matlab signal processing toolbox - miami...

MATLAB Signal

Processing Toolbox

Greg Reese, Ph.D

Research Computing Support Group

Academic Technology Services

Miami University

October 2013

MATLAB Signal

Processing Toolbox

© 2013 Greg Reese. All rights reserved 2

Toolbox

3

Toolbox

• Collection of code devoted to solving

problems in one field of research

• Can be purchased from MATLAB

• Can be purchased from third parties

• Can be obtained for free from third parties

Toolbox

4

MATLAB Signal Processing Toolbox

• Code for solving problems in signal

processing (!)

• Sold by MATLAB

• Part of both Miami’s student and

faculty license

Overview

5

MATLAB divides Signal Processing

Toolbox as follows

• Waveforms – Pulses, modulated signals, peak-to-peak and RMS

amplitude, 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, Hilbert

transform, cepstrum, Walsh-Hadamard transform

Overview

6

• Analog and Digital Filters – Analog filter design, frequency transformations, FIR and IIR

filters, filter analysis, filter structures

• Spectral Analysis – Nonparametric and parametric spectral estimation, high

resolution spectral estimation

• Parametric Modeling and Linear

Prediction – Autoregressive (AR) models, linear predictive coding (LPC),

Levinson-Durbin recursion

• Multirate Signal Processing – Downsampling, upsampling, resampling, anti-aliasing filter,

interpolation, decimation

Overview

7

Will look very briefly at

• Analog and Digital Filters

• Spectral Analysis

• Parametric Modeling and Linear Prediction

• Multirate Signal Processing

Will look in more depth at

• Waveforms

• Convolution and Correlation

Analog and Digital Filters

8

Toolbox especially good for those serious

about 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 transformation


9

Digital 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, Hilbert


10

Digital Filter Design interactively (GUI)

• Filterbuilder – specify desired

characteristics 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/phase

response, pole-zero plots


11

SPTool – composite of four tools

1. Signal Browser – analyze signals

2. FDATool

3. FVTool (Filter Visualization Tool) –

analyze filter characteristics

4. Spectrum Viewer – spectral analysis


12

Digital Filter Analysis

• Magnitude and phase response, impulse

response, group delay, pole-zero plot

Digital Filter Implementation

• Filtering, direct form, lattice, biquad, state-

space structures

Spectral Analysis

13

Nonparametric Methods – Periodogram, Welch's overlapped segment

averaging, multitaper, cross-spectrum,

coherence, spectrogram

Parametric Methods – Yule-Walker, Burg, covariance, modified

covariance

Subspace Methods – Multiple signal classification (MUSIC),

eigenvectorestimator, pseudospectrum

• Windows – Hamming, Blackman, Bartlett, Chebyshev,

Taylor, Kaiser

Parametric Modeling and

Linear Prediction

14

Parametric Modeling

• AR, ARMA, frequency response

modeling

Linear Predictive Waveforms

• Linear predictive coefficients (LPC), line

spectral frequencies (LSF), reflection

coefficients (RC), Levinson-Durbin

recursion

Multirate Signal Processing

15

Multirate signal processing

– Downsampling, upsampling, resampling,

anti-aliasing filter, interpolation, decimation

Waveforms

16

Waveforms part of toolbox lets you create

many commonly used signals, which you

can use to study models programmed in

MATLAB

Uses of waveforms

• Testing – E.g., have simple waveform and can analytically

determine model’s output. Use toolbox to create that

waveform, run it through MATLAB model, confirm

result

Waveforms

17

Uses of waveforms

• Simulation

– Most of time can’t get analytical output

– Make waveform of known characteristics

and study model’s response

• Modeling of real signals

– Create waveform that looks like the real

signal

Waveforms

18

Time vectors

Digital signals usually sampled from

analog at fixed intervals Δt . Want time

axis with N points

0 1Δt 2Δt … (N-2)Δt (N-1)Δt

Waveforms

19

For – tStart: starting time

– tEnd: ending time

– N: number of points

– deltaT: 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 5.5 5.6 5.7 5.8 5.9

