Fourier Transform
Chapter 4 Notation for some useful
functions
Chapter 4. Notation for some useful functions
1. Rectangular function, (x) 2. Triangle function, (x)
3. Various exponentials & Gaussian & Rayleigh curves 4. Heaviside’s unit step function, H(x)
5. Sign function, sgn x
6. Filtering or interpolation function, sinc x 7. Pictorial representation
Rectangle function of unit height and base, (x)
2 1 1
2) 1 2
(1
2 0 1
) (
x x x x
Fig. 4. 1
Example of Function Segmentation by (x)
x x x x
x f
2 1
2 1 2
1
2 1
1 cos 0 )
(
Fig. 4. 2
] / ) [( x c b h
Displaced Rectangle function
of height h and base b, centered on x=c
Any segment of a given function can be selected by multiplication by a displaced rectangle function
In the frequency domain, multiplication by (x) is an expression of ideal low-pass filtering
Fig. 4. 3
The triangle function of unit height and area, (x)
• (x) is the self-convolution of (x)
• (x) is used for giving compact notation for polygonal functions (continuous functions consisting of linear segments
• (x) of height h, base b, and area hb/2
1 1 1
) 0
(
x
x x x
) /
( x
21b h
Fig. 4. 4
Various Exponential Functions
(a) rising exponential (b) falling exponential
(c) truncated falling exponential (d) double-sided falling exponential
Fig. 4. 5
The Gaussian Functions, exp(-x
2)
Fig. 4.6 The Gaussian function and the probability ordinate
(2)1/2 exp(21 x2)) exp(
x2• Normalizing the Gaussian function for certain advantage
- The central ordinate and the area under the curve are unity
• The Fourier transform of the Gaussian function is also Gaussian.
The Gaussian function in statistics
• The Gaussian distribution
– “normal(error) distribution with zero mean”
– Normalized so that the area and the standard deviation are unity.
• The probability ordinate is
) 2 exp(
1 2
2 1 x
) 2 / 2 exp(
1 2
2
xWhen the standard deviation is , it is
and the area under the curve remains unity. The central
ordinate is equal to 0.3989/
The error integral erf x
The complementary error integral erfc x The probability integral (x)
x t dt
x 0
2) 2 exp(
erf
erf 2 )
2 exp(
) 1
( 21 2 x
dt t
x x
x
x x 1 erf erfc
• In this book, exp(- x
2) is used extensively because of its symmetry under the Fourier transform.
• Integral of exp(- x
2) is related to erf x.
] 2 2 [
1 2 erf 1
2 1 2 ) 1
exp(
] 2 2 [
erf 1 2 ) 1
exp(
2 0
2
x x
dt t
x x
dt t
x x
The customary dispersion parameters of exp(-x
2)
• Probability error = 0.2691= 0.6745
• Mean absolute error (mean of |x|) =
-1 = 0.3183 = 0.7979• Standard deviation (mean of x2) = (2)-1/2 =0.3989 =
• Width to half-peak = 0.9394= 2.355
• Equivalent width = 1.000 = 2.5066
Fig. 4. 6
The Gaussian distribution in two dimensions
• In two dimensions the Gaussian distribution generalizes to
)]
(
exp[
x2 y2with symmetry under the Fourier transformation, with unit central ordinate, and with unit volume.
• The version used in statistics, for arbitrary standard deviations
x,
y, is
)]
/ 2
/ ( 2 exp[
1 2 2 2 2
y x
y x
y
x
• Under conditions of circular symmetry, and putting x
2+y
2=r
2, the 2D probability ordinate becomes
) 2 / 2 exp(
1 2 2
2
rRayleigh’s Distribution
• The probability R(r)dr of finding the radial distance in the range r to r+dr is 2rdr times previous equation
.
) 2 / exp(
) (
2 ) 2 / 2 exp(
) 1 (
2 2
2
2 2
2
r r rR
rdr r
dr r R
• It occurred in the problem of the drunkard’s walk discussed by Rayleigh
.
- the drunkard always falls down after taking one step
- the direction of each step bears no relation to the previous step - After a long time, the probability of finding him at (x,y) is a
two-dimensional Gaussian function
- The probability of finding him at a distance r from the origin is given by a Rayleigh distribution.
Some infinite integrals
) 4 exp(
22
x x dx
2 )
exp(
21 2
t dt 1 )
exp(
2
t dt
exp( t )
2dt
2 1 2
)
exp(
At dt A
2 ) 1
0
exp(
2
x x dt
Sequences of Gaussian functions(Fig 4.7)
(a) The sequence exp(- x
2)
as 0
Useful for multiplying with functions whose integrals do not converge.
The limiting member of the sequence is unity.
(b) The sequence | |
-1exp(- x
2/
2)
Useful for recovering ordinary functions, in cased of impulsive behavior, by convolution.
• The sequences of Gaussian functions play a special role in connection with transforms in the limit.
• The properties which make the Gaussian functions useful in connection.
• Its derivatives are all continuous.
