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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Four Due Wednesday, October 26 1. (10 points) Solving the wave equation An infinite string is stretched along the x-axis and is given an initial displacement described by a function f ( x ). It is then free to vibrate. The displacement u ( x,t ) at a time t > 0 and at a point x on the string is described by the wave equation 2 u t 2 = 2 u x 2 . One often includes physical constants in the equation, e.g, the speed of the wave, but these are suppressed to keep things simple. Assume that u ( x, 0) = f ( x ) and u t ( x, 0) = 0 (zero initial velocity) and use the Fourier transform to show that u ( x,t ) = 1 2 ( f ( x + t ) + f ( x- t )) . This is dAlemberts (famous) solution to the wave equation. 2. (20 points) Cross Correlation The cross-correlation (sometimes just called correlation) of two real-valued signals f ( t ) and g ( t ) is defined by ( f ? g )( x ) = Z - f ( y ) g ( x + y ) dy . ( f ? g )( x ) is often described as a measure of how well the values of g , when shifted by x , correlate with the values of f . It depends on x ; some shifts of g may correlate better with f than other shifts. To get a sense of this, think about when ( f ? g )( x ) is positive (and large) or negative (and large) or zero (or near zero). If, for a given x , the values f ( y ) and g ( x + y ) are tracking each other both positive or both negative then the integral will be positive and so the value ( f ? g )( x ) will be positive. The closer the match between f ( x ) and g ( x + y ) (as y varies) the larger the integral and the larger the cross-correlation. In the other direction, if, for example, f ( y ) and g ( x + y ) maintain opposite signs as y varies (so are negatively correlated) then the integral will be negative and ( f ? g )( x ) < 0 The more the negatively they are correlated the more negative ( f ? g )( x ). Finally, it might be that the values of f ( y ) and g ( x + y ) jump around as y varies; sometimes positive and sometimes negative, and it may then be that in taking the integral the values 1 cancel out, making ( f * g )( x ) near zero. One might say one does say that f and g are uncorrelated if ( f ? g )( x ) = 0 for all x . (a) Cross-correlation is similar to convolution, with some important differences. Show that f ? g = f- * g = ( f * g- )- . Is it true that f ? g = g ? f ? (b) Cross-correlation and delays Show that f ? ( b g ) = b ( f ? g ) , where ( b f )( t ) = f ( t- b ). Why does this make sense, intuitively? What about ( b f ) ?g ? 3. (20 points) The autocorrelation of a real-valued signal f ( t ) with itself is defined to be ( f ? f )( x ) = Z - f ( y ) f ( x + y ) dy ....
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