Synchronized Swept-Sine: Theory Application
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DIGITAL SIGNAL PROCESSING LAB Cycle I 1. Generation of
Generation of Sinusoidal waveform / signal based on recursive difference Objective: To wite the MATlab code to find the DFT / IDFT of given signal.
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Abstract—This paper examined a novel approach to the DSP of sinusoids with MATLAB using the Discrete Fourier. Transform. A sinusoid is a mathematical curve
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a = sin(2*pi*60*t) the code returns something bad What am i doing wrong? How can i generate a sin wave with different frequencies?
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It looks like 'df' in the above code can be set to 'pi/8' to represent time shift in original signal Other than that everything looks fine
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f = -fs/2:fs/(N-1):fs/2; z = fftshift(fft(x));What is the code for MATLAB for sine wave?
The below code is developed to generate sin wave having values for amplitude as '1' and liner frequency as '10'. f = 10; t = 0:0.1:10000; st = sin(2*3.141516*f*t);How to use MATLAB for DFT?
To plot the magnitude and phase in degrees, type the following commands: f = (0:length(y)-1)*100/length(y); % Frequency vector subplot(2,1,1) plot(f,m) title('Magnitude') ax = gca; ax. XTick = [15 40 60 85]; subplot(2,1,2) plot(f,p*180/pi) title('Phase') ax = gca; ax. XTick = [15 40 60 85];The FFT
1create_figure('sin wave', 1, 2);2plot(t, y)3xlabel('Time (sec)')4subplot(1, 2, 2)5yf = fft(y); % Fourier transform.6yf = abs(yf/L); % When X is complex, ABS(X) is the complex modulus (magnitude) of the elements of X.7f = Fs*(0:(L-1))/L; % define frequencies for each element of yf.8plot(f, yf)
Julius O. Smith III (jos@ccrma.stanford.edu)
Center for Computer Research in Music and Acoustics (CCRMA)Department of Music, Stanford University
Stanford, California 94305
August 11, 2002
Page ii
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atContents
1 Introduction to the DFT 1
1.1 DFTDefinition............................ 1
1.2 MathematicsoftheDFT ...................... 2
1.3 DFTMathOutline.......................... 5
2 Complex Numbers 7
2.1 FactoringaPolynomial........................ 7
2.2 TheQuadraticFormula ....................... 8
2.3 ComplexRoots............................ 10
2.4 FundamentalTheoremofAlgebra.................. 11
2.5 ComplexBasics............................ 12
2.5.1 TheComplexPlane ..................... 13
2.5.2 MoreNotationandTerminology .............. 15
2.5.3 ElementaryRelationships .................. 15
2.5.4 Euler"s Formula ........................ 16
2.5.5 De Moivre"s Theorem .................... 17
2.6 Numerical Tools in Matlab . .................... 17
2.7 Numerical Tools in Mathematica . . ................ 24
3 Proof of Euler"s Identity 27
3.1 Euler"s Theorem........................... 27
3.1.1 PositiveIntegerExponents ................. 27
3.1.2 PropertiesofExponents................... 28
3.1.3 The Exponent Zero . . .................... 28
3.1.4 NegativeExponents ..................... 28
3.1.5 RationalExponents ..................... 29
3.1.6 RealExponents........................ 30
iiiPage iv CONTENTS
3.1.7 A First Look at Taylor Series . . .............. 31
3.1.8 Imaginary Exponents . . .................. 32
3.1.9 Derivatives off(x)=a
x ................... 323.1.10 Back toe........................... 33
3.1.11SidebaronMathematica................... 34
3.1.12 Back toe
jθ .......................... 343.2 Informal Derivation of Taylor Series . . .............. 36
3.3 Taylor Series with Remainder . . .................. 38
3.4 FormalStatementofTaylor"sTheorem............... 39
3.5 Weierstrass Approximation Theorem . . .............. 40
3.6 Differentiability of Audio Signals .................. 40
4 Logarithms, Decibels, and Number Systems 43
4.1 Logarithms .............................. 43
4.1.1 ChangingtheBase...................... 45
4.1.2 Logarithms of Negative and Imaginary Numbers..... 45
4.2 Decibels . . .............................. 46
4.2.1 PropertiesofDBScales ................... 47
4.2.2 Specific DB Scales ...................... 48
4.2.3 DynamicRange........................ 54
4.3 LinearNumberSystemsforDigitalAudio............. 55
