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## The Fourier Transform as a Unitary Transformation

Fourier's integral theorem expresses a linear transformation, say , when applied to the space of square integrable functions. From this perspective one has

 (244)

Furthermore, this transformation is one-to-one because Fourier's theorem says that its inverse is given by

 (245)

That maps square integrable functions into square integrable functions is verified by the following computation, which gives rise to Parseval's identity: For we have

Thus we obtain Parseval's identity (= completeness relation'', see Eq.(1.17) on page ). The only proviso is (a) that the function be square-inegrable and (b) that its Fourier transform be given by the Fourier transform integral.

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

whenever .

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,

 (246)

take square integrable functions into square integrable functions, the basis elements'' are not square integrable. Instead, they are Dirac delta function'' normalized, i.e.,

Thus they do not belong to . Nevertheless linear combinations such as Eq.(2.46) are square integrable, and that is what counts.

Exercise 23.1 (THE FOURIER TRANSFORM: ITS EIGENVALUES)
The Fourier transform, call it , is a linear one-to-one operator from the space of square-integrable functions onto itself. Indeed,

Note that here and are viewed as points on the common domain of and .

(a)
Consider the linear operator and its eigenvalue equation

What are the eigenvalues and the eigenfunctions of ?

(b)
Identify the operator ? What are its eigenvalues?

(c)
What are the eigenvalues of

Exercise 23.2 (THE FOURIER TRANSFORM: ITS EIGENVECTORS)

Recall that the Fourier transform is a linear one-to-one transformation from onto itself.
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 .
Consider the SUBSPACE spanned by these vectors, namely

(a)
Show that is finite dimensional.
What is ?
(Hint: Compute , etc. in terms of )
(b)
Exhibit a basis for .
(c)
It is evident that is a (unitary) transformation on .
Find the representation matrix of , relative to the basis found in part b).
(d)
Find the secular determinant, the eigenvalues and the corresponding eigenvectors of .
(e)
For , exhibit an alternative basis which consists entirely of eigenvectors of , each one labelled by its respective eigenvalue.
(f)
What can you say about the eigenvalues of viewed as a transformation on as compared to which acts on a finite-dimensional vector space?

Exercise 23.3 (EQUIVALENT WIDTHS)

Suppose we define for a square-integrable function and its Fourier transform

the equivalent width as

and the equivalent Fourier width as

(a)
Show that

is independent of the function , and determine the value of this
(b)
Determine the equivalent width and the equivalent Fourier width for the unnormalized Gaussian

and compare them with its full width as defined by its inflection points.

Exercise 23.4 (AUTO CORRELATION SPECTRUM)

Consider the auto-correlation 22

 (247)

of the function whose Fourier transform is

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.

Exercise 23.5 (MATCHED FILTER)

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 .
(a)
Compute the total energy

of the cross correlation in terms of the Fourier amplitudes

and

(b)
Consider the family of forcing functions

and the corresponding family of normalized cross correlations (i.e. the corresponding responses of the system)

Show that
(i)
is the peak intensity, i.e., that

(Nota bene: The function corresponding to is called the auto correlation function of ). Also show that
(ii)
equality holds if the forcing function has the form

Lecture 14

#### Footnotes

...\space 22
Not to be confused with the convolution integral Eq.(2.56) on page

Next: Fourier Transform via Parseval's Up: The Fourier Integral Previous: The Fourier Integral Theorem   Contents   Index
Ulrich Gerlach 2010-12-09