** **

** **

It is a remarkable fact that any stable sequence x(n_{1},
n_{2}) 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(n_{1}, n_{2}). The relationships between x(n_{1}, n_{2})
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(n_{1}, n_{2}). The function _{} is called the
discrete-space Fourier transform, or Fourier transform for short, of x(n_{1},
n_{2}). Equation (1.31b) shows how complex exponentials _{} are specifically
combined to form x(n_{1}, n_{2}). The sequence x(n_{1},
n_{2}) 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(n_{1}, n_{2}) 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 w_{1 }and w_{2}, although x(n_{1}, n_{2}) is a function of discrete
variables n_{1} and n_{2} . In addition, _{} is always periodic with a
period of 2∏ x 2∏ ; that is, _{} for all w_{1 }and w_{2}. We can also show that the Fourier
transform converges uniformly for stable sequences. The Fourier transform of
x(n_{1}, n_{2}) is said to **converge uniformly** when _{} is finite and

_{}
for all w_{1} and w_{2
}(1.37)

When the Fourier transform of x(n_{1}, n_{2})
converges uniformly, _{} is an analytic function and is
infinitely differentiable with respect to w_{1} and w_{2}.

A sequence x(n_{1}, n_{2})
is said
to be an eigenfunction of a system T if T[x(n_{1}, n_{2})]=k.x(n_{1},
n_{2})** **for some scalar k. Suppose we use a complex exponential _{}** **as an input
x(n_{1}, n_{2}) to an LSI system with impulse response h(n_{1},
n_{2}). The output of the system y(n_{1}, n_{2}) 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(n_{1},n_{2}). 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)