1. Math.375 V- Interpolation 3 >Sounds and Fourier< >analysis<
Vageli Coutsias

2. A telephone number
aka “name that tune”
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3. A single dial pulse
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4. [y,Fs,bits]=wavread('tele_tones_1.wav'); x=y(744:1000); sz=length(x);
% script sounds1
[y,Fs,bits]=wavread('tele_tones_1.wav');
x=y(744:1000); sz=length(x);
td = linspace(0,sz,sz)';
plot(td,x)
sound(x,Fs)
Caution! This is not
a periodic signal. We
needed to chop it in
order to analyze it
as a periodic time
series. But chopping
introduces its own
“Gibbs phenomenon”.
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5. f is a periodic function:
In practise, a time series of values of
f is available; a periodic extension of
f is assumed so that if the f-values at
the two end-points are different, the
function behaves as if it had a jump
there. A Gibbs phenomenon results
introducing “noise”in the computed
spectrum
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6. of x, 2x, and x/pi is y = [x 2*x x/pi]
For multichannel data, each column of a matrix represents one channel. Each row of such a matrix then corresponds to a sample point. A three-channel signal y that consists
of x, 2x, and x/pi is y = [x 2*x x/pi]
Y(1)
Y(2)
Y(3)
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7. Consider generating data with a 1000 Hz sample frequency.
begin with a vector representing a time base.
Consider generating data
with a 1000 Hz sample frequency.
An appropriate time vector is
t = (0:0.001:1)';
Given t you can create a sample signal y consisting of two sinusoids, one at 50 Hz and one at 120 Hz with twice the amplitude.
y = sin(2*pi*50*t) + 2*sin(2*pi*120*t);
The new variable y, formed from vector t, is also 1001 elements long. You can add normally distributed white noise to the signal and graph the first fifty points using
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8. %a sample signal with its time vector t = (0:0.001:1)';
y = sin(2*pi*50*t) + 2*sin(2*pi*120*t);
randn('state',0);
yn = y + 0.5*randn(size(t));
plot(t(1:50),yn(1:50))
% z = [t t.^2 square(4*t)];
% a sawtooth wave of prescribed properties
fs = 10000;
t = 0:1/fs:1.5;
x = sawtooth(2*pi*50*t);
plot(t,x), axis([ ])
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9. Real form
Continuous time forms
Complex form
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10. Real form
Complex form
Discrete time forms
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11. The DFT and its inverse
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12. function y = DFT(x,isign) %isign=+/-1
% +1 point sample -> freq. domain coeffs.
% -1 freq.dom.coef.-> point values
n = length(x); y = x(1)*ones(n,1);
if n>1
m=n/2; w=exp(isign*pi*sqrt(-1)/m);
v = w.^(0:n-1)';
for k = 2:n
z = rem((k-1)*(0:n-1)',n )+1;
y = y + v(z)*x(k); end
if isign == 1; y = y/m; end
end
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13. function y = FFTRecur(x) %n=2^k, x col. n = length(x); if n == 1
y = x;
else
m = n/2;
yT = FFTRecur(x(1:2:n));
yB = FFTRecur(x(2:2:n));
d = exp(-2*pi*sqrt(-1)/n).^(0:m-1)';
z = d.*yB;
y = [ yT+z ; yT-z ];
end
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14. FFTrecur is of order
DFT is of order
CSInterp is of order
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15. [y,Fs,bits]=wavread('tele_tones_1.wav'); a=input('starting point')
% script interface
[y,Fs,bits]=wavread('tele_tones_1.wav');
a=input('starting point')
n=input('length of series (=power of 2)')
x=y(a:a+n); m=n/2;
y = fft(x(1:n));
P2 = struct('a',real(y(1:m+1))/m,'b',
-imag(y(2:m))/m);
P2.a(1)=P2.a(1)/2;P2.a(m+1)=P2.a(m+1)/2;
Bvals = linspace(0,n/Fs,n+1)';
Gvals = CSeval(P2,n/Fs,Bvals);
plot(Bvals(1:100),Gvals(1:100)-x(1:100))
