UNIVERSITY OF TORONTO Department of Electrical and Computer Engineering ECE356S Spring 1999 Problem Set #6

Due: March 18, 1999

1. (a) The sequence x_{k}, −∞ < k < ∞, is given by

x_{k} = (\frac{1}{2})^{k} k ≥ 0
= (\frac{1}{3})^{−k} k < 0

Determine the Fourier transform X(e^{iω}) of x_{k}.

(b) Carry out the Fourier inversion integral for the Fourier transform X(e^{iω}) in 1(a), and verify that the sequence is indeed x_{k}.

2. (a) Suppose a sequence x_{k}, k ≥ 0, has a z-transform X(z) = \frac{1}{(1 − az^{−1})(1 − bz^{−1})}

with region of convergence given by |z| > max(|a|, |b|). Compute x_{k} using the inversion integral.

(b) Consider X(z) in part (a) as a ratio of polynomials in z^{−1} and expand X(z) in a partial fraction expansion

X(z) = α_{1} \frac{1}{z^{−1} − a^{−1}} + α_{2} \frac{1}{z^{−1} − b^{−1}}  (2.1)

Determine α_{1} and α_{2}. Using the fact that a^{k} has z-transform \frac{1}{1 − az^{−1}} , |z| > |a|,

determine x_{k} using (2.1), and verify that it is the same as that determined in 2(a).

3. (a) If the sequence {x_{k}} has a z-transform X(z), with region of convergence |z| > R_{x}, determine the z-transform of

(i) {kx_{k}} (ii) {k^{2}x_{k}}

(b) If X(z) and Y (z) are the z-transforms of {x_{k}} and {y_{k}}, with regions of convergence |z| > R_{x} and |z| > R_{y} respectively, show that {x_{k}y_{k}} has a z-transform given by

\frac{1}{2πi} ∮ X (\frac{z}{v} ) Y (v)\frac{dv}{v}

and determine the associated region of convergence.

(c) What are the regions of convergence of the 2 sequences in 3(a)? Explain your answers.

4. Sometimes it is necessary to consider z-transforms for sequences defined on −∞ < k < ∞. The z-transform for such a sequence x_{k} is defined to be

X(z) = ∑_{−∞}^{∞} x_{k}z^{−k}

It is sometimes referred to as the bilateral z-transform. It exists if the 2 series

∑_{−∞}^{−1} x_{k}z^{−k} and ∑_{0}^{∞} x_{k}z^{−k}

both converge. Since the first sum converges for |z| sufficiently small while the second sum converges for |z| sufficiently large, the region of convergence is an annulus.

(a) Let

x_{k} = 3^{k} k < 0
= 2^{k} k ≥ 0

Determine the bilateral z-transform of x_{k}, and determine its region of convergence.

(b) Given the z-transform X(z) = \frac{1}{z−4}, determine the sequence x_{k} if

(i) the region of convergence is |z| < 4 (Hint: Expand in a power series in \frac{z}{4}); (ii) the region of convergence is |z| > 4 (Hint: Expand in a power series in 4z^{−1}).