown in Figure 8.1-A. The transmitted signal for the first user is given by s 1 (t)=A ∞  k=−∞ b 1 (k)a 1 (t)p(t −kT) where b 1 (k) is the binary data sequence, a 1 (t) is the spreading code, which is a periodic sequence of unit amplitude positive and negative rectangular pulses (chips) of duration T c , T = NT c is the symbol duration, and N is the processing gain. The j th code pulse has amplitude a j i = a i (t)forjT c ≤ t ≤ (j +1)T c ,andp(t)isa unit energy pulse in the interval 0 ≤ t ≤ T . The received signal is r(t) L  l=1 c 1l (t −τ 1l )s 1l (t −τ 1l )+ K u  m=2 L  l=1 c ml (t −τ ml )s ml (t −τ ml )+n(t) where L is the number of paths, n(t) is the AWGN with power spectral density N 0 in the real and imaginary parts, and K u is the number of users. The channel complex gain c ml (t) represents the Rayleigh or Rician fading for the l th path of the m th user, with an autocorrelation function [4] R c (τ)=σ 2 g  K 1+K + 1 1+K J 0 (2πf D τ)  where K is the ratio between the line of sight power and the scattered power, and the paths are assumed independent and with identical distributions. The output of the normalizing matched filter, representing the finger of the rake receiver, for the first path of the first user, with impulse response a 1 (−t)p ∗ (−t)/( √ N 0 ), and assuming τ 11 = 0, will be given by r 11 (k)=u 11 (k)b 1 (k)+ L  l=2 u 1l (k)I 1l (k)e jφ 1l (k) + K u  m=2 L  l=1 u ml (k)I ml (k)e jφ ml (k) + n 11 (k) CARRIER RECOVERY 189 where the Gaussian noise samples n 11 (k) are white with unit variance, and complex symbol gain u ml (k) has mean E[u ml (k)] =  γ s ml  K K +1 and variance σ 2 u ml = γ s ml 1 K +1 where the average SNR for path l of user m is given by γ s ml = E s ml N 0 and I ml (k) is the interference from path l of user m to path 1 of user 1. For the SAD scheme E s ml = E b ml J − 1 J where E b ml is the energy per bit, and γ b ml = E b ml N 0 8.2.2 Design of Modulator and Demodulator There are several issues that must be taken into consideration for the proper design of the symbol-aided modulation/demodulation system. TheRateofAid-Symbols A proper choice of the value of J is of paramount importance for SAD system design. Increasing J will result in increasing the throughput, but at the same time it will increase the processing delay and the carrier-phase estimation error in both the known symbols and the data symbols. Guidelines for the choice of J are given below. The value of J is determined from the bandwidth, rate of fading or of Doppler, or in general, from the rate of change of the phenomenon that introduces the uncertainty (and change) in the carrier phase. If we assume that the fading rate (or other rate of change) is R f , then the sampling period T fs and sampling rate R fs =1/T fs of the channel observations must satisfy the Nyquist condition R fs = 1 T fs ≥ 2R f For notational convenience, define J max = R s 2R f where R s is the symbol rate. J max corresponds to the sampling of the fading phenomenon at exactly the Nyquist rate, presented in a more convenient form 190 CDMA: ACCESS AND SWITCHING A. B. MPSK-Mod & Symbol or Pilot Insertion Pulse Shaping Channel AWGN & Fading Other User Interf . Rake Receiver Decision Data Σ Delay 11 τ 12 τ L1 τ T- Channel Estimation Delay T- Channel Estimation Delay T- Channel Estimation Decoder Carrier Removal complex signal SAD PAD SAD PAD SAD PAD · · · · · · Figure 8.1 A. The SAD/PAD CDMA system B. A rake receiver for SAD (or PAD in dotted lines). corresponding to the rate which known symbols are inserted for a given data and fading rates in order to fully capture the variation in the fading (or other phenomenon). We expect that R f << R s that is, the rate of change of the carrier phase is much slower than the symbol rate of the system. For example, we may have R s = 64 kbps (kHz) while