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,

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

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*.

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