Digital Filter Design using MATLAB

Digital Filter Design using MATLAB Dr. Tony Jacob Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati April 11, 2015 Dr. Tony Jacob IIT Guwahati April 11, 2015 1 / 30

Outline of the Talk Review of LTI Systems Filter Specifications Linear Phase FIR Filters FIR Filter Design Hands on MATLAB Session Dr. Tony Jacob IIT Guwahati April 11, 2015 2 / 30

What is an LTI System? Dr. Tony Jacob IIT Guwahati April 11, 2015 3 / 30

Impulse Response Description: Convolution Dr. Tony Jacob IIT Guwahati April 11, 2015 4 / 30

Transform Domain Description: Fourier Transform Dr. Tony Jacob IIT Guwahati April 11, 2015 5 / 30

Transform Domain Description: Fourier Transform Dr. Tony Jacob IIT Guwahati April 11, 2015 6 / 30

Transform Domain Description: z Transform Dr. Tony Jacob IIT Guwahati April 11, 2015 7 / 30

Difference Equation Description Dr. Tony Jacob IIT Guwahati April 11, 2015 8 / 30

Conversion between Descriptions Dr. Tony Jacob IIT Guwahati April 11, 2015 9 / 30

Properties of LTI Systems Dr. Tony Jacob IIT Guwahati April 11, 2015 10 / 30

What is a Filter? An LTI system processes a signal x[n] by amplifying or attenuating the sinusoids in its Fourier representation (DTFT) X (e jω ) by the complex factor H(e jω ). This inspires the terminology that X (e jω ) is filtered by H(e jω ) to produce Y (e jω ). Dr. Tony Jacob IIT Guwahati April 11, 2015 11 / 30

Ideal Filters Categories of Ideal Filters Ideal Low Pass Filters Ideal High Pass Filters Ideal Band Pass Filters Ideal Band Stop Filters Dr. Tony Jacob IIT Guwahati April 11, 2015 12 / 30

Low Pass Filter Specifications Dr. Tony Jacob IIT Guwahati April 11, 2015 13 / 30

Low Pass Filter Specifications Dr. Tony Jacob IIT Guwahati April 11, 2015 14 / 30

Low Pass Filter Specifications Dr. Tony Jacob IIT Guwahati April 11, 2015 15 / 30

FIR Filters: Linear Phase Dr. Tony Jacob IIT Guwahati April 11, 2015 16 / 30

FIR Filters: Linear Phase Dr. Tony Jacob IIT Guwahati April 11, 2015 17 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 18 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 19 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 20 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 21 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 22 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 23 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 24 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 25 / 30

Design Method 1: Windowing Dr. Tony Jacob IIT Guwahati April 11, 2015 26 / 30

Other Design Methods Frequency Sampling Optimal Equiripple Filters Dr. Tony Jacob IIT Guwahati April 11, 2015 27 / 30

Exercises Familiarize yourselves with the commands: boxcar, bartlett, hann, hamming, blackman, kaiser. Design a digital FIR low pass filter with the following specifications: ω p = 0.2π, ω s = 0.3π, R p = 0.25dB and A s = 50dB. Try various window functions for this pupose. Determine the impulse response and provide a plot of the frequency response of the designed filter. Familiarize yourselves with designfilt command. Launch FDATool and FVTool and familiarize yourselves with their features. Dr. Tony Jacob IIT Guwahati April 11, 2015 28 / 30

References 1 Vinay K. Ingle and John G. Proakis, Digital Signal Processing Using MATLAB, 3rd Ed., Cengage Learning, 2012. 2 Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing, 3rd Ed., Pearson Education, 2010. 3 John G. Proakis and Dmitris K. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th Ed., Prentice Hall, 2006. 4 Dmitris K. Manolakis and Vinay K. Ingle, Applied Digital Signal Processing, Cambridge University Press, 2011. Dr. Tony Jacob IIT Guwahati April 11, 2015 29 / 30

Online Courses 1 Alan V. Oppenheim, 6.007 Signals and Systems, MIT OCW. 2 Alan V. Oppenheim, 6.008 Digital Signal Processing, MIT OCW. 3 Alan V. Oppenheim and Thomas A. Baran, 6.341x Discrete-Time Signal Processing, edx. 4 Paolo Prandoni and Martin Vetterli, Digital Signal Processing, Coursera. Dr. Tony Jacob IIT Guwahati April 11, 2015 30 / 30

Questions? Dr. Tony Jacob IIT Guwahati April 11, 2015 31 / 30

Discrete Fourier transform (DFT). >> ts = 0.0001 ; >> fs = 1/ts ; >> N = 30001 ; >> T = (N-1)*ts ; >> t = 0:ts:T; >> x = 2*sind(360* 2000* t); >> noise = 1*sind(360* 4000* t); >> SIG = x + noise + 1; >> FFT = fftshift(fft(sig)); % fft finds DFT using FFT algorithm. % fftshift shifts the DTF to the scale pi to pi >> f = (-fs/2) : (fs/ (N-1)) : (fs/2); >> figure;plot (f, (abs(fft)).^2) >> PSD = FFT.*conj(FFT); >> figure;plot (f, PSD) TRY >> psd(sig)