4.3.1 PulseCodeModulation(PCM)............... 55
4.3.2 Binary Integer Fixed-Point Numbers . . .......... 55
4.3.3 Fractional Binary Fixed-Point Numbers .......... 60
4.3.4 HowManyBitsareEnoughforDigitalAudio? ...... 60
4.3.5 WhenDoWeHavetoSwapBytes?............. 62
4.4 LogarithmicNumberSystemsforAudio.............. 63
4.4.1 Floating-PointNumbers................... 63
4.4.2 Logarithmic Fixed-Point Numbers............. 65
4.4.3 Mu-LawCompanding .................... 66
4.5 Appendix A: Round-Off Error Variance .............. 67
4.6 Appendix B: Electrical Engineering 101 .............. 68
5 Sinusoids and Exponentials 71
5.1 Sinusoids ............................... 71
5.1.1 ExampleSinusoids...................... 72
5.1.2 WhySinusoidsareImportant................ 73
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atCONTENTS Page v
5.1.3 In-Phase and Quadrature Sinusoidal Components . . . . . 74
5.1.4 Sinusoids at the Same Frequency . . ............ 75
5.1.5 Constructive and Destructive Interference . ........ 76
5.2 Exponentials ............................. 79
5.2.1 WhyExponentialsareImportant.............. 79
5.2.2 AudioDecayTime(T60) .................. 80
5.3 ComplexSinusoids.......................... 81
5.3.1 Circular Motion....................... 81
5.3.2 Projection of Circular Motion ................ 82
5.3.3 Positive and Negative Frequencies . ............ 83
5.3.4 The Analytic Signal and Hilbert Transform Filters . . . . 83
5.3.5 Generalized Complex Sinusoids............... 87
5.3.6 SampledSinusoids ...................... 88
5.3.7 Powers ofz.......................... 89
5.3.8 Phasor & Carrier Components of Complex Sinusoids . . . 89
5.3.9 Why Generalized Complex Sinusoids are Important . . . 91
5.3.10 Comparing Analog and Digital Complex Planes . . . . . . 93
5.4 Mathematica for Selected Plots................... 96
5.5 Acknowledgement........................... 97
6 Geometric Signal Theory 99
6.1 TheDFT ............................... 99
6.2 SignalsasVectors........................... 100
6.3 VectorAddition............................ 101
6.4 Vector Subtraction . . ........................ 102
6.5 Signal Metrics . ............................ 102
6.6 TheInnerProduct.......................... 107
6.6.1 LinearityoftheInnerProduct ............... 108
6.6.2 NormInducedbytheInnerProduct............ 109
6.6.3 Cauchy-SchwarzInequality ................. 110
6.6.4 TriangleInequality...................... 111
6.6.5 Triangle Difference Inequality ................ 111
6.6.6 VectorCosine......................... 111
6.6.7 Orthogonality . ........................ 112
6.6.8 ThePythagoreanTheoreminN-Space........... 112
6.6.9 Projection........................... 113
6.7 SignalReconstructionfromProjections .............. 114
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atPage vi CONTENTS
6.7.1 AnExampleofChangingCoordinatesin2D ....... 115
6.7.2 GeneralConditions...................... 118
6.7.3 Gram-Schmidt Orthogonalization .............. 122
6.8 Appendix:MatlabExamples .................... 123
7 Derivation of the Discrete Fourier Transform (DFT) 129
7.1 The DFT Derived .......................... 129
7.1.1 Geometric Series . ...................... 129
7.1.2 Orthogonality of Sinusoids .................. 130
7.1.3 Orthogonality of the DFT Sinusoids . . .......... 133
7.1.4 NormoftheDFTSinusoids................. 133
7.1.5 AnOrthonormalSinusoidalSet............... 133
7.1.6 The Discrete Fourier Transform (DFT) .......... 134
7.1.7 Frequencies in the "Cracks"................. 135
7.1.8 Normalized DFT . ...................... 138
7.2 TheLength2DFT.......................... 139
7.3 MatrixFormulationoftheDFT................... 140
7.4 MatlabExamples........................... 142
7.4.1 Figure7.2........................... 142
7.4.2 Figure7.3........................... 143
7.4.3 DFTMatrixinMatlab ................... 144
8 Fourier Theorems for the DFT 147
8.1 TheDFTanditsInverse....................... 147
8.1.1 NotationandTerminology.................. 148
8.1.2 Modulo Indexing, Periodic Extension . . .......... 148
8.2 SignalOperators ........................... 150
8.2.1 FlipOperator......................... 150
8.2.2 ShiftOperator ........................ 150
8.2.3 Convolution.......................... 153
8.2.4 Correlation .......................... 156
