Implications of Linear-Time-Invariance
Using the basic properties of
linearity and time-invariance, we will
derive the
convolution representation which gives an algorithm for implementing the
filter
directly in terms of its
impulse response. In other words,

Figure:
Input and output signals for the filter
. (a) Input impulse
. (b) Output
impulse response
. (c) Input delayed-impulse
. (d) Output delayed-impulse response
.
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The
convolution formula plays the role of the
difference equation when
the impulse response is used in place of the difference-equation
coefficients as a filter representation. In fact, we will find that,
for
FIR filters (nonrecursive,
i.e., no feedback), the difference
equation and convolution representation are essentially the same
thing. For recursive filters, one can think of the convolution
representation as the difference equation with all feedback terms
``expanded'' to an infinite number of feedforward terms.
An outline of the derivation of the convolution formula is as follows:
Any signal

may be regarded as a superposition of impulses at
various amplitudes and arrival times,
i.e., each sample of

is
regarded as an impulse with amplitude

and delay

. We can
write this mathematically as

. By the
superposition principle for
LTI filters, the filter output is simply
the superposition of impulse
responses 
, each having a scale factor and time-shift given by
the amplitude and time-shift of the corresponding input impulse.
Thus, the sample

contributes the signal

to
the convolution output, and the total output is the sum of such
contributions, by superposition. This is the heart of
LTI filtering.
Before proceeding to the general case, let's look at a simple example
with pictures. If an impulse strikes at time

rather than at
time

, this is represented by writing

. A
picture of this delayed impulse is given in Fig.
5.2c. When

is fed to a time-invariant filter, the output will be
the impulse response

delayed by 5 samples, or

. Figure
5.2d shows the response of the example filter of
Eq.

(
5.3) to the delayed impulse

.
In the general case, for time-invariant filters we may write
where

is the number of samples delay. This equation states that
right-shifting the input impulse by

points merely right-shifts the
output (impulse response) by

points. Note that this is just a
special case of the definition of time-invariance, Eq.

(
4.5).
If
two impulses arrive at the filter input, the first at time

,
say, and the second at time

, then this input may be expressed
as

. If, in addition, the amplitude of the
first impulse is 2, while the second impulse has an amplitude of 1,
then the input may be written as

. In
this case, using
linearity as well as time-invariance, the
response of the general LTI filter to this input may be expressed as
For the example filter of Eq.

(
5.3), given the input

(pictured in Fig.
5.3a),
the output may be computed by scaling, shifting, and adding together
copies of the impulse response

. That is, taking the impulse
response in Fig.
5.2b, multiplying it by 2, and adding it to the
delayed impulse response in Fig.
5.2d, we obtain the output
shown in Fig.
5.3b. Thus, a weighted sum of impulses produces
the
same weighted sum of impulse
responses.
Figure 5.3:
Input impulse pair and corresponding output
for the filter
. (a) Input: impulse of
amplitude 2 plus delayed-impulse
. (b)
Output:
.
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Next Section: Convolution RepresentationPrevious Section: Impulse Response Example