It is a remarkable fact that any stable sequence x(n1,
n2) can be obtained by appropriately combining complex exponentials
of the form .The
function
,which
represents the amplitude associated with the complex exponential
,can be obtained from
x(n1, n2). The relationships between x(n1, n2)
and
are
given by
(1.35a)
(1.35b)
(1.35c)
Equation (1.35a) shows how the amplitude associated with the exponential
can
be determined from x(n1, n2). The function
is called the
discrete-space Fourier transform, or Fourier transform for short, of x(n1,
n2). Equation (1.31b) shows how complex exponentials
are specifically
combined to form x(n1, n2). The sequence x(n1,
n2) is called the inverse discrete-space Fourier transform
or inverse Fourier transform of
. The consistency of (1.31a) and (1.31b)
can be easily shown by combining them.
From (1.35), it can be seen that is in general complex, even
though x(n1, n2) may be real. It is often convenient to
express
in
terms of its magnitude
|
| and phase
or in terms
of its real part
and
imaginary part
as
(1.36)
From (1.35), it can also be seen that is a function of
continuous variables w1 and w2, although x(n1, n2) is a function of discrete
variables n1 and n2 . In addition,
is always periodic with a
period of 2∏ x 2∏ ; that is,
for all w1 and w2. We can also show that the Fourier
transform converges uniformly for stable sequences. The Fourier transform of
x(n1, n2) is said to converge uniformly when
is finite and
for all w1 and w2
(1.37)
When the Fourier transform of x(n1, n2)
converges uniformly, is an analytic function and is
infinitely differentiable with respect to w1 and w2.
A sequence x(n1, n2)
is said
to be an eigenfunction of a system T if T[x(n1, n2)]=k.x(n1,
n2) for some scalar k. Suppose we use a complex exponential as an input
x(n1, n2) to an LSI system with impulse response h(n1,
n2). The output of the system y(n1, n2) can be
obtained as
(1.38)
From (1.38), is an eigenfunction of any LSI system
for which
is well defined and
is the Fourier
transform of h(n1,n2). The function
is
called the frequency response of the LSI system. The fact that
is an
eigenfunction of an LSI system and that
is the scaling
factor by which
is
multiplied when it is an input to the LSI system simplifies system analysis
for a sinusoidal input. For example. the output of an LSI system with frequency
response
when the input is
can be obtained as
follows:
(1.39)