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Homework 7 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

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Exercise 1

Consider an On-Off Keying system where s 1 t=Acos2π f c t+θ s 1 t A 2 f c t θ for 0≤t≤T 0 t T and s 2 t=0 s 2 t 0 for 0≤t≤T 0 t T . The channel is ideal AWGN with zero mean and spectral height N 0 2 N 0 2 .

  1. Assume θθ is known at the receiver. What is the average probability of bit-error using an optimum receiver?
  2. Assume that we estimate the receiver phase to be θ ^ θ ^ and that θ ^ ≠θ θ ^ θ . Analyze the performance of the matched filter with the wrong phase, that is, examine P e ¯ P e ¯ as a function of the phase error.
  3. When does noncoherent become preferable? (You can find an expression for the P e ¯ P e ¯ of noncoherent receivers for OOK in your textbook.) That is, how big should the phase error be before you would switch to noncoherent?

Exercise 2

Proakis and Salehi, Problems 9.4 and 9.14

Exercise 3

A coherent phase-shift keyed system operating over an AWGN channel with two sided power spectral density N 0 2 N 0 2 uses s 0 t=A p T tcos ω c t+ θ 0 s 0 t A p T t ω c t θ 0 and s 1 t=A p T tcos ω c t+ θ 1 s 1 t A p T t ω c t θ 1 where ∀i,i∈01:| θ i |≤π3 i i 0 1 θ i 3 , are constants and that f c T=integer f c T integer with ω c =2π f c ω c 2 f c .

  1. Suppose θ 0 θ 0 and θ 1 θ 1 are known constants and that the optimum receiver uses filters matched to s 0 t s 0 t and s 1 t s 1 t . What are the values of P e 0 P e 0 and P e 1 P e 1 ?
  2. Suppose θ 0 θ 0 and θ 1 θ 1 are unknown constants and that the receiver filters are matched to s ^ 0 t=A p T tcos ω c t s ^ 0 t A p T t ω c t and s ^ 1 t=A p T tcos ω c t+π s ^ 1 t A p T t ω c t and the threshold is zero.

    Hint:

    Use a correlation receiver structure.
    What are P e 0 P e 0 and P e 1 P e 1 now? What are the minimum values of P e 0 P e 0 and P e 1 P e 1 (as a function of θ 0 θ 0 and θ 1 θ 1 )?

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