digital filter design using matlab · digital filter design using matlab dr. tony jacob department...

Digital Filter Design using MATLAB

Dr. Tony Jacob

Department of Electronics and Electrical EngineeringIndian 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


What is an LTI System?


Impulse Response Description: Convolution


Transform Domain Description: Fourier Transform


Transform Domain Description: z Transform


Difference Equation Description


Conversion between Descriptions


Properties of LTI Systems


What is a Filter?

An LTI system processes a signal x [n] by amplifying or attenuatingthe sinusoids in its Fourier representation (DTFT) X (e jω) by thecomplex factor H(e jω).

This inspires the terminology that X (e jω) is filtered by H(e jω) toproduce Y (e jω).


Ideal Filters

Categories of Ideal Filters

Ideal Low Pass Filters

Ideal High Pass Filters

Ideal Band Pass Filters

Ideal Band Stop Filters


Low Pass Filter Specifications


FIR Filters: Linear Phase


Design Method 1: Windowing


Other Design Methods

Frequency Sampling

Optimal Equiripple Filters


Exercises

Familiarize yourselves with the commands: boxcar, bartlett, hann,hamming, blackman, kaiser.

Design a digital FIR low pass filter with the following specifications:ωp = 0.2π, ωs = 0.3π, Rp = 0.25dB and As = 50dB. Try variouswindow functions for this pupose. Determine the impulse responseand provide a plot of the frequency response of the designed filter.

Familiarize yourselves with designfilt command.

Launch FDATool and FVTool and familiarize yourselves with theirfeatures.


References

1 Vinay K. Ingle and John G. Proakis, “Digital Signal Processing UsingMATLAB”, 3rd Ed., Cengage Learning, 2012.

2 Alan V. Oppenheim and Ronald W. Schafer, “Discrete-Time SignalProcessing”, 3rd Ed., Pearson Education, 2010.

3 John G. Proakis and Dmitris K. Manolakis, “Digital SignalProcessing: Principles, Algorithms, and Applications”, 4th Ed.,Prentice Hall, 2006.

4 Dmitris K. Manolakis and Vinay K. Ingle, “Applied Digital SignalProcessing”, Cambridge University Press, 2011.


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-TimeSignal Processing, edX.

4 Paolo Prandoni and Martin Vetterli, Digital Signal Processing,Coursera.


Questions?


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.

digital filter design using matlab · digital filter design using matlab dr. tony jacob department...

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