R f = 64 or 128 Hz (or a value in the range of 30 Hz to 200 Hz). Denoting the symbol duration as T s =1/R s ,wehavethatT f >> T s , and define J as the ratio T fs /T s , i.e. J =[T fs /T s ]=[R s /R fs ] ≤ [R s /(2R f )] = J max Therefore, J max corresponds to sampling at exactly the Nyquist rate, and J ≤ J max corresponds to oversampling; for example J = J max /4 corresponds to sampling at four times the Nyquist rate, while J = J max /8 corresponds to sampling at eight times the Nyquist rate. TheideaistouseasufficientlysmallJ so that oversampling at rate R fs =(J max /J) · (2R f ) CARRIER RECOVERY 191 captures the change in the phenomenon and reduces the noise in the estimates of the phase (by a factor of J max /J through smoothing, as we will see next) but still maintains the throughput loss (equal to 1/J) within acceptable values. A simple analysis of the SAD technique was presented in reference [5]. This is an approximate analysis assuming perfect filtering, but it provides an intuitive understanding of the problem and it helps identify the optimum J.Itisdoneby simply taking into consideration the power loss due to reference insertion, expressed as L r ≈ J +1 J =1+ 1 J and the amount of increase in the noise due to the noisy reference (assuming perfect filtering and interpolation) which is given by L n ≈ 1+ J J max Thus the total loss compared to coherent system is given by L t (dB)=L r (dB)+L n (dB) The optimum choice of J given J max can be obtained by calculating the minimum achievable loss; we can easily get J opt ≈  J max L t (J opt ) ≈ (1 + 1 √ J max ) 2 Then the conclusion is that the performance of any SAD system can be no better than L t dBs below (worse than) the performance of a coherent system (which assumes the perfect knowledge of the fading phase). For example, for J max = 50, J opt ≈ 7, and L t ≈ 1.15dB. For the SAD scheme, the demodulation will delay the data symbols by JM symbols, where M is half the order of the smoothing filter. Clearly, decreasing J will produce a shorter delay, but as we mentioned, it will decrease the throughput, and the estimation error will be increased. The Smoothing Filter The bandwidth of the smoothing filter is another important issue. This filter is a digital filter that estimates u ml (k) of the unknown symbol samples. Decreasing J (which is equivalent to oversampling) will enable the filter to better estimate u ml (k) by removing more noise, and allow easier tracking of the relatively slower fading. Two approaches are addressed here. The first is to derive the optimal Wiener filter for every unknown data point within the frame of length J, which means that filtering and interpolation are done simultaneously (in a sliding window manner). The second approach is to use a single filter for filtering all the known symbols, which is also a Wiener filter, and then to linearly interpolate the resulting output in order to obtain all unknown data symbols. The difference in performance between the two approaches is evaluated below. 192 CDMA: ACCESS AND SWITCHING 8.3 BER Analysis for SAD The first step for calculating the performance is to calculate the interference (see r 11 (k) in Section 8.2.1). The best way to proceed is to calculate I ml (k)foragiven code selection, and hence calculate the mean square power of the interference averaged over τ ml . A very good approximation is to follow references [6] and [1], and to assume a random signature sequence of length N. This approximation is very accurate if the system uses long (period) codes like the IS-95 system [3], and has sufficiently large N and K u . Following references [6] or [1], we can calculate E {I ml (k)} =0 E  I 2 ml (k)  = 2 3N In this section the analysis of the SAD system will be presented for both the optimum Wiener filter and linear interpolation