Convolution - Signal Processing of Random Physiological Signals - page 85

CHAPTER 10

Convolution

The convolution integral dates back to 1833. It is one of the methods used to determine
the output response of a system to a specific input if the system transfer function is
known. Convolution applies the superposition principal. Conceptually,

1. the input signal is represented as a continuum of impulses;

2. the response of the system to a single impulse is obtained;

3. the response of the system to each of the elementary impulses, representing the

input, is computed; and then

4. the total input response is obtained by superposition.

Often signals are represented in terms of elementary linear basis functions, but not
as a continuum of impulses. Therefore let us examine the representation of an arbitrary
time function by a continuum of impulses. Let us begin with the approximation of a
function f ( t ) in the interval T by rectangles (rectangular pulses) whose height

is equal to the function at the center of the rectangular pulse, and the width is

− T to +

T

(Fig. 10.1).
The equation describing the impulse representation of the continuous signal is
given by (10.1).

+ T

f ( t ) = f ( τ ) δ ( t − τ ) ∂τ from − T < t < T (10.1)

− T

Since the convolution is used in the time domain analysis of a filter response, let us