In terms of their mathematical definitions, convolution and correlation are closely related.
Given deterministic signals x[n], h[n], and y[n], convolution is
defined by Equation 3.1
and cross-correlation as in
Problem 5.5. (Note that the sum in
Equation 3.1 could
just as well have been written with n as the dummy variable to produce y[k].)
In this laboratory exercise we will review your understanding of the difference between
convolution and correlation.
Easy: Each time you press the Generate two signals
button, a real signal x1[n] of random length L will be
generated. This signal and its time-reversed version
x2[n] = x1[L - n] will then be used to compute
a convolution and a cross-correlation. The two signals, x1[n] and
x2[n] will be displayed in the top row. The two results will be
displayed in the bottom row. Each time you repeat the experiment you will generate an
x1[n] with a new random length.
Which of the two signals displayed in the bottom row is the convolution and
which is the cross-correlation?
To what extent does your confidence in your answer depend upon the random choice
of L?
The horizontal time axis units are not displayed in the convolution and
correlation results. Why would displaying them make the problem trivial?
Demanding: In the experiment above, we always used the same form for
x1[n] and changed only its length. Now we will randomly choose both
x1[n] from a family of signals as well as its length. Thus each time you press the
Generate two signals button, a real signal
x1[n] will be chosen at random with a random length L.
This signal and its time-reversed version x2[n] =
x1[L - n] will then be used to compute a convolution and a
cross-correlation.
Which of the two signals displayed in the bottom row is the convolution and which
is the cross-correlation?
To what extent does your confidence in your answer depend upon the random choice
of the signal and the random choice of L?
Very Demanding: In this next experiment, we will use two signals, x1[n]
and x2[n] each of which has been chosen at random from a family of
signals. Each signal has the same randomly chosen length L. Thus each time you press the
Generate two signals button, two real signals
x1[n] and x2[n] will be chosen at random, each
with the same random length L. These two signals will be used to compute a convolution
and a cross-correlation.
Which of the two signals displayed in the bottom row is the convolution and which
is the cross-correlation?
To what extent does your confidence in your answer depend upon the random choice
of the signal and the random choice of L?