Characterizing the Number of Crossings in a Random Process, Slides of Structural Analysis

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Failure of randomly vibrating systems-

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2

Number of times

( ) crosses

in 0 to

An integer valued random variableGiven the complete description of

can we characterize

This is known as the level crossing problem.

N

T

X t

T

X t

N

T







 

 









 

0 0  

,^

T T

N

T

N T

X

t

X

t

dt

n

t dt

n

t

X

t

X

t

 

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



 

 





XX

n

t

X

t

X

t

x p

x t dx









 









2 2

,^

exp

x x

x

n

t













Stationary Gaussianrandom process







 ^











2 2

2

0

2

2

(^00)

2

2 0

2

,^

exp

,^

exp

x

x^

x

x^

XX

x^

XX

n

n

XX

n

t

S

d

S

d

S

d

n

t

  

  

Spectral moments





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











Poisson model for

•The threshold level

is high (so that crossing is a rare event)

•Crossing times are mutually independent•

)is a Poisson random variable

, 0,

exp

k

N

T

N

T

T

P

N

T

k

k

Assumptions













2 2

rate of crossing of level

If

( ) is a stationary gaussian random process with zero mean

,^

exp

x x

x

T

n

t

X t n

t



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Narrow band and broad band processes  









 









 

^







^



 

^















2

2

Example 1

cos

;^

~ Rayleigh and

~

0, 2

;

cos

cos

0

cos

cos

cos

is a stationary random process

2

Check

1

e

2

xx

XX

XX

x t

P

t^

P

U

P

x t

P

t^

P

t

x t

x t

P

t^

P

t

P x t S

P

R

S















































 



























  









Ideal narrow band process





^















2

2

1 2

xp

cos

cos

2

1

2

cos

cos

(ok)

2

XX

i^

d

S

d

P

P

d

P































 

 

















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 

 

 





























1 2 2

1

1

2

3

2

2

2

2

2

Realistic narrow band processes

lim

,^

exp

cos

sin

xx

d

d

t t t^

t XX

n

n

mx

cx

kx

w t

w t

w t w t

I

I

R

t

t

m

S

H

I m

H

i





 



 

 























Example 2

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10

0

2

4

6

8

10

12

14

16

18

20

0 0.07 0.06 0.05 0.04 0.03 0.02 0.

frequency rad/s

PSD

0

2

4

6

8

10

12

14

16

18

20

(^2) 1.8 1.6 1.4 1.2 (^1) 0.8 0.6 0.4 0.2 0

frequecny rad/s

PSD

0

2

4

6

8

10

12

14

16

18

20

(^2) 1.8 1.6 1.4 1.2 (^1) 0.8 0.6 0.4 0.2 0

frequecny rad/s

PSD

Ideal broadbandprocess (White noise)

Realistic narrowbandprocess

Realistic broadbandprocess (band limited whitenoise)

0

2

4

6

8

10

12

14

16

18

20

0 0.07 0.06 0.05 0.04 0.03 0.02 0.

frequecny rad/s

PSD

Ideal narrowbandprocess

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0

1

2

3

4

5

6

7

8

9

10

(^43210) -1 -2 -3 -

time s

Sample of an ideal narrow band processx(t)









cos

x t

a

t









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13

0

1

2

3

4

5

6

7

8

9

10

(^43210) -1 -2 -3 -

time s

x(t) Sample of a band limited process

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1

(^3210) -1 -2 -3 -

time s

x(t)

Sample of a band limited process

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1

(^3210) -1 -2 -3 -

time s

x(t)

Broad band process

Crossing with +

ve

slope can be followed by several extrema

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  ^







zero mean, stationary, narrow band,

Gaussian random processConsider peaks above level

in the interval 0 to

.

Peak

1

Peak>

P

X

t

T

P

P

P









[Heuristic

Distribution of peaks for a narrow band proces

approach]

s









Number of peaks above levelRelative frequency de

eak>

Total number of peaks

Total number of times the level

is crossed with positive slope in 0-T

Total n

fin

umb

itio

er of

n

zer









o crossings with po

( ) is assumed to

sitive

be a n

slope in 0 to

arrow band p

T

rocess

X t

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











 













2

2

2 2

2 2

2

2

2

Peak>

exp

is Gaus

exp

Peak>

exp

exp

sian

x x^

x

x x

x

p

x

p

x^

x

n

t

P

n

t

X

t

P

P

p

 





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0

1

2

3

4

5

6

7

8

9

10

(^43210) -1 -2 -3 -

time s

x(t)

Gaussian narrow band process





2 2

1

1

exp

;

2

2

X

x

x

x

p

x

x









 

 













2

2

2

exp

;

2

p

x^

x

p























 









Summary

Heuristic basis

0

1

2

3

4

0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.

x, peak

pdfs of X, peak

Gaussian

Rayleigh

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