Basics of MATLAB and Beyond

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-0.5000-0.3750-0.2500-0.125000.12500.25000.3750We combine fftshift and fftfreq to plot the two-sided FFT:plot(fftfreq(.5,8),fftshift(abs(Y)))axis([-.5 .5 0 7])zeroaxesLet us do a slightly more realistic example. We simulate some datarecorded at a sampling frequency of 1 kHz, corresponding to a time stepdt = 1/1000 of a second. The Nyquist frequency is, therefore, 500 Hz.Suppose there is a 100 Hz sinusoid contaminated by noise. We simulatethe data, calculate the FFT, and plot the results as follows:dt = 1/1000;t = dt:dt:200*dt;sine = sin(2*pi*100*t);y = sine + randn(size(t));Y = fft(y);f = fftfreq(500,length(Y));clfsubplot(211)stairs(t,y)hold onstairs(t,sine-4)boxxlabel(’Time (seconds)’)subplot(212)stairs(f,fftshift(abs(Y)))boxxlabel(’Frequency (Hz)’)c○ 2000 by CRC Press LLC
The top trace in the top plot is the noisy data, and the bottom trace isthe original pure sinusoid. The lower plot clearly shows the frequency at100 Hz.Two GUI-based FFT demos can be accessed by typing demo at theprompt. Select the “Signal Processing” option, then choose the “DiscreteFourier Transform” or the “Continuous Fourier Transform”.Exercise 5 Extend the ideas in the previous example to twodimensions, as would be the case, for example, if you made measurementsin space and time, rather than time alone. Generatea two-dimensional sinusoid and explore its FFT. (Answer onpage 185.)18 Power SpectrumThe power spectrum (or power spectral density, or PSD) is a measureof the power contained within frequency intervals. The problem is thatwe only have a finite set of samples of the true signal so we can neverhave perfect knowledge about its power spectrum. A common way toestimate a PSD is to use the square of the FFT of the samples. Thesquare of the FFT is called the periodogram. The workhorse of matlab’speriodogram-based spectral estimation is the spectrum function(in the Signal Processing Toolbox). We illustrate using data similar tothe previous example of a noisy sinusoid, but we take more samples. APSD estimate can be found by typing:dt = 1/1000;t = dt:dt:8192*dt;sine = sin(2*pi*100*t);y = sine + randn(size(t));clfspectrum(y)The frequency scale is normalised to the Nyquist frequency. The middleline is the PSD estimate and the two dashed lines are the 95% confidenceintervals. Typing help spectrum reveals that there are manyparameters that you can adjust when calculating the power spectrum.matlab’s spectrum function uses the Welch method of PSD estimation, 6which divides a long signal into a number of smaller blocks, calculates6 See Alan V. Oppenheim and Ronald W. Schafer, Digital Signal Processing,Prentice-Hall, 1975, p. 553. An excellent general treatment of PSD estimation isalso given in William Press, Brian Flannery, Saul Teukolsky and William Vetterling,Numerical Recipes, Cambridge University Press, 1989.c○ 2000 by CRC Press LLC
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BASICS OFMATLAB®and Beyondc○ 200
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Library of Congress Cataloging-in-P
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typing pwd (print working directory
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ContentsIBasics of MATLAB1 First St
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20 Time-Frequency Analysis21 Line A
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39Answers to Exercises (Part I)40 A
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Set → Getting Started with MATLAB
Page 16 and 17: 1.5 The Colon OperatorTo generate a
Page 18 and 19: matlab will recall the last command
Page 20 and 21: a = zeros(2,3)a =0 0 00 0 0>> b = o
Page 22 and 23: [a a(a)]ans =1 2 3 1 4 74 5 6 2 5 8
Page 24 and 25: a + bans =11 3344 6677 993.8 Transp
Page 26 and 27: 4.2 Adding PlotsWhen you issue a pl
Page 28 and 29: x = [[1 2 3 4]’ [2 3 4 5]’ [3 4
Page 30 and 31: 4.7 AxesSo far we have allowed matl
Page 32 and 33: v =1122>> v*uans =1 2 0 11 2 0 12 4
Page 34 and 35: max returns a vector containing the
Page 36 and 37: s = 100*spiral(3)s =700 800 900600
Page 38 and 39: We can plot the surface z as a func
Page 40 and 41: The contour function plots the cont
Page 42 and 43: m = gray(8);colormap(m)imagesc(1:10
Page 44 and 45: hold onplot3(x(ind),y(ind),z(ind),
Page 46 and 47: z = 5*ones(3,3)z =5 5 55 5 55 5 5>>
Page 48 and 49: contour(x,y,z,30);You may have noti
Page 50 and 51: variables that are local to themsel
Page 52 and 53: SwitchThe basic form of a switch st
Page 54 and 55: i =i =123136Vectorised Codematlab i
Page 56 and 57: 9.1 MATLAB FormatTo save all the va
Page 58 and 59: 11 StartupEach time you start matla
Page 60 and 61: Where p i is the population for yea
Page 62 and 63: NaNs by leaving them off the plot.
