Convolution

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Convolution

Convolution is the term given to the mathematical technique for
determining a system output given an input signal and the
system impulse response. It is written as

(1)

When you start the convolution tool

you will find that you can select signals, run animations, and control
the display properties. The current input signal and system
impulse response are displayed - you can change the data in
these graphs by clicking and dragging with the mouse or you can
select pre-defined signals using the menus. It is helpful if you
ensure that the signal and system response are coloured differently.

The other graphs show the time-reversed input signal and the system
response, the product of the time-reversed input signal
(due to the

in equation (1)) and the system response, and their convolution. The
product and convolution data will change as the time-reversed
input signal is moved relative to the system impulse response.

When the system impulse response is a unit pulse and the input is a
ramp function, you can test out the convolution process by
clicking on the

symbol in the combined time-reversed system impulse response and
signal graph and dragging the time reversed input signal
backwards and forwards through the system impulse response.
You can also let the tool do this for you by clicking on the
double arrow button.

As the ramp signal is shifted through the unit pulse, the product
graph clearly shows that the convolution is calculating the area
under a selected portion of the ramp signal. The convolution
graph plots the increasing, and then decreasing area of this
product. The system with this impulse response is known as a
finite time integrator as it performs integration over a finite
length of the input signal. Experiment with other input signals
such as sine waves, other unit pulses, impulses, etc.

Convolution with an impulse

Change the system response to a single impulse at 2ms. As you perform
the convolution, as before, you will find that the output is a
copy of the input signal, except that it is delayed by 2ms. In
this instance, the impulse is performing a sampling function of
the input signal as described by

Examination of the product will reveal that the integrand,

, is also an impulse. In fact, the impulse is a scaled unit impulse,
the scaling resulting in an impulse of area

, hence the final convolution result of

Thus, analytically we find that the system response of a unit impulse
results in the output of the system being a copy of the input
signal, with the only difference being a time shift.

As a heuristic argument as to why this should be the case, think of
the system response as the system output when the input signal
is an impulse function at time

When the system response is also an impulse, then the input signal
shape doesn't alter. If that impulse is at a non-zero time
value, then there is a time shift between the input and the
output of the system. (For real systems any time shift must
represent a delay, otherwise the system would predict its own
input signal)!

A system that sums two delayed versions of the input signal is
represented by two impulses in the system response. Experiment
with the effects that you can obtain by examining system
responses consisting of impulses.

Next: Discrete Convolution

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