8-fourier

# 8-fourier - Chapter 8 Fourier Analysis We all use Fourier...

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Chapter 8 Fourier Analysis We all use Fourier analysis every day without even knowing it. Cell phones, disc drives, DVDs, and JPEGs all involve fast ﬁnite Fourier transforms. This chapter discusses both the computation and the interpretation of FFTs. The acronym FFT is ambiguous. The ﬁrst F stands for both “fast” and “ﬁnite.” A more accurate abbreviation would be FFFT, but nobody wants to use that. In Matlab the expression fft(x) computes the ﬁnite Fourier transform of any vector x . The computation is fast if the integer n = length(x) is the product of powers of small primes. We discuss this algorithm in section 8.6. 8.1 Touch-Tone Dialing Touch-tone telephone dialing is an example of everyday use of Fourier analysis. The basis for touch-tone dialing is the Dual Tone Multi-Frequency (DTMF) system. The program touchtone demonstrates how DTMF tones are generated and decoded. The telephone dialing pad acts as a 4-by-3 matrix (Figure 8.1). Associated with each row and column is a frequency. These basic frequencies are fr = [697 770 852 941]; fc = [1209 1336 1477]; If s is a character that labels one of the buttons on the keypad, the corre- sponding row index k and column index j can be found with switch s case ’*’, k = 4; j = 1; case ’0’, k = 4; j = 2; case ’#’, k = 4; j = 3; otherwise, d = s-’0’; j = mod(d-1,3)+1; k = (d-j)/3+1; end February 15, 2008 1

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2 Chapter 8. Fourier Analysis 1209 1336 1477 697 770 852 941 Figure 8.1. Telephone keypad. A key parameter in digital sound is the sampling rate. Fs = 32768 A vector of points in the time interval 0 t 0 . 25 at this sampling rate is t = 0:1/Fs:0.25 The tone generated by the button in position (k,j) is obtained by superimposing the two fundamental tones with frequencies fr(k) and fc(j) . y1 = sin(2*pi*fr(k)*t); y2 = sin(2*pi*fc(j)*t); y = (y1 + y2)/2; If your computer is equipped with a sound card, the Matlab statement sound(y,Fs) plays the tone. Figure 8.2 is the display produced by touchtone for the ’1’ button. The top subplot depicts the two underlying frequencies and the bottom subplot shows a portion of the signal obtained by averaging the sine waves with those frequencies. The data ﬁle touchtone.mat contains a recording of a telephone being dialed. Is it possible to determine the phone number by listening to the signal generated? The statement load touchtone loads both a signal y and a sample rate Fs in the workspace. In order to reduce ﬁle size, the vector y has been saved with 8-bit integer components in the range - 127 y k 127. The statement
8.1. Touch-Tone Dialing 3 400 600 800 1000 1200 1400 1600 0 0.5 1 f(Hz) 1 0 0.005 0.01 0.015 -1 -0.5 0 0.5 1 t(seconds) Figure 8.2. The tone generated by the 1 button.

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8-fourier - Chapter 8 Fourier Analysis We all use Fourier...

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