Study of Signals, Systems and Transforms. Design of FIR and IIR Filters. Advanced tools for Signal Processing

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A Hands-on Training Session on MATLAB for Signal Processing In Connection with the FDP on Electronic Design Tools @ GCE Kannur 11 th 15 th December 2017 Resource Person : Dr. A.

1 A Hands-on Training Session on MATLAB for Signal Processing In Connection with the FDP on Electronic Design GCE Kannur 11 th 15 th December 2017 Resource Person : Dr. A. Ranjith Ram Associate Professor, ECE Dept. Govt. College of Engineering Kannur Cell : arr@gcek.ac.in Objectives : Study of Signals, Systems and Transforms Design of FIR and IIR Filters Advanced tools for Signal Processing Study of Signals, Systems and Transforms 30 Minutes Design of FIR and IIR Filters 30 Minutes Advanced tools for Signal Processing 30 Minutes 2 Dr. A. Ranjith Ram arr@gcek.ac.in 1

2 Outline Signals in MATLAB Environment Correlation & Convolution Laplace Transform & Z-Transform Circular Convolution and Parseval s Theorem Systems Impulse Response & Frequency Response Transfer Function and Pole-Zero Plots Design of FIR and IIR Filters Advanced Tools : wvtool, fvtool, fdatool and sptool 3 ECE Toolboxes of Interest Signal Processing Toolbox Communications System Toolbox Computer Vision System Toolbox DSP System Toolbox Wavelet Toolbox Image Acquisition Toolbox Image Processing Toolbox Data Acquisition Toolbox Fuzzy Logic Toolbox Symbolic Math Toolbox Statistics Toolbox Phased Array System Toolbox Neural Network Toolbox Optimization Toolbox RF Toolbox Control System Toolbox 4 Dr. A. Ranjith Ram arr@gcek.ac.in 2

3 Useful Built-in Functions exp() filter() dirac() conv() sinc() cconv() erf() impz erfc() step() gamma() freqz() xcorr() fir1() corr() firpm() laplace() butter() ztrans() buttord() fft() cheby1() dct() cheby2() hilbert() sptool() 5 Signal Input to MALAB Three methods of inputting signals to MATLAB : generated by using MATLAB functions itself from memory read an audio file read a speech file from I/O devices input from a sensor input from an instrument Input from a microphone 6 Dr. A. Ranjith Ram arr@gcek.ac.in 3

4 Signal Input Examples x = rand(1, 200); [x fs] = wavread( aud.wav ); S = load( filemname ); import menu import a file 7 Signal Output from MATLAB Three methods of outputting signals from MATLAB : directly showing in the command window to memory create and write an audio file from a MATLAB variable write an array of samples from a MATLAB variable to I/O devices output to a DAC output to a loudspeaker output to a display 8 Dr. A. Ranjith Ram arr@gcek.ac.in 4

$Signal Output Examples sprintf( The signal is %d', x) or display(x) fid = fopen('exp.txt','w'); fprintf(fid,'%6.2f %12.8f\n',y); fclose(fid); wavwrite(s,fs, au.$

5 Signal Output Examples sprintf( The signal is %d', x) or display(x) fid = fopen('exp.txt','w'); fprintf(fid,'%6.2f %12.8f\n',y); fclose(fid); wavwrite(s,fs, au.wav ) save( filename, var) save menu / sound(s,fs) 9 Generating Basic Analog Signals >> t = 1:20; >> a = 0.2; >> x = exp(a*t); >> plot(x); Define the time span Define other parameters Generate the signal and plot it Generate & plot an analog signum function Generate & plot an analog sinc function Generate & plot an analog gaussian function 10 Dr. A. Ranjith Ram arr@gcek.ac.in 5

