Homework 10

A transfer function $L(s)$ is state-space realizable if there exists finite dimensional $(A, B, C, D)$ so that

$L(s) = C(sI - A)^{-1}B + D$

Prove the following statement: A transfer function $L(s)$ is realizable if and only if it is a proper rational fraction (that is, $L(s) = \frac{P(s)}{Q(s)}$ where both the numerator $P$ and the denominator $Q$ are polynomials, and with degree $P$ less than or equal to the degree of $Q$).

We say that two systems $(A, B, C, D)$ and $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ are zero-state equivalent if the induced transfer functions are equal, that is

$C(sI - A)^{-1}B + D = \tilde{C}(sI - \tilde{A})^{-1}\tilde{B} + \tilde{D}$

Let $P$ be invertible and let us define a transformation through $\tilde{x} = Px$. Then, we can write the state-space model $(A, B, C, D)$ in terms of $\tilde{x}$ as

$\frac{d\tilde{x}}{dt} = \tilde{A}\tilde{x}(t) + \tilde{B}u(t), \qquad \tilde{y}(t) = \tilde{C}\tilde{x}(t) + \tilde{D}u(t),$

with $\tilde{A} = PAP^{-1}, \tilde{B} = PB, \tilde{C} = CP^{-1}, \tilde{D} = D$. In this case, we say that $(A, B, C, D)$ and $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ are algebraically equivalent.

Show that algebraic equivalence implies zero-state equivalence but not vice versa.

Algebraically express and draw the realizations of the transfer function

$H(s) = \frac{s^2 + 2s + 1}{(s+1)(s^2 - 4)}$

in the controllable canonical realization, observable canonical realization, and modal canonical realization forms.

Consider the following continuous-time system:

$y(t) = a(u(t-1) + y(t-1)), \quad t \in \mathbb{R}$

Find the transfer function, from $u$ to $y$, of this system and note that this transfer function is not rational.

Consider

$\frac{d}{dt}x(t) = Ax(t) + Bu(t), \qquad y(t) = Cx(t)$

Suppose that we apply piece-wise constant control actions $u$ which are varied only at the discrete time instances given with $\{t : t = kT, k \in \mathbb{Z}_+\}$ so that $u(t) = u(kT)$ for $t \in [kT, (k+1)T)$. Show that with $x_k := x(kT)$ and $u_k := u(kT)$, we arrive at

$x_{k+1} = A_d x_k + B_d u_k$

where

$A_d = e^{AT}$

$B_d = \int_0^T e^{A\tau} B \, d\tau$

Show also that if $A$ is invertible, we can further write

$B_d = \int_0^T e^{A\tau} B \, d\tau = A^{-1}(e^{AT} - I)B$

Consider an impulse train defined by:

$w_P(t) = \sum_{n \in \mathbb{Z}} \delta(t + nP)$

so that the distribution that we can associate with this impulse train would be defined by:

$\overline{w_P}(\phi) = \sum_{n \in \mathbb{Z}} \phi(nP),$

for $\phi \in \mathcal{S}$.

a) Show that $\overline{w_P}$ is a distribution.

b) [Optional] Show that

$\hat{w}_P(\phi) = \int \frac{1}{P} w_{1/P}(t) \phi(t) \, dt,$

that is, the $\mathcal{F}_{CC}$ of this train is another impulse train.

a) Typically human voice has a bandwidth of 4kHz. Suppose we wish to store a speech signal with bandwidth equal to 4kHz with a recorder. Since the recorder has finite memory, one needs to sample the signal. What is the maximum sampling period (in seconds) to be able to reconstruct this signal with no error. Explain how the reconstruction from the samples can be done.

b) Consider a discrete-time signal $\{x(n)\}$ with a bandwidth $B$. A discrete-time sampler samples this signal with a period $N$ such that the sampled signal satisfies

$x_p(n) = \begin{cases} x(n) & \text{if } 0 \equiv n \mod N, \\ 0 & \text{else.} \end{cases}$

Following this, a decimator is applied to the system to obtain the signal:

$x_d(n) = x_p(nN).$

This new signal is stored in a storage device such as a recorder. Later, the original signal is attempted to be recovered from the storage device. What should the relation between $B$ and $N$ be such that, such a recovery is possible?

Identify the steps such that $\{\tilde{x}(n)\}$ is recovered from $\{x_d(n)\}$.

Consider a discrete-time signal $\{h(n)\}$ with DCFT as:

$\hat{h}(f) = 1_{(f \in ([0, \frac{1}{4}] \cup (\frac{3}{4}, 1)))} \quad f \in [0, 1).$

Determine the DCFT of the signal $g(n) = h(3n), n \in \mathbb{Z}$.

a) Let $m$ be a real-valued signal with a bandwidth B. One can use a transformation, known as the Hilbert transform, to further compress the signal: This compression makes use of the fact that the Fourier transform of a real signal is conjugate symmetric and that it suffices to transmit only the positive frequency band (from which the negative band can be recovered).

Let $\bar{x}$ denote the Hilbert transform of a signal $x$ in $L_2(\mathbb{R}; \mathbb{R})$. We will only characterize this transform with the following information: The CCFT of the Hilbert transform of a signal is given by:

$\hat{\bar{x}}(f) = -i \, \text{sign}(f) \hat{x}(f).$

Using this relation, prove that the Hilbert transform of a signal is orthogonal to the signal itself.

b) Briefly describe the (double-sideband) Amplitude Modulation (AM) and the Frequency Modulation (FM) techniques for radio communications.

Using the result in part a), one can further suppress the bandwidth requirement for the double-sideband Amplitude Modulation (AM) technique, leading to a single-sideband AM signal.