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# Homework 7 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

Summary: (Blank Abstract)

## 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 0tT 0 t T and s 2 t=0 s 2 t 0 for 0tT 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,i01:| 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:

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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#### Definition of a lens

##### 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.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

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

##### What are tags?

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