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16. [z,Fs,bits]=wavread('tele_tones_1.wav'); a=input('starting point')
% script aliasing
[z,Fs,bits]=wavread('tele_tones_1.wav');
a=input('starting point')
n=input('length of series (=power of 2)')
stride=input('stride (=power of 2) < length)')
x=z(a:stride:a+n); m=n/2/stride;
y = fft(x(1:n/stride));
P2 = struct('a',real(y(1:m+1))/m,'b',-imag(y(2:m))/m);
P2.a(1)=P2.a(1)/2;P2.a(m+1)=P2.a(m+1)/2;
Bvals = linspace(0,n/Fs,n+1)';
Gvals = CSeval(P2,n/Fs,Bvals);
plot(Bvals(1:100*stride),Gvals(1:100*stride)
-z(1:100*stride))
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17. [y,Fs,bits]=wavread('tele_tones_1.wav'); a=input('starting point')
% script spectrum
[y,Fs,bits]=wavread('tele_tones_1.wav');
a=input('starting point')
n=input('length of series (=power of 2)')
x=y(a:a+n); m=n/2;) % sound(x,Fs)
y = fft(x(1:n));
P2 = struct('a',real(y(1:m+1))/m,'b',
-imag(y(2:m))/m);
P2.a(1)=P2.a(1)/2;P2.a(m+1)=P2.a(m+1)/2;
E(2:m)=sqrt(P2.a(2:m).^2+P2.b(1:m-1).^2);
E(1)=abs(P2.a(1));E(m+1)=abs(P2.a(m+1));
plot(E(1:500))
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18. for i=1:m+1
if (E(i)>.1)
i-1
end
x(1:4096);
i = 78, 135 1
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19. a = 6000
n = 4096
i = 78,135 2
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20. a = 13000
n = 4096
i = 78, 122 3
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21. a = 19000
n = 4096
i =70, 122 4
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22. a = 25500
n = 4096
i = 78, 135
a = 38000
n = 4096
i = 78, 135 5 7
XXY-ZXXX
a = 32000
n = 4096
I = 78,135 6
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23. APPENDIX a FFT Discrete Fourier transform.
FFT(X) is the discrete Fourier transform (DFT) of vector X. For
matrices, the FFT operation is applied to each column. For N-D
arrays, the FFT operation operates on the first non-singleton
dimension.
FFT(X,N) is the N-point FFT, padded with zeros if X has less
than N points and truncated if it has more.
FFT(X,[],DIM) or FFT(X,N,DIM) applies the FFT operation
across the dimension DIM.
See also IFFT, FFT2, IFFT2, FFTSHIFT.
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24. For length N input vector x, the DFT
is a length N vector X, with elements:
The inverse DFT (computed by IFFT)
is given by
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25. %an example of use: FFT, DFT, FFTRecur
n=512;m=n/2;
t = (0:(1/n):1)'; kk=linspace(0,512,513)';
y = sin(2*pi*50*t) + 2*cos(2*pi*120*t);
z1=fft(y(1:n)); % FFT
x1=real(ifft(z1));
z2=FFTRecur(y(1:n)); % FFTRecur
P2=struct('a',real(z2(1:m+1))/m,'b',imag(z2(2:m))/m);
P2.a(1)=P2.a(1)/2;P2.a(m+1)=P2.a(m+1)/2;
x2 = CSeval(P2,1,t);
z3=DFT(y(1:n),1); % DFT
P3=struct('a',real(z3(1:m+1))/m,'b',imag(z3(2:m))/m);
P3.a(1)=P3.a(1)/2;P3.a(m+1)=P3.a(m+1)/2;
x3 = CSeval(P3,1,t);
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26. Reference\MatlabFunctionReference\ File I/O \Audio and Audio/Video
APPENDIX b
HELP:
Reference\MatlabFunctionReference\
File I/O \Audio and Audio/Video
General
SPARCstation-Specific Sound Functions
Microsoft WAVE Sound Functions
Audio Video Interleaved (AVI) Functions
Microsoft Excel Functions
Lotus123 Functions
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27. wavplay wavread wavrecord wavwrite
Microsoft WAVE Sound Functions
wavplay
Play sound on PC-based audio output device
wavread
Read Microsoft WAVE (.wav) sound file
wavrecord
Record sound using PC-based audio input device
wavwrite
Write Microsoft WAVE (.wav) sound file
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28. Wavread Read Microsoft WAVE (.wav) sound file
Graphical Interface
As an alternative to auread, use the Import Wizard. To activate the Import Wizard, select Import Data from the File menu.
