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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 x, y ( ) = Z  x ( t + ) y * ( t ) dt = Z  x ( t + ) y * ( t ) dt = Z  x ( + ) y * ( ) d = R x,y ( ) on substituting = t . (b) Writing x ( t + ) = Z  X ( f )exp( j 2 ( t + ) f ) df in the definition of R x,y ( ), we get R x,y ( ) = Z  x ( t + ) y * ( t ) dt = Z  Z  X ( f )exp( j 2 ( t + ) f ) df y * ( t ) dt = Z  X ( f )exp( j 2 f ) Z  y ( t )exp( j 2 ft ) dt * df = Z  [ X ( f ) Y * ( f )]exp( j 2 f ) df that is, R x,y ( ) X ( f ) Y * ( f ). (c) Since R x,y ( ) = 0 for all , we have that F{ R x,y ( ) } = 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 ( ) = Z  x ( t + ) x * ( t ) dt = Z  x ( t + ) x ( t ) dt is a realvalued function. Fur thermore, R x ( ) = Z  x ( t ) x ( t ) dt = Z  x ( ) x ( + ) d = R x ( ) (upon making a change of variable = t ), and thus R x ( ) is a realvalued even function of . Next, note that F{ R x ( ) } = 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 + )] 2 dt 0. On expanding out the integrand and using the properties developed above, we get 2[ R x (0) R x ( )] 0, that is, R x (0) R x ( ) R x (0)....
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 Spring '09

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