Homework 5

Consider a linear system described by the relation:

$y(n) = \sum_{m \in \mathbb{Z}} h(n,m)u(m), \quad n \in \mathbb{Z}$

for some $h : \mathbb{Z} \times \mathbb{Z} \to \mathbb{C}$.

a) When is such a system causal?

b) Show that such a system is time-invariant if and only if it is a convolution system.

Let $x(t) \in \mathbb{R}^N$ and $t \ge 0$ and real-valued. Recall that the solution to the following differential equation:

$x'(t) = Ax(t) + Bu(t),$ $y(t) = Cx(t),$

with the initial condition $x(t_0) = x_0$ is given by

$x(t) = e^{A(t-t_0)}x_{t_0} + \int_{\tau=t_0}^{t} e^{A(t-\tau)}Bu(\tau)d\tau, \quad t \ge 0$

(a) Suppose that $x(t_0) = 0$ and all eigenvalues of $A$ have their real parts as negative and $\|u\|_{\infty} < \infty$. Let $t_0 \to -\infty$. Show that if one is to represent $x(t) = (h * u)(t)$, we have

$h(t) = Ce^{At}B1_{\{t \ge 0\}}.$

(b) Alternatively, we could skip the condition that the eigenvalues of $A$ have their real parts as negative, but require that $x(0) = 0$ and $u(t) = 0$ for $t < 0$. Express the solution as a convolution

$y(t) = (h * u)(t),$

and find $h(t)$.

Let $x(n) \in \mathbb{R}^N$ and $n \in \mathbb{Z}$. Consider a linear system given by

$x(n+1) = Ax(n) + Bu(n)$ $y(n) = Cx(n), \quad n \ge 0$

with the initial condition $x(n_0) = 0$ for some $n_0$.

(a) Suppose all the eigenvalues of $A$ are strictly inside the unit disk in the complex plane and $\|u\|_{\infty} < \infty$. Let $n_0 \to -\infty$. Express the solution $y(n)$ as a convolution

$y(n) = (h * u)(n),$

and find that

$h(n) = CA^{n-1}B1_{\{n \ge 1\}}.$

(b) Alternatively, we could skip the condition that the eigenvalues of $A$ are strictly inside the unit disk in the complex plane, but require that $n_0 = 0$ so that $x(0) = 0$ and also $u(n) = 0$ for $n < 0$. Express the solution as a convolution

$y(n) = (h * u)(n),$

and find $h(n)$.

Let $\mathcal{T}$ be a linear system mapping $l_1(\mathbb{Z}; \mathbb{R})$ to $l_1(\mathbb{Z}; \mathbb{R})$. Show that if this system is linear and continuous, it can be written so that $y = \mathcal{T}(u)$;

$y(n) = \sum_{m \in \mathbb{Z}} h(n,m)u(m), \quad n \in \mathbb{Z}$

for some $h : \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$.

Remark. Building on this discussion, one takes the representation above as a definition of a linear system: As noted earlier, in our course and in standard terminology in engineering and applied science, we generally say that a discrete-time (DT) system is linear if the input output relation can be written as:

$y(n) = \sum_{m=-\infty}^{\infty} h(n,m)u(m).$

In the above, $h(n,m)$ is called the kernel of the system. The value $h(n,m)$ reveals to effect of an input at time $m$ to the output at time $n$.

Consider a continuous-time system described by the equation:

$\frac{dy(t)}{dt} = ay(t) + u(t), \quad t \in \mathbb{R},$

where $a < 0$.

a) Find the impulse response of the system. Is the system bounded-input-bounded-output (BIBO) stable?

b) Suppose that the input to this system is given by $\cos(2\pi f_0 t)$. Let $y_{f_0}$ be the output of the system. Find $y_{f_0}(t)$.

c) If exists, find

$\lim_{f_0 \to \infty} y_{f_0}(t),$

for all $t \in \mathbb{R}$.

Let $x(t) \in \mathbb{R}^N$. Consider a system defined with the relation:

$x'(t) = Ax(t) + Bu(t), \qquad y(t) = Cx(t) + Du(t), \qquad t \ge t_0,$

for some fixed $t_0$. Consider an input $u(t) = e^{st}, t \ge t_0$ for some $s \in \mathbb{C}$. Suppose $s$ is not an eigenvalue of $A$. Show that

$y(t) = Ce^{A(t-t_0)}\left( x(t_0) - (sI - A)^{-1}e^{st_0}B \right) + \left( C(sI-A)^{-1}B + D \right)e^{st}, \qquad t \in \mathbb{R}_+$

The first term is called the transient response of the system and the second term is called the steady-state response.

If $A$ is a stable matrix, with all its eigenvalues inside the unit circle, the first term decays to zero as $t$ increases (or with fixed $t$, as $t_0 \to -\infty$). Alternatively, if we set $t_0 = 0$ and write

$x(0) = (sI - A)^{-1}B,$

the output becomes

$y(t) = \left( C(sI-A)^{-1}B + D \right)e^{st}, \qquad t \ge 0$

The map $C(sI-A)^{-1}B + D$ is called the transfer function of the system.

Hint: Prove and use the relation, for $s$ not an eigenvalue of $A$:

$e^{At}(sI - A)^{-1}e^{-At} = (sI - A)^{-1}$

a) Following your class notes, study the impulse response for the RC circuit and the RLC circuit.

b) Following the lecture notes, study the Bode plots of some common frequency response functions.