MATLAB Signal Processing Toolbox

MATLAB SignalProcessing ToolboxGreg Reese, Ph.DResearch Computing Support GroupAcademic Technology ServicesMiami UniversityOctober 2013

MATLAB SignalProcessing Toolbox 2013 Greg Reese. All rights reserved2

ToolboxToolbox Collection of code devoted to solvingproblems in one field of research Can be purchased from MATLAB Can be purchased from third parties Can be obtained for free from third parties3

ToolboxMATLAB Signal Processing Toolbox Code for solving problems in signalprocessing (!) Sold by MATLAB Part of both Miami’s student andfaculty license4

OverviewMATLAB divides Signal ProcessingToolbox as follows Waveforms– Pulses, modulated signals, peak-to-peak and RMSamplitude, rise time/fall time, overshoot/undershoot Convolution and Correlation– Linear and circular convolution, autocorrelation,autocovariance, cross-correlation, cross-covariance Transforms– Fourier transform, chirp z-transform, DCT, Hilberttransform, cepstrum, Walsh-Hadamard transform5

Overview Analog and Digital Filters– Analog filter design, frequency transformations, FIR and IIRfilters, filter analysis, filter structures Spectral Analysis– Nonparametric and parametric spectral estimation, highresolution spectral estimation Parametric Modeling and LinearPrediction– Autoregressive (AR) models, linear predictive coding (LPC),Levinson-Durbin recursion Multirate Signal Processing– Downsampling, upsampling, resampling, anti-aliasing filter,interpolation, decimation6

OverviewWill look very briefly at Analog and Digital Filters Spectral Analysis Parametric Modeling and Linear Prediction Multirate Signal ProcessingWill look in more depth at Waveforms Convolution and Correlation7

Analog and Digital FiltersToolbox especially good for those seriousabout their filter design!Analog filters Standard filters– Bessel, Butterworth, Chebyshev, Elliptic Filter transforms– Low pass to: bandpass, bandstop, or highpass– Change cutoff frequency of lowpass Analog to digital filter conversion– Bilinear transformation8

Analog and Digital FiltersDigital Filter Design with functions Standard filters– Butterworth, Chebyshev, Elliptic FIR and IIR design– Low pass to: bandpass, bandstop, or highpass– Change cutoff frequency of lowpass Objects for specification of filters– Arbitrary, lowpass, highpass, bandpass, Hilbert9

Analog and Digital FiltersDigital Filter Design interactively (GUI) Filterbuilder – specify desiredcharacteristics first, then choose filter type– Butterworth, Chebyshev, Elliptic FDATool (Filter Design and Analysis Tool)– Quickly design digital FIR or IIR filters– Quickly analyze filters, e.g., magnitude/phaseresponse, pole-zero plots10

Analog and Digital FiltersSPTool – composite of four tools1. Signal Browser – analyze signals2. FDATool3. FVTool (Filter Visualization Tool) –analyze filter characteristics4. Spectrum Viewer – spectral analysis11

Analog and Digital FiltersDigital Filter Analysis Magnitude and phase response, impulseresponse, group delay, pole-zero plotDigital Filter Implementation Filtering, direct form, lattice, biquad, statespace structures12

Spectral AnalysisNonparametric Methods– Periodogram, Welch's overlapped segmentaveraging, multitaper, cross-spectrum,coherence, spectrogramParametric Methods– Yule-Walker, Burg, covariance, modifiedcovarianceSubspace Methods– Multiple signal classification (MUSIC),eigenvectorestimator, pseudospectrum Windows– Hamming, Blackman, Bartlett, Chebyshev,Taylor, Kaiser13

Parametric Modeling andLinear PredictionParametric Modeling AR, ARMA, frequency responsemodelingLinear Predictive Waveforms Linear predictive coefficients (LPC), linespectral frequencies (LSF), reflectioncoefficients (RC), Levinson-Durbinrecursion14

Multirate Signal ProcessingMultirate signal processing– Downsampling, upsampling, resampling,anti-aliasing filter, interpolation, decimation15

WaveformsWaveforms part of toolbox lets you createmany commonly used signals, which youcan use to study models programmed inMATLABUses of waveforms Testing– E.g., have simple waveform and can analyticallydetermine model’s output. Use toolbox to create thatwaveform, run it through MATLAB model, confirmresult16