Waveforms

20

• If have starting time, ending time, interval

(tStart, tEnd, deltaT) >> tStart = 5;

>> tEnd = 5.9;

>> deltaT = 0.1;

>> t = tStart:deltaT:tEnd

t = 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

• If have starting time, ending time, number of

points (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 5.6 5.7 5.8 5.9

Waveforms

21

In multichannel processing, a number of

signals are gathered at the same time

• Will assume all sampled at same time and

same rate

Signal processing toolbox, and MATLAB

in general, treats each column of a

matrix (2D array) as an independent column vector

Waveforms

22

Example >> M = [ 1:3; 4:6; 7:9; 10:12 ]

M = 1 2 3

4 5 6

7 8 9

10 11 12

>> mean( M )

ans =

5.5000 6.5000 7.5000

Result is mean of each column

Waveforms

23

TIP

Make time vector be a column vector

• Any vectors created from time vector

will also be column vectors and so

can be processed more easily >> t = t >> y = abs( t - 3 )

t = 1 y = 2

2 1

3 0

4 1

5 2

6 3

Column vector begeteth column vector

Waveforms

24

TIP

repmat (replicate matrix) is general

purpose function to make large matrix by

replicating small one

Trick - quick way to replicate column vector,

i.e., to make an m x n matrix T out of an m x

1 column vector v, is

T = v(:,ones(1,n))

>> v = (1:5) >> v(:,ones(1,3))

v = 1 ans = 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

Waveforms

25

TIP

Trick - quick way to replicate row vector,

i.e., to make an m x n matrix T out of an 1 x

n row vector v, is

T = v(ones(m,1),:)

>> v = 1:3 >> T = v( ones(6,1), : )

v = 1 2 3 ans = 1 2 3

2 1 2 3

3 1 2 3

4 1 2 3

5 1 2 3

1 2 3

Waveforms

26

TIP

Can use preceding two tips to make

multichannel signal, e.g.,

Simulate the multichannel signal

sin(2πt), sin(2πt/2), sin(2πt/3),

sin(2πt/4) sampled for one second

at one-tenth second per sample

Waveforms

27

TIP >> t = (0:0.1:1)'

t = 0

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

Waveforms

28

TIP >> T = t(:,ones(1,4))

T = 0 0 0 0

0.1000 0.1000 0.1000 0.1000

0.2000 0.2000 0.2000 0.2000

0.3000 0.3000 0.3000 0.3000

0.4000 0.4000 0.4000 0.4000

0.5000 0.5000 0.5000 0.5000

0.6000 0.6000 0.6000 0.6000

0.7000 0.7000 0.7000 0.7000

0.8000 0.8000 0.8000 0.8000

0.9000 0.9000 0.9000 0.9000

1.0000 1.0000 1.0000 1.0000

Waveforms

29

TIP >> M = T ./ C

M =

0 0 0 0

0.1000 0.0500 0.0333 0.0250

0.2000 0.1000 0.0667 0.0500

0.3000 0.1500 0.1000 0.0750

0.4000 0.2000 0.1333 0.1000

0.5000 0.2500 0.1667 0.1250

0.6000 0.3000 0.2000 0.1500

0.7000 0.3500 0.2333 0.1750

0.8000 0.4000 0.2667 0.2000

0.9000 0.4500 0.3000 0.2250

1.0000 0.5000 0.3333 0.2500

Waveforms

30

TIP >> signal = sin( 2*pi*M )

signal =

0 0 0 0

0.5878 0.3090 0.2079 0.1564

0.9511 0.5878 0.4067 0.3090

0.9511 0.8090 0.5878 0.4540

0.5878 0.9511 0.7431 0.5878

0.0000 1.0000 0.8660 0.7071

-0.5878 0.9511 0.9511 0.8090

-0.9511 0.8090 0.9945 0.8910

-0.9511 0.5878 0.9945 0.9511

-0.5878 0.3090 0.9511 0.9877

-0.0000 0.0000 0.8660 1.0000

Waveforms

31

Impulse • Use to compute impulse response

of linear, time-invariant system >> t = (0:0.1:1.9)';