8.2.5 StretchOperator....................... 157
8.2.6 Zero Padding......................... 157
8.2.7 RepeatOperator....................... 158
8.2.8 Downsampling Operator . .................. 160
8.2.9 AliasOperator ........................ 162
8.3 EvenandOddFunctions....................... 165
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atCONTENTS Page vii
8.4 The Fourier Theorems ........................ 167
8.4.1 Linearity ........................... 167
8.4.2 ConjugationandReversal.................. 167
8.4.3 Symmetry........................... 169
8.4.4 ShiftTheorem ........................ 171
8.4.5 ConvolutionTheorem .................... 173
8.4.6 DualoftheConvolutionTheorem ............. 175
8.4.7 Correlation Theorem . .................... 175
8.4.8 PowerTheorem........................ 176
8.4.9 Rayleigh Energy Theorem (Parseval"s Theorem) . . . . . 176
8.4.10StretchTheorem(RepeatTheorem) ............ 177
8.4.11 Downsampling Theorem (Aliasing Theorem)....... 177
8.4.12 Zero Padding Theorem .................... 179
8.4.13BandlimitedInterpolationinTime............. 180
8.5 Conclusions.............................. 182
8.6 Acknowledgement........................... 182
8.7 Appendix A: Linear Time-Invariant Filters and Convolution . . . 182
8.7.1 LTIFiltersandtheConvolutionTheorem......... 183
8.8 AppendixB:StatisticalSignalProcessing ............. 184
8.8.1 Cross-Correlation....................... 185
8.8.2 Applications of Cross-Correlation . . ............ 186
8.8.3 Autocorrelation ........................ 187
8.8.4 Coherence........................... 190
8.9 AppendixC:TheSimilarityTheorem ............... 191
9 Example Applications of the DFT 193
9.1 Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and
theFFT................................ 1939.1.1 Example1:FFTofaSimpleSinusoid ........... 193
9.1.2 Example2:FFTofaNot-So-SimpleSinusoid....... 196
9.1.3 Example 3: FFT of a Zero-Padded Sinusoid ........ 200
9.1.4 Example4:BlackmanWindow............... 202
9.1.5 Example5:UseoftheBlackmanWindow......... 204
9.1.6 Example 6: Hanning-Windowed Complex Sinusoid . . . . 206
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atPage viii CONTENTS
A Matrices 213
A.0.1 Matrix Multiplication . . .................. 214 A.0.2 Solving Linear Equations Using Matrices......... 217B Sampling Theory 219
B.1 Introduction.............................. 219 B.1.1 ReconstructionfromSamples-PictorialVersion ..... 220 B.1.2 ReconstructionfromSamples-TheMath......... 221 B.2 AliasingofSampledSignals..................... 222 B.3 Shannon"sSamplingTheorem.................... 225 B.4 AnotherPathtoSamplingTheory ................. 228 B.4.1 What frequencies are representable by a geometric sequence?228 B.4.2 Recovering a Continuous Signal from its Samples..... 230 DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atList of Figures
2.1 An example parabola defined byp(x)=x
2 +4........... 102.2 Plotting a complex number as a point in the complex plane. . . . 14
4.1 Windowed sinusoid (top) and its FFT magnitude (bottom). . . . 52
5.1 Anexamplesinusoid.......................... 73
5.2 In-phaseandquadraturesinusoidalcomponents. ......... 75
5.3 A comb filter with a sinusoidal input................ 78
5.4 Comb filter amplitude response when delayτ=1sec. ...... 78
5.5 The decaying exponentialAe
-t/τ ................... 795.6 Growinganddecayingexponentials. ................ 80
5.7 Acomplexsinusoidanditsprojections. .............. 82
5.8 Creation of the analytic signalz(t)=e
jω 0 t from the real sinusoid x(t) = cos(ω 0 t) and the derived phase-quadrature sinusoidy(t)= sin(ω 0 t), viewed in the frequency domain. a) Spectrum ofx.b) Spectrum ofy. c) Spectrum ofjy. d) Spectrum ofz=x+jy.. 865.9 Exponentially decaying complex sinusoid and its projections. . . 89
5.10 Generalized complex sinusoids represented by points in thesplane. 94
ixPage x LIST OF FIGURES
5.11 Generalized complex sinusoids represented by points in thezplane. 95
6.1 A length 2 signalx=(2,3) plotted as a vector in 2D space. . . . 101
6.2 Geometricinterpretationofalength2vectorsum. ........ 101
6.3 Vector sum with translation of one vector to the tip of the other. 102
6.4 Geometric interpretation a difference vector. . . .......... 102
6.5 Geometricinterpretationofasignalnormin2D. ......... 105
