54)
In order to provide some insight, let us fix the dimension of the problem such that there are two
receive antennas,M2, and there are three taps in the linear equalizer, K3. With this we can write
the covariance block matrix as
(6.55)
where Rd(k) represents the autocorrelation of the pulse-shaped desired signal. And the crosscovariance
block matrix as
(6.56)
Rd (T ) # hx(k) # hy(k T )
RdQT
2 R# hxQk
T
2 R# hy(k T )
s2s
hx(k T ) # hy(k T ) V RdQT
2 R# hx(k) # hyQk
T
2 R
s2s
hxQt
T
2 R# hyQk
T
2 R
RdQT
2 R# hx(k T) # hyQk
T
2 R Rxy F ss
2hx(k) # hy(k)
RdQT
2 R# hxQk
T
2 R# hy(k)
Rd (T ) # hx(k T ) # hy(k)
Rxx
s2s Zhx(k)Z2 FRdQT
2 R# hx(k) # hxQk
T
2 R
Rd (T) # hx(k) # hx(k T)
RdQT
2 R# hx(k) # hxQk
T
2 R
s2s PhxQk
T
2 RP2
RdQT
2 R# hxQk
T
2 R# hx(k T )
Rd (T ) # hx(k) # hx(k T )
RdQT
2 R# hxQk
T
2 R# hx(k T )
s2s
Zhx(k T )Z2 V s2n
I
SNR
E5Zs(k) # hx(k) Z26 E5Znx(k) Z26
s2s
s2n
# E5Zhx(k) Z26
s2s
s2n
# E5Zhy(k) Z26
y(k) s(k) # hy(k) ny(k)
x(k) s(k) # hx(k) nx(k)
306 CHAPTER SIX
In Fig. 6.12, we plot the symbol and fractionally spaced STE (K 5 taps) performance versus the
rank of the covariance matrix or the signal subspace.
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