The MAP symbol detector computes symbol estimates using the MAP rule
ˆ
x[i] = arg max
x∈{+1,−1}
P(x[i] =x|y), i= 0,1, . . . , N −1, (2.84)
where, using Bayes’ rule, the a posteriori probabilities can be computed from
P(x[i] =x|y) = X
x:x[i]=x
p(y|x)P(x), x∈ {+1,−1}. (2.85)
Here p(y|x) is the likelihood function and P(x) is the a priori probability. Note that the marginal probability, p(y), does not have to be included in this form of the equation. Hence, MAP detection can be thought of as a process that takes a series of observations, y, and bit-wise a priori probabilities, {P(x[i])}i, and computes bit-wise a posteriori
probabilities,{P(x[i]|y)}i, as shown in the block diagram model in Figure 2.10.
MAP Detector a posteriori probabilities a priori probabilities observations y P( | )x y P( )x Likelihood Calculation Posterior Calculation P( | )y x
Figure 2.10: The MAP detection process in block diagram form, which takesa priori prob- abilities and observations as input and produces a posteriori probabilities as output
In the BCJR equalization algorithm of Section 2.3.1, the a posteriori probabilities are formed from the transition probabilities, γi(Sr, Ss), computed from (2.56), i.e.,
γi(Sr, Ss) =P(x[i] =xr,s)p(y[i]|v[i] =vr,s) (2.86)
where:
• p(y[i] | v[i] = vr,s) is the likelihood function, and can be interpreted as “local”
evidence about which branch in the trellis was traversed; and
• P(x[i] =xr,s) is the a priori information, which accounts for any prior knowledge
about the probability of trellis branch being traversed.
In the separate equalization and decoding strategy of Section 2.3, the equalizer does not have any a priori information available, the symbols are assumed to be i.d.d (P(x[i] = +1) =P(x[i]) =−1) = 1/2), and the transition probabilities,γi(Sr, Ss), are computed
solely from the observed data y[i]. However, the performance of the BCJR algorithm can be greatly improved if good a priori information is available. In turbo equalization, the a posteriori probabilities from the MAP FEC decoder are fed back and used as a priori information by the MAP equalizer. This is performed in an iterative process where the symbol and data bit estimates become more accurate as the quality of the a priori
information improves over a number of iterations.
MAP Equalizer channel output y[i] MAP FEC Decoder extrinsic information extrinsic information a priori probabilities (intrinsic information) a priori probabilities (intrinsic information) (code bit) a posteriori probabilities a posteriori probabilities Deinterleaver Interleaver -1 L 1(b[i]|y) l1(b[i]|y) l 2(b[i]|p) l 1(c[i]|y) l 2(c[i]|p) L 2(d[i]|p) L 2(c[i]|p) hard decision d[i] ^ data bit estimates (data bit) a posteriori probabilities
Figure 2.11: Block diagram of a turbo equalization receiver.
When designing the feedback loop structure, it is important to consider the effect that soft information generated from one bit in one of the constituent algorithms (equalizer or decoder) will have on the other bits in the other constituent algorithm.
When processing soft information input to the equalizer or the decoder, it is assumed that the soft information about each bit (or channel symbol) is independent. This assumption reduces the complexity of the equalizer and decoder algorithms. However, if the decoder formulates its soft information about a given bit, based on soft information provided to it from the equalizer about exactly the same bit, then the equalizer cannot consider this information to be independent of its channel observations. In effect, this would create a feedback loop in the overall process of length two: the equalizer informs the decoder about a given bit; and then the decoder simply reiterates to the equalizer what it already knows.
To avoid short cycles in the feedback, local minima, and limit cycle behavior in the iterative process, when soft information is passed between constituent algorithms, such information is never formed based on the information passed into the algorithm concerning the same bit. Essentially, this amounts to the equalizer only telling the decoder new information about a given bit based on information it gathered from distant parts of the received signal. Similarly, the decoder only tells the equalizer information it gathered from distant parts of the encoded bit stream. This consideration leads to the concept of extrinsic and intrinsic information [53].
For the optimal receiver in Section 2.2.2, it was shown (from (2.47)-(2.49)) that the
a posteriori LLR, Λ(d[i]|y), can be separated into extrinsic LLR, λ(d[i]|y), and the intrinsic (a priori) LLR, λ(d[i]). Also, that λ(d[i]|y) does not depend on λ(d[i]). In the case of the (BCJR) MAP equalization algorithm, the same functional relation can be applied to the output a posteriori LLRs in order to separate the two contributions. That is, the a posteriori LLRs, Λ(b[i]|y), can be split into:
• extrinsic information,λ(b[i]|y) = Λ(b[i]|y)−λ(b[i]); and
• intrinsic information, λ(b[i]).
It is essential to the performance of turbo decoding algorithms that only extrinsic information is passed between the constituent decoders.
The block diagram of a turbo equalization receiver is shown in Figure 2.11. The two MAP algorithms form the core of the turbo equalizer. The MAP equalizer operates on channel observations and a priori information about individual bits, while the MAP FEC decoder operates on a priori information only. (In Figure 2.11, the observation input of the MAP decoder is grounded to indicate that it is not used). Only the extrinsic information is fed back in the iterative loop.