6 Generating Basic Discrete Signals >> n = 1:20; >> a = 0.2; >> x = exp(a*n); >> stem(x); Define sampling instants Define other parameters Generate the signal and plot it Generate & plot a discrete signum function Generate & plot a discrete sinc function Generate & plot a discrete gaussian function 11 Correlation of two Signals Cross Correlation Function : c = xcorr(x,y) where x and y are having a length M (M > 1) and c is a 2*M 1 sequence If x and y are of different length, the shortest one is zero-padded c will be a row vector if x is a row vector, and c will be a column vector if x is a column vector >> x = 1:5 >> y = 9:-1:5 >> c = xcorr(x,y) yields c = [ ] 12 Dr. A. Ranjith Ram arr@gcek.ac.in 6

7 Autocorrelation function ACF is given by c = xcorr(x), where x is a vector when x is an M x N matrix, is a large matrix with 2*M 1 rows whose N^2 columns contain the cross-correlation sequences for all combinations of the columns of x The zero th lag of the output correlation is in the middle of the sequence, at element or row M >> x = 1:5 >> c = xcorr(x) yields c = [ ] 13 Convolution of two Signals Convolution and polynomial multiplication : conv(x,y) c = conv(x, y) convolves vectors x and y c is a vector of length length(x) + length(y) 1 If x and y are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials >> x = 1:5 >> y = 9:-1:5 >> c = conv(x,y) yields c = [ ] 14 Dr. A. Ranjith Ram arr@gcek.ac.in 7

Transforms Laplace & Z The Laplace Transform gives the representation of a continuous time signal/system in s domain The Z Transform gives the representation of a discrete signal/system in z domain

8 Transforms Laplace & Z The Laplace Transform gives the representation of a continuous time signal/system in s domain The Z Transform gives the representation of a discrete signal/system in z domain One can say that t is transformed to s and n is transformed to z. Both s & z are complex variables Both needs no numerical computation, but an integration/summation of a signal/sequence To cope with such a situation, MATLAB supports another type of variable the symbolic variable 15 The Symbolic Math Toolbox In MATLAB, symbolic variables are also possible, which do not possess any values, but only symbolic in nature Symbolic Math Toolbox provides functions for solving and manipulating symbolic math expressions and performing variable-precision arithmetic One can analytically perform differentiation, integration, simplification, transforms, and equation solving. Also, one can generate code for MATLAB and Simulink from symbolic math expressions. >> syms x >> diff(sin(x^2)) >> ans = 2*x*cos(x^2) 16 Dr. A. Ranjith Ram arr@gcek.ac.in 8

9 Laplace Transform Laplace Transform : laplace(); Inverse LT : ilaplace() >> syms t n >> f = t^2; >> disp(f); >> F = laplace(f); >> disp(f); >> disp('inverse Laplace transform is') >> f1 = ilaplace(f); >> disp(f1); 17 Z-Transform Z-Transform : ztrans(); Inverse ZT : iztrans() >> x = 0.5^n; >> disp(x); >> X = ztrans(x); >> disp(x); >> disp('inverse Z-Transform is'); >> x1 = iztrans(x); >> disp(x1); 18 Dr. A. Ranjith Ram arr@gcek.ac.in 9

10 Discrete Fourier Transform Discrete (Fast) Fourier Transform : fft(); Inverse DFT : ifft() fft(x) is the discrete Fourier transform (DFT) of vector x For matrices, the fft operation is applied to each column fft(x, N) is the N-point FFT, padded with zeros if x has less than N points and truncated if it has more. >> n = 1:50; >> x = cos((pi/8)*n) + cos((pi/20)*n); plot(x) >> X = fft(x); stem(abs(x)) >> xcap = ifft(x); plot(xcap) 19 Circular Convolution Modulo-N circular convolution : cconv() c = cconv(x, y, N) circularly convolves vectors x and y. c is of length N. If omitted, N defaults to length(x) + length(y) 1 When N = length(x) + length(y) 1, the circular convolution is equivalent to the linear convolution computed by conv() >> a = [ ]; >> b = [ ]; >> c = cconv(a,b,11) >> cref = conv(a,b) 20 Dr. A. Ranjith Ram arr@gcek.ac.in 10

