A block diagram of the standard iterative receiver with a parallel interference canceller (PIC-STD) is shown in Fig. 6.6. In order to simplify the presentation we assume that an HLST architecture with separate error control coding in each layer is used. In addition, the same convolutional codes with BPSK modulation are selected in each layer.
In the first iteration, the PIC detectors are equivalent to a bank of matched filters. The detectors provide decision statistics of thenT transmitted symbol sequences. The decision statistics in the first iteration, for antennaiand timet, denoted byyti,1, is determined as
yti,1=hHi r (6.30)
Figure 6.6 Block diagram of an iterative receiver with PIC-STD
wherehHi is theith row of matrixHH. These decision statistics are passed to the respective decoders, which generate soft estimates on the transmitted symbols.
In the second and later iterations, the decoder soft output is used to update the PIC detector decision statistics.
The decision statistics in thekth iteration at timet, for transmit antenna i, denoted by yti,k, is given by
yti,k=hHi (r−HXˆki−1) (6.31) where Xˆki−1 is an nT ×1 column matrix with the symbol estimates from the (k−1)th iteration as elements, except for theith element which is set to zero. It can be written as
ˆ
xki−1=(xˆt1,k−1, . . . ,xˆti−1,k−1,0,xˆti+1,k−1, . . . ,xˆtnT,k−1)T (6.32) The detection outputs for layeri for a whole block of transmitted symbols form a vector, yi,k, which is interleaved and then passed to the i-the decoder.
The decoder in thekth iteration calculates the log-likelihood ratios (LLR) for antennai at timet, denoted byλi,kt and given by
λi,kt =log P (xti,k=1|yi,k)
P (xti,k= −1|yi,k) (6.33)
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whereP (xti,k = j|yi,k), j = 1,−1, are the symbol a posteriori probabilities (APP). The LLR can be calculated by the iterative MAP algorithm (Appendix 5.1).
The symbol a posteriori probabilitiesP (xti,k=j|yi,k), j =1,−1, can be expressed as P (xti,k=1|yi,k)= eλi,kt
1+eλi,kt
(6.34) P (xti,k= −1|yi,k)= 1
1+eλi,kt
(6.35) The estimates of the transmitted symbols in (6.32) are calculated by finding their mean
ˆ
xi,kt =1ãP (xti,k=1|yi,k)+(−1)ãP (xti,k= −1|yi,k) (6.36) By combining Eqs. (6.36), (6.34) and (6.35), we express the symbol estimates as functions of the LLR
ˆ
xti,k=eλi,kt −1 eλi,kt +1
(6.37) When the LLR is calculated on the basis of the a posteriori probabilities, it is obtained as
λi,kt =log
m,m=Ms−1
m,m=0,xit=1
αj−1(m)pt(xti =1)exp
−
j n
l=(j−1)n
(yli,k−xli)2 2(σi,k)2
βj(m)
m,m=Ms−1
m,m=0,xit=−1
αj−1(m)pt(xti = −1)exp
−
j n
l=(j−1)n
(yli,k−xli)2 2(σi,k)2
βj(m)
(6.38) whereλi,kt denotes the LLR ratio for thepth symbol within thejth codeword transmitted at timet =(j−1)n+pandnis the code symbol length.m andmare the pair of states connected in the trellis,xti is thetth BPSK modulated symbol in a code symbol connecting the statesm andm,yi,kt is the detector output in iterationk, for antennai, at timet,(σi,k)2 is the noise variance for layeriand iterationk,Ms is the number of states in the trellis and α(m)andβ(m) are the feed-forward and the feedback recursive variables, defined as for the LLR (Appendix 5.1).
In computing the LLR value in (6.38) the decoder uses two inputs. The first input is the decision statistics,yti,k, which depends on the transmitted signalxti. The second input is the a priori probability on the transmitted signalxit, computed as
pt(xti =l)= 1
√2π σe−
(yi,k t −làit )2
2σ2 , l=1,−1 (6.39)
whereàit is the mean of the received amplitude after matched filtering, given byàit =hHi hi. Aspt(xti =l)in (6.39) depends also on xti, the inputs to the decoder in iterationk, where k >1, are correlated. This causes the decision statistics mean value, conditional onxti, to be biased [20][27]. The bias always has a sign opposite ofxti. That is, the bias reduces the useful signal term and degrades the system performance. This bias is particularly significant for a large number of interferers.
