Decision-Feedback Differential Detection in Fading Channels
9.4.1 Decision-Feedback Differential Detection in
Flat-Fading Channels
The coherent detection methods discussed in Section 9.3 require
explicit or implicit estimation of the fading channel, which in turn requires
the transmission of pilot or training symbols. In this section we discuss
decision-feedback differential detection in flat-fading channels, which does not
require channel estimation. Consider again the signal model (9.19). Assume that the
transmitted symbols {b[i]} are the outputs of a differential encoder:
Equation 9.49
where {d[i]} is a sequence of PSK information symbols. In simple
differential detection, the complex plane is divided into M disjoint sectors, where M is the size of the PSK signaling alphabet. The
detected information symbol 
is determined by the sector into
which the complex number (r[i]r[i – 1]*) falls. Such a simple differential
detection rule incurs a 3 dB performance loss compared with coherent detection
in AWGN channels [396]. In flat-fading channels,
however, it exhibits an irreducible error floor in the high-SNR region [222]. For example, for
binary DPSK we have
Equation 9.50
where r
is the correlation coefficient between the fading gains at two consecutive
symbol intervals.
Multiple-symbol decision-feedback differential detection was
developed in [441].
This method makes use of the correlation function of the channel and can
significantly reduce the error floor of simple differential detection. In
multiple-symbol differential detection [101, 102, 179, 304], an observation interval of
length N is introduced. Define the following
quantities:
Similar to (9.24)-(9.25), we can write the
log-likelihood function as
Equation 9.51
where
Equation 9.52
Equation 9.53
The maximum-likelihood decision metric thus becomes
Equation 9.54
Since T-1 is
symmetric, so is T. That is, if we denote
T = [ti,j], then
ti,j =
tj,i.
Hence we can write
Equation 9.55
where (9.55) follows
from (9.49). In decision-feedback
differential detection, the previous information symbols d[i - 1], d[i - 2], ..., d[i - N + 2] in (9.55)
are assumed to take values given by the previous decisions (i.e., 
, and we
will make a decision on the current information symbol d[i] to minimize the
cost function r(d[i]) above. To
that end, such a decision rule can be simplified to [441]
Equation 9.56
Based on (9.56), we
arrive at the following decision rule: Divide the complex plane into M disjoint sectors and determine 
by the
sector into which the complex number
Equation 9.57
falls. The multiple-symbol decision-feedback differential
detection algorithm is summarized as follows.
Algorithm 9.2:
[Multiple-symbol decision-feedback differential detection] Given the decision memory order N, fading statistics SN, and signal-to-noise
ratio Es/s2:
-
Compute the feedback filter
coefficients from 
.
-
Estimate the initial information
symbols 
by simple differential
detection.
-
For i = N, N + 1, ..., estimate
according to (9.55).
The corresponding multiple-symbol decision-feedback
differential receiver structure is shown in Fig. 9.1, where the coefficients of the feedback filter
are given by the metric coefficients tj = t0,j, 1 
j N - 1. Note that when N
= 2, this receiver reduces to the simple differential detector.
Simulation Examples
For all simulation results presented below, a differential QPSK
constellation is used. The feedback metric coefficients are obtained from the
sample autocorrelation of the simulated fading process. Figures 9.2, 9.3,
and 9.4 show the BER performance of the
decision-feedback differential detector in flat-fading channels with normalized
Doppler frequencies BdT equal to 0.003,
0.0075, and 0.01, respectively. It is seen that the error floors of the simple
differential detector (N = 2) are reduced by the
decision-feedback differential detector (DF-DD) with N = 3 and N = 4.
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