# (see ) Consider the diversity receiver shown below where antipodal signals V are received via..

(see ) Consider the diversity receiver shown below where
antipodal signals V are received via two paths. Because of differing
attenuation (or fading) along the two paths, the signals are received as V1
and V2 by branches 1 and 2 of the diversity receiver, respectively. Each
branch has

a. Show that the summed output v is a Gaussian variable of
mean value (A1 V1 +A2 V2) and variance A1 2 _1 2 + A2 2 _2 2 .

b. Show that the effective SNR is given by (V1 + KV2) 2
/(_1 2 + K2 _2 2 ), where K=A2/A1. K = 0 and K = _ correspond
to the single-receiver case. Show that the diversity system

(see ) Consider the diversity receiver shown below where
antipodal signals V are received via two paths. Because of differing
attenuation (or fading) along the two paths, the signals are received as V1
and V2 by branches 1 and 2 of the diversity receiver, respectively. Each
branch has

a. Show that the summed output v is a Gaussian variable of
mean value (A1 V1 +A2 V2) and variance A1 2 _1 2 + A2 2 _2 2 .

b. Show that the effective SNR is given by (V1 + KV2) 2
/(_1 2 + K2 _2 2 ), where K=A2/A1. K = 0 and K = _ correspond
to the single-receiver case. Show that the diversity system improves the output
SNR and error probability performance compared to a singlereceiver.

c. Show that the optimum choice of the gain ratio K is given
by Kopt = V2 _2 1 V1 _2 2 , which is equivalent to setting A1 = kV1
_2 1, A2 = kV2 _2 2, where k is an arbitrary constant. The optimum
combining system is hence MRC, where each receiver input is weighted by the
ratio of the signal voltage to the noise variance measured at that input. Show
that the output SNR is given by the sum of the SNRs of two branches, V2 1
_2 1 + V2 2 _2 2.

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