Review of Fourier Analysis - Introduction to Imaging from Scattered Fields - page 190

the dIRAC deltA FunCtIon
This function is very convenient and widely used and best understood in
terms of the following short derivation. If

+∞ +∞ +∞

∫ ∫ ∫

fx () = Fk e ()

− ik x ( )

d k = fxe ()

ik x ( )

d x �

�

e

− ( ik x ) d

k (A.4)

x x x

x x x

−∞ −∞ −∞

then this gives us the opportunity to define the delta function by writing

+∞ +∞ +∞

∫ ∫ ∫

fx () = fx ()( ′ δ xxx ′ − ) d ′ = f xe ()

ik x
x

( )

d xe �

�

− ik x ( ) d

k

(A.5)

x

x

−∞ −∞ −∞

leading to the identity

+∞

∫

δ( xx ′ −=

−∞

) e

ik xkx ( ′ − ) d

k

(A.6)

x x

x

∫

Fx {( )} δ ⇒ δ () xe

− ix ω

d xe =

− i ω 0

= 1

(A.7)

∞ ∞

∫ ∫

If f = e it

−ω 0 then F

() ω = e

it ω

× et

− it ω

d = e

−− i ( ωω ) t

d t = δ ωω ( − )

0 0

0

− ∞ − ∞

If f then = cosω 0 t

∞

1
2

∫

F () ω = ( e

it ω

+ eet

− it ω

)

− it ω

d (A.8)

0 0

−∞

∞ ∞

1
2

1
2

∫ ∫

= e

−− i ( ωω ) t

d t + e

−+ i ( ωω ) t

d t (A.9)

0 0

−∞ −∞

1
2

1
2

(A.10)

= δω ω ( − ) + δωω ( + )

0 0

examples of Some notations
Using the similarity or scaling theorem, that is,

1 u

a

� -

‚

() F.T ¨→ gax ¨¨ G (A.11)

| a |

F.T.

¨¨ ( ) which has equidistant
real zeros at every u / a = m π, where the rect( x ) function is defined in Figure
A.1 and the Fourier transform of this function is illustrated in Figure A.2.
Fourier transform symmetries can be found in Figure A.3.

then, rect()is ax 1 / a units wide ¨→ 1 / a sinc ua /