Homework 8

a) Compute the (two-sided) Z-transform of

$x(n) = 4^{n-1} 1_{\{n-1 \geq 0\}}$

Note that you should find the Region of Convergence as well.

b) Compute the (two-sided) Laplace-transform of

$x(t) = e^{5t} 1_{\{t \geq 0\}}$

Find the regions in the complex plane, where the transforms are finite valued.

c) Show that the one-sided Laplace transform of $\cos(\alpha t)$ satisfies

$\mathcal{L}_+\{\cos \alpha t\} = \frac{s}{s^2 + \alpha^2}, \quad \text{Re}\{s\} > 0$

d) Compute the inverse Laplace transform of

$\frac{s^2 + 9s + 2}{(s-1)^2(s+3)}, \quad \text{Re}\{s\} > 1$

Hint: Use partial fraction expansion and the properties of the derivative of a Laplace transform.

Find the inverse Z-transform of:

$X(z) = \frac{3 - \frac{5}{6}z^{-1}}{(1 - \frac{1}{4}z^{-1})(1 - \frac{1}{3}z^{-1})}, \quad |z| > 1$

Let $x(n) = 0$ for $n < 0$ and $|x(n)| \leq Mr^n$ for some $M, r \in \mathbb{R}$. Show that

$\lim_{|z| \to \infty} X(z) = x(0)$

a) Prove the final value theorem for the Laplace transforms. That is, if $\lim_{t \to \infty} x(t) =: M < \infty$, then

$\lim_{t \to \infty} x(t) = \lim_{s \downarrow 0} s X_+(s)$

b) Read (you do not need to write anything on your assignment) the rules of converse differentiation (and their proofs in the lecture notes). That is,

(i) Suppose that $\limsup_{n \to \infty} |x(n)|^{1/n} \leq R$ for some $R \in \mathbb{R}$. Let $y(n) = -nx(n)$. Then,

$\mathcal{Z}_+(y)(z) = z \frac{d}{dz}(\mathcal{Z}_+(x))(z)$

for $|z| > R$.

(ii) Let $|x(t)| \leq Me^{at}$ for some $M, a \in \mathbb{R}$. Let $y(t) = -tx(t)$. Then, for $s$ with $\text{Re}\{s\} > a$, we have

$\mathcal{L}_+(y)(s) = \frac{d}{ds}(\mathcal{L}_+(x))(s)$

Let $P(z)$ and $Q(z)$ be polynomials in $z$. Let the transfer function of a discrete-time LTI system be given by

$H(z) = \frac{P(z)}{Q(z)}$

a) Show that the system is causal (non-anticipative) if and only if $\frac{P(z)}{Q(z)}$ is a proper fraction (that is the degree of the polynomial in the numerator cannot be greater than the one of the denominator).

b) Show that the system is BIBO stable if and only if the Region of Convergence of the transfer function contains the unit circle. Thus, for a system to be both causal and stable, what are the conditions on the roots of $Q(z)$?

c) Repeat the above for a continuous-time system and with the Laplace transform instead of the Z-transform.

d) What does it mean to be a minimum-phase system? Why do we call such systems minimum-phase?