# HW01 - University of Illinois Spring 2011 ECE 361 Problem...

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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 1: Solutions Decision-Making and Matched Filtering 1. [Crosscorrelations and Autocorrelations] (a) R ˆ x, ˆ y ( τ ) = Z ∞-∞ ˆ x ( t + τ )ˆ y * ( t ) dt = Z ∞-∞ x ( t- λ + τ ) y * ( t- λ ) dt = Z ∞-∞ x ( μ + τ ) y * ( μ ) dμ = R x,y ( τ ) on substituting μ = t- λ . (b) Writing x ( t + τ ) = Z ∞-∞ X ( f )exp( j 2 π ( t + τ ) f ) df in the definition of R x,y ( τ ), we get R x,y ( τ ) = Z ∞-∞ x ( t + τ ) y * ( t ) dt = Z ∞-∞ Z ∞-∞ X ( f )exp( j 2 π ( t + τ ) f ) df y * ( t ) dt = Z ∞-∞ X ( f )exp( j 2 πfτ ) Z ∞-∞ y ( t )exp(- j 2 πft ) dt * df = Z ∞-∞ [ X ( f ) Y * ( f )]exp( j 2 πfτ ) df that is, R x,y ( τ ) ↔ X ( f ) Y * ( f ). (c) Since R x,y ( τ ) = 0 for all τ , we have that F{ R x,y ( τ ) } = 0 for all f . Hence it must be that X ( f ) Y * ( f ) = 0 for all f , that is, for each f , at least one of X ( f ) and Y ( f ) must be zero. In other words, the supports of X ( f ) and Y ( f ) are disjoint sets. In more engineering terms, the signals x ( t ) and y ( t ) are uncorrelated if and only if they occupy non-overlapping frequency bands . (d) R x (0) = Z ∞-∞ x ( t ) x * ( t ) dt = Z ∞-∞ | x ( t ) | 2 dt is the energy in the signal x ( t ) and is thus greater than zero except for (uninteresting) zero-energy signals. Next, if x ( t ) is real-valued, then x * ( t ) = x ( t ), and so R x ( τ ) = Z ∞-∞ x ( t + τ ) x * ( t ) dt = Z ∞-∞ x ( t + τ ) x ( t ) dt is a real-valued function. Fur- thermore, R x (- τ ) = Z ∞-∞ x ( t- τ ) x ( t ) dt = Z ∞-∞ x ( λ ) x ( λ + τ ) dλ = R x ( τ ) (upon making a change of variable λ = t- τ ), and thus R x ( τ ) is a real-valued even function of τ . Next, note that F{ R x ( τ ) } = S x ( f ) = | X ( f ) | 2 is a real-valued function of f , and since Fourier trans- forms of real functions have conjugate symmetry (that is, X (- f ) = X * ( f )), we have that S x (- f ) = | X (- f ) | 2 = | X * ( f ) | 2 = | X ( f ) | 2 = S x ( f ), and so S x ( f ) is a real-valued even func- tion of f . (e) Obviously, Z ∞-∞ [ x ( t ) ± x ( t + τ )] 2 dt ≥ 0. On expanding out the integrand and using the properties developed above, we get 2[ R x (0) ± R x ( τ )] ≥ 0, that is,- R x (0) ≤ R x ( τ ) ≤ R x (0)....
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HW01 - University of Illinois Spring 2011 ECE 361 Problem...

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