Fourier Transforms

FOURIER TRANSFORMS

1 INTRODUCTION

2 INTEGRAL TRANSFORMS

3 FOURIER INTEGRAL THEOREM

4 FOURIER TRANSFORMS AND ITS PROPERTIES

5 CONVOLUTION THEOREM AND PARSEVAL’S THEOREM

6 FOURIER SINE AND COSINE TRANSFORMS

1 Introduction

This unit starts with integral transforms and presents three well-known integral transforms, namely, Complex Fourier transform, Fourier sine transform, Fourier cosine transform and their inverse transforms. The concept of Fourier transforms will be introduced after deriving the Fourier Integral Theorem. The various properties of these transforms and many solved examples are provided in this chapter. Moreover, the applications of Fourier Transforms in partial differential equations are many and are not included here because it is a wide area and beyond the scope of the book.

2 Integral Transforms

The integral transform f(s) of a function f(x) is defined by

if the integral exists and is denoted by I{f(x)}. Here, K(s,x) is called the kernel of the transform. The kernel is a known function of „s‟ and „x‟. The function f(x) is called the  inverse transform of f(s). By properly selecting the kernel in the definition of general integral transform, we get various integral transforms.

The following are some of the well-known transforms:

(i) Laplace Transform

(ii) Fourier Transform

(iii) Mellin Transform

(iv) Hankel Transform

where J_n(sx)   is the Bessel function of the first kind and order 'n'.

3 FOURIER INTEGRAL THEOREM

If f(x) is defined in the interval (-ℓ,ℓ), and the following conditions 

(i) f(x) satisfies the Dirichlet‟s conditions in every interval (-ℓ,ℓ),  

which is known as the Fourier integral of f(x).

Note:

When f(x) satisfies the conditions stated above, equation (3) holds good at a point of continuity. But at a point of discontinuity, the value of the integral is (1/ 2) [f(x+0) + f(x-0)] as in the case of Fourier series.

Fourier sine and cosine Integrals

The Fourier integral of f(x) is given by

When f(x) is an odd function, f(t) coslt is odd while f(t) sinlt is even. Then the first integral of (4) vanishes and, we get

which is known as the Fourier sine integral.

Similarly, when f(x) is an even function, (4) takes the form

which is known as the Fourier cosine integral.

Complex form of Fourier Integrals

The Fourier integral of f(x) is given by

Since cos l(t –x) is an even function of l, we have by the property of definite integrals

Similarly, since sin l(t –x) is an odd function of l, we have

which is the complex form of the Fourier integral.

4 Fourier Transforms and its properties

Fourier Transform

We know that the complex form of Fourier integral is

The function F(s), defined by (1), is called the Fourier Transform of f(x). The function f(x), as given by (2), is called the inverse Fourier Transform of F(s). The equation (2) is also referred to as the inversion formula.

Properties of Fourier Transforms

(1) Linearity Property

If  F(s) and G(s) are Fourier Transforms of f(x) and g(x) respectively, then

F{a f(x) + bg(x)} = a F(s) + bG(s),

where a and b are constants.

= a F(s) + bG(s) i.e,

F{a f(x) + bg(x)} = a F(s) + bG(s)

(2) Shifting Property

(i)                If  F(s) is the complex Fourier Transform of f(x), then

F{f(x-a)} = e^isa F(s).

(ii) If F(s) is the complex Fourier Transform of f(x), then

(3) Change of scale property

If F(s) is the complex Fourier transform of f(x), then

F{f(ax)} =1/a F(s/a), a ¹0.

Put  ax = t, so that dx = dt/a.

(4) Modulation theorem.

If F(s) is the complex Fourier transform of f(x),

Then  F{f(x) cosax} = ½{F(s+a) + F(s-a)}.

(5) n^th derivative of the Fourier Transform

If F(s) is the complex Fourier Transform of f(x),

Then F{xⁿ f(x)} = (-i)ⁿ dⁿ/dsⁿ .F(s).

(6) Fourier Transform of the derivatives of a function.

If  F(s) is the complex Fourier Transform of f(x),

Then, F{f-isF(s) if„(x)}f(x)®0as x=® ±¥.

