Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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The Fourier transform is a generalization of the complex Fourier series in the limit as Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev. Replace the discreteAn with the continuous  F(k) dk while letting n/L → k Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev. Then change the sum to an integral, and the equations become

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev          Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev
Here,

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

is called the forward (-i) Fourier transform, and

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev             Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

is called the inverse (+i) Fourier transform. The notation Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is introduced in Trott (2004, p. xxxiv), and Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202).

Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev instead of the oscillation frequency v. However, this destroys the symmetry, resulting in the transform pair

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

To restore the symmetry of the transforms, the convention

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 sometimes used (Mathews and Walker 1970, p. 102).

In general, the Fourier transform pair may be defined using two arbitrary constants a and b as

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The Fourier transform F(k) of a function f(x) is implemented the Wolfram Language as FourierTransform[f, x, k], and different choices of a and b can be used by passing the optional FourierParameters-> {a, b} option. By default, the Wolfram Language takes FourierParameters as (0,1). Unfortunately, a number of other conventions are in widespread use. For example, (0,1) is used in modern physics, (1,-1) is used in pure mathematics and systems engineering, (1,1) is used in probability theory for the computation of the characteristic function, (-1,1) is used in classical physics, and (0, -2π) is used in signal processing. In this work, following Bracewell (1999, pp. 6-7), it is always assumed that a = 0 and b = -2π unless otherwise stated. This choice often results in greatly simplified transforms of common functions such as 1, Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev, etc.

Since any function can be split up into even and odd portions E(x) and O(x),

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

a Fourier transform can always be expressed in terms of the Fourier cosine transform and Fourier sine transform as

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (19)

A function f(x) has a forward and inverse Fourier transform such that

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (20)

provided that

1.. Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

2. There are a finite number of discontinuities.

3. The function has bounded variation. A sufficient weaker condition is fulfillment of the Lipschitz condition

(Ramirez 1985, p. 29). The smoother a function (i.e., the larger the number of continuous derivatives), the more compact its Fourier transform.

The Fourier transform is linear, since if f(x) and g(x) have Fourier transforms F(k) and G(k), then

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Therefore,

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev          Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The Fourier transform is also symmetric sinceFourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev.

Let f * g denote the convolution, then the transforms of convolutions of functions have particularly nice transforms,

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev    Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The first of these is derived as follows:

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev    Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

where  .x" ≡ x - x'

There is also a somewhat surprising and extremely important relationship between the autocorrelation and the Fourier transform known as the Wiener-Khinchin theorem. Let Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevdenote the complex conjugate of f, then the Fourier transform of the absolute square of F(k) is given by

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (33)

The Fourier transform of a derivative f'(x) of a function f(x) is simply related to the transform of the function f(x)  itself. Consider

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (34)

Now use integration by parts

 

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev    (35)

with

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev   Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

and 

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

then

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (40)

The first term consists of an oscillating function times f(x). But if the function is bounded so that

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (41)

(as any physically significant signal must be), then the term vanishes, leaving

 

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

This process can be iterated for the nth derivative to yield

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(44)

The important modulation theorem of Fourier transforms allows Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev to be expressed in terms of Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev as follows,

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Since the derivative of the Fourier transform is given by

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        (49)

it follows that

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev               (50)

Iterating gives the general formula

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The variance of a Fourier transform is

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        (53)

and it is true that

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev            (54)

If f(x) has the Fourier transform Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev, then the Fourier transform has the shift property

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(57)

If f(x) has a Fourier transform Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev, then the Fourier transform obeys a similarity theorem.

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (58)

so f(a x) has the Fourier transform

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (59)

The "equivalent width" of a Fourier transform is

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The "autocorrelation width" is

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

where f * g denotes the cross-correlation of f and Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev and  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  is the complex conjugate.

Any operation on f(x) which leaves its area unchanged leaves F(0) unchanged, since

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev          (64)

The following table summarized some common Fourier transform pairs.

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

In two dimensions, the Fourier transform becomes

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Similarly, the n-dimensional Fourier transform can be defined for Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev by

Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev   Fourier Transform - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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