# 1.3    THE FOURIER TRANSFORM

## 1.3.1 The Fourier Transform Pair

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)