In general, the Fourier transform of the n^th derivative of f(x) is given by

F{f ⁿ(x)} = (-is)ⁿ F(s),

provided-1 ‟the derivatives frst®„n±¥.   vanish   as   x

Property (7)

If F(s) is the complex Fourier Transform of f(x), then F

Property (8)

If F(s) is the complex Fourier transform of f(x),

Note:  If F{f(x)} = F(s), then

F{f(-x)} = F(-s).

Example 1

Find the F.T of f(x) defined by

f(x) = 0       x<a

= 1    a<x<b

= 0    x>b.

The F.T of f(x) is given by

Example 2

Find the F.T of f(x) = x  for  |x | =< a

= 0    for |x | > a.

Example 3

Find the F.T of   f(x) = e^iax ,  0 < x < 1

= 0 otherwise

The F.T of f(x) is given by

Note:

If the F.T of f(x) is f(s), the function f(x) is called self-reciprocal. In the above example e ^{-x 2/ 2} is self-reciprocal under F.T.

Example 5

Find the F.T of              

f(x) = 1 for |x|<1.

= 0 for |x|>1.

5 Convolution   Theorem . and   Parseval’s Theorem

The convolution of two functions f(x) and g(x) is defined as

Convolution Theorem for Fourier Transforms.

The Fourier Transform of the convolution of f(x) and g(x) is the product of their Fourier Transforms,

i.e, F{f(x) * g(x)} = F{f(x).F{g(x)}.

Proof:

F{f(x) * g(x)} = F{(f*g)x)}

Hence,  F{f(x) * g(x)} = F{f(x).F{g(x)}.

Parseval’s identity   for   Fourier   Transforms

If  F(s) is the F.T of f(x), then

Proof:

By convolution theorem, we have

F{f(x) * g(x)} = F(s).G(s).

Therefore,  (f*g) (x) = F^-1{F(s).G(s)}.

Example 6

Find the F.T of f (x) = 1-|x|for |x|<1.

= 0    for |x|> 1

Example 7

Find the F.T of f(x) if

f(x) =

= 1 for |x|<a

=  0 for |x|>a>0.

6 Fourier sine and cosine transforms:

Fourier sine Transform

We know that the Fourier sine integral is

The function F_s(s), as defined by (1), is known as the Fourier sine transform of f(x). Also the function f(x), as given by (2),is called the Inverse Fourier sine transform of F_s(s) .

Fourier cosine transform

Similarly, it follows from the Fourier cosine integral

The function F_c(s), as defined by (3), is known as the Fourier cosine transform of f(x). Also the function f(x), as given by (4),is called the Inverse Fourier cosine transform of F_c(s) .

Properties of Fourier sine and cosine Transforms

If F_s(s) and F_c(s) are the Fourier sine and cosine transforms of f(x) respectively, the following properties and identities are true.

(1) Linearity property

F_s [a f(x) + b g(x) ] = a F_s { f(x) } + b F_s { g(x) }.

and  F_c [a f(x) + b g(x) ] = a F_c { f(x) } + b F_c { g(x) }.

(2) Change of scale property

F_s [ f(ax) ] = (1/a) F_s [ s/a ].

and   F_c [ f(ax) ] = (1/a) F_c [ s/a ].

(3) Modulation Theorem

i.           F_s [ f(x) sinax ] = (1/2) [ F_c (s-a) - F_c (s+a)].

ii.         F_s [ f(x) cosax ] = (1/2) [ F_s (s+a) + F_s (s-a)].

iii.      F_c[ f(x) cosax ] = (1/2) [ F_c (s+a) + F_c (s-a) ].

iv.      F_c[ f(x) sinax ] = (1/2) [ F_s (s+a) - F_s (s-a) ].

Proof

The Fourier sine transform of   f(x)sinax  is given by

Similarly, we can prove the results (ii), (iii) & (iv). 

Similarly, we can prove the second identity and the other identities follow by setting g(x) = f(x) in the first identity.

Property (5)

If  F_s(s) and F_c(s) are the Fourier sine and cosine transforms of f(x) respectively, then

Example 8

Find the Fourier sine and cosine transforms of e^-ax and hence deduce the inversion formula.

The Fourier sine transform of  f(x) is given by