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PS01 - University of Illinois Spring 2011 ECE 361 Problem...

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University of Illinois Spring 2011 ECE 361: Problem Set 1 Decision-Making and Matched Filtering Due: Thursday January 27 at 9:30 a.m. Reading: Lecture Notes: Lectures 1-4 1. [Crosscorrelations and Autocorrelations] The crosscorrelation function for complex-valued signals x ( t ) and y ( t ) is defined as R x,y ( τ ) = Z -∞ x ( t + τ ) y * ( t ) dt. (a) Show that the crosscorrelation function for the delayed signals ˆ x ( t ) = x ( t - λ ) and ˆ y ( t ) = y ( t - λ ) is also R x,y ( τ ) regardless of the value of the delay λ . (b) Show that R x,y ( τ ) X ( f ) Y * ( f ). (c) x ( t ) and y ( t ) are said to be uncorrelated if R x,y ( τ ) = 0 for all τ . What does this imply about the supports of X ( f ) and Y ( f )? Note: The support of X ( f ) is the set { f : X ( f ) 6 = 0 } . (d) The autocorrelation function of a signal x ( t ) is R x,x ( τ ) which is often written as R x ( τ ). Show that R x (0) 0, and that if x ( t ) is a real-valued function, then R x ( τ ) and its Fourier transform S x ( f ) both are real-valued even functions of τ and f respectively. S x ( f ) is called the (two-sided) power density spectrum of x ( t ) or more commonly (and inaccurately) the (two-sided) power spectral density of x ( t ). (e) Show that if x ( t ) is a real-valued signal, then - R x (0) R x ( τ ) R x (0) for all τ . (f) Suppose that x ( t ) is the input and y ( t ) the output of a linear time-invariant system with impulse response h ( t ) and transfer function H ( f ). Show that S y ( f ) = S x ( f ) | H ( f ) | 2 and thus R y ( t ) can be thought of as the result of passing the signal R x ( t ) through a linear system whose impulse response is R h ( t ).
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