11 Parseval s Theorem The energy of a signal/sequence is a constant, irrespective of the domain Steps: Find out the energy in time domain Transform the signal to frequency domain Find out the energy of the spectra Both values should be the same >> x = [ ]; >> e = sum(x.^2); % Energy from samples >> X = fft(x); >> E = sum(abs(x).^2)/8; % Energy from spectra 21 Discrete Time Systems Impulse Response Discrete Impulse Response : impz(b,a) b : numerator vector; a : denominator vector >> b=[ ]; >> a=[ ]; >> h = impz(b,a); % Impulse Response, h(n) >> stem(h); >> grid on; >> title('impulse Response'); >> ylabel('response'); >> xlabel('n'); 22 Dr. A. Ranjith Ram arr@gcek.ac.in 11

12 Systems Transfer Function >> sys = tf(b,a) creates a continuous-time transfer function SYS with numerator b and denominator a. >> sys = tf(b,a,ts) creates a discrete-time transfer function with sampling time TS (set TS = -1 if the sampling time is undetermined). >> b = [ ]; >> a = [ ]; >> sys = tf(b,a) sys = s^ s^ s^ s^ s s^ s^ s^ s^ s Systems Frequency Response [H,W] = freqz(b,a,n) returns the N-point complex frequency response vector H and the N-point frequency vector W in radians/sample of the digital system >> b = [ ]; >> a = [ ]; >> freqz(b,a) 24 Dr. A. Ranjith Ram arr@gcek.ac.in 12

Alternate Method >> b = [0.0013 0.0064 0.0128 0.0128 0.0064 0.0013]; >> a = [1-2.9754 3.8060-2.5453 0.8811-0.

13 Alternate Method >> b = [ ]; >> a = [ ]; >> h = impz(b,a); N = 1024; >> sh = fft(h,n); >> f = [0:N/2-1]*2/N; % Frequency Normalization >> subplot(2 1 1) >> plot(f,20*log10(abs(sh(1:n/2)))); >> p=unwrap(angle(sh))*180/pi; >> subplot(2 1 2) >> plot(f,p(1:n/2)); 25 TF & Frequency Response Transfer Function H(z), in 3-D plot Amplitude Response H(ω), in 1-D plot 26 Dr. A. Ranjith Ram arr@gcek.ac.in 13

14 Pole-Zero Plot Pole-zero plot of a digital system : zplane() >> b = [ ]; >> a = [ ]; >> zplane(b,a); >> title('pole-zero Plot'); >> grid on; 27 FIR Digital Filters FIR Digital filters can be designed using one of the following methods Window Method Frequency Sampling Method Optimal Design (Min-max) Method MATLAB functions : Window Method : fir1() Frequency Sampling Method : fir2() Optimal Design (Min-max) Method : firpm() 28 Dr. A. Ranjith Ram arr@gcek.ac.in 14

15 FIR Filter Design Window Method FIR filters are linear phase discrete time systems Since the FIR systems are having b n s only (a n s are zeros except a 0 ), the task is in finding b n s the numerator coefficients FIR filters can be designed classically by windowing technique Here the impulse response of the ideal filter is windowed to get a finite duration sequence If h d (n) is the impulse response of the desired (ideal) filter, the designed (actual) impulse response is given by h(n) = h d (n). w(n) where w(n) is the window function The transition band of the filter directly determines its order The selection of the window is based on the minimum stop band attenuation required 29 FIR Filter Design Window Method b = fir1(n,wn) designs an N'th order lowpass FIR digital filter It returns the filter coefficients in length N+1 vector b The cut-off frequency wn must be between 0 < wn < 1.0, with 1.0 corresponding to half the sample rate The filter b is real and has linear phase. The normalized gain of the filter at wn is 6 db b = fir1(n,wn,'high') designs an N'th order high-pass filter If wn is a two-element vector, wn = [w1 w2], fir1 returns an order N band-pass filter with pass-band w1 < w < w2 One can also specify b = fir1(n,wn,'bandpass') If wn = [w1 w2], b = fir1(n,wn,'stop') will design a band-stop filter 30 Dr. A. Ranjith Ram arr@gcek.ac.in 15