Discrete Fourier transform (DFT). >> ts = 0.0001 ; >> fs = 1/ts ; >> N = 30001 ; >> T = (N-1)*ts ; >> t = 0:ts:T; >> x = 2*sind(360* 2000* t); >> noise = 1*sind(360* 4000* t); >> SIG = x + noise + 1; >> FFT = fftshift(fft(sig)); % fft finds DFT using FFT algorithm. % fftshift shifts the DTF to the scale pi to pi >> f = (-fs/2) : (fs/ (N-1)) : (fs/2); >> figure;plot (f, (abs(fft)).^2) >> PSD = FFT.*conj(FFT); >> figure;plot (f, PSD) TRY >> psd(sig)

Filters IIR Filters >> [ b, a ] = butter( N, fc, 'low') % N = Order, fc = filter cutoff frequency Note: fc is the normalised frequency in the range 0 1. If sampling frequency is Fs, 1 is mapped to Fs/2. Choose low high stop pass accordingly. For stop or pass, fc = [ fc1, fc2 ] upper & lower cutoff frequencies. Eg; >> [b, a] = butter (10, 0.5,'low'); % for a lowpass filter >> [b, a] = butter ( 16, [ 0.5, 0.8], stop ) % for a band stop filter. >> [N, fc] = buttord( fp, fs, Rp, Rs) % For finding optimal filter order. Note: Rp, Rs are the pass & stop band ripples in db and for LPF: fp =.1, fs =.2 HPF: fp =.2, fs =.1 BPF: fp = [.2.7], fs = [.1.8] BSF: fp = [.1.8], fs = [.2.7] The above written format must be followed for your design

Filters IIR Filters Use cheby1, cheby2, ellip functions instead of butter to meet your various design requirements. >> Y = filter ( b, a, X); % to filter the data X ; Y receives the filtered data. a and b are digital filter coeffs generated by any of the previously discussed filter functions. >> freqz ( b, a ) % plots Z domain frequency response >> [ H,W ] = freqz (b, a, N) % gives response in H and N point freq scale in W. >> freqs (b, a ) % plots S domain frequency response >> [ H,W ] = freqs (b, a, N) % gives response in H and N freq point scale in W. TRY >> fvtool ( b, a )

Filters FIR Filters In FIR, it is possible to design filters of linear phase. That is group delay is a constant. Types of FIR filters.

Filters FIR Filters How to find b0, b1, b2,.. bn using Matlab? >> b = fir1( N, Wn, low ) % for an FIR lowpass filter Note: use high, stop, bandpass as per requirement. N must be even for fir1(. ). Matlab will show an error otherwise. Wn is the normalised frequency in range 0 1; 1 maps to Fs/2. Wn must be of the form [ W1 W2] for stop and pass filters. >>Y = filter ( b, 1, X); % to filter the data X ; Y receives the filtered data. b is the digital filter coeffs generated by fir1 (.) since there are no poles, a is taken as 1.

Filters FIR Filters Choosing Windows in the design of FIR filters >> b = fir1( N, Wn, low, window) % specify the required window. Matlab uses hamming as the default window Note: Window along with its parameters need to be specified. Matlab provides the following windows bartlett - Bartlett window. barthannwin- Modified Bartlett-Hanning window. blackman- Blackman window. blackmanharris- Min 4-term Blackman-Harris bohmanwin- Bohman window. chebwin - Chebyshev window. flattopwin - Flat Top window. gausswin- Gaussian window. hamming - Hamming window. hann - Hann window. kaiser - Kaiser window. triang- Triangular window. parzenwin - Parzen window. rectwin - Rectangular window. taylorwin- Taylor window. tukeywin - Tukey window. nuttallwin- Nuttall defined min 4-term Blackman-Harris

Filters FIR Filters Choosing Windows in the design of FIR filters >> b = fir1( N, Wn, low, kaiser ( N+1, 4) ) % here beta of kaiser = 4. How to specify a window of your design? How to caliberate one? >> wintool % allows you to design your own windows. % also you can tune the available matlab windows. Note: The length of the window must be chosen carefully to match the length of the filter ; other wise Matlab will give an error. >> b = fir1( N, Wn, low, yourwindow) % yourwindow must be available in the Workspace.

Filters FIR Filters Design FIR filters with desired frequency response >> b = fir2( N, F, A ) % F is the frequency sample points. % A is the corresponding amplitude at those freqs Note: vector in F must be normalised; first and last samples of the vector must be 0 and 1 respectively. For example : >> F = [0 : 1: 21] ; >> F = F./ 21; % this is done for normalizing the vector F >> A = [0 : 1 : 10 ] ; >> A = [A, fliplr ( A ) ] ; >> b = fir2 (100, F, A) ; >> fvtool ( b, 1 )

Filters Filters design tool for instant designing >> fdatool % allows user to quickly design or analyse a filter Note: Options are available for choosing all types of filters in both IIR and FIR categories. The designed filters can be exported to the workspace in various forms. For easy usage, the design can be ported as filter objects.

Let s filter some real time data using the filters designed in Matlab.