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# Exercises on Systems and Density

Module by: Behnaam Aazhang. E-mail the author

Summary: Exercises

## Exercise 1

Consider the following system

Assume that N τ N τ is a white Gaussian process with zero mean and spectral height N 0 2 N 0 2 .

If b b is "0" then X τ =A p T τ X τ A p T τ and if b b is "1" then X τ =(A) p T τ X τ A p T τ where p T τ={1  if  0τT0  otherwise   p T τ 1 0 τ T 0 . Suppose Prb=1=Prb=0=1/2 b 1 b 0 12 .

1. Find the probability density function Z T Z T when bit "0" is transmitted and also when bit "1" is transmitted. Refer to these two densities as f Z T H 0 z f Z T H 0 z and f Z T H 1 z f Z T H 1 z , where H 0 H 0 denotes the hypothesis that bit "0" is transmitted and H 1 H 1 denotes the hypothesis that bit "1" is transmitted.
2. Consider the ratio of the above two densities; i.e.,
Λz=f Z T H 0 zf Z T H 1 z Λ z f Z T H 0 z f Z T H 1 z
(1)
and its natural log lnΛz Λ z . A reasonable scheme to decide which bit was actually transmitted is to compare lnΛz Λ z to a fixed threshold γγ. ( Λz Λ z is often referred to as the likelihood function and lnΛz Λ z as the log likelihood function). Given threshold γγ is used to decide b ^ =0 b ^ 0 when lnΛzγ Λ z γ then find Pr b ^ b b ^ b (note that we will say b ^ =1 b ^ 1 when lnΛz<γ Λ z γ ).
3. Find a γγ that minimizes Pr b ^ b b ^ b .

## Exercise 2

Proakis and Salehi, problems 7.7, 7.17, and 7.19

## Exercise 3

Proakis and Salehi, problem 7.20, 7.28, and 7.23

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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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Any individual member, a community, or a respected organization.

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