Syntax
y = wavread('filename')
[y,Fs,bits] = wavread('filename')
[...] = wavread('filename',N)
[...] = wavread('filename',[N1 N2])
[...] = wavread('filename','size')
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29. [y,Fs,bits] = wavread('filename')
Description
wavread supports multichannel data, with up to 16 bits per sample.
y = wavread('filename')
loads a WAVE file specified by the string filename, returning the sampled data in y. The .wav extension is appended if no extension is given. Amplitude values are in the range [-1,+1].
[y,Fs,bits] = wavread('filename')
returns the sample rate (Fs) in Hertz and the number of bits per sample (bits) used to encode the data in the file.
[...] = wavread('filename',N)
returns only the first N samples from each channel in the file.
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30. [...] = wavread('filename',[N1 N2])
returns only samples N1 through N2 from each
channel in the file.
siz = wavread('filename','size')
returns the size of the audio data contained in the file
in place of the actual audio data, returning the vector
siz = [samples channels].
See Also
auread, wavwrite
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31. Wavwrite Write Microsoft WAVE (.wav) sound file
Syntax wavwrite(y,'filename')
wavwrite(y,Fs,'filename')
wavwrite(y,Fs,N,'filename')
Description
wavwrite supports multi-channel 8- or 16-bit WAVE data.
wavwrite(y,'filename') writes a WAVE file specified by the
string filename. The data should be arranged with one channel per column.
Amplitude values outside the range [-1,+1] are clipped prior to writing.
Wavwrite(y,Fs,'filename') specifies the sample
rate Fs, in Hertz, of the data.
Wavwrite(y,Fs,N,'filename') forces an N-bit file
format to be written, where N <= 16.
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32. Sound Convert vector into sound Syntax sound(y,Fs) sound(y)
sound(y,Fs,bits)
Description
sound(y,Fs) sends the signal in vector y (with sample frequency
Fs) to the speaker on PC and most UNIX platforms. Values in y are
assumed to be in the range . Values outside that range are clipped. Stereo
sound is played on platforms that support it when y is an n-by-2 matrix.
sound(y) plays the sound at the default sample rate or 8192Hz.
sound(y,Fs,bits) plays the sound using bits number of
bits/sample, if possible. Most platforms support bits = 8 or bits = 16.
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33. Appendix c
% p10.m - polynomials and corresponding equipotential curves
(N.Trefethen, Spectral Methods in Matlab)
N = 16; clf
for i = 1:2
if i==1, s = 'equispaced points';
x = *(0:N)/N; end
if i==2, s = 'Chebyshev points';
x = cos(pi*(0:N)/N); end
p = poly(x);
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34. xx = -1:.005:1; pp = polyval(p,xx);
% Plot p(x) over [-1,1]:
xx = -1:.005:1; pp = polyval(p,xx);
subplot(2,2,2*i-1)
plot(x,0*x,'.','markersize',13), hold on
plot(xx,pp), grid on
set(gca,'xtick',-1:.5:1), title(s)
% Plot equipotential curves:
subplot(2,2,2*i)
plot(real(x),imag(x),'.','markersize',13), hold on
axis([ ])
xgrid = -1.4:.02:1.4; ygrid = -1.12:.02:1.12;
[xx,yy] = meshgrid(xgrid,ygrid); zz = xx+1i*yy;
pp = polyval(p,zz); levels = 10.^(-4:0);
contour(xx,yy,abs(pp),levels), title(s), colormap(1e-6*[1 1 1]);
end
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35. 9/18/2018

36. ,112
a =14000,n =4096
2 65,124
a =41000,n =4096
3 65,137
a =65000,n =4096
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37. 4 72,112
a=134000,n=4096
5 72,124
a=134000,n =4096
6 72,137
a =163000,n =4096
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38. 7 79,112
a =198000,n =4096
8 79,124
a =231500,n =4096
9 79,137
a =267500,n =4096
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39. * 87,112
a =345000,n =4096
0 87,124
a =299500,n =4096
# 87,137
a =363000,n =4096
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40. frequencies in Hertz
687.5
781.25
843.75
937.5
* #
The matrix
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41. n=4096 65 72 79 87
* #
The matrix
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