and by replacing the temporary variable k’ by k, we get
That is the order of convolution is reversed. Thus we have two formulae of convolution:
and
In practive we usually let the longer sequence stay fixed, and shift the shorter one.
The commutative characteristic of convolution means that we can swap the input signal with the impulse response of a system without affecting the output. This idea is depicted in
[link] .
Commutation between input signal and impluse response gives the same output
Associativity
It can be shown that
[link] shows the system meaning of the associativity, where two systems in series (in cascade) can be replaced by only one whose impulse response is the convolution of the two individual impulse responses.
Impluse response of two systems in cascade
Two systems in cascade have impulse responses
Find the overall impulse response.
Solution
First
and
should be smaller than 1 to ensure the convergence of the sequences. Notice that both impulse responses are causal. The overall impulse response is
The actual limits of summation are
and
(see
[link] later), hence
Using the formula of finite geometric series.
here
, we get
Distributivity
It can be shown
The system meaning is illustrated in
[link] where two systems connected in parallel can be replaced by one whose impulse response is the sum of the two ones.
Impulse response of two systems in parallel
Impulse response for causal system and signal
Since impulse response is a characterization (among other characterizations) of systems. As such, the causality of a system would be reflected on its impulse response. From the convolution
Figure the output at instant
is:
In order the output signal
does not depend on future
values of input signal
, the second term of above equation should be zero, i.e.
for
. As k is a dummy variable, we conclude
Thus, the causality of a system implies that its impulse response is zero and vice versa. The output at time
is now the first term of the equation
For any time n,
Had the convolution
been used, the result would be
In above, only the causality of the system is considered. Now, the imput signal is also causal, the result is
And equivalently
Notice that the summation limits for both cases are the same and the upper limit increases with n, also that the output signal
at time n depends only the convolution summation up to n and independent of future values after n. So we can write the upper limit as
without effecting the result.
Input signal and impulse response are respectively
Find the output signal by analysis computation.
Solution
Notice that both
and
are causal and of infinite duration. The given condition
ensures the convergence of
. We choose to evaluate
, using
[link]
So, the result is.
The output signal
does not go to
but grows asymptotically to the finite value of
(see
Example ).
System identification
In DSP sometimes we need to determine a system, assumed LTI (or LSI), when we know the input signal and output signal, e.g. by setting up an experiment. This problem is called
system identification . Specifically, we must determine the impulse response of the system, and then the signal difference equation if necessary.
Adaptive filters using FIR filters are often used to identify unknown DSP systems. In control theory, also, sytem identification is a familiar problem.
For causal systems the output is given by the reduced convolution (
[link] ) which is repeated here
At
giving
provided
. At
, We can write
giving
In order to identify an unknown DSP system (hardware or software) we applied a signal x(n) and obtained the output y(n) as follows.
Determine the impulse response.
Solution
Note that both input and output signals are causal. We proceed the evaluation of the impulse response as follows.
Continuing we will see that for . Thus the system impulse response is
The system is causal as expected.
Another method is to transform the problem to the z-domain (chapter 4).
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