The bias effect can be eliminated by estimating the mean of the transmitted symbols based on the a posteriori extrinsic information ratio instead of the LLR [16][20]. The extrinsic information represents the information on the coded bit of interest calculated from the a priori information on the other coded bits and the code constraints. The EIR does not include the metric for the symbolxti that is being estimated. That is
λi,kt,e=log
m,m=Ms−1
m,m=0,xit=1
αj−1(m)pt(xti =1)exp
−
j n
l=(j−1)n,l=t
(yli,k−xli)2 2(σi,k)2
βj(m)
m,m=Ms−1
m,m=0,xti=−1
αj−1(m)pt(xti = −1)exp
−
j n
l=(j−1)n,l=t
(yli,k−xli)2 2(σi,k)2
βj(m)
(6.40) whereλi,kt,edenotes the EIR for thepth symbol within thejth codeword transmitted at time t=(j−1)n+p,yti,k is the detector output in iterationk, for antennai,α(m)andβ(m) are defined as for the LLR (Appendix 5.1). However, excluding the contribution of the bit of interest reduces the extrinsic information SNR, which leads to a degraded system performance.
A decision statistics combining (DSC) method is effective in minimizing these effects.
In the iterative parallel interference canceller with decision statistics combining (PIC-DSC) [20], shown in Fig. 6.7, a DSC module is added to the PIC-DSC structure. The decision statistics of the PIC-DSC is generated as a weighted sum of the current PIC output and the DSC output from the previous operation. In each stage, except in the first one, the PIC output is passed to the DSC module. The DSC module performs recursive linear combining of the detector output in iterationkfor layer i, denoted byyi,k, with the DSC output from the previous iteration for the same layer, denoted by yci,k−1. The output of the decision statistics combiner, in iterationkand for layeri, denoted byyci,k, is given by
yci,k=pi,k1 yi,k+pi,k2 yi,kc −1 (6.41) wherepi,k1 andp2i,k are the DSC weighting coefficients in stage k, respectively. They are estimated by maximizing the signal-to-noise plus interference ratio (SINR) at the output of DSC in iterationkunder the assumption thatyi,k andyci,k−1are Gaussian random variables with the conditional means ài,k and ài,kc −1, given that xi is the transmitted symbol for antennai, and variances(σi,k)2 and(σci,k−1)2, respectively.
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Figure 6.7 Block diagram of an iterative receiver with PIC-DSC
Coefficientspi,k1 andpi,k2 can be normalized in the following way
E{yci,k} =pi,k1 ài,k+p2i,kài,kc −1=1 (6.42) The SINR at the output of the DSC for layeri and in iterationkis then given by
SINRi,k= 1
(pi,k1 )2(σi,k)2+2p1i,k
1−p1i,kài,k ài,k−1c
ρk,ki −1σi,kσci,k−1+
1−p1i,kài,k ài,k−1c
2
(σci,k−1)2 (6.43) whereρk,ki −1is the correlation coefficient for layeri, between the detector output in thekth and(k−1)th iterations defined as
ρk,ki −1=E{(yi,k−ài,kxi)(yci,k−1−ài,kc −1xi)|xi}
σi,kσci,k−1 (6.44)
The optimal combining coefficient is given by p1 opti,k =
ài,k
(ài,kc −1)2(σci,k−1)2− 1
ài,kc −1
ρk,ki −1σi,kσci,k−1
(σi,k)2−2 ài,k
ài,kc −1ρk,ki −1σi,kσci,k−1+
ài,k ài,kc −1
2
(σci,k−1)2
(6.45)
In the derivation of the optimal coefficients we assume that àk,i, àk,ic −1, (σk,i)2 and (σck,i−1)2are the true conditional means and the true variances of the detector outputs.
The parameters required for the calculation of the optimal combining coefficients in Eq. (6.45) are difficult to estimate, apart from the signal variances.
However, in a system with a large number of interferers, which happens when the num- ber of transmit antennas is large relative to the number of receive antennas, and for the APP based symbol estimates, the DSC inputs in the first few iterations are low correlated.
Thus, it is possible to combine them, in a way similar to receive diversity maximum ratio combining.
Under these conditions, the weighting coefficient in this receiver can be obtained from Eq. (6.45) by assuming that the correlation coefficient is zero and neglecting the reduction of the received signal conditional mean caused by interference. The DSC coefficients are then given by
pi,k1 = (σci,k−1)2
(σci,k−1)2+(σi,k)2 (6.46) The DSC output, in the second and higher iterations, with coefficients from (6.46) can be expressed as
yci,k= (σci,k−1)2
(σci,k−1)2+(σi,k)2yi,k+ (σi,k)2
(σi,k)2+(σci,k−1)2yci,k−1 i >1 (6.47) The complexity of both PIC-STD and PIC-DSC is linear in the number of transmit antennas.
We demonstrate the performance of an HLST scheme with separateR=1/2, 4-state con- volutional component encoders, the frame size ofL=206 symbols and BPSK modulation.