case. 8.3.1 Optimum Wiener Filtering The best performance that can be expected from the SAD technique (for a given filter length) can be obtained from Wiener filters. We will obtain its performance in this section for multipath Rician fading channels. Cavers [2] was the first to perform this analysis for Rayleigh fading channels. The phase reference of the l th path of the m th user for the unknown symbols is obtained from v ml (k)=h † (k)r ml = M  i=−M h ∗ (i, k)r ml (iJ) where the dagger denotes conjugate transpose, and r ml is the vector formed from r ml (iJ), the samples of the output of a matched filter of a finger of the rake receiver, −M ≤ i ≤ M is the index of the known (SAD) symbols, and 1 ≤ k ≤ J − 1. Note that there will be J − 1 different filters used. The Wiener filter equation will be given by ˜ Rh(k)=w(k) where ˜ R is the autocorrelation matrix of size 2M + 1 defined by ˜ R = 1 2 E[r ml r ml ] and the J − 1 vectors are w(k)= 1 2 E[u ∗ ml (k)r ml ] The channel is Rician as described by R c (τ) in Section 8.2.1. Perfect power control is assumed, such that γ b ml is constant for all users and is denoted by γ bL . It is assumed CARRIER RECOVERY 193 that all the paths are identical, and so γ bL = γ b L where γ b = Lγ b ml is the total average SNR for every bit from all the paths. Now we can obtain ˜ R and w(k)from R ij = γ bL K +1 J − 1 J  K + J 0  π (i −j)J J max  +  γ bL J − 1 J 2(K u ∗ L − 1) 3N +1  δ i,j w i (k)= γ bL K +1 J − 1 J  K + J 0  π (iJ − k) J max  where δ i,j is the Kronecker delta. The Rake receiver is shown in Figure 8.1-B, which is the maximal ratio combiner with noisy reference. From reference [7] we can calculate the probability of error using Pe = Q 1 (a, b) −I 0 (ab)e [− 1 2 (a 2 +b 2 )] + I 0 (ab)e [− 1 2 (a 2 +b 2 )] [2/(1 − µ)] 2L−1 L−1  k=0  2L −1 k  1+µ 1 −µ  k + e [− 1 2 (a 2 +b 2 )] [2/(1 − µ)] 2L−1 × L−1  n=1 I n (ab) L−1−n  k=1  2L −1 k  ·   b a  n  1+µ 1 −µ  k −  a b  n  1+µ 1 −µ  2L−1−k  where a 2 = L 2     E{r} σ r − E{v} σ v     2 and b 2 = L 2     E{r} σ r + E{v} σ v     2 Q 1 (a, b)=  ∞ b xe [− 1 2 (a 2 +x 2 )] I 0 (ax)dx and µ = σ 2 rv σ v σ r where for the SAD scheme, σ 2 rv = w † (k)h(k) − K K +1 J − 1 J γ bL S h (k) σ 2 v = w † (k)h(k) − (E{v}) 2 σ 2 r = γ bL J − 1 J 1 K +1 +1+γ bL J − 1 J 2[K u ∗ L − 1] 3N E{v} =  K K +1 J − 1 J γ bL S 2 h (k) E{r} =  K K +1 J − 1 J γ bL 194 CDMA: ACCESS AND SWITCHING where S h (k)=  M i=−M h(i, k), while for coherent demodulation we have v ml (k)= u ml (k) σ 2 rv = 1 K +1 γ bL σ 2 v = 1 K +1 γ bL σ 2 r = γ bL 1 K +1 +1+γ bL 2[K u ∗ L − 1] 3N E{v} =  K K +1 γ bL E{r} =  K K +1 γ bL and for differential modulation we have v ml (k)=r ml (k − 1) σ 2 rv = 1 K +1 γ bL J 0  π J max  σ 2 v = γ bL 1 K +1 +1+γ bL 2[K u ∗ L − 1] 3N σ 2 r = γ bL 1 K +1 +1+γ bL 2[K u ∗ L − 1] 3N E{v} =  K K +1 γ bL E{r} =  K K +1 γ bL Filtering Followed by Interpolation The second approach is to design a single filter to filter the known samples and then linearly interpolate the output to estimate the channel at the unknown samples. A Wiener filter can still be used to maximize the effective SNR at k = iJ. Following the same Wiener optimization approach as before, we can obtain h for k =0and use it. Following the filter, the interpolator linearly interpolates the estimates of the carrier (or fading) inphase and quadrature components of the data symbols between each two successive known symbols (0 and J). We consider the case of 1 ≤ k ≤ J −1 without any loss of generality. We have v(k)= k J v(J)+ J − k J v(0) The expression for the probability of error P e (given above) will be used again to calculate the performance, where now CARRIER RECOVERY 195 σ 2 rv = M  i=−M h(i)γ bL J − 1 J 1 K +1  K + J − k J · J 0  π iJ − k J max  + k J J 0  π (i +1)J − k J max  − K K +1 J − 1 J γ bL S h (k) σ 2 v = M  