UNIVERSITY OF TORONTO Department of Electrical and Computer Engineering ELE 356S Spring 1999 Problem Set #9

Review Problems

1. A discrete-time linear time-invariant system has an impulse response sequence

\begin{aligned}h_{k} & = (\frac{1}{2})^{k} & k ≥ 0 \\ & = 0 & k < 0 \end{aligned}

Using z-transforms, find the output of the system if the input is a unit step, i.e.

\begin{aligned}u_{k} & = 1 & k ≥ 0 \\ & = 0 & k < 0 \end{aligned}

2. Sketch the Nyquist plot of the system described by the transfer function P (s) = \frac{s^{2} + 1}{(s^{2} + 4)(s + 4)}.

Based on the Nyquist plot, determine the range of values of the gain K for which the feedback system

\frac{s^{2} + 1}{(s^{2} + 4)(s + 4)}

K

-

is stable.

3. Suppose a low pass digital filter is to be designed so that the maximum ripple δ is .05%, with a transition width Δ = 0.1π. If the maximum allowable length of the filter is 71, can the above design specs be met using a nonrecursive filter designed with the Kaiser window?