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Problem 1. The z-transform of a discrete-time filter h(k) at a 1-Hz sample rate is given by:
H(z) = 1 + (1/2)z
−1
[1 − (1/2)z−1][1 + (1/3)z−1
(a) Let u(k) and y(k) be the discrete input and output of this filter. Find the difference equation
relating u(k) and y(k).
(b) Find the natural frequency and the damping coefficient for both of the filter’s poles.
(c) Is this filter stable or unstable? Why?
Problem 2. Use the z-transform to solve the following difference equation:
y(k) − 3y(k − 1) + 2y(k − 2) = 2u(k − 1) − 2u(k − 2)
where:
u(k) =
k, k ≥ 0
0, k < 0
y(k) = 0
Problem 3. Consider a lead compensator with the following transfer function, which has been designed
to add about 60◦ of phase at ω1 = 3 rad/sec:
H(s) = s + 1
0.1s + 1
(a) Assume a sampling period of T = 0.25 sec, and compute and plot in the z-plane the pole and
zero locations of the digital implementation of H(s) obtained using (1) Tustin’s method, and (2)
matched pole-zero method. For each case, compute the amount of phase lead provided by the
compensator at z1 = e
jω1T
.
(b) For the frequency range ω = 0.1 to ω = 100 rad/sec, plot the magnitude Bode diagrams for H(s)
as well as of each of the equivalent digital implementations you obtained in Part (a), and compare
the three magnitude plots. (Hint: Magnitude Bode plots are given by
H(z)
=
H(e
jωT )
.
These homework problems are compiled using the different textbooks listed on the course syllabus
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