Question

) Given a digital filter with the following numerator and denominator coefficient vectors, respectively: Num= 8.831, 16.234, 25.118, 16.234, 8.831, and Den= 1.0000, 1.7342, 2.5277, 1.5502, 0.7998. (a) Find the poles and zeros of the filter transfer function. Is the filter stable? (b) Plot the magnitude and phase of H(e jω) of the filter. Hence, determine the type of filter (LP, HP, BP, or Notch) it represents. (c) Determine the 3-dB bandwidth of the filter. (d) If the sampling period is T = 0.5 µsec, determine the 3-dB bandwidth in Hz.

Answer 1

We have the transfer function

zeros of H(z) are

z=

-0.4711 + 0.8821i
-0.4711 - 0.8821i
-0.4480 + 0.8940i
-0.4480 - 0.8940i
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poles of H(z) are

z=

-0.3770 + 0.8654i
-0.3770 - 0.8654i
-0.4901 + 0.8108i
-0.4901 - 0.8108i

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%matlab

w=-pi:0.001*pi:pi;
z=exp(1i.*w);
num=8.831.*z.^4+16.234.*z.^3+25.118.*z.^2+16.234.*z+8.831;
den=1.0000.*z.^4+1.7342.*z.^3+2.5277.*z.^2+1.5502.*z+0.7998;
H=num./den;
subplot(211);
plot(w,20.*log10(abs(H)));
xlabel('\Omega');
ylabel('20log_{10}(H(e^{j\Omega})');
subplot(212);
plot(w,angle(H).*(180/pi));
xlabel('\Omega');
ylabel('\angle{H(e^{j\Omega}}');

-----------------------

----------------------------

From the magnitude plot it can be inferred that the filter is a NOTCH FILTER.

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c)

From the magnitude plot, the -3dB frequencies are

therefore

3-dB bandwidth is

-------------\

d)

analog frequencies are related to digital frequencies by the relation

therefore

analog band width