Digital Signal Processing Reference
In-Depth Information
The hard-decision majority vote (HDMV) method described above works
well in high-SNR environments,
54
,
55
where samples that originated from sig-
nal plus noise and samples that stem from noise only are easily distinguished.
A notable loss in efficiency is incurred, however, in low-SNR environments.
Decision rules based on rank ordering are motivated by the observation
53
that the cumulative distributions satisfy
F
S
+
N
(
x
)
≤
F
N
(
x
)
(3.25)
for all
x
. That is, the samples in the correct row are
stochastically larger
than the
samples in the remaining rows. The maximum rank sum receiver (MRSR)
56
exploits this property by rank ordering the samples in the decoded matrix
x
and forming a rank matrix
r
, whose elements are the crisp ranks
of the corresponding entries in
x
, i.e.,
r
k,
={
r
k,
}
. Thus, accord-
ing to Equation 3.25, hits are typically assigned high ranks, whereas noise
samples are assigned low ranks: The decision is made by summing the ranks
across each row, and choosing the row with the largest sum as the correct
row:
is the rank of
x
k,
L
H
k
: max
k
r
k,
.
(3.26)
=
1
The main shortcoming of this approach is that the real-valued energy sam-
ples in
x
are replaced by integer ranks that reflect only the relative values of
the received energies but not their spread. As an example, consider the integer
rank matrix pertaining to
x
, Figure 3.30c, where bright tones correspond to
high ranks and dark tones correspond to low ranks. Comparing Figure 3.30a
and Figure 3.30c, it can be observed that, although the two strongest hits are
clearly recognizable in both matrices, in the integer rank matrix the weaker
hits are difficult to distinguish from the nonhits.
The more general fuzzy ordering-based FRO detectors
50-52
are capable of
making a reliable distinction between hits and nonhits due to the clustering
property of fuzzy ranks.
38
,
57
Extending the MRSR detector to the FRO detector
is a straightforward fuzzy extension,
L
H
k
: max
k
r
k,
,
(3.27)
=
1
for
k
,Q
. Figure 3.30d shows the effect of fuzzy ranking when
applied to the decoded matrix. The consideration of sample spread results in
a clear identification of hits and nonhits indicated by bright and dark gray
tones, respectively.
To further illustrate the performance of the various detectors, consider the
simplified Rayleigh fading channel.
56
In this model the entries in the decoded
matrix obey exponential distributions whose mean value reflects the presence
or absence of a signal. More specifically, the samples in a correct row are
=
1
,
2
,
...
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