>> impulse = zeros( size(t) );

>> impulse( 1 ) = 1;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Impulse

Waveforms

32

Step • Use to model switch turning on

>> t = (-1:0.1:0.9)';

>> step = [ zeros(10,1); ones(10,1) ];

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Step

Waveforms

33

Ramp • Use to model something gradually turning on

>> t = (0:0.1:1.9)';

>> ramp = t;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Ramp

Autocorrelation

34

Autocorrelation – the detection of a delayed

version of a signal

• In temporal signal, delay often called “lag”

• In spatial signal, delay often called

“translation” or “offset”

• Delayed signal may also be scaled

Can also think of autocorrelation as

similarity of a signal to itself as a function of

lag

Autocorrelation

35

Autocorrelation and cross-correlation

• Common in both signal processing

and statistics

• Definitions and uses are different

• When looking for help on these

topics, make sure you’re looking at

a signal-processing source

Autocorrelation

36

Examples 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 pulsars

Autocorrelation

37

Examples of spatial autocorrelation

• Optical Character Recognition (OCR) – reading

text from images of writing/printing

• X-ray diffraction – helps recover the "Fourier

phase information" on atom positions

• Statistics – helps estimate mean value

uncertainties when sampling a heterogeneous

population

• Astrophysics – used to study and characterize

the spatial distribution of galaxies

Autocorrelation

38

Examples of optical autocorrelation

• Measurement of optical spectra and of very

short-duration light pulses produced by lasers

• Analysis of dynamic light scattering data to

determine particle size distributions

• The small-angle X-ray scattering intensity of

some systems related to the spatial

autocorrelation function of the electron density

• In optics, normalized autocorrelations and

cross-correlations give the degree of

coherence of an electromagnetic field

Autocorrelation

39

Typical use

• Blip sent to object

• Small blip reflected from

object back to sender

• Use autocorrelation to detect

small blip at some lag

• Know velocity of blip in medium so total

distance blip traveled is

distance = velocity * lag

• Distance is round trip, so

object distance/2 away

Autocorrelation

40

Autocorrelation

• Multiply and sum. Result is autocorrelation at

that point

• Slide over one, multiply and sum

10 10 10

0 0 0 0 0 1 1 1 0 0

x x x

10∙0 + 10∙0 + 10∙0 = 0

10 10 10

x x x

10∙0 + 10∙0 + 10∙0 = 0

Autocorrelation

41

Autocorrelation

• Repeat, sliding in both directions until have

covered all positions

• What happens when go

past end?

10 10 10

0 0 0 0 0 1 1 1 0 0

x x x

10∙0 + 10∙0 + ?

?

Autocorrelation

42

When go past end, have two options

Zero – padding – put zeros on both end of both

signals. Can either imagine they are there or

actually extend arrays in memory and put in

zeros

Will discuss second option later

0 10 10 10 0

0 0 0 0 0 0 0 1 1 1 0 0 0 0

x x x

10∙0 + 10∙0 + 10∙0 = 0

Autocorrelation

43

Suppose the real discrete-time signal x[n] has

L points and the real discrete-time signal y[n]

has N points, with L ≤ N. The autocorrelation of

x and y is

𝑅𝑥𝑦 𝑚 = 𝑥 𝑛 + 𝑚 𝑦[𝑛]

𝐿−𝑚−1

𝑛=0

for m = -(N-1), -(N-2), …, -1, 0, 1, 2, …, L-1

Autocorrelation

44

Aside

• 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 p

Example

• Unit impulse

𝑥 𝑛 = 1 for 𝑛 = 00 for 𝑛 ≠ 0

-6 -5 -4 -3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

n

x[n]

x[n+5]

x[n-2]

Autocorrelation

45

TRY IT

At time n=0 a transmitter sends out a

pulse of amplitude ten and duration 3.

At time time n=5 it gets back the

reflected pulse with the same duration

but one tenth the amplitude. What is the

autocorrelation?

