Convolution - Signal Processing of Random Physiological Signals - page 87

Input: Impulse Filter Response a & b Combined

1/ t

t

(a) (b) (c)

FIGURE 10.3 : Low-pass filter response. (a) It shows a single unity pulse of t duration and

1 t magnitude. (b) It shows the output response of the low-pass filter shown in Fig. 10.2 to the

input pulse, (a). (c) It shows the combined input and output signals

/

becomes (10.2).

1

t
RC

RC e −

h ( t ) = (10.2)

What Fig. 10.4 implies is that at the instant the impulse occurs a current flows in
the circuit and is supposed to charge the capacitor to a voltage of 1/RC, instantaneously.
However, the instantaneous charging that theoretically occurs in the instant that the
impulse is applied is not physically possible because of the principle of conservation of
momentum. As a note of information, the LaPlace transformation of the delta function,
δ

, is unity (1); therefore, the frequency domain is a better and easier method by which
to analyze the response to an impulse function.
It is important to understand the time/frequency relationships that exist when
functions are transformed into another domain. For example, the product (or Modulation )

h ( t )

FIGURE 10.4 : Low-pass filter response to a delta function as t 0

➡