evan.md

Problem 1 LTI Properties (26 pts)

[24 pts] Classify the following systems, with input $x(t)$ (or $x[n]$) and output $y(t)$ (or $y[n]$). In each column, write "yes", "no", or "?" if the property is not edcidable with the given information. (+1 for correct, 0 for blank, -0.5 for incorrect). (For 1d, you are given the system is known to be linear and time=invariant.) For 1b and 1d, 2 test input cases are given.

Let $\Pi(t) = u(t + \frac12) - u(t - \frac12)$.

System	Causal	Linear	Time-invariant	BIBO stable
a. $y(t) = 2x(t-1)-5$	yes	no	yes	yes
b. \begin{cases} x & \text{if input }x(t)=0 \ tu(t) & \text{if input }x(t)=u(t-1)\end{cases}	?	?	?	no
c. $y(t) = x(t)[\cos(2\pi t)u(t)]$	yes	yes	no	yes
d. $y(t) = \begin{cases}0 &\text{if input }x(t)=0 \ u(t) & \text{if input (x(t) = u(t)\end{cases}$	yes	YES	YES	yes
e. $y(t) = \int_{r=-\infty}^{\infty}x(\tau)\Pi (t-\tau)d\tau$	no	yes	yes	yes
f. $y(t) = x(t)\cdot [1-\delta (t + 100)]$	yes	yes	no	no
g. $y[n] =\mathcal{Z}^{-1}\left{\frac{z^2}{z+1}\right} * x[n]$	no	yes	yes	no

e) $y(t) = x(t) * pi(t)$ f) $x(t) - x(t)\delta(t+100) = x(t) - x(t + 100)\delta(t+100)$

Problem 2 Short Answers (40 pts)

Answer each part independently. Note $\Pi(t) = u(t+\frac12) - u(t-\frac12)$.

[4 pts] a. Complete the table with the approprate type of Fourier transform to use (FS, FT, DTFT, or DFT) on a signal of each type.

	aperiodic in time	periodic in time
continuous time
discrete time

[3 pts] b. $X(j\omega) = \cos(\omega/2) + 1$. Find $x(t)$.

[4 pts] c. A periodic signal $x(t) = p(t) * \sum_{n=-\infty}^\infty \delta(t-2n)$, where $\mathcal{F}{p(t)}=P(j\omega)=\cos(\omega/2)+1$.

[7 pts] d. A periodic signal $x(t)$ has period 4 seconds and Fourier Series coefficients $a_k = \frac{\sin(k\pi/4)}{k\pi}$. Find the time average power $\frac{1}{T}\int_T x^2(t)dt$.

[9 pts] e. Initial and final value. i. Given $X(s) = \frac{s+3{s^2+3s+2}$. Find $x(0^+)$. ii. Given causal $X(z) = \frac{z^{-2} + 2z^{-3}}{1-2z^{-1}+\frac54z^{-2}-\frac14z^{-3}}$. Find $\lim_{n\rightarrow \infty}x[n]$. iii. Given causal $X(z) = \frac{z^{-2} + 2z^{-3}}{1-2z^{-1}+\frac54z^{-2}-\frac14z^{-3}}$. Find $x[0]$.

[5 pts] f. Given $X(s) = \frac{s+5}{s^3+3s^2+2s}$. Find $x(t)$.

[8 pts] g. Sketch $y(t) = 3\pi \cdot u(t+1)* \cos(\pi t)u(t)$.

Problem 3. Digital Filter (12 pts)

A continuous time filter has impulsse response $h(t) = e^{-\pi t/2} u(t)$. [5 pts] a. The filter is sampled such that $h[n] = h(nT_s)$ where the sampling rate $T_s = 1$ sec. Find the Z transform of $h[n]$. [5 pts] b. Sketch $|H(e^{j\omega})|$, labelling maximum and minimum amplitude. (Maximum nd minimum may be left as functions of $e$).