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)
   

   A note about MATLAB help:

   Remember that MATLAB has on-line help available. For example, typing help conv will describe the convolution command and how to use it. Also, if you type hthelp on the Suns, then a hypertext version of help will be opened in a new window. This might be useful to view the available commands and follow links to related commands.

Thank you, and have a good weekend!