• It dies away more rapidly than any power of x
for all n.
0 lim
2
x n
x
x e
Heaviside’s unit step function, H(x)
• Used in the representation of simple discontinuities x
x x x
H
0
0 0
1 ( 0 )
(
21(a) The unit step function
• It represents voltage which are suddenly switched on or forces which begin to act at a definite time and are
constant thereafter
Fig. 4. 8
Example of the decomposition (x) with step functions
• Any function with a jump can be decomposed into a continuous function plus a step function
(b) The functions whose sum is (x)
) (
) (
)
(
21
21 x H x H x
Fig. 4. 9
H(x) representing the switching
• The unit step function provides a convenient way of representing the switching on
F4.11 A voltage E which appears at t=t0 represented in step function notation by EH(t-t0)
Fig. 4. 10
Fig 4.12 The Ramp function R(x) = xH(x)
• (F/m)R(t) represents
– The velocity of a mass m to which a steady force FH(t) has been applied
– The current in a coil of inductance m across which the potential difference is FH(t)
) ( )
( '
' ) ' ( )
(
x H x
R
dx x
H x
R
x
H(x) for simplifying integrals
• Step-function notation plays a role in simplifying
integrals with variable limits of integration by reducing the integrand to zero in the range beyond the original limits
) (
* ) (
' ) ' (
) ' ( '
) ' ( )
(
' ) ' (
) ' ( '
) ' (
x H x
H
dx x
x H x H dx
x H x
R
dx x
x H x f dx
x f
x x
• The shaded area is , or a value of the convolution of f(x) with H(x)
Convolution with H(x) means integration (Fig 4.13)
x f (x')dx'
)]
(
* ) ( [ )
(
' ) ' ( '
) ' (
) ' ( )
(
* ) (
x f x
dx H x d
f
dx x
f dx
x x
H x f x
f x
H
x
Define H(0)
• Usually H(0)=1/2 from
2 1 )]
( )
( lim [
) 0 ( )
0 ( '
0 )
0 ( )
0 ( ' , 1 ) 0 ( )
0 ( '
) ( )
( '
2 1 2
1
0
x
x x
R x
x H R
R
H R
H R
x H x
R
x
• Sometimes H(0)=0 from (see Fig 4.14), )]
( ) 1
[(
lim )
ˆ (
/0
e H x
x
H
x
Fig 4.14 (a) The H(x), (b) approximation to H(x), the integral of (b)
0 0 0
) 1 (
) ( )
ˆ ( )
(
0 0 0
) 1 ˆ (
0
0 2 1
x x x
where
x x
H x
H
x x x
H
Difference between H(x) & (!!SKIP!!) H ˆ x ( )
x u
du u H e
R(x)
x H x
R
) ( ) 1
( lim
since
) ˆ ( )
( '
/ 0
0 )
0 ˆ ( )
0 (
' H
R
• R’(0) regarded as the limiting slope at x=0 of approximation in Fig. 4. 14(c)
Fig. 4. 14
Continuous Approximation to H(x)(I) (!!SKIP!!)
• All approach H(x) as a limit for all x as 0
0 0 )
1 (
) 1
( 2
1
sinc 1 Si
2 1
) 1 (
2 erfc 1
arctan 1
2 1
/ 2
1
/ 2
1
1
1
/
1 2 2
x x e
e
u du
u du x
du x e
x
x x x
x x
u
Continuous Approximation to H(x)(II) (!!SKIP!!)
• All approach H(x) as a limit as 0 for all x except x=0
x x x e
f
x) 1
) 0 ,
(
( )/) ( ) (
) ( )
, (
lim
21 1 00
f x H x e x
In this case
since f(0, )=1 - e
-1for all
• A further example which approaches H(x) as 0
u du
x
1 21
The sign function, sgn x
Relation to H(x)
0 1
0 sgn 1
x x x
1 ) ( 2
sgn x H x
0 sgn
lim
A x dx A A
Fig. 4. 15
The filtering or interpolating function, sinc x
n=nonzero integer
x x x
s
inc sin
0 sin
1 0
sin
n c c
sin c x dx 1
Property
• The unique properties of sinc x go back to its spectral character
– It contains components of all frequencies up to a certain limit and none beyond.
– The spectrum is flat up to the cutoff frequency.
– sinc x and (s) are a Fourier transform pair(cutoff frequency = 0.5)
– When sinc x enters into convolution it performs ideal low-pass filtering
Sine integral Si x
u du x u
Si
0xsin
) sin (
) sin (
0
x Si dx x d
c
x du Si
u
x
c
) ( 2
sin 1 sin
)
( Si x
du u c x
c x
H
x
Relations
Fig. 4. 16
Fig. 4. 17
• Squaring cannot generate frequencies higher than the sum- frequency of any pair of sinusoidal constituents
2
2
sin
sinc
x x x
1 sinc
0 sinc
1 0
sinc
2 2 2
x
n
n=nonzero integer• Property
r r J
2
)
1
