Representation of signals and systems using MATLAB

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Representation of signals and systems using
MATLAB
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Power of MATLAB

• Matrix oriented numerical operations
• MATLAB integrates
• mathematical computing
• programming language
• graphics
• Library of MATLAB mathematical functions
• Possibility of creating new functions

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The purpose of this talk is to illustrate by
examples the application of MATLAB functions in
signal processing.
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Continuous and discrete systemsbasic
characteristics
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Impulse response h (n) Frequency response H
(ej?) Transfer function H (z) Difference
equation Block diagram
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Continuous signal in MATLAB
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Continuous signal can be represented as a
sequence of instantaneous values in discrete time
intervals tnT
Continuous signal Serbian word slika
gtgt load slika gtgt t01/8000(length(slika)-1)/8000
gtgt plot(t,slika)
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Continuous signal Serbian word slika,
spectrogram
gtgt specgram(slika,256,8000,256,128)
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Continuous signal Serbian word slika, spectrum
gtgt SL,f,tspecgram(slika,256,8000,256,128)gtgt
waterfall(t,f,abs(SL))gtgt axis(0,0.7,0,4000,0,4)
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Continuous system in MATLAB
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Continuous system
Transfer function
gtgt z,p,Cellipap(5,0.5,40) gtgt ACpoly(z) A
0.0508 0 0.2669 0
0.3092 gtgt Bpoly(p) B 1.0000 1.1536
2.0624 1.4702 0.9642 0.3092
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Continuous system complex s plane
gtgt splane(z,p)
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Continuous system magnitude response
gtgt Hafreqs(Cpoly(z),poly(p),w)gtgt
plot(w,(abs(Ha)))
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Continuous system electrical scheme
MATLAB DrawFilt Toolbox
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Discrete signal in MATLAB
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• Discrete signal time domane representation
• Sequence xn

Generated signal
gtgtxsin(2pi5(0127)/128) gtgtstem(0127,x)
Imported signal, vowel e
gtgtload e gtgtxee(10011128) gtgtstem(xe)
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• Discrete signal frequency domain representation
  spectrum

gtgtxsin(2pi5(0127)/128) gtgtXX,ffreqz(x,1,40
96,8000) gtgtplot(f,abs(XX))
Segment of the vowel e (Serbian) gtgtXefreqz(xe,1
,4096,8000) gtgtplot(f,abs(Xe))
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Discrete signal representation in frequency
domain Discrete Fourier Transform, DFT
sequence Xk
gtgtxsin(2pi5(0127)/128) gtgtXfft(x)
gtgtstem(abs(X))
Segment of the vowel e (Serbian) gtgtXEfft(xe) gt
gtstem(abs(XE))
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Discrete system in MATLAB
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Discrete systems

• Transfer function of an IIR (Infinite Impulse
  Response) system

• Transfer function of an FIR (Finite Impulse
  Response) system

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Discrete system IIR
Transfer Function
gtgt A,Bellip(5,0.5,40,0.4) A 0.0528
0.0797 0.1295 0.1295 0.0797 0.0528 B
1.0000 -1.8107 2.4947 -1.8801
0.9537 -0.2336
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Discrete system IIR
Pole-zero location in the complex z-plane gtgt
zplane(A,B)
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Discrete system IIR
Impulse response gtgt impz(A,B,50)
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Discrete system IIR
Magnitude response gtgt H,ffreqz(A,B,1024,1)gtgt
plot(f,abs(H))
Magnitude
Normalized Frequency
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Discrete system block diagram
MATLAB DrawFilt Toolbox
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Example of design and analysis of a discrete
system using MATLAB
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System design
Specifications F08000 Hz Fp1500 Hz Fs2000
Hz ?p0.01 ?s0.01 Linear phase
Specifications can be met with an optimal FIR
filter.
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Design and analysis of an FIR system
gtgt N,fo,mo,w remezord( 1500 2000, 1 0,
0.01 0.01, 8000 )gtgt a remez(N,fo,mo,w)gtgt
stem(0N,a)
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Design and analysis of an FIR system
gtgt zplane(a,1)
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Design and analysis of an FIR system
gtgt H,ffreqz(a,1,1024,8000)gtgt plot(f,abs(H))
Magnitude
Frequency Hz
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Processing of discrete signal using MATLAB
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Discrete system processing of the signal
gtgt A,Bellip(5,0.5,40,0.4) gtgt
yefilter(A,B,xe)
A,B
ye(n)
xe(n)
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Discrete system processing of the signal
A,B
xe(n)
ye(n)
Spectrogram vowel e
gtgt specgram(xe)
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Discrete system processing of the signal
A,B
ye(n)
xe(n)
Spectrogram, vowel e at the output of the
system
gtgt specgram(ye)
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Multirate systemsusing MATLAB
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Multirate systemsbasic processing