In simulations decoding is performed by a MAP algorithm. The HLSTC withnT transmit andnR receive antennas is denoted as an(nT, nR)HLSTC. The channel is modelled as a frequency flat slow Rayleigh fading channel. The results are shown in the form of the frame error rate (FER) versusEb/N0. The SNR is related toEb/N0as
SN R=ηEb/N0 (6.48)
where η = RmnT is the spectral efficiency and m is the number of bits per modulation symbol. Figure 6.8 compares the performance of the PIC-STD with EIR and LLR based symbol estimates and the PIC-DSC for a (6,2) HLSTC. The spectral efficiency of the HLSTC isη=3 bits/s/Hz. The results show that for the PIC-STD with LLR based symbol estimates the error floor is higher than for the other two schemes. With 8 iterations the error floor for the PIC-STD(LLR) appears at FER of 0.1, while for the PIC-STD (EIR) the error floor is about 0.04. However, the PIC-DSC receiver has an error floor below 0.007.
Figure 6.9 shows the performance for the HLSTC code withnT =4 transmit andnR=2 receive antennas. The spectral efficiency of the HLSTC is η = 2 bits/s/Hz. The relative performance of the three receivers, PIC-STD(LLR), PIC-STD(EIR) and PIC-DSC, is the same as in the previous figure. Note that in both (4,2) HLSTC and (6,2) HLSTC the FER in second iteration for the PIC-STD (LLR) is better than FER for the PIC-STD (EIR).
Generally, if the number of interferers is low, the receiver with LLR symbol estimates converges faster than the receiver with EIR symbol estimates. This can be explained by the fact that under low interference the bias effect is negligible and the LLR estimates have a lower variance relative to the EIR estimates resulting in a faster convergence.
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Figure 6.8 FER performance of HLSTC with nT =6,nR =2,R=1/2, BPSK, a PIC-STD and PIC-DSC detection on a slow Rayleigh fading channel
Figures 6.8 and 6.9 show the performance results for various PIC receivers when the PIC output variance in iteration i is estimated assuming that the receiver ideally recovers the transmitted symbols, as
(σmvi,k)2= 1 L
L
t=1
(yti,k−àitxit)2 (6.49) where xti is the transmitted symbol, àit = hiHhi is the nominal mean of the received amplitudes after the maximum-ratio-combining (MRC) andyti,kis PIC output. The variance of the DSC output is estimated in the same way. The variance in (6.49) is called a measured variance.
In a real system the transmitted symbols are not known at the receiver. The variance can be calculated by using the symbol estimate of the transmitted symbol from the previous
Figure 6.9 FER performance of HLSTC withnT =4,nR=2,R=1/2, BPSK, the PIC-STD and the PIC-DSC detection on a slow Rayleigh fading channel
decoder output as
(σevi,k)2= 1 L
L
t=1
(yti,k−ài,txˆti,k−1)2 (6.50) wherexˆti,k−1 is a symbol estimate in iteration k−1. The variance in (6.50) is called an estimated variance.
Figures 6.10 and 6.11 compare the performance of the PIC-DSC with a measured variance as in Eq. (6.49) (PIC-DSC mv) and estimated variance as in Eq. (6.50) (PIC-DSC ev) for a(6,2)and a (8,2)HLSTC. Clearly, until the number of interfering layers relative to the number of receive antennas becomes very high as in the example of the(8,2)HLSTC, the differences between the performance of the PIC-DSC mv and PIC-DSC ev is not large.
Figure 6.12 compares the performance of the iterative PIC-STD and iterative PIC-DSC decoder for a (4,4) HLSTC code with a rate R = 1/2, 4 state convolutional component code, BPSK modulation on a slow Rayleigh fading channel.
Figure 6.13 illustrates the performance of an HLSTC (4,4) system on a two-path Rayleigh fading channel with PIC-STD detection. As the results show, the error rate is very close to the one achieved in an interference free system. This proves that the PIC-STD receiver is also able to remove the interference coming from frequency selective fading. The overall performance is better than on a single path Rayleigh fading channel due to a diversity gain.
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Figure 6.10 Effect of variance estimation for an HLSTC withnT = 6,nR =2, R= 1/2, BPSK and a PIC-DSC receiver on a slow Rayleigh fading channel
Figure 6.11 Effect of variance estimation on an HLSTC withnT = 8,nR = 2,R = 1/2, BPSK and PIC-DSC detection on a slow Rayleigh fading channel
Figure 6.12 FER performance of HLSTC withnT =4,nR =4, R= 1/2, BPSK, PIC-STD and PIC-DSC detection on a slow Rayleigh fading channel
Figure 6.13 Performance of an HLSTC(4,4),R=1/2 with BPSK modulation on a two path slow Rayleigh fading channel with PIC-STD detection
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