Prof. Rich Kozick
ELEC 320, Fall 1997
Convolution: Computation and Application to Concert Halls
The main objective in lab this week is to practice computing the convolution
sum for discrete-time systems and the convolution integral for
We will also learn about the impulse function and impulse response
of continuous-time systems.
Lab Reports: Each pair of students is required to
submit a report explaining your answers to items 2 through 6.
This "report" can be hand-written, and is actually more like
a homework assignment.
The objective is for you to practice with convolution
computations, and to show all of the steps in your solutions.
- What questions do you have on the concepts of
impulse function, impulse response, and convolution?
Some notes on the continuous-time impulse function are
and we will demonstrate how to measure the impulse response of an
RC circuit in lab.
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!
- Find the impulse response h(t) of an ideal integrator,
i.e. a system whose output y(t) is the integral of the input
as in .
- Convolve a unit step function with itself: y(t) = u(t) * u(t),
and sketch y(t).
- Solve the three convolution problems on the
Note that the answers are provided.
Be sure that you understand the steps that lead to the answers.
- 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
systems that transmit bits (0s and 1s) from one place to another.
[GRAPHIC NOT AVAILABLE IN HTML FILE -- SEE PAPER VERSION]
- An application of convolution:
The MATLAB script
passes digitized music through discrete-time
systems with various impulse responses and then plays the resulting
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
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!
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
- 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?
- Work on the problems in the
Homework 14 assignment.
All reports are due on Tuesday, October 6 at 8 AM.
Thank you, and have fun!