Fourier's integral theorem expresses a linear transformation, say , when applied to the space of square integrable functions. From this perspective one has
Remark 1: The fact that the Fourier transform is a one-to-one linear transformation from the linear space to the linear space is summarized by saying that the Fourier transform is an ``isomorphism''.
Remark 2: The line of reasoning leading to Parseval's identity also leads to
Remark 3: The above two remarks imply that the Fourier transform is a unitary transformation in . Unitary transformations are ``isometries'' because they preserve lengths and inner products. One says, therefore, that the space of functions defined on the spatial domain is ``isometric'' to the space of functions defined on the Fourier domain. Thus the Fourier transform operator is a linear isometric mapping. This fact is depicted by Figure 2.6
Note, however, that even though the Fourier transform and its inverse,
Thus they do not belong to . Nevertheless linear combinations such as Eq.(2.46) are square integrable, and that is what counts.
What are the eigenvalues and the eigenfunctions of ?
Recall that the Fourier
is a linear one-to-one transformation from
Let be an element of .
Let , the Fourier transform of , be defined by
It is clear that
are square-integable functions, i.e. elements of .
Suppose we define for a square-integrable function and its Fourier transform
the equivalent width as
and the equivalent Fourier width as
is independent of the function , and determine the value of this
and compare them with its full width as defined by its inflection points.
Consider the auto-correlation 22
Compute the Fourier transform of the auto correlation function and thereby show that it equals the ``spectral intensity'' (a.k.a. power spectrum) of whenever is a real-valued function. This equality is known as the Wiener-Khintchine formula.
Consider a linear time-invariant system. Assume its response to a specific driving force, say , can be written as
Here , the ``unit impulse response' (a.k.a. ``Green's function'', as developed in CHAPTER 4 and used in Section 4.2.1), is a function which characterizes the system completely. The system is said to be matched to the particular forcing function if
(Here the bar means complex conjugate.) In that case the system response to a generic forcing function is
A system characterized by such a unit impulse response is called a matched filter because its design is matched to the particular signal . The response is called the cross correlation between and .
of the cross correlation in terms of the Fourier amplitudes
and the corresponding family of normalized cross correlations (i.e. the corresponding responses of the system)