Design two lowpass FIR filters with M = 50 and omega_c = 0.6 pi, one using a Ham

Question

Design two lowpass FIR filters with M = 50 and omega_c = 0.6 pi, one using a Hamming window, the other with a Rectangular window. Repeat the process with two FIR bandpass filters (M = 50) using cutoff frequencies at omega_c1 = 0.3 pi and omega_c2 = 0.6 pi. Discussion questions to consider: How close is the designed filter to an ideal filter? Is the Hamming-window design better? Why? How many zeros are found in the z-plane? Where are they located? How do the zero locations relate to the passbands and stopbands of the filters?

Explanation / Answer

close all,

clear all,

clc,

CutOFF_Freq = 0.6;

Filter_Order = 50;

Filter_Type = 'low';

window = 'haming';

b = fir1(Filter_Order,CutOFF_Freq,Filter_Type,'haming','normalization','scale'); % DEfault Window - Haming

figure, freqz(b,1,512);

str = strcat('Low Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

figure, zplane(b);

str = strcat('Z-Plane --> Low Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

Filter_Type = 'low';

window = 'rectwin';

b2 = fir1(Filter_Order,CutOFF_Freq,Filter_Type,'rectwin','normalization','scale'); % DEfault Window - Haming

figure, freqz(b2,1,512);

str = strcat('Low Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

figure, zplane(b);

str = strcat('Z-Plane --> Low Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

Filter_Type = 'bandpass';

window = 'haming';

b = fir1(Filter_Order,[0.3 0.6],Filter_Type,'rectwin','normalization','scale'); % DEfault Window - Haming

figure, freqz(b,1,512);

str = strcat('Band Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);   title(str);

figure, zplane(b);

str = strcat('Z-Plane --> Band Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

Filter_Type = 'bandpass';

window = 'rectwin';

b = fir1(Filter_Order,[0.3 0.6],Filter_Type,'rectwin','normalization','scale'); % DEfault Window - Haming

figure, freqz(b,1,512);

str = strcat('Band Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);   title(str);

figure, zplane(b);

str = strcat('Z-Plane --> Band Pass Filter, Order = ',num2str(Filter_Order),' Window = ',window);    title(str);

Discussion:

Band-pass filter has maximum gain in pass band region.Therefore, lots of poles opposite the imaginary axis should be around the band pass frequency.

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