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PS-6-2011-Solutions

PS-6-2011-Solutions - EE 261 The Fourier Transform and its...

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EE 261 The Fourier Transform and its Applications Fall 2011 Solutions to Problem Set Six 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? (b) Download the file fm.mat from the course site. It contains g , a guitar playing the G above middle C - approximately 392 Hz; t , the time vector; and fs , the sampling frequency. Listen to the note and plot its spectrogram: 2
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load fm.mat; soundsc(g,fs); spectrogram(g, 256, 64, 256, fs, ’yaxis’); Approximately how many frequencies make up this note? Let’s say we modeled this note by the function g ( t ) = N summationdisplay n =1 A n ( t ) sin(2 πf n t ) where the amplitudes A n ( t ) are of the form A n e t/t n so that the note dies down after some time. About how many parameters ( f n , A n , t n ) would we need? Early pioneers in computer music synthesized notes in exactly this way, a tech- nique now known as additive synthesis . The large number of parameters needed to characterize each note of each instrument led musicians to look for other ways to produce musical tones. The rest of this problem will explore an approach de- vised by John Chowning in the 1970s that uses frequency modulation synthesis to dynamically change the signal’s frequency content over time. Chowning’s model was easy to implement as he was able borrow many ideas from FM circuits built by communications engineers. The first digital implementation of Chowning’s re-
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