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Exercises on Systems and Density

Module by: Behnaam Aazhang. E-mail the author

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Summary: Exercises

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Exercise 1

Consider the following system

Figure 1
Figure 1 (HW4Fig.png)

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=1if0≤τ≤T0otherwise 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 lnΛz≥γ 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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