3.21 Extra Exercises
3.1. Develop and list the source code of a Matlab/Octave routine that implements
the overlap-save method using a structure similar to Listing 3.7.
3.2. Using FilterPro as in Section 3.10.1.0 or similar software, design a second-order
lowpass Butterworth filter
with passband frequency of 5 kHz. Show the schematics. Adopt commercial
components with 10% of tolerance. What is the exact value of the passband
frequency? Does it coincide with 5 kHz? After that, execute another design to obtain
,
trying to improve the attenuation using an approximation other than Butterworth’s.
Comparing
and ,
what can you say on how good are their magnitudes and phases?
3.3. Design a digital Butterworth filter
using the bilinear transform. The cutoff and rejection filters are
and ,
respectively. The linear gain at DC should be 1 and the minimum gain at the passband
should be 0.95. The maximum gain at the rejection band is 0.01. This filter can be
obtained with the commands:
1Gs=20*log10(0.01) 2Gp=20*log10(0.95) 3Wp=1/4 4Ws=2/3 5[n, Wc] = buttord(Wp, Ws, -Gp, -Gs) 6[b, a]=butter(n,Wc)
a) What are the gain and phase at frequencies
and
rad? b) What is the difference equation that corresponds to ?
3.4. The bilinear transform Eq. (3.71) was obtained by expanding the numerator and
denominator of
using a first order Taylor expansion. Compare it with the expansion of
.
What are the reasons for using the former expansion? An alternative point of view is to
calculate the series expansion of
for the matched Z-transform .
Check the expansion for
at [ url3iit].
3.5. Study the algorithm used to design [b, a]=butter(2,1/4) including the pre-warping
stage.
3.6. Using the function fir1, compare the FIR filters with cutoff frequency
rad for the windows Hamming, Hann and rectangular. The commands can be
and freqz can help. How do they compare with respect to: attenuation, decay, sidelobe and
width of the main lobe.
3.7. Observe that fir1 in Matlab uses a modified cutoff frequency
. Instead
of
being the frequency of half the power at passband, for fir1
is the
frequency of half the gain at passband. In other words, considering a filter with gain
at DC, fir1 provides a filter
with gain (which corresponds
to dB), instead of the
commonly adopted dB
value of . For the
Hamming window, design an alternative to fir1 that allows the user to specify the cutoff frequency as the
one in which .
3.8. The Kaiser window is very flexible. Study the project below that uses
Hz, passband
Hz, and
rejection band .
The minimum attenuation at rejection band is 40 dB and the maximum ripple
at the passband is 5% (i. e., the filter can go from 1 to 0.95 at the passband).
1[n,w,beta,ftype]=kaiserord([1000,1200],[1,0],[0.05,0.01],11025); 2freqz(fir1(n,w,ftype,kaiser(n+1,beta),'noscale'),1,[],11025);
Design an IIR Butterworth that meets the requirements and compare the order with the
Kaiser FIR filter.
3.9. Generate realizations (waveforms) of a Gauss-Markov process. Recall that a first-order Gauss-Markov
process obeys , where
is an i. i. d. Gaussian noise.
Generate realizations of
with samples each, using
with unitary power, and
for the following values of :
.
Provide a qualitative description of the difference between
and
. Calculate the
autocorrelation of
in terms of
(or obtain it from references such as [JN84] or the Web).
3.10. Study the anti-aliasing filters at [ url3min].
3.11. Using FilterPro, execute the project of a stopband using default options, which should
lead to an order 6 and cutoff frequency of 1 kHz. a) What does it mean “Min GBW reqd.” in
the end of the “Schematic”? b) how to interpret the Group Delay plot? c) Why each stage of
SOS has f0, BW and Q? d) Use “Edit” to re-do the project using Chebyshev with a
maximum of 1 dB ripple in the passband instead of Bessel. How did that impact the “Group
delay”?
3.12. Study the advantages and disadvantages when comparing Butterworth, Chebyshev and
Bessel responses.
3.13. When cascading the 2nd order stages, why the stages with lower quality factor
are
placed first by Filter Pro?
3.14. Compare the MFB and Sallen-Key topologies. Can you find a more sophisticated
topology in the literature? If yes, describe it.
3.15. Briefly explain what sensitivity means.