6.6 Lengthofvectorsinsum. ...................... 105
6.7 Length of a difference vector. . . .................. 105
6.8 Example of two orthogonal vectors forN=2............ 112
6.9 Projection ofy
ontoxin2Dspace.................. 1147.1 TheNroots of unity forN=8. .................. 131
7.2 Complex sinusoids used by the DFT forN=8........... 132
7.3 Frequency response magnitude of a single DFT output sample. . 137
7.4 Graphicalinterpretationofthelength2DFT............ 139
8.1 Illustration ofxandFlip(x)forN= 5 and two different domain
interpretations: a)n?[0,N-1]. b)n?[-(N-1)/2,(N-1)/2]. ................. 1518.2 Successive one-sample shifts of a sampled periodic sawtooth wave-
form having first period [0,1,2,3,4]. ................ 1528.3 Illustration of convolution ofy=[1,1,1,1,0,0,0,0] and its "matched
filter" h=[1,0,0,0,0,1,1,1] (N=8)................... 154 DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atLIST OF FIGURES Page xi
8.4 Illustration ofStretch
3 (x). .................... 1578.5 Illustration of frequency-domain zero padding:
a) Original spectrumX=[3,2,1,1,2] plotted over the domain k?[0,N-1] whereN= 5 (i.e., as the spectral array would normally exist in a computer array). b)ZeroPad 11 (X). c) The same signalXplotted over the domaink?[-(N-1)/2,(N-1)/2] which is more natural for interpreting negative
frequencies. d)ZeroPad 11 (X).......................... 1598.6 Illustration ofRepeat
2 (x). ..................... 1608.7 Illustration ofRepeat
3 (X). a) Conventional plot ofX. b) Plot ofXover the unit circle in thezplane. c)Repeat 3 (X). .......................... 1618.8 Illustration ofSelect
2 (x). The white-filled circles indicate the retained samples while the black-filled circles indicate the dis- carded samples. ............................ 1628.9 Example of aliasing due to undersampling in time. ........ 163
8.10 Illustration of aliasing in the frequency domain.
a)Repeat 3 (X) from Fig. 8.7c. b) First half of the original unit circle (0 toπ) wrapped around the new, smaller unit circle (which is magnified to the original size). c) Second half (πto 2π), also wrapped around the new unit circle. d) Overlay of components to be summed. e) Sum of components (the aliased spectrum). f) Both sum and overlay. . .................... 1648.11Thefilterinterpretationofconvolution. .............. 182
8.12FIRsystemidentificationexampleinMatlab............ 188
DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atPage xii LIST OF FIGURES
9.1 Sampled sinusoid atf=f
s /4. a) Time waveform. b) Magnitude spectrum. c) DB magnitude spectrum. . .............. 1959.2 Sinusoid at Frequencyf=0.25 + 0.5/N. a) Time waveform. b)
Magnitude spectrum. c) DB magnitude spectrum. . . ...... 1989.3 Time waveform repeated to show discontinuity introduced by pe-
riodicextension(seemidpoint).................... 1999.4 Zero-Padded Sinusoid at Frequencyf=0.25 + 0.5/N.a)Time
waveform. b) Magnitude spectrum. c) DB magnitude spectrum. 2019.5 The Blackman Window. a) The window itself in the time do-
main. b) DB Magnitude spectrum of the Blackman window. c) Blackman-window DB magnitude spectrum plotted over frequen- cies [-0.5,0.5). ............................ 2039.6 Effect of the Blackman window on the sinusoidal data segment. . 205
9.7 A length 31 Hanning Window ("Raised Cosine") and the win-
dowed sinusoid created using it. Zero-padding is also shown. The sampled sinusoid is plotted with '*" using no connecting lines. You must now imagine the continuous sinusoid threading through the asterisks. . .............................. 2079.8 Spectral Magnitude, linear scale. .................. 209
9.9 Spectral Magnitude, dB scale. . . .................. 210
9.10Spectralphase............................. 212
B.1 How sinc functions sum up to create a continuous waveform from discrete-time samples. . . ...................... 220 B.2 The sinc function. .......................... 221 DRAFTof "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, July 2002. The latest draft is available on-line atPreface
This book is an outgrowth of my course entitled "Introduction to Digital Audio Signal Processing and the Discrete Fourier Transform (DFT) 1 which I have given at the Center for Computer Research in Music and Acoustics (CCRMA) everyquotesdbs_dbs12.pdfusesText_18
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