Autocorrelation

46

TRY IT

Sent

Received

Rxy(0)=0

Rxy(1)=0

Rxy(2)=0

10 10 10

0 0 0 0 0 1 1 1 0 0

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(0) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(1) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(2) = ?

Autocorrelation

47

TRY IT

Rxy(-1)=0

Rxy(-2)=0

Rxy(-3)=10

Rxy(-4)=20

Rxy(-5)=30

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-1) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-2) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-3) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-4) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-5) = ?

Autocorrelation

48

TRY IT

Rxy(-6)=20

Rxy(-7)=10

Rxy(-8)=0

Rxy(-9)=0

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-6) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-7) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-8) = ?

10 10 10

0 0 0 0 0 1 1 1 0 0

Rxy(-9) = ?

Autocorrelation

49

TRY IT

Put 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 5

0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 1 1 1 0 0

Autocorrelation

50

TRY IT >> plot( -m, R, ‘o’ )

Note shape is a triangle, not a rectangle,

which is shape of pulse. Autocorrelation

detects signal of given shape – it does not

replicate signal

-2 0 2 4 6 8 100

5

10

15

20

25

30

Lag m

Auto

corr

ela

tion(m

)

Autocorrelation

51

MATLAB considers what we’re doing

to be cross-correlation

• Concept is same as what described

here for autocorrelation

• If one array shorter than another,

MATLAB pads shorter one with zeros

until both same length

Autocorrelation

52

To compute cross-correlation of

vectors x and y in MATLAB, use

c = xcorr( x, y )

where

• c is a vector with 2N-1 elements

• N is length of longer of x and y

If m is lag as previously defined, c(k)

is autocorrelation for lag m = k - N

Autocorrelation

53

TRY IT

Let’s do previous graphical

autocorrelation with MATLAB

>> 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 1

10 10 10

0 0 0 0 0 1 1 1 0 0

Autocorrelation

54

TRY IT

Example of finding signal buried in noise

1. Make a sine wave with a period of 20 and

amplitude of 100 >> wave = 100 * sin( 2*pi*(0:19)/20 );

2. Reset random number generator

(so we all get the same random numbers) >> rng default

3. Make 500 points of noise with randn and

variance 75% of wave amplitude noisyWave = 75 * randn( 1, 500 );

Autocorrelation

55

TRY IT

4. Pick a random spot to place the wave,

ensuring that the whole wave fits in ix = randi( [ 1, 481 ] );

5. Add the wave to the noise noisyWave(ix:ix+19)=noisyWave(ix:ix+19)+wave;

Plot wave in noise. Is wave visible? plot( noisyWave )

6. Compute autocorrelation

>> c = xcorr( wave, noisyWave );

0 50 100 150 200 250 300 350 400 450 500-300

-200

-100

0

100

200

300

Autocorrelation

56

TRY IT

7. Find max of autocorrelation and

calculate lag from that >> [~,maxIx] = max( c )

maxIx = 269

>> m = maxIx – 500

m = -231 % shift 231 to right

8. Show random spot where wave added

to noise. Match? >> ix

ix = 231

Very close match! m should be -230

Autocorrelation

57

TRY IT

9. For grins, plot autocorrelation >> lags = -499:499;

>> plot( lags, c )

Why is almost

all of right

size zero? • Right side

corresponds to

shifting left and once

shift wave more than

20, rest of wave

is zeros -500 -400 -300 -200 -100 0 100 200 300 400 500-8

-6

-4

-2

0

2

4

6

8x 10

4

Correlation

58

Questions?

Convolution

59

convolution

• Uses – Polynomial multiplication

– LTI response

– Joint PDF

• Linear and circular – Explain, show when equivalent (padding), good for

computing convolution with fft. Do example with

tic,toc, time-domain convolution vs. fft,ifft, see cconv

Convolution

60

Applications of convolution

• Acoustics – reverberation is the convolution of

the original sound with echoes from objects

surrounding the sound source

• Computational fluid dynamics – the large eddy

simulation (LES) turbulence model uses

convolution to lower the range of length scales

necessary in computation and thereby reducing

computational cost

• Probability – probability distribution of the sum

of two independent random variables is the

convolution of their individual distributions

Convolution

61

Applications of convolution

• Spectroscopy – line broadening can be due to

the Doppler effect and/or collision broadening.

