Numerical Computing with MatLab, Fourier Analysis.pdf

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 putation 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. putation is fast if the integer n = length(x) is the product
of powers of small primes. We discuss this algorithm in section .
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 ). 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
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697
770
852
941
1209 1336 1477
Figure . Telephone keypad.
A key parameter in digital sound is the sampling rate.
Fs = 32768
A vector of points in the time interval 0 ≤ t ≤ at this sampling rate is
t = 0:1/Fs:
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 puter is equipped with a sound card, the Matlab statement
sound(y,Fs)
plays the tone.
Figure is the displ