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hw6 - EE 261 The Fourier Transform and its Applications...

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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 πf ( 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. We’ll take a closer look at how this done later in this problem. Let’s start by taking the case where we modulate a pure tone, x ( t ) = A cos(2 πf c t + k sin 2 πf m t ) . Here’s a plot for 0 t 2 with A = 1, f c = 5, k = 2, and f m = 20. 1
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What is the spectrum? Remember Bessel functions, introduced in an earlier problem? The answer depends on these. Let’s 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 π integraldisplay 2 π 0 cos( x sin θ - ) dθ . You showed in Problem Set 1 that e ix sin θ = summationdisplay n = −∞ J n ( x ) e inθ and e ix cos θ = summationdisplay n = −∞ i n J n ( x ) e inθ . (a) Show the Fourier series relationship exp(2 πif c t + ik sin(2 πf m t )) = summationdisplay n = −∞ J n ( k ) exp(2 πi ( 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 )[ δ ( s - ( f c + nf m )) + δ ( s + ( f c + nf m )] . Hint: What is the real part of the Fourier series relationship?
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