Lab Handout 09: FIR Filter Design Using MATLAB

Lab Handout 09: FIR Filter Design Using MATLABCourse Code: EEE 420

Course Name: Digital Signal Processing

Electrical & Electronic Engineering , Eastern Mediterranean University

Instructor: Asst. Prof. Dr. Erhan A. nce

E-mail: erhan.ince@emu.edu.trExt: 2778Lab Assistant: Mr. Burin zmen

E-mail: burcin.ozmen@emu.edu.trExt: 2771

LINEAR-PHASE FIR FILTERShapes of impulse and frequency responses and locations of system function zeros of linear-phase FIR filters will be discussed.

Let , be the impulse response of length (or duration) M. Then the system function is

which has poles at the origin z = 0 (trivial poles) and zeros located anywhere in the z-plane. The frequency response function is

Type-1 Linear-Phase FIR FiltersSymmetrical impulse response and M odd. In this case , is an integer, and , . Then we can show that

where sequence is obtained from as

Since then, we have

where is an amplitude response function and not a magnitude response function.

function [Hr,w,a,L]=Hr_Type1(h);

% Computes Amplitude response Hr(w)of a Type-1 LP FIR filter

% -----------------------------------------------------------

% [Hr,w,a,L]=Hr_Type1(h)

% Hr = Amplitude Response

% w = 500 frequencies between [0,pi] over which Hr is computed

% a = Type-1 LP filter coefficients

% L = Order of Hr

% h = Type-1 LP filter impulse response

%

M = length(h);

L= (M-1)/2;

a = [h(L+1) 2*h(L:-1:1)]; % 1x(L+1) row vector

n = [0:1:L] % (L+1)x1 column vector

w = [0:1:500]'*pi/500;

Hr = cos(w*n)*a';Type-2 Linear-Phase FIR FiltersSymmetrical impulse response and M even. In this case , is not an integer, and , . Then we can show that

where

Hence,

where is an amplitude response function. Note that we get

regardless of or . Hence we cannot use this type (i.e. symmetrical and M even.) for highpass or bandstop filters.

Type-3 Linear-Phase FIR FiltersAntisymmetric impulse response and M odd. In this case , is an integer, and , , and . Then we can show that

where

Hence,

Note that at and we have , regardless of or . Furthermore, , which means that is purely imaginary. Hence this type of filter is not suitable for designing a lowpass or a highpass filter. However, this behavior is suitable for approximating ideal Hilbert transformers and differentiators.

function [Hr,w,c,L]=Hr_Type3(h);

% Computes Amplitude response Hr(w) of a Type-3 LP FIR filter

% -----------------------------------------------------------

% [Hr,w,c,L]=Hr_Type3(h)

% Hr = Amplitude Response

% w = frequencies between [0,pi] over which Hr is computed

% b = Type-3 LP filter coefficients

% L = Order of Hr

% h = Type-3 LP filter impulse response

%

M = length(h);

L = (M-1)/2;

c = [2*h(L+1:-1:1)];

n = [0:1:L];

w = [0:1:500]'*pi/500;

Hr = sin(w*n)*c';Type-4 Linear-Phase FIR FiltersAntisymmetric impulse response and M even. This case is similar to Type-2. We have

where

and

Note that at and we have and . Hence, this type is also suitable for designing digital Hilbert transformers and differentiators.

Assignments

1. MATLAB functions for Type-1 and Type-3 is given. Write a MATLAB function to compute the amplitude response and the location of the zeros of for Type-2 and Type-4.

2. Let

Determine the Type of the Linear-Phase FIR filters. Find the amplitude response and locations of the zeros of . Also plot the impulse response, amplitude response, coefficients, and the zero locations.

WINDOW DESIGN TECHNIQUESSummary of commonly used window function characteristics.

Window

NameTransition ApproximateWidth Exact ValuesMin. Stopband Attenuation

Rectangular

21 dB

Bartlett

25 dB

Hanning

44 dB

Hamming

53 dB

Blackman

74 dB

MATLAB provides several routines to implement window functions discussed above table.

w = boxcar(n) returns the n-point rectangular window in array w. w = triang(n) returns the n-point Bartlett (Triangular) window in array w. w = hanning(n) returns the n-point symmetric Hanning window in a column vector in array w .

w = hamming(n) returns the n-point symmetric Hamming window in a column vector in array w. w = blackman(n) returns the n-point symmetric Blackman window in a column vector in array w.

w = kaiser(n,beta) returns the beta-valued n-point Kaiser window in array w.Using these routines, we can use MATLAB to design FIR filters based on the window technique, which also requires an ideal lowpass impulse response as shown below.

function hd=ideal_lp(wc,M);

% Ideal LowPass filter computation

% --------------------------------

% [hd] = ideal_lp(wc,M);

% hd = ideal impulse response between 0 to M-1

% wc = cutoff frequency in radians

% M = length of the ideal filter

%

alpha = (M-1)/2;

n = [0:1:(M-1)];

m = n - alpha +eps; % add smallest number to avoi divided by zero

hd = sin(wc*m)./(pi*m);

To display the frequency domain plots of digital filters, MATLAB provides the freqz routine. Using this routing, we can developed a modified version, called freqz_m, which returns the magnitude response in absolute as well as dB scale, the phase response, and the group delay response as shown below.

function [db,mag,pha,grd,w] = freqz_m(b,a)

% Modified version of freqz subroutine

% ------------------------------------

% [db,mag,pha,grd,w] = freqz_m(b,a)

% db = relative magnitude in dB computed over 0 to pi radians

% mag = absolute magnitude computed over 0 to pi radians

% pha = Phase response in radians over 0 to pi radians

% grd = Group delay over 0 to pi radians

% w = 501 frequency samples between 0 to pi radians

% b = numerator polynomial of H(z) (for FIR: b=h)

% a = denominator polynomial of H(z) (for FIR: a=[1])

%

[H,w] = freqz(b,a,1000,'whole') ;

H = (H(1:1:501))'; w = (w(1:1:501))';

mag = abs(H);

db = 20*log10((mag+eps)/max(mag));

pha = angle(H);

grd = grpdelay(b,a,w);

Assignment 33. Design a digital FIR lowpass filter with the following specifications:

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