Frequency response of digital filter

I need to calculate the frequency response of a butter worth low pass digital filter.

I calculated the transfer function of a butterworth low pass filter to be:

Y(s)/X(s) =39.476/(s^2+8.886 s+39.476)

I then used bilinear transform to obtain the 'z' transfer funtion:

Y(z)/X(z) =(0.0198(z^2+2z+1))/(z^2-1.564 z+0.644)

I now need to use matlab to obtain the frequency response of this digital filter in the digital and analgue domain. then check if they match around the break/cutoff frequency.

I assume I need to use 'Freqs' and 'Freqz' but I am not sure if i am getting the correct results. Could somone assist? (my current code below)

a=[1,1.564,0.644]

b=[0.0198,0.0396,0.0198]

w=logspace(-2,3)

sys=tf(b,a)

h = freqs(b,a,w );

mag = mag2db(abs(h ));

phase = angle(h );

phasedeg = phase*180/pi ;

subplot(2,1,1), semilogx(w,mag), grid on

xlabel 'Frequency (rad/s)', ylabel 'Magnitude (dB)'

subplot(2,1,2), semilogx(w,phasedeg), grid on

xlabel 'Frequency (rad/s)', ylabel 'Phase (degrees )'

pause

[h,w] = freqz(b,a,'whole',2001);

mag = mag2db(abs(h ));

phase = angle(h );

phasedeg = phase*180/pi ;

subplot(2,1,1), semilogx(w,mag), grid on

xlabel 'Frequency (rad/s)', ylabel 'Magnitude (dB)'

subplot(2,1,2), semilogx(w,phasedeg), grid on

xlabel 'Frequency (rad/s)', ylabel 'Phase (degrees )'

Answer 1

[Mahesh Taparia]

Hi Joshua

In your code, you are using same transfer function for ‘s’ and ‘z’ domain. You specified Y(s)/X(s) which you did not used. It should be as following

a=[36.476]; 
b=[1,8.886,39.476]; 
sys=tf(a,b); 

To do bilinear transformation you can use MATLAB in built function ‘bilinear’ as follows:

[num,den]=bilinear(a,b,Fs); 

where Fs=sampling rate. For more information of bilinear function, you can refer to the documentation here.

Now to analyze the frequency response of filter in analog and in digital domain you can use ‘freqs’ and ‘freqz’ command respectively (like you did in your code).

Moreover, you can also use ‘fvtool’ function to find frequency response of a digital filter. For more information on ‘fvtool’ you can refer to the documentation here

Answer 2

[Zubair Zebary]

If 𝑥(𝑛) = 0.5 𝛿(𝑛) + 𝛿(𝑛 − 1) + 0.5 𝛿(𝑛 − 2). Plot the magnitude and angle of 𝐻(𝑒 𝑗𝑊) in the period of [0 2π].