Page 64 and 65: clfpolar(t,gdb)In this case you mus
Page 68 and 69: the periodograms of the blocks, and
Page 70 and 71: plot(y(1:1000))axis([0 1000 -5 5])z
Page 72 and 73: comet(x,y)(You can get a three-dime
Page 74 and 75: By clicking within the panner box a
Page 76 and 77: x = linspace(-1,1,8192);Fs = 1000;y
Page 78 and 79: 23.3 Graphical Object Hierarchymatl
Page 80 and 81: Now we need to get the handles of a
Page 82 and 83: Beyond the BasicsPreludeThis part o
Page 84 and 85: clfgplot(A,[x y])axis off(Try zoomi
Page 86 and 87: str =you’re the one>> str = ’
Page 88 and 89: strrep(str,’go’,’am’)ans =I
Page 90 and 91: ans =3.14159e+100Some additional te
Page 92 and 93: eval(str)v =1 2 3 4 5The eval(str)
Page 94 and 95: Inline objects, like every other ma
Page 96 and 97: a(1,1) = {[1 2 3]}a =[1x3double]or>
Page 98 and 99: t = {’help’ spiral(3) ; eye(2)
Page 101 and 102: meteo.Temperatureans =24ans =19.000
Page 103 and 104: 3 >> a = [1 2 ;4 5 6;7 8 9]a =1 2 3
Page 105 and 106: [x,y] = meshgrid(1:5,1:3)x =1 2 3 4
Page 107 and 108: ans(:,:,2) =999And to sum over “p
Page 109 and 110: An Application of RGB ImagesTo see
Page 111 and 112: You can see that over the 8 time st
Page 113 and 114: name>> staff(2,1,2).name(5:9)ans =B
Page 115 and 116: q =7 8 96 1 25 4 3>> save saved_dat
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fid = fopen(’asc.dat’);The fope
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display and the enclosed area is th
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MarkerEdgeColor = auto(and so on)In
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text(-.7,f(-.7),’f(X)’)But we h
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plt(x,f(x/2),’--’)Let us try ca
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Example: findobjThe findobj command
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3. Add the property you want to set
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is numbered from top to bottom. The
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axis ijUsually images like this do
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32.2 Tick Marks and Labelsmatlab’
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clfaxes(’pos’,[.2 .1 .7 .4])x =
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property to [0 top] in the call to
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object’s HorizontalAlignment and
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the required symbols. The m-file ti
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1. Use graphical elements that serv
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uicontrol(’Callback’,’ezplot(
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uicontrol(’Pos’,[110 280 60 19]
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’String’,’0.5’,’Style’,
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function exradio(action)if nargin =
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fail. One place to put variables th
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34.7 Fast DrawingFor fast drawing o
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controlled by guide and which are a
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InvertHardcopy: [ {on} | off ]Paper
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Another option is to use an image f
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yur is the y coordinate of the uppe
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\end{center}\caption{This is the fi
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x 294x1 ...y 294x1 ...z 294x1 ...Gr
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diagram. Such a triangular grid can
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xt = x([1 9 10 1]);yt = y([1 9 10 1
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x = [0 0 .5 0.5 .5 1 11 .5 .5 .5];y
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fac2 = [1 2 61 6 31 4 61 6 5];The r
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t = linspace(0,2*pi,20);x = cos(t);
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N = 20;dt = 2*pi/N;t = 0:dt:(N-1)*d
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colouring of the sphere itself. To
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material dullThe default is materia
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time taken by this routine to calcu
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first row of a by taking the column
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38.3 Debuggingmatlab has a suite of
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10: y = linspace(ycentre - L,ycentr
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We change variables:p ′ = C + Bx.
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Exercise 6 (Page 59)You can generat
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33 ! 36 $ 39 ’ 42 * 45 - 48 034 "
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R = 1; N = 200;[x,y] = meshgrid(lin
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(The plot might take a few seconds
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% Reproduce the layers at the diffe
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