ELEC 320, Fall 1998
Prof. Rich Kozick

## Homework 10

Date Assigned: Friday, September 25, 1998
Date Due: Monday, September 28, 1998

1. Reading: Please read Sections 3.1 and 3.2 on convolution for discrete-time systems and signals. For next week, please read Sections 3.3 and 3.4 on convolution for continuous-time systems and signals. Also, please review Section 1.2 on signals, with particular attention to the impulse function delta(t), step function u(t), and time-shifting operations on signals.

2. Lab 1: Remember that reports for Lab 1 are due at 9 AM on Wednesday, September 30.

3. The questions on the "half-sheets" of paper that I gave you on Wednesday fell into three main categories: (1) linear, time-invariant systems, (2) difference equations, and (3) MATLAB. I hope that class today and Monday will tell you more about difference equations, and that Lab 2 will help answer your MATLAB questions. Below is a practice problem for classifying systems according to linearity and time-invariance. You do not have to submit a solution, but please work the problem for Monday, and I will look at your solutions and answer your questions during class.

Consider the following three systems, where x(t) is the input signal and y(t) is the output signal.

1. y(t) = 2 x(t) + 1
2. Differentiator: y(t) = dx / dt
3. Half-wave rectifier: y(t) = x(t) if x(t) >= 0, y(t) = 0 if x(t) < 0.

Sketch the output y1(t), y2(t), y3(t), y4(t), and y5(t) from each system when the input is x1(t), x2(t), x3(t), x4(t), and x5(t) defined as follows:

• x1(t) in the graph to the right
• x2(t) = -2 x1(t)
• x3(t) = x1(t-3)
• x4(t) = 1 for all t
• x5(t) = x1(t) + x4(t)

I would suggest that you sketch the input and output signal for each case. What can you conclude about linearity and time-invariance for each system?

4. Please solve item 3 on the Homework 9 assignment. Submit your solution on Monday, September 28.

5. If you would like to practice with discrete-time convolution, here is a problem that I will ask you to solve next week for homework. Find and sketch the sequence y[n] = x[n] * h[n], where x[n] and h[n] are defined below and * denotes convolution.

[GRAPHIC NOT AVAILABLE IN HTML FILE -- SEE PAPER VERSION]

You can check your result using MATLAB and the conv command as follows:

```>> h = [1.5, 1.5, 1.5];
>> x = [2, 2, 2, 2, 2];
>> y = conv(x, h);
>> n = 0:length(y)-1;
>> stem(n, y)
```