The effect due to both is the convolution of the

two effects

• Electronic music – imposition of a rhythmic

structure on a sound done by convolution

• Image processing – blurring, sharpening, and

edge enhancement done by convolution

• Numerical computation – can multiply

polynomials quickly with convolution

Convolution

62

Convolution finds many applications

because it is central to linear, time-

invariant systems and many things

can be modeled by such systems

Convolution

63

Linear – a linear system obeys two principles – Principle of superposition – the output to a sum of

inputs is equal to the sum of the outputs to the

individual inputs

– Scaling – the output to the product of an input and a

constant is the product of the constant and the output

to the input alone

– In other words, for a linear system L,

a L{ x(t) } + bL{ x(t) } = L{ ax(t) + bx(t) }

Convolution

64

Suppose you put some input into a

system and get some output. If you put

in the same input at a later time, if the

system is time invariant, the output will

be the same as the original output

except it will occur at that later time – In other words, for a time-invariant system S,

If y(t) = S{ x(t) }, then y(t-t0) = S{ x(t-t0) }

Time-invariance and linearity are independent. A

linear system can be time-invariant or not. A time-

invariant system can be linear or not.

Convolution

65

Example

Change machine at a laundromat.

Put in dollar bills, press button, get out

quarters

Linear?

1. Put $1 in, press button, get 4 quarters out


3. (output from $1) + (output from $2) = 12

quarters

4. Put $3, press button, get 12 quarters out

5. Two outputs equal, so system linear

Convolution

66

Example

Time invariant?



An hour later



Outputs identical except for same delay as

input, so system is time invariant

Convolution

67

(discrete) impulse: δ 𝑛 = 1 for 𝑛 = 00 for 𝑛 > 0

impulse response –response h[n] of a system S when the input is an impulse,

i.e.,

h[n] = S { δ 𝑛 }

Convolution

68

Major fact

The output of a linear, time-invariant

(LTI) system to any input is the

convolution of that input with the

system’s impulse response

Other words:

• The impulse response of an LTI system

completely characterizes that system

• The impulse response of an LTI system

specifies that system

Convolution

69

Graphical view of convolving two signals

• Pick one signal

• Flip it 180° around left edge

• Position right element of flipped signal

over left element of unflipped signal

1 0 2 2 5 -2 -2 0 0 1

3 2 1 1 2 3

3 2 1

1 2 3

Convolution

70

Graphical view

• Multiply corresponding elements and sum

3

3∙1

x

1 2 3

1 0 2 2 5 -2 -2 0 0 1

Convolution

71

Graphical view

• Slide right, multiply, sum

3 2

2∙1 + 3∙0

x

1 2 3

x

1 0 2 2 5 -2 -2 0 0 1

Convolution

72

Graphical view

• Repeat until “fall off” right side

3 2 7

1∙1 + 2∙0 + 3∙2

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

73

Graphical view

3 2 7 10

1∙0 + 2∙2 + 3∙2

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

74

Graphical view

3 2 7 10 21

1∙2 + 2∙2 + 3∙5

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

75

Graphical view

3 2 7 10 21 6

1∙2 + 2∙5 + 3∙(-2)

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

76

Graphical view

3 2 7 10 21 6 -5

1∙5 + 2∙(-2) + 3∙(-2)

