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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 1: Solutions DecisionMaking 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 nonoverlapping 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) zeroenergy signals. Next, if x ( t ) is realvalued, 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 realvalued 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 realvalued even function of τ . Next, note that F{ R x ( τ ) } = S x ( f ) =  X ( f )  2 is a realvalued 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 realvalued 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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