16 FIR Filter Design (Contd ) b = fir1(n,wn,window) designs an N-th order FIR filter using the N+1 length window vector of the impulse response If empty or omitted, fir1 uses a Hamming window of length N+1 For a complete list of available windows, see the help for the window function Kaiser and Chebwin can be specified with an optional trailing argument For example, b = fir1(n,wn,kaiser(n+1,4)) uses a Kaiser window with beta = 4 b = fir1(n,wn,'high',chebwin(n+1,r)) uses a Chebyshev window with r decibels of relative side-lobe attenuation 31 FIR Filter Design (Contd ) For filters with a gain other than zero at fs/2, e.g., high-pass and band-stop filters, N must be even. Otherwise, N will be incremented by one In this case the window length should be specified as N+2 By default, the filter is scaled so the center of the first pass band has magnitude exactly one after windowing Use a trailing 'noscale' argument to prevent this scaling, e.g., b = fir1(n,wn,'noscale') b = fir1(n,wn,'high','noscale') b = fir1(n,wn,wind,'noscale') We can also specify the scaling explicitly, e.g. fir1(n,wn,'scale') 32 Dr. A. Ranjith Ram arr@gcek.ac.in 16

17 FIR Filter Design example % Design a 48th-order FIR band-pass filter with % pass-band 0.35 <= w <= 0.65 b = fir1(48,[ ]); freqz(b,1,512) 33 If N (Order) is not Given? Mitra's book for Digital Signal Processing quotes Kaiser with a simple estimate N = [-20 * log10 (sqrt (rp * rs)) - 13] / [14.6 (ws - wp) / 2π] where rp and rs are the pass-band ripple peak and stop-band ripple peak values, ws and wp are the stop-band and pass-band edges (which are normalized to 2π, where 2π = fs). So, N is a direct function of the ripple peaks and an inverse function of the normalized transition band width) If rp and rs are both 0.01 and (ws - wp) / 2π = 0.1 then N = [-20 * log10 (sqrt (10 ^ -4)) - 13] / [1.46] = [-20 * -2-13] / 1.46 = 27/1.46 = 18.5 or, rounded up to Dr. A. Ranjith Ram arr@gcek.ac.in 17

18 Optimal Design of FIR Filters Parks-McClellan optimal equi-ripple FIR filter design : firpm() b = firpm(n,f,a) returns a length N+1 linear phase (real, symmetric coefficients) FIR filter which has the best approximation to the desired frequency response described by F and A in the mini-max sense F is a vector of frequency band edges in pairs, in ascending order between 0 and 1 1 corresponds to the Nyquist frequency or half the sampling frequency At least one frequency band must have a non-zero width A is a real vector, the same size as F which specifies the desired amplitude of the frequency response of the resultant filter B 35 Optimal Design of FIR Filters The desired response is the line connecting the points (F(k), A(k)) and (F(k+1), A(k+1)) for odd k; firpm treats the bands between F(k+1) and F(k+2) for odd k as transition bands or don't care regions Thus the desired amplitude is piecewise linear with transition bands The maximum error is minimized. For filters with a gain other than zero at f s /2, e.g., high-pass and band-stop filters, N must be even. Otherwise, N will be incremented by one Alternatively, you can use a trailing 'h' flag to design a type 4 linear phase filter and avoid incrementing N. b = firpm(n,f,a,w) uses the weights in W to weight the error 36 Dr. A. Ranjith Ram arr@gcek.ac.in 18

19 Optimal Design of FIR Filters >> % Example of a length 31 low-pass filter >> h=firpm(30,[ ]*2,[ ]); >> freqz(h,1,512) gives the frequency response : 37 IIR Filters IIR systems are not having any linear phase property Occur as having recursive structures Since the IIR systems are having both a n s and b n s, the task is in finding both of the numerator and denominator coefficients Classical analog filter design theory could be used for designing IIR filters Hence there are many approximations Butterworth, Chebyshev, Elliptical, etc. Butterworth approximation is having a smooth passband as well as a stopband But a Chebyshev approximation is either the passband having a ripple (Type-I) or the stopband having a ripple (Type-II) 38 Dr. A. Ranjith Ram arr@gcek.ac.in 19

