6 (x) lim (a/ r ) exp -a2x2. (2.12)
a-u
The delta function possesses these fundamental properties:
6 (x) = 0 for x 0 (2.13)
Co
f 6 (x)dx = f6(x)dx = 1 (2.14)
CO CO
6 (x) = 6(-x) (2.15)
6 (ax) = (1/a) 6(x) a A 0 (2.16)
f f(x) 6(x-a)dx = f(a). (2.17)
The Fourier transform of the delta function is unity. This property
provides a useful tool when studying systems in which an output is
dependent on the input to the system. When an impulse is the input to
the system, the input spectrum is unity at all frequencies. The
spectrum of the output must then correspond to the gain or attenuation
of the system. This frequency response of the system is the Fourier
transform of the output when an impulse is the input. The output of
the system is the impulse response. Thus, the impulse response and
the frequency response of the system are Fourier transform pairs. To
determine the output of a system for a given input, multiply the
Fourier transform of the input by the frequency response of the system
and take the inverse Fourier transform of the result. The convolution
property shows an equivalent operation is to convolve the input with
the impulse response of the system.
O(u,v) = I(u,v) H(u,v)
(2.18)