MR combining Consider the system shown in Figure 7.15, where the signal s is coherently sensed by two receivers. The measurements y1 and y2 can be expressed as follows:

y1 y2

 

¼

h1 h2

 

s þ

1 1

 

;

where the channel gains are given by h1 ¼ aej1 ; and h2 ¼ a2ej2 . Suppose the noise 1 and 2 are zero-mean white noise with variance
2

.

(a) We are to use a linear receiver y¼w1y1þw2y2 to maximize the SNR at the receiver. Assume the signal s can be modeled as a white stationary process with zero mean and variance
2s

. Prove that theSNRat the receiver can be expressed as SNR ¼

2s jj w1h1 þ w2h2 jj 2

2v

ðw21þ w22

Þ

:

(b) Find the coeffecients w1 and w2 that maximize the SNR. (Hint: use the Cauchy–Schwartz inequality, which states that the inner product of two vectors is less than or equal to the product of the two vectors’ squared norms.)