Problems

 Compute the magnitude and phase of $(1-j) \bigg(\frac{1}{2} + j \frac{\sqrt{3}}{2}\bigg)$ Compute the magnitude and phase of $e^{j\pi/2} (1+j) (1+3j)$ Compute the magnitude and phase of $j e^{j \pi/3}$ Compute the magnitude and phase of $e^{j \pi/4} + e^{j 3 \pi/4}$ Compute the magnitude and phase of $(1+3j)^2$ Compute the magnitude and phase of $(1-3j)/(1+3j)^2$ Compute the magnitude and phase of $e^{j \pi/5} \times e^{j 2 \pi/5} \times e^{j 3 \pi/5} \ldots e^{j 9 \pi/5}$ Compute the magnitude and phase of $e^{j \pi/5} \times e^{j 2 \pi/5} \times e^{j 3 \pi/5} \ldots e^{j 9 \pi/5} \times e^{j 10 \pi/5}$ Let $z_1 = 1, z_2 = -\frac{1}{2} + j \frac{\sqrt{3}}{2}, z_3 = -\frac{1}{2} - j \frac{\sqrt{3}}{2}$ [a)] What are $z_1^3, z_2^3$ and $z_3^3$? [b)] Show that $z_3 = z_2^2$ [c)] Show that $z_1 + z_2 + z_3 = 0$ Can you now see why $z_1,z_2,z_3$ can be called the cube roots of unity. They are usually expressed as $1, \omega, \omega^2$. Part c shows that the sum of the cube roots of unity is zero. In one of the homework problems, we will show that this true for $n$th roots of unity for any $n$. Let $z_1 = 2 e^{j\pi/4}$ and $z_2 = 8 e^{j\pi/3}$. Find and express your answer in Cartesian and polar form [a)] $2z_1-z_2$ [b)] $\frac{1}{z_1}$ [c)] $\frac{z_1}{z_2^2}$ [d)] $\sqrt[3]{z_2}$ Prove that $\int e^{ax} \ \cos(bx) \ dx = \frac{e^{ax}}{a^2+b^2} \left(a \cos(bx) + b \sin(bx) \right)$ You can use integration tricks you learned in your calculus class to solve this problem. That is not the point of the exercise. Try using Euler's formula and then using integration of exponentials to see if you can solve the problem.

 OWN 1.25d Is the even part of $x(t) = \cos \left( 4 \pi t\right) \ u(t)$ periodic? If it is, what is the time period? OWN 1.25e Is the even part of $x(t) = \sin \left( 4 \pi t\right) \ u(t)$ periodic? If it is, what is the time period? OWN 1.26b Is the discrete-time signal $x[n] = \cos \left(\frac{n}{8}\right)$ periodic? If it is, what is the time period? OWN 1.26c Is the discrete-time signal $x[n] = \cos \left(\frac{\pi n^2}{8}\right)$ periodic? If it is, what is the time period?

 OWN 1.27a Is the system $y(t) = x(t-2)+x(2-t)$ memoryless, linear, stable, time-invariant and causal? OWN 1.27c Is the system $y(t) = \int_{-\infty}^{2t} x(\tau) \ d \tau$ memoryless, linear, stable, time-invariant and causal? OWN 1.27d Is the system $y(t) = \begin{cases} 0, \ \ t < 0 \\ x(t)+x(t-2), t \geq 0 \end{cases}$ memoryless, linear, stable, time-invariant and causal? OWN 1.27g Is the system $y(t) = \frac{d}{dt}x(t)$ memoryless, linear, stable, time-invariant and causal? Consider the system defined by $y(t) = x(t) \ u(t)$. Is this linear, time invariant, stable, invertible?

 OWN 2.4 Compute the convolution of (1) \begin{align} x[n] = \left\{ \begin{array}{ll} 1, & 3 \leq n \leq 8 \\ 0, & \hbox{otherwise.} \end{array} \right. \ \hbox{and} \ h[n] = \left\{ \begin{array}{ll} 1, & 4 \leq n \leq 15 \\ 0, & \hbox{otherwise.} \end{array} \right. \end{align} OWN 2.6 Compute the convolution of $x[n] = \left(\frac{1}{3} \right)^{-n} \ u[-n-1]$ and $h[n] = u[-n-1]$

Bandpassfromoldexam
DTFS
Phasefunction
Two systems
Problem5fromoldexam - There is a small mistake in the solutions to the last part. The correct answer is 450Hz,500Hz,550 Hz

Review done on Sunday, December 10, 2017 before the final exam

 Compute Discrete-time Fourier transform of $x[n] = \sin \left( \frac{\pi n/5}{\pi n}\right) \ \cos \left(\frac{5 \pi n}{2} \right)$
page revision: 43, last edited: 21 Aug 2021 10:45