course - EE 261 The Fourier Transform and its Applications...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 10

course - EE 261 The Fourier Transform and its Applications...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online