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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 xaxis 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 d’Alembert’s (famous) solution to the wave equation. 2. (20 points) Cross Correlation The crosscorrelation (sometimes just called correlation) of two realvalued 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 crosscorrelation. 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) Crosscorrelation 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) Crosscorrelation and delays Show that f ? ( τ b g ) = τ b ( f ? g ) , where ( τ b f )( t ) = f ( t b ). Why does this make sense, intuitively? What about ( τ b f ) ?g ? 3. (20 points) The autocorrelation of a realvalued signal f ( t ) with itself is defined to be ( f ? f )( x ) = Z ∞∞ f ( y ) f ( x + y ) dy ....
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This note was uploaded on 01/10/2012 for the course EE 216 taught by Professor Harris,j during the Fall '09 term at Stanford.
 Fall '09
 Harris,J

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