20 IIR Filters Transfer Function Zero-pole-gain to second-order sections model conversion : zp2sos() [sos, g] = zp2sos(z,p,k) finds a matrix sos in second-order sections form and a gain g which represent the same system H(z) as the one with zeros in vector z, poles in vector p and gain in scalar k The poles and zeros must be in complex conjugate pairs sos is an L by 6 matrix with the following structure : [b 01 b 11 b 21 1 a 11 a 21 b 02 b 12 b 22 1 a 12 a b 0L b 1L b 2L 1 a 1L a 2L ] 39 IIR Transfer Function (Contd ) Each row of the sos matrix describes a 2nd order transfer function H(z) : (b0k + b1k z^-1 + b2k z^-2) (1 + a1k z^-1 + a2k z^-2), where k is the row index. g is a scalar which accounts for the overall gain of the system If g is not specified, the gain is embedded in the first section The second order structure thus describes the system H(z) as: H(z) = g H1(z) H2(z)... HL(z) Embedding the gain in the first section when scaling a direct-form II structure is not recommended and may result in erratic scaling. To avoid embedding the gain, use zp2sos with two outputs. 40 Dr. A. Ranjith Ram arr@gcek.ac.in 20

21 IIR Transfer Function (Contd ) Zero-pole to transfer function conversion : zp2tf() [b,a] = zp2tf(z,p,k) forms the transfer function b(s)/a(s), given a set of zero locations in vector z, a set of pole locations in vector p, and a gain in scalar k Vectors b and a are returned with numerator and denominator coefficients in descending powers of s. Zero-pole to state-space conversion : zp2ss() [A,B,C,D] = zp2ss(z,p,k) calculates a state-space model x = Ax + Bu y = Cx + Du, the A,B,C,D matrices are returned in block diagonal form 41 IIR Filter Design Butterworth Butterworth digital and analog filter design : butter() [b, a] = butter(n,wn) designs an N th order lowpass Butterworth filter and returns the filter coefficients in length N+1 vectors b and a The coefficients are listed in descending powers of z. The cutoff frequency Wn must be 0 < wn < 1, with 1 corresponding to half the fs If wn is a two-element vector, wn = [w1 w2], butter returns an order 2N bandpass filter with passband w1 < w < w2 [b,a] = butter(n,wn,'high') designs a highpass filter; [b,a] = butter(n,wn,'low') designs a lowpass filter and [b,a] = butter(n,wn,'stop') is a bandstop filter if wn = [w1 w2] 42 Dr. A. Ranjith Ram arr@gcek.ac.in 21

22 IIR Filter Design butter & buttord When used as [z,p,k] = butter( ), the zeros and poles are returned in length N column vectors z and p, and the gain in scalar k When used with four left-hand arguments, as in [A,B,C,D] = butter(...), state-space matrices are returned butter(n,wn,'high','s') designs an analog HP Butterworth filter. In this case, wn is in [rad/s] and it can be > 1 Butterworth filter order selection : buttord() [N, wn] = buttord(wp,ws,rp,rs) returns the order N of the lowest order digital Butterworth filter which has a passband ripple of no more than rp db and a stopband attenuation of at least rs db 43 IIR Filter Design Buttord wp and ws are the passband and stopband edge frequencies, normalized from 0 to 1 (where 1 corresponds to pi radians/sample). e.g., Lowpass: wp = 0.1, ws = 0.2 & Bandstop: wp = [0.1.8], ws = [ ] buttord also returns wn, the Butterworth natural frequency (or, the 3 db frequency) to use with butter to achieve the specifications [N, wn] = buttord(wp, ws, rp, rs, 's') does the computation for an analog filter, in which case wp and ws are in radians/second When rp is chosen as 3 db, the wn in butter is equal to wp in buttord 44 Dr. A. Ranjith Ram arr@gcek.ac.in 22

