Butteworth Filter Design

As the first step, the various commonly used functions are discussed briefly. After that programs are given.

buttord

This functiion returns the order and natural cut-off frequency of the butterworth filter. Its input are passband freq, stopband freq, passband attenuation and stoopband attenuation. The freq must be normalised.

[N,Wn] = buttord(omp,oms,alphap,alphas);
N is the order, Wn is the cut-off.

butter
This is used to get the system function. The numeratoe and denomiantor values are obtained in an array.

[B,A] = butter(N,Wn);


abs

This gives the absolute value of a complex quantity.
m = abs(X);

angle

This will return the phase angle associated with a complex quantity.
an = angle(X);

subplot

This function is very commonly used one. It working can be described through an example.
subplot(3,1,1) means the entire window is divided into a 3X1 matrix and the active portion at the present time is the first one.

plot

It performs the plotting function of qnty. vs other.
plot(Y,X) plots Y as function of X. ie, Y on y axis and X on xaxis.

grid

It will draw grid lines to the plot.

Lowpass Filter

%To design a Butterworth Lowpass Filter
%specifications
clear all
alphap = 4; %Passband Attenuation
alphas = 30; %Stopband attenuation
fp = 400; %Passband frequency in Hz
fs = 800; %Stopband frequency in Hz
F = 2000; %Sampling Frequency in Hz
omp = 2*fp/F;
oms = 2*fs/F;

%To find cut-off frequency and order of filter
[N,Wn] = buttord(omp,oms,alphap,alphas); %N is the order of filter, and Wn is its natural frequency

%System function of filter
[B,A] = butter(N,Wn);
w = 0:0.01:pi;
[h,om] = freqz(B,A,w,'whole'); %h is the complex value of filter corres. to freq w
m = abs(h); %Absolute value of h
an = angle(h); %Angle of h

subplot(2,1,1);
plot(om/pi,20*log(m));
grid;
xlabel('Normalised frequency');
ylabel('Gain in dB');

subplot(2,1,2);
plot(om/pi,an);
grid;
xlabel('Normalised frequency');
ylabel('Phase in radian');


Bandpass Filter

%To design a Butterworth Bandpass Filter
%Specifications
clear all
alphap = 2; %Passband Attenuation
alphas = 20; %Stopband Attenuation
wp = [0.2*pi,0.4*pi]; %Passband Frequency
ws = [0.1*pi,0.5*pi];

%to find cut-off frq and order
[n,wn] = buttord(wp/pi,ws/pi,alphap,alphas);

%System function
[b,a] = butter(n,wn);
w = 0:0.01:pi;
[h,ph] = freqz(b,a,w);
m = 20*log10(abs(h));
an = angle(h);

subplot(2,1,1);
plot(ph/pi,m);
grid;
ylabel('Gain in dB');
xlabel('Normalised Frequency');

subplot(2,1,2);
plot(ph/pi,an);
grid;
ylabel('Phase in radian');
xlabel('Normalised frequency');


Bandreject Filter

%To design a Butterworth bandreject filter
%Specification
clear all
alphap = 2; %Passband attenuation
alphas = 20; %Stopband attenuation
ws = [0.2*pi,0.4*pi]; %Stopband frequency in rad/s
wp = [0.1*pi,0.5*pi]; %Passband frequency in rad/s
%To find cut-off frequency and order
[n,wn] = buttord(wp/pi,ws/pi,alphap,alphas);
%To find system function
[b,a] = butter(n,wn,'stop'); %[b,a] is [numerator,denominator]
w = 0:0.01:pi;
[h,ph] = freqz(b,a,w);
m = 20*log(abs(h));
an = angle(h);
%Plotting
subplot(2,1,1);
plot(ph/pi,m,'r');
grid;
ylabel('Gain in dB');
xlabel('Normalised frequency');
subplot(2,1,2);
plot(ph/pi,an,'b');
grid;
ylabel('Phase in rad');
xlabel('Normalised frequency');



Highpass Filter

%To design a Highpass Butterworth filter
%Specification
clear all
alphap = 4; %Passband attenuation
alphas = 30; %Stopband attenuation
fs = 400; %Stopband frequency in Hz
fp = 800; %Passband frequency
F = 2000;
omp = 2*fp/F; %Normalised
oms = 2*fs/F; %Normalised
%To find cut-off and order
[n,wn] = buttord(omp,oms,alphap,alphas)
%System function
[b,a] = butter(n,wn,'high');
w = 0:0.01:pi;
[h,om] = freqz(b,a,w);
m = 20*log(abs(h));
an = angle(h);
%Plotting
subplot(2,1,1);
plot(om/pi,m);
grid;
ylabel('Gain in db');
xlabel('Normalised frequency');
subplot(2,1,2);
plot(om/pi,an);
grid;
ylabel('Phase in rad');
xlabel('Normalised frequency');



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