Random sine function

( ) sinx t c t

stochastic phase angle, for example uniform on [0,2 ]

deterministic amplitude

deterministic circle frequency

c

f()

2

Random sine function

x and deterministic, uniform on (0,2)

2 2

0 0

1{ ( )} ( ) ( ) sin( ) 0

2x t x t f d x t d

22 2 2 2

2 2

0 0

1{ ( )} ( ) ( ) sin ( )

2 2

xx t x t f d x t d

( ) sinx t x t

1.0 1.0

Two important integrals

0.5

( )f y

2y y

( )f y

2

22 2

0 0

sin 0 sin 0.5 2y dy y dy

1( ) sin( ) ( )k k k k

N

k

x t x t x t

sum of sine functions with random phase angle:

{ ( )} { ( )} 0kx t x t 2 2 21 { ( )} { ( )}

2k kx t x t x

0

2

0

10

1

1,2,3, 10 ; 4 ; 2 ; 5

exp ; 2 ; 0.1

( ) sin ; 0 50 ; random phase angle

k k k

k k

k k k

k

k T k T

x a b a b

x t x t t

t

( )x t

0 5 10 15 20 25 30 35 40 45 50

10

8

6

4

2

0

2

4

6

8

10

Process with 11 sine functions

k=1,11, xk = 1.0

1= 4.0, 2 = 4.2. 10 = 6.0 rad/s

1

( ) cos sin sin cosk k k k k k

N

k

x t x t x t

0

2( ) sink k

T

A x t t dtT

0

2( ) cosk k

T

B x t t dtT

2 2 2k k kx A B

2 2 21

1{ ( )}

2k k

N

k

x t A B

T T

x

t

2kx

k

2kx

k2

T

4

T

base time T

2

T

base time 2T

k

2( ) 2k k kS x 2 2

lim kxxx

S

Definition of the variance spectrum

1

( ) cos sin sin cosk k k k k k

N

k

x t x t x t

0

2( ) sink k

T

A x t t dtT

0

2( ) cosk k

T

B x t t dtT

2 2 2k k kx A B

2 2 21

1{ ( )}

2k k

N

k

x t A B

2

10

2 2( ) lim withkxx k k k

xS

T

2 2

0

1( ) ( ) ( )

2k xx k xxx x S S d

2 2

2 2

1 1( ) ( ) cos sin withi tx

T T

T T

S x t e dt x t t i t dt T

*( ) ( )xx x xS S ST

0

( )ii xxm S d

The standard deviation

0( )x m

central frequency:

0 1 0m m

2

0 2 0m m

width of the spectrum

2

1

0 2

1m

qm m

moments of the spectrum

a) narrow-band process

b) wide-band process

c) process with two distinctive frequencies

0 02T

16%x

x

x

t

t

xxS

xxS

0

0

21

xxS

0

t

0.5q

0.6q

q

Fig. A.2: Stochastic process as a series of stochastic variables.

( )x t

t t

( )x t

( )x t t

( )x t n t

t t n t t

Alternative Approach (Annex)

random variable f(x) x x

two randomvariabels f(x,y) x x y y covxy

n random variables f(x) x x covxy

random process f(x(t)) x(t) x(t) covxy(t1,t2)

stationary random proc f(x) x x Rxx(t)

Description of continuous processes

Gaussian process:

Mean value for every point in time

Covariance for every two points in time

General process:

Multidimensional probability distribution for every

set X(t1), X(t2), X(t3), .

Stationary Gaussian process:

Mean value X

Autocovariance function RXX()

Spectrum

0

2( ) ( )cosxx xxS R d

0

( ) ( )cos( )xx xxR S d

2

0

(0) ( )x xx xxR S d

2 1 2 1

2 1 0 2 1 0 1 0

2 1 0 2 1 0 1 0

2 1 2 1

2 1 2 2 1 2

2 1 2

( ) ( ) ( ) ( )

1( ) ( ) cos ( ) ( ) ( )

21

( ) ( )sin ( ) ( ) ( )2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

nn

n

R a R S a S

R R S S S

R R S S Si

dR R S i S

d

R R R S S S

R R g t dt S

1

2 1 2 1

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

Hereby is the Fourier transform of ( ) defined by:

( ) ( )

complex gonjugate of ( ) real function

i t

G S

R R g t dt S G S

G g t

G g t e dt

G Gg t

Table A.1: Properties of Fourier transforms.

( ) 1S

1

( ) 2 ( )S

2area

0 0( ) ( ) ( )S

1area

0 0

( )S

1area

1

( )S

1area

0 1 2

( )R

1area

( ) ( )R

( )R

( ) 1R

1

( )R

0( ) cosR

1

0

( )

sin cos

R

( )R

( )R

1

( )

sin1

R

1( ) ( ) ( ) iR S R e dt

1

2

3

4

5

( )R e

( )R 2 2

1 2( )S

1

0

( )

cos

R

e

( )R

1

2 22 2

0 0

1( )S

0 0

00 0( ) cos sin4

R e

0 0

4

0

22 2 2 2 2

0 0

( )4

S

2

0 1

6

( ) sinx t c t

sine function

X= 0

2( ) ( )1

cov ( ) cos2

x t x t xxR c

2 21 1(0) ; 22 2

x xx xR c c

Fig. A.4: Auto-covariancefunction of with uniform on .

( )xxR

212c

c and deterministic, uniform on (0,2)

SXX() = c2 (-k)

( ) sinx t c t

1 2 1 2( ) ( ) 1 ( ) 2 ( )cov {[ ( ) ] [ ( ) ]}x t x t x t x tE x t x t

1 2

2

( ) ( ) 1 2 1 2cov { ( ) ( )} { sin sin }x t x t E x t x t E c t t

1 2

2

( ) ( ) 1 2

2

0

cov sin sin2

x t x t

ct t d

1 2

2 2

( ) ( ) 1 2 1 2

2 2

1 2 1 2

2

1 2

2 2

0 0

2 2

0 0

1 1cov cos cos 2

2 2 2 2

cos sin 24 8

1cos

2

x t x t

c ct t d t t d

c ct t t t

c t t

1

2

sin

k

k

k k k k

N

k k k

k

S

a S

y a t

random

generator

Sk

Generation of a random process

To Remember (1):

Construction of the variance spectrum

x

t

Sx () = (1/) 0 T x(t) exp (it) dt

Sxx() = ( /T) Sx Sx*

2

0

(0) ( )x xx xxR S d

Sk

To Remember (2):