i=−M M  j=−M h(i)h(j)γ bL J − 1 J 1 K +1   J − k J  2 +  k J  2   K + J 0  π (i −j)J J max  + h(i)h(j)γ bL J − 1 J 1 K +1 J − k J k J  2K + J 0  π (i −j − 1)J J max  + J 0  π (i −j +1)J J max   + h(i)h(j)   J − k J  2 +  k J  2  1 + γ bL J − 1 J 2(K u ∗ L − 1) 3N  δ i,j + h(i)h(j) J − k J k J ·  1+γ bL J − 1 J 2(K u ∗ L − 1) 3N  (δ i,j−1 + δ i,j+1 ) − (E{v}) 2 The other parameters (E{v},σ 2 r ,E{r},S h (k)) are like those given in the previous section for the optimum Wiener filter case. 8.4 Pilot-Aided Demodulation 8.4.1 System Model The transmitted signal for user 1 is given by s 1 (t)=A ∞  k=−∞ ((A p a p 1 (t)+b 1 (k)a 1 (t))p(t −kT) where A p is the pilot amplitude, a p 1 (t) is the pilot spreading code, and all the other parameters are as described for the SAD scheme. a p 1 (t)anda 1 (t) could easily be made orthogonal through the use of code concatenation. The orthogonality will be maintained for every path because both codes will pass through the same channel. 196 CDMA: ACCESS AND SWITCHING The received signal will be given by r(t)= L  l=1 c 1l (t −τ 1l )s 1l (t −τ 1l )+ K u  m=2 L  l=1 c ml (t −τ ml )s ml (t −τ ml )+n(t) The output of the normalizing matched filter, representing the finger of the rake receiver, for the first path of the first user, with impulse response a 1 (−t)p ∗ (−t)/( √ N 0 ) assuming equal energy pulses and BPSK modulation, will be given by r 11 (k)=u 11 (k)(b 1 (k)+I 11 p (k)) + L  l=2 u 1l (k)I 1l (k)e jφ 1l (k) + K u  m=2 L  l=1 u ml l(k)I ml (k)e jφ m l + n 11 (k) where the Gaussian noise samples n 11 (k) are white with unit variance. I 11 p (k)isthe interference from the pilot signal to the data signal from the same path. As mentioned above, each user’s data and pilot codes are assumed orthogonal, and so I 11 p (k)=0. For the pilot-aided scheme, for fair comparison, the energy per bit E b will be the sum of the pilot energy E p and data energy E d ,whichmeansE b = E p + E d .Inthe following, we will denote the power in the pilot as a fraction of the power of the data signal, and so we can write E p = PE b . The complex symbol gain u ml (k) has mean E[u ml (k)] =  γ d ml  K K +1 and variance σ 2 u ml = γ d ml 1 K +1 where γ d ml = E d ml N 0 and I ml (k) is the interference from path l of user m to path 1 of user 1, including the interference from both the data and pilot signals. The same expression can be obtained for the pilot fingers of the rake, but with E p replacing E d . The output of the first finger of the first user pilot Rake will be denoted by r 11p (k). 8.4.2 Design of Modulator and Demodulator There are several issues that must be taken into consideration in the proper design of a pilot-aided modulation/demodulation system. Filter Length The filter length is of great importance for the performance of the PAD scheme. If the fading is very slow relative to the data rate, an averaging filter could be used; this [...]... one and six users J = 7, Jmax = 50, P = 1, N = 31, L = 1 Filter length = 31 for both schemes 202 CDMA: ACCESS AND SWITCHING of users Ku = 1, 6, and the processing gain (number of chips per symbol) is N = 31 The filters for the SAD and PAD schemes are of length 29 J = 7, P = 1/7 and Jmax = 50 It was noticed and previously mentioned [2] that if the optimum filter is used for every point between two known... information for both the data and code bits Following the first iteration, the probability of the bits is calculated according to p(x(k) = 1) = eL(k) /(1 + eL(k) ) 200 CDMA: ACCESS AND SWITCHING and the channel estimate is adapted after every iteration according to M h(l)r1l (l + m)[2p(l + m) − 1] cl (m) = ˆ l=−M,l=0 where p(k) = 1 for known symbols The new Lcl (k)’s are then calculated and a new iteration is... over DPSK Figures 8.4-A and -B show that oversampling is very important for the SAD