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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Six Due Friday, November 11 1. (35 points) Frequency Modulation and Music A frequency modulated (FM) signal is one whose frequency is a function of time: x ( t ) = A cos(2 f ( t )) . FM signals are central to many scientific fields. Most notably, they are used in com- munications where f ( t ) = f c t + k integraldisplay t m ( t ) dt. Here f c is the carrier frequency (typically a large value is necessary for the physics of wave propagation), k is a constant known as the frequency modulation index , and m ( t ) is the function with the information and doing the modulating. You set your receiver to f c to listen to the signal. Another famous (and profitable) application of FM signals is in the digital synthesis of music, pioneered by John Chowning at Stanford. Well take a closer look at how this done later in this problem. Lets start by taking the case where we modulate a pure tone, x ( t ) = A cos(2 f c t + k sin 2 f m t ) . Heres a plot for 0 t 2 with A = 1, f c = 5, k = 2, and f m = 20. 1 What is the spectrum? Remember Bessel functions, introduced in an earlier problem? The answer depends on these. Lets recall: The Bessel equation of order n is x 2 y + xy + ( x 2- n 2 ) y = 0 . A solution of the equation is the Bessel function of the first kind of order n , given by the integral J n ( x ) = 1 2 integraldisplay 2 cos( x sin - n ) d . You showed in Problem Set 1 that e ix sin = summationdisplay n = J n ( x ) e in and e ix cos = summationdisplay n = i n J n ( x ) e in . (a) Show the Fourier series relationship exp(2 if c t + ik sin(2 f m t )) = summationdisplay n = J n ( k ) exp(2 i ( f c + nf m ) t ) . Use this result to show that the Fourier transform of x ( t ) is F x ( s ) = A 2 summationdisplay n = J n ( k )[ ( s- ( f c + nf m )) + ( s + ( f c + nf m )] ....
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This note was uploaded on 12/04/2011 for the course EE 263 at Stanford.