MIMO equalization A discrete time two by two MIMO system is shown in Figure 7.14, where s1(n), s2(n) are source sequences, and y1 and y2 are measurement sequences. The channel transfer functions are:

h11ðzÞ ¼ 1 þ az1; h22ðzÞ ¼ 1 þ bz1; h12ðzÞ ¼ cz1; h21ðzÞ ¼ dz1:

Besides the interference from the other source, the measurements are corrupted by the white Gaussian noise sequences

(n) with zero mean and variance 2v

.

(a) We define the vectors y ¼

y1ðnÞ

y2ðnÞ

y1ðn 1Þ

y2ðn 1Þ

2 664 3

775; s

¼

s1ðnÞ

s2ðnÞ

s1ðn 1Þ

s2ðn 1Þ

s1ðn 2Þ

s2ðn 2Þ

2 6666664 3

7777775

;

¼

1ðnÞ

2ðnÞ

1ðn 1Þ

2ðn 1Þ

2 664 3

775

:

Verify that y can be expressed in terms of s and as y ¼ Cs þ

:

(b) Assume the random sources s1 and s2 are stationary and white. Their mean and covariance are given by

E½sðnÞ ¼
0 0
; Rs ¼ E½sðnÞsHðnÞ ¼
2s 1 0 0 2s 2
:
Find the covariance matrix of y defined as Ry¼E[yyH].

(c) We estimate the source from measurements y based on the following scheme:
s ^
ðnÞ ¼
s ^
1ðnÞ
s ^
2ðnÞ
" #
¼ Wy in whichWis a 24 matrix. DetermineWin terms of C, Rs, and 2 such that the mean square error of estimation is minimized.

Transcribed Image Text:

S h Y2 V2 Figure 7.15 MR combining.