scheme The optimum J was obtained previously, and it should be followed It also shows that CARRIER RECOVERY 203 Figure 8.3 BER versus the number of users, for (A) Rician (K = 10), (B) Rayleigh fading J = 7, Jmax = 50, P = 1/7, L = 1, 4 and SNR very large Filter order = 11 204 CDMA: ACCESS AND SWITCHING Figure 8.4 BER... c(m) = ˆ h(l)r11t (l + m) l=−M,l=0 and the iterations are repeated in a similar way to the SAD scheme 8.6 Performance Results Figures 8.2-A and -B, show the BER over Rayleigh and Rician fading channels, respectively, when the optimum Wiener filters are used and for different numbers CARRIER RECOVERY 201 Figure 8.2 BER in (A) Rician (K = 10), (B) Rayleigh fading and for one and six users J = 7, Jmax = 50,... /N0 = 6 dB 208 CDMA: ACCESS AND SWITCHING the filter design is very important for fast fading Rayleigh channels, while averaging could be used for very slow Rician fading channels Figures 8.5-A, -B and 8.6-A, -B indicate the performance of the coded system In these graphs, the order of the filter used for the SAD scheme is 6, while the PAD scheme uses a filter length of 19 For the SAD-ID and PAD-ID schemes,... for the PAD scheme is more than that for the SAD scheme, and so at high SNR the SAD scheme is better It can also be shown that the iterative decoding algorithm used for both SAD and PAD schemes, and denoted by SAD-ID and PAD-ID, improves the performance of both systems significantly For the Rician and 4-path Rayleigh fading it is clear that the SAD and PAD schemes are almost identical The higher error... exact BER floor for six users, but we increased the number of users until this floor was within our accuracy and the same phenomenon was noticed as for 1-path Rayleigh fading Figures 8.7-A and -B shows the effect of changing P and J for SAD and PAD As shown, for conventional SAD and PAD, the optimum J and P obtained for the uncoded scheme are still optimum 8.7 Conclusions We conclude that the choice of filtering... coding and decoding’ Proc ICC’93, May 1993 [9] J Hagenauer ‘The Turbo Principle: Tutorial Introduction and State of the Art’ International Symposium on Turbo Codes and Related Topics, Brest, France, September 1997, pp 1–9 [10] H ElGamal, M Khairy and E Geraniotis ‘Iterative Decoding and Channel Estimation of DS/CDMA over Slow Rayleigh Fading Channels’ PIMRC 98, Boston, MA, 1998 [11] M Khairy and E Geraniotis... minimum achievable loss; we can easily get Popt ≈ Lt (Jopt ) ≈ 1 Jmax 1 1+ √ Jmax 2 198 CDMA: ACCESS AND SWITCHING Note the similarity of this result to the one given for the SAD scheme Ideally, the two schemes will have the same performance The question is, in practical situations where there is only finite length filtering and other user interference, which one of the two schemes will be better The other... Jmax = 50, P = 1/7, N = 31 (B) L = 4, Ku = 1, Jmax = 50, P = 1/7, N = 31 206 CDMA: ACCESS AND SWITCHING Figure 8.6 BER versus Eb /N0 for coded signal, in Rician fading (K = 10), L = 1 (A) Ku = 1, J = 7, Jmax = 50, P = 1/7, N = 31 (B) Ku = 6, J = 7, Jmax = 50, P = 1/7, N = 31 CARRIER RECOVERY 207 Figure 8.7 The effect of J and P on the BER for (A) Rayleigh fading, L = 1, Jmax = 50, Ku = 6, N = 31, Eb . schemes. 202 CDMA: ACCESS AND SWITCHING of users K u =1, 6, and the processing gain (number of chips per symbol) is N = 31. The filters for the SAD and PAD schemes are of length 29. J =7,P =1/ 7and J max =. probability of the bits is calculated according to p(x(k)=1)=e L(k) /(1 + e L(k) ) 200 CDMA: ACCESS AND SWITCHING and the channel estimate is adapted after every iteration according to ˆc l (m)= M  l=−M,l=0 h(l)r 1l (l. Section 8.4 we present the system model, the design and the BER analysis CDMA: Access and Switching: For Terrestrial and Satellite Networks Diakoumis Gerakoulis, Evaggelos Geraniotis Copyright