Homework 7

CCFT is a map from $\mathcal{S}$ to $\mathcal{S}$. One typical example is the Gaussian signal $e^{-at^2/2}$ for some $a > 0$: Show that for $a > 0$, the CCFT of

$\phi(t) = e^{-at^2/2}$

is equal to

$\hat{\phi}(f) = Ke^{-2\pi^2 f^2/a},$

for some $K$ and conclude that $\hat{\phi}$ is also a Schwartz signal.

Hint: Take the CCFT of $\phi$, obtain the integral as $M(f)e^{-2\pi^2 f^2/a}$ where $M(f)$ is a function of $f$. Then show that $M(f)$ does not depend on $f$, by taking its derivative with respect to $f$ and observing that its derivative is equal to 0. Set $K := M(f)$.

Show that with $\mathcal{F}_{CC}$ denoting the Continuous-Continuous Fourier transform, show that

$\Big( \mathcal{F}_{CC}(u * v) \Big)(f) = \hat{u}(f)\hat{v}(f), \qquad f \in \mathbb{R}$

for $u \in \mathcal{S}$ and $v \in \mathcal{S}$. Here, $\mathcal{F}_{CC}(u) =: \hat{u}$ and $\mathcal{F}_{CC}(v) =: \hat{v}$.

Consider the R-C circuit considered in class with the equations:

$\frac{dV_C(t)}{dt} = -\frac{1}{RC}V_C(t) + \frac{1}{RC}u(t)$

a) Viewed as a linear time-invariant system, where $u$ is the input and $V_C$ is the output, find the impulse response and the frequency response.

b) Qualitatively, plot the Bode diagram.

Consider the R-L-C circuit considered in class, with the dynamics

$L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{1}{C}Q = u(t)$

Note $V_C = Q/C$.

a) Viewed as a linear time-invariant system, where $u$ is the input and $V_C$ is the output, find the impulse response and the frequency response.

b) Qualitatively, plot the Bode diagram in the setup when $R$ is very small.

Consider a linear time invariant (LTI) system characterized by:

$y^{(1)}(t) = -ay(t) + u(t), \qquad t \in \mathbb{R}$

with $a > 0$.

a) Find the impulse response of this system.

b) Find the frequency response of the system.

c) Let $u(t) = e^{-t}1_{\{t \ge 0\}}$. Find $y(t)$.

Consider a continuous time LTI system with a frequency response

$\hat{h}(f) = 1_{\{|f| < f_0\}} \quad f \in \mathbb{R}$

a) Find the impulse response of the system. You can also think of this as the output of the system when the input is the delta signal representing the $\bar{\delta}$ distribution.

b) Find the CCFT of the output, when the input is given by

$u(t) = e^{-t}\cos(f_1 t)1_{\{t \ge 0\}}$

Let a non-anticipative LTI system be given by:

$y(n) = \frac{3}{4}y(n-1) - \frac{1}{8}y(n-2) + u(n)$

a) Compute the frequency response of this system.

b) Compute the impulse response of the system.

c) Find the output when the input is

$u(n) = \left(\frac{1}{2}\right)^n 1_{\{n \ge 0\}}$

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

$\hat{\overline{w_P}}(\phi) = \int \frac{1}{P} w_{\frac{1}{P}}(t)\phi(t)dt,$

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

Compute the Fourier transform of the unit-step distribution defined with

$\bar{U}(\phi) = \int 1_{\{t \ge 0\}} \phi(t)dt, \qquad \phi \in \mathcal{S}.$