WaveformsUses of waveforms Simulation– Most of time can’t get analytical output– Make waveform of known characteristicsand study model’s response Modeling of real signals– Create waveform that looks like the realsignal17

WaveformsTime vectorsDigital signals usually sampled fromanalog at fixed intervals Δt . Want timeaxis with N points01Δt2Δt (N-2)Δt (N-1)Δt18

WaveformsFor––––tStart: starting timetEnd: ending timeN: number of pointsdeltaT: sampling interval If have starting time, number of points,interval, (tStart, N, deltaT): deltaT 0.1; N 10; t0 5; t t0 deltaT * (0:N-1)t 5.0 5.1 5.2 5.3 5.4 5.55.65.75.85.919

Waveforms If have starting time, ending time, interval(tStart, tEnd, deltaT) tStart 5; tEnd 5.9; deltaT 0.1; t tStart:deltaT:tEndt 5.0 5.1 5.2 5.3 5.45.55.65.75.85.9 If have starting time, ending time, number ofpoints (tStart, tEnd, N) tStart 5; tEnd 5.9; N 10; t linspace( tStart, tEnd, N )t 5.0 5.1 5.2 5.3 5.4 5.5 5.65.75.85.920

WaveformsIn multichannel processing, a number ofsignals are gathered at the same time Will assume all sampled at same time andsame rateSignal processing toolbox, and MATLABin general, treats each column of amatrix (2D array) as an independentcolumn vector21

WaveformsExample M [ 1:3; 4:6; 7:9; 10:12 ]M 123456789101112 mean( M )ans 5.50006.50007.5000Result is mean of each column22

WaveformsTIPMake time vector be a column vector Any vectors created from time vectorwill also be column vectors and socan be processed more easily t tt 123456 y abs( t - 3 )y 2101Column vector begeteth column vector2323

TIPWaveformsrepmat (replicate matrix) is generalpurpose function to make large matrix byreplicating small oneTrick - quick way to replicate column vector,i.e., to make an m x n matrix T out of an m x1 column vector v, isT v(:,ones(1,n)) v (1:5)v 12345 v(:,ones(1,3))ans 11122233344455524

WaveformsTIPTrick - quick way to replicate row vector,i.e., to make an m x n matrix T out of an 1 xn row vector v, isT v(ones(m,1),:) v 1:3v 1 2 32345 T v( ones(6,1), : )ans 12312312312312312325

WaveformsTIPCan use preceding two tips to makemultichannel signal, e.g.,Simulate the multichannel signalsin(2πt), sin(2πt/2), sin(2πt/3),sin(2πt/4) sampled for one secondat one-tenth second per sample26

WaveformsTIP t (0:0.1:1)'t .90001.000027

TIPWaveforms T t(:,ones(1,4))T 000.40000.50000.60000.70000.80000.90001.000028

TIPWaveforms M T ./ CM 500.10000.12500.15000.17500.20000.22500.250029

TIPWaveforms signal sin( 2*pi*Msignal 30

WaveformsImpulse Use to compute impulse responseof linear, time-invariant system t (0:0.1:1.9)'; impulse zeros( size(t) ); impulse( 1 ) 811.21.41.61.8231

WaveformsStep Use to model switch turning on t (-1:0.1:0.9)'; step [ zeros(10,1); ones(10,1) 0.200.20.40.60.8132

WaveformsRamp Use to model something gradually turning on t (0:0.1:1.9)'; ramp 1.41.61.8233

AutocorrelationAutocorrelation – the detection of a delayedversion of a signal In temporal signal, delay often called “lag” In spatial signal, delay often called“translation” or “offset” Delayed signal may also be scaledCan also think of autocorrelation assimilarity of a signal to itself as a function oflag34

AutocorrelationAutocorrelation and cross-correlation Common in both signal processingand statistics Definitions and uses are different When looking for help on thesetopics, make sure you’re looking ata signal-processing source35

AutocorrelationExamples of autocorrelation of digital signals Radar – determine distance to object Sonar – determine distance to object Music– Determine tempo– Detect and estimate pitch Astronomy – Find rotation frequency of pulsars36

AutocorrelationExamples of spatial autocorrelation Optical Character Recognition (OCR) – readingtext from images of writing/printing X-ray diffraction – helps recover the "Fourierphase information" on atom positions Statistics – helps estimate mean valueuncertainties when sampling a heterogeneouspopulation Astrophysics – used to study and characterizethe spatial distribution of galaxies37