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

77

Graphical view

3 2 7 10 21 6 -5 -6

1∙(-2) + 2∙(-2) + 3∙0

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

78

Graphical view

3 2 7 10 21 6 -5 -6 -2

1∙(-2) + 2∙0 + 3∙0

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

79

Graphical view

3 2 7 10 21 6 -5 -6 -2 3

1∙0 + 2∙0 + 3∙1

x

1 2 3

x x

1 0 2 2 5 -2 -2 0 0 1

Convolution

80

Graphical view

• As with correlation, ignore elements that have

fallen off or pad bottom array with zeros

3 2 7 10 21 6 -5 -6 -2 3 2

1∙0 + 2∙1

x

1 2 3

x

1 0 2 2 5 -2 -2 0 0 1

Convolution

81

Graphical view

3 2 7 10 21 6 -5 -6 -2 3 2 1

1∙1

1 2 3

x

1 0 2 2 5 -2 -2 0 0 1

Convolution

82

So convolution of

with

gives

3 2 7 10 21 6 -5 -6 -2 3 2 1

3 2 1

1 0 2 2 5 -2 -2 0 0 1

Convolution

83

Note that when we convolved with

the shorter

signal was not completely on top of the longer

one for the first two and last two elements of the

longer. For these cases, the multiply-and-add

convolution computation was missing either one

or two terms, so those four calculations are not

valid and should be ignored. Bad values at the

left and right of a convolution are known as edge

effects.

3 2 1

1 0 2 2 5 -2 -2 0 0 1

Convolution

84

In general, if the shorter of two signals

in a convolution has M elements, you

should ignore the first and last M-1

elements in the result

Convolution

85

Mathematical definition

Suppose we have two signals – x[n] has M

points and y[n] has N points, M ≤ N. The

convolution of x[n] and y[n] is

𝑤 𝑛 =

𝑥 𝑛 − 𝑘 𝑦 𝑘 for 𝑛 = 0,1,2, …𝑀 − 1

𝑛

𝑘=0

𝑥 𝑛 − 𝑘 𝑦[𝑘]

𝑛

𝑘=𝑛−𝑀+1

for 𝑛 = 𝑀,𝑀 + 1,…𝑁 − 1

𝑥 𝑛 − 𝑘 𝑦 𝑘 for 𝑛 = 𝑁,𝑁 + 1,…𝑁 + 𝑀 − 2

𝑛

𝑘=𝑛−𝑀+1

n = 0, 1, 2, … N+M-2

Convolution

86

Compute a convolution in MATLAB with

w = conv( u, v )

where

u and v are vectors. The output vector w

has length length(u)+length(v)-1

Convolution

87

TRY IT

We figured out graphically that

convolved with

gave

Try it in MATLAB >> u = [ 3 2 1 ];

>> v = [ 1 0 2 2 5 -2 -2 0 0 1 ];

>> w = conv( u, v )

w = 3 2 7 10 21 6 -5 -6 -2 3 2 1

3 2 7 10 21 6 -5 -6 -2 3 2 1

3 2 1

1 0 2 2 5 -2 -2 0 0 1

Convolution

88

Typically convolution involves a data

signal and another signal. The second

signal, called a kernel, is

• Also called a filter

• Usually much shorter than the data signal

• Designed to produce a desired effect on

the data

Convolution

89

TRY IT

To introduce a time lag of m units

into the data signal, i.e., to shift it

to the right by m, use a kernel of m

zeros followed by a one.

Example

Introduce a lag of two into the signal

>> kernel = [ 0 0 1 ];

>> w = conv( kernel, v )

w = 0 0 1 0 2 2 5 -2 -2 0 0 1

1 0 2 2 5 -2 -2 0 0 1

Convolution

90

TRY IT

Approximate the derivative by

replacing a point with the difference

between itself and the previous point

Example

Approximate the derivative of

>> u = [ 1 -1 ];

>> w = conv( u, v )

w = 1 -1 2 0 3 -7 0 2 0 1 -1

1 0 2 2 5 -2 -2 0 0 1

Convolution

91

TRY IT

Reduce noise in a signal by replacing

a point with a weighted average of

itself and the previous two points

Example

>> kernel = (1/9)*[ 5 3 1 ];

>> w = conv( u, v );

w*9 = 5 3 11 16 33 7 -11 -8 -2 5 3 1

1 0 2 2 5 -2 -2 0 0 1

Convolution

92

Questions?

Signal Processing

Toolbox

93

Questions?

94

The End

matlab signal processing toolbox - miami...

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