23 IIR Filter Design Butterworth Example >> % For data sampled at 1000 Hz, design a lowpass >> % filter with less than 3 db of ripple in the >> % passband, defined from 0 to 40 Hz, and at least >> % 60 db of attenuation in the stopband, defined >> % from 150 Hz to the Nyquist frequency (500 Hz) >> Wp = 40/500; >> Ws = 150/500; >> [n,wn] = buttord(wp,ws,3,60); % Gives the order >> [b,a] = butter(n,wn); % Butterworth filter design >> freqz(b,a,512,1000);% Plots the frequency response 45 IIR Filter Design Chebyshev [N,wp] = cheb1ord(wp,ws,rp,rs) returns the order N of the lowest order digital Chebyshev Type-I filter which has a passband ripple of no more than rp db and a stopband attenuation of at least rs db [b,a] = cheby1(n,r,wp) designs an Nth order lowpass digital Chebyshev filter with r decibels of peak-to-peak ripple in the passband [N,ws] = cheb2ord(wp,ws,rp,rs) returns the order N of the lowest order digital Chebyshev Type-II filter which has a passband ripple of no more than rp db and a stopband attenuation of at least s db [b,a] = cheby2(n,r,wst) : N th order LP digital Chebyshev filter with the stopband ripple r db down and stopband-edge frequency wst 46 Dr. A. Ranjith Ram arr@gcek.ac.in 23

24 1-D Filtering Process One-dimensional digital filtering : filter() y = filter(b,a,x) filters the data in vector x with the filter described by vectors a and b to create the filtered data y The filter is a Direct Form II Transposed implementation of the standard difference equation, a(1)y(n) = b(1)x(n) + b(2)x(n-1) (nb+1)x(n-nb) a(2)y(n-1)... a(na+1)y(n-na) If a(1) is not equal to 1, filter normalizes the filter coefficients by a(1). filter() always operates along the first non-singleton dimension, namely dimension 1 for column vectors and non-trivial matrices, and dimension 2 for row vectors 47 1-D Filtering Process Example >> wp=0.3; ws=0.4; rp=1; rs=10; % Specifications >> [n,wn] = buttord(wp,ws,rp,rs); % Order selection >> [b,a] = butter(n,wn); % IIR filter design >> figure(1); freqz(b,a); >> title('the frequency response of LP IIR filter'); >> t=1:100; >> sig1 = sin(0.1*pi*t); % Signal inside the PB >> sig2 = sin(0.6*pi*t); % Signal inside the SB >> sig = sig1 + sig2; >> filsig = filter(b,a,sig); figure(2); plot(filsig) 48 Dr. A. Ranjith Ram arr@gcek.ac.in 24

bartlett(64); >> w2 = hamming(64); >> wvtool(w1,w2); 49 Filter Visualization

25 Window Visualization Tool wvtool >> % Analysis of a single window >> w = chebwin(64,100); >> wvtool(w); >> % Analysis of multiple vectors >> w1 = bartlett(64); >> w2 = hamming(64); >> wvtool(w1,w2); 49 Filter Visualization Tool fvtool [b,a] = butter(5, 0.5); h1 = fvtool(b, a); 50 Dr. A. Ranjith Ram arr@gcek.ac.in 25

suite of four tools : Signal Browser, Filter Design and Analysis Tool (fdatool), fvool, and Spectrum Viewer.

26 Filter Design & Analysis Tool fdatool fdatool; % Lanches fdatool 51 Signal Processing Tool sptool sptool opens the sptool window which allows you to import, analyze, and manipulate signals, filters, and spectra in a GUI environment sptool is a suite of four tools : Signal Browser, Filter Design and Analysis Tool (fdatool), fvool, and Spectrum Viewer. These tools provide access to many of the signal, filter, and spectral analysis functions in the toolbox. 52 Dr. A. Ranjith Ram arr@gcek.ac.in 26

27 Thanks 53 Dr. A. Ranjith Ram 27

Filters. Phani Chavali

Filters. Phani Chavali Filters Phani Chavali Filters Filtering is the most common signal processing procedure. Used as echo cancellers, equalizers, front end processing in RF receivers Used for modifying input signals by passing