1. Turbo equalizer inputs:
a) observation sequence, y= [y[0], y[1], . . . , y[N −1] ]T
b) channel coefficients, h0, h1, . . . , hL
2. Initialization: initialize the MAP equalizer a priori information to all zeros, i.e., λ2(b|p) = [0N×1]
3. Recursively compute (for a predetermined number of iterations):
Λ1(b |y) = MAP Equalizer(λ2(b|p) ) λ1(b |y) = Λ1(b|y)−λ2(b|p)
λ1(c|y) = Deinterleaver(λ1(b|y))
Λ2(c|p) = MAP FEC Decoder(λ1(c|y) ) λ2(c|p) = Λ2(c|p)−λ1(c|y)
λ2(b|p) = Interleaver(λ2(c|p) )
4. Turbo equalizer output: compute the data bit estimates, ˆd[i], from the probabilities, Λ2(d[i]|y), using:
ˆ
d[i] = sgn (Λ2(d[i]|y)), i= 0,1, . . . , M −1
Table 2.3: Turbo equalization algorithm
The interleaver and deinterleaver are incorporated into the iterative loop to further disperse the direct feedback effect. In particular, the BCJR algorithm creates output that is locally highly-correlated, but the use of an interleaver can largely suppress the correlations between neighboring symbols.
The operation of the turbo equalization receiver is shown in Table 2.3. The notation:
Λ1(b|y) = MAP Equalizer(λ2(b|p))
represents the generation of APP LLRs, Λ1(b|y), by the MAP Equalizer from observa- tions, y, and a priori LLRs, λ2(b|p), using the BCJR algorithm described in Table 2.1.
Similarly, the notation:
Λ2(c|p) = MAP FEC Decoder(λ1(c|y))
represents the generation of APP LLRs, Λ1(b |y), by the MAP FEC Decoder from the
a priori LLRs, λ2(c|y), using the BCJR algorithm described in Table 2.2.
While the turbo equalization algorithm presented is based on two MAP algorithms, any pair of equalization and FEC decoding algorithms that make use of soft information can be used as constituent algorithms in the turbo equalizer.
For example, the linear MMSE equalizer in Section 2.3.2 can usea priori information about the transmitted symbol x[i] to compute symbol statistics E{x[i]} and Var{x[i]}
(using (2.71)-(2.72)) which are then incorporated into the MMSE filter, (2.69)-(2.70), to compute symbol estimate, ˆx[i], and APP LLR, Λ1(x[i] | y). As with the MAP equalization algorithm, the APP LLR is computed the constraint that Λ1(x[i]|y) is not a function of thea priori LLR,λ2(b[i]|p), at the same indexi. This helps to avoid short feedback cycles, and is equivalent to extracting only the extrinsic part of the information in the iterative scheme [53]. Note also that there are several low-complexity alternatives for re-estimating ˆx[i], e.g. [53], [121], [122], [33], [120], [98], [139].
Figure 2.12 shows the performance of the turbo equalization scheme (of Figure 2.11 and Table 2.3) for the ISI channel model of Figure 2.5. The scheme is evaluated for an input data block length (M) of 512 bits with forward error correction performed by the rate-1/2 convolutional encoder of Figure 2.1b, resulting in a coded block length (N) of 1024 bits. The coded bits are scrambled using a random interleaver and mapped onto BPSK symbols.
Figure 2.12a shows the effect of receiver iterations for a turbo equalizer using the MAP symbol detector of Section 2.3.1, while Figure 2.12b shows the effect of receiver iterations for a turbo equalizer using the MMSE linear equalizer of Section 2.3.2. In both cases, FEC decoding is performed using the BCJR algorithm of Section 2.3.3. Note that zero-iterations represents the first pass when there is noa-priori information available for APP equalizer–this is equivalent to the separate equalization and decoding scheme (with soft information) evaluated in Section 2.3.4. The ISI-free bound represents the lower BER performance bound of the underlying rate-1/2 code used over an ISI-free channel, i.e., the performance bound for the evaluated system.
Turbo Equalization using MAP Symbol Detection
SNR (dB)
-2 0 2 4 6
Bit Error Rate
10-5 10-3 10-2 10-1 100 10-4 ISI-Free Bound 0 Iterations 1 Iteration 2 Iterations 10 Iterations
(a)Performance of turbo equalization using MAP symbol detection
SNR (dB)
-2 0 2 4 6
Bit Error Rate
10-5 10-3 10-2 10-1 100 10-4
Turbo Equalization using Linear MSSE Equalizer
ISI-Free Bound 0 Iterations 1 Iteration 2 Iterations 10 Iterations
(b) Performance of turbo equalization using linear MMSE equalization
Figure 2.12: Performance of turbo equalization after 0, 1, 2, and 10 iterations using: (a) MAP symbol detection; and (b) linear MMSE equalization.
Both schemes show significant BER performance gain over the iterations, with performance approaching the ISI-free bound after 10 iterations. It is observed that turbo equalization using MAP symbol detection provides superior performance compared to the MMSE linear equalizer based scheme, but at the cost of additional computational complexity. However, it is noted that for larger block lengths, M, the performance of linear MMSE equalizer approaches that of the MAP detector [53], [121].