## 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,

*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

*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

*same*weighted sum of impulse

*responses*.

**Next Section:**

Convolution Representation

**Previous Section:**

Impulse Response Example