• Decimation
• Interpolation
• Efficient filtering EMF Toolbox
• Filter banks

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Multirate systems
Signal compression down-sampling
gtgt xsin(2pi5(063)/128) gtgt xdx(1463)
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Multirate systems
Signal expansion up-sampling
L
xd(n)
xu(n)
gtgt xuzeros(1,4length(xd))gtgt
xu(14length(xu))xd
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Multirate systems
Interpolator, example 1
gtgt yinterp(xd,4)
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Multirate systems
Interpolator, example 2
gtgt A,Bellip(5,0.5,40,0.20)gtgt
yfilter(A,B,xu)
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Modulations and demodulations using MATLAB
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• Modulations
• gtgt y modulate(x,fc,fs,METHOD,opt)
• Amplitude modulations amdsb-sc, amdsb-tc, amssb
• Frequency modulation fm
• Phase modulations ppm, pwm
• Quadrature amplitude modulation qam

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• Demodulations
• gtgt y demod(x,fc,fs,METHOD,opt)
• Amplitude modulations amdsb-sc, amdsb-tc, amssb
• Frequency modulations fm
• Phase modulations ppm, pwm
• Quadrature amplitude modulation qam

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Illustrative example SSB modulation of the
sequence representing Serbian word
zadivljenost gtgt load zadivljenost gtgt zad
zadivljenost gtgt y modulate(zad,0.25,1,'amssb')

Time-domain presentation
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Illustrative example Serbian word zadivljenost
spectrum analysis
gtgt load zadivljenost gtgt zad zadivljenost gtgt
Zspecgram(zad,256,1,256,128) gtgt
n08000/618000-8000/61 f01/2560.5 gtgt
waterfall(n',f',abs(Z)), axis(0,8000,0,0.5,0,2)
Frequency-domain presentation
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Illustrative example Serbian word zadivljenost
spectrum analysis of the SSB modulated signal gtgt
y modulate(zad,0.25,1,'amssb') gtgt
ZYspecgram(y,256,1,256,128) gtgt
n08000/618000-8000/61 f01/2560.5 gtgt
waterfall(n',f',abs(ZY)), axis(0,8000,0,0.5,0,2
)
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Illustration of solving a complex design problem
in MATLAB
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Solution to be presented Design of high-speed
low-sensitivity IIR filtersimplemented as tapped
cascaded interconnection of all-pass sub-filters
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Basic references 1 T. Saramäki and M. Renfors,
A novel approach for the design IIR filters as a
tapped cascaded interconnection of identical
allpass subfilters, in Proc. 1987 IEEE Int.
Symp. Circuits Syst. (Philadelphia,
Pennsylvania), vol. 2, pp. 629?632, May 1987.
2 H. Johansson and L. Wanhammar, High-speed
recursive filter structures composed of identical
all-pass subfilters for interpolation,
decimation, and QMF banks with perfect magnitude
reconstruction, IEEE Trans. Circuits and
Systems-II Analog and Digital Signal Processing,
vol. 46, no. 1, pp. 16?28, January 1999. 3 Lj.
Milic and M. Lutovac, "High speed IIR filters for
QMF banks," TELSIKS, Ni, Yugoslavia, 2001, pp.
171-174. 4 M. D. Lutovac and Lj. Milic, Design
of optimal halfband FIR filters with minimum
phase, ETRAN Conference, June 2003, Herceg-Novi,
Montenegro, Avilable http//kondor.etf.bg.ac.yu/
lutovac/pdf/etan03lm.pdf
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• FDFDesignMATLAB program for designing high-speed
  half-band digital filters (Lutovac and Milic,
  2003.)
• Start-up prototype filter is a minimum phase
  half-band FIR filter 1-3
• Transformation based on all-pass sub-filters
  from an IIR prototype 1-3
• For solving numerical problems in computing
  coefficients of the half-band FIR prototype, new
  original expressions have been derived using
  Mathematica 4

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Transformation of an FIR filter using the
all-pass sectionsfrom an IIR prototype
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Start program FDFDesign
gtgtFDFDesign
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gtgtFDFDesign n2, fp0.2
Half-band filter design
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Programmable power-complementary filter
pairs Using the half-band filter solution from
FDFDesign, and simple approach given in 5 and
6, low-pass/high-pass power-complementary
filter pairs with a programmable crossover
frequency can be easily obtained. 5 L. Milic,
and T. Saramaki, Three classes of IIR
complementary filter pairs with an adjustable
crossover frequency, Circuits and Systems, 2003.
ISCAS '03. Proceedings of the 2003 International
Symposium on, Volume 4, pp. 145 148, May 25 -
28, 2003. 6 L. Milic, and T. Saramaki,
Power-complementary IIR filter pairs with an
adjustable crossover frequency, Facta
Universitatis, Ser. Elec. Energ. vol. 16, pp.
295-304, Dec.2003.
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Example of programmable power-complementary filter
pair. New filter pair is obtained by computing
only three constants in all-pass branches.
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Thank you for your attention!