AutocorrelationExamples of optical autocorrelation Measurement of optical spectra and of veryshort-duration light pulses produced by lasers Analysis of dynamic light scattering data todetermine particle size distributions The small-angle X-ray scattering intensity ofsome systems related to the spatialautocorrelation function of the electron density In optics, normalized autocorrelations andcross-correlations give the degree ofcoherence of an electromagnetic field38

AutocorrelationTypical use Blip sent to object Small blip reflected fromobject back to sender Use autocorrelation to detectsmall blip at some lag Know velocity of blip in medium so totaldistance blip traveled isdistance velocity * lag Distance is round trip, soobject distance/2 away39

AutocorrelationAutocorrelation Multiply and sum. Result is autocorrelation atthat point Slide over one, multiply and sum10101010xxxx000001110010 0 10 0 10 0 10 00 040

AutocorrelationAutocorrelation Repeat, sliding in both directions until havecovered all positions What happens when gopast end?10101000000111xxx00?10 0 10 0 ?41

AutocorrelationWhen go past end, have two optionsZero – padding – put zeros on both end of bothsignals. Can either imagine they are there oractually extend arrays in memory and put inzeros00000000111101010xxx0000010 0 10 0 10 0 0Will discuss second option later42

AutocorrelationSuppose the real discrete-time signal x[n] hasL points and the real discrete-time signal y[n]has N points, with L N. The autocorrelation ofx and y is𝐿 𝑚 1𝑅𝑥𝑦 𝑚 𝑥 𝑛 𝑚 𝑦[𝑛]𝑛 0for m -(N-1), -(N-2), , -1, 0, 1, 2, , L-143

AutocorrelationAside For p 0, x[n-p] is x[n]shifted to the right by p For p 0, x[n p] is x[n]shifted to the left by pExample Unit impulse1x[n]x[n 5]x[n-2]0.80.60.40.20-6-5-4-3-2-10123n1 for 𝑛 0𝑥𝑛 0 for 𝑛 044

AutocorrelationTRY ITAt time n 0 a transmitter sends out apulse of amplitude ten and duration 3.At time time n 5 it gets back thereflected pulse with the same durationbut one tenth the amplitude. What is theautocorrelation?45

AutocorrelationTRY Rxy(0) ?0011100Rxy(0) 01100Rxy(1) 01100Rxy(2) 0Rxy(1) ?0001Rxy(2) ?10000000146

AutocorrelationTRY ITRxy(-1) 0010101000001101010Rxy(-2) ?00011101010Rxy(-3) ?00111010100111101010111Rxy(-2) 000Rxy(-3) 10000Rxy(-4) 200000Rxy(-1) ?Rxy(-5) 30000001101100000Rxy(-4) ?00Rxy(-5) ?0047

AutocorrelationTRY ITRxy(-6) 20000Rxy(-6) ?101010011001010101001010000Rxy(-7) 100001Rxy(-7) ?00Rxy(-8) 00011Rxy(-8) ?0001Rxy(-9) 000000111Rxy(-9) ?10101010101048

AutocorrelationTRY ITPut it together m 2:-1:-9; R [ 0 0 0 0 0 10 20 30 20 10 0 0 ]; [ ,maxIndex] max( R )maxIndex 8 m(maxIndex)ans -5 % max when shifted right by 50000011101234567080949

AutocorrelationTRY IT plot( -m, R, ‘o’ )Note shape is a triangle, not a rectangle,which is shape of pulse. Autocorrelationdetects signal of given shape – it does notreplicate signal3025Autocorrelation(m)20151050-2024Lag m681050

AutocorrelationMATLAB considers what we’re doingto be cross-correlation Concept is same as what describedhere for autocorrelation If one array shorter than another,MATLAB pads shorter one with zerosuntil both same length51

AutocorrelationTo compute cross-correlation ofvectors x and y in MATLAB, usec xcorr( x, y )where c is a vector with 2N-1 elements N is length of longer of x and yIf m is lag as previously defined, c(k)is autocorrelation for lag m k - N52

AutocorrelationTRY ITLet’s do previous graphicalautocorrelation with MATLAB1010100000011100 x [ 10 10 10 ]; y [ 0 0 0 0 0 1 1 1 0 0 ]; c xcorr( x, y ); [ ,maxCix] max( c )maxCix 5 m maxCix - length( y )m -5 % Move 5 to right from element 153