The main objective in lab this week is to practice computing the convolution sum for discrete-time systems and the convolution integral for continuous-time systems. We will also learn about the impulse function and impulse response of continuous-time systems. Please work in groups of two or three students.

1. Please show your MATLAB programs from Lab 2 to Professor Kozick or a lab assistant. Do you have questions about using MATLAB?

2. What questions do you have on the concepts of impulse function, impulse response, and convolution for discrete-time systems? Can you do Homework 11?

3. Some notes on the continuous-time impulse function are attached, and we will demonstrate how to measure the impulse response of an RC circuit in lab.

   Important point: Linear, time-invariant (LTI) systems are very nice. The impulse response h(t) of the system can be measured fairly easily. Then, the system response to any other input signal x(t) is obtained by convolving the input signal with h(t), i.e. the output is y(t) = x(t) * h(t) . This is why convolution is important!

4. Find the impulse response h(t) of an ideal integrator, i.e. a system whose output y(t) is the integral of the input x(t), as in .

5. Convolve a unit step function with itself: y(t) = u(t) * u(t), and sketch y(t).

6. Solve the three convolution problems on the attached sheet. Note that the answers are provided. Be sure that you understand the steps that lead to the answers.

7. Consider the signals s1(t), s2(t) and filters with impulse response h1(t), h2(t) as shown below. Compute the output of each filter due to each input. That is, compute the four convolutions y11(t) = s1(t)*h1(t), y12(t) = s1(t)*h2(t), y21(t) = s2(t)*h1(t), y22(t) = s2(t)*h2(t).

   These signals and filters are commonly used in digital communication systems that transmit bits (0s and 1s) from one place to another.

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8. An application of convolution:

   The MATLAB script mus.m passes digitized music through discrete-time systems with various impulse responses and then plays the resulting music. Simulations of this type are used to understand how an audio speaker or a listening room with a certain impulse response will affect the music that is heard in the room. The impulse response can be measured easily in practice in order to obtain a model for a listening room.

   Run the Matlab script mus.m on a Sun computer. The original music will be played, followed by the music convolved with g(t) = (2 pi 300) exp(-2 pi 300 t), and then the music convolved with a different function h(t). The impulse responses g(t) and h(t) will be plotted on your screen. The program takes a while to run, so be patient!

   You can listen to the results here, without running the MATLAB program:

   • Original music
   • Result of filtering with g(t)
   • Result of filtering with h(t)

   Please write brief answers to the following questions. You don't have to submit any plots.

   • What effect does convolution with g(t) have on the music, i.e. what is different about the music after the processing? Can you explain this effect from the shape of g(t)? (Hint: Does g(t) resemble the impulse response of an RC circuit? What type of filter does g(t) describe, and what is the cutoff frequency?)
   • What effect does convolution with h(t) have on the music? Can you relate this effect to the shape of h(t)? What physical mechanism might give rise to an effect like this in a concert hall?

All reports are due on Friday, October 9 at 9 AM.

Thank you, and have fun!