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assignment1v2

# assignment1v2 - COSC 4214 Digital Communications Assignment...

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COSC 4214: Digital Communications Assignment # 1: Signals and Spectra Due Date: Quiz will be held on September 28, 2006 1. Determine the energy spectral density G x ( f ) of a square pulse = elsewhere. 0 1 ) ( 2 2 T T t t x Also, calculate the normalized energy E x in the pulse. 2. Determine which of the following functions satisfy the properties of autocorrelation functions. a. τ = τ elsewhere. 0 1 1 1 ) ( x b. ) 2 sin( ) ( ) ( 0 τ π + τ δ = τ f x . c. |). exp(| ) ( τ = τ x d. τ τ = τ elsewhere. 0 1 1 1 ) ( x 3. Determine which of the following functions satisfy the properties of power spectral density functions. a. ). 2 ( cos ) ( ) ( 2 f f f X π + δ = b. ). 10 ( 10 ) ( δ + = f f X c. ( ) 10 2 exp ) ( π = f f X . d. ( ) ( ) 10 2 exp ) ( 2 π = f f X . 4. Consider a random process given by ( ) φ + π = t f A t x 0 2 cos ) ( , where A and are constants and φ is a random variable that is uniformly distributed over (0, 2 π ). If x ( t ) is an ergodic process, the time averages of x ( t ) in the limit as t

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assignment1v2 - COSC 4214 Digital Communications Assignment...

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