# ii - random signals/ .ii - random signals/ processes ... random processes are characterized by joint

Post on 11-Jul-2018

223 views

Embed Size (px)

TRANSCRIPT

11/10/12 EC3500.FallFY13/MPF - Section II 1

II - Random Signals/ Processes [p. 3] Random signal/sequence definition [p. 4] Signal mean, variance, autocorrelation & autocovariance, normalized cross-correlation [p. 15] Statistical characterization of random signals

I.I.D. Random process Stationarity

Wide sense stationarity (wss)

Jointly wide sense stationarity (jointly wss)

Correlation & cross-correlation for stationary RPs

Signal average

Ergodicity

[p. 34] Specific examples of RPs:

Binary signal, white noise, colored noise, Bernoulli process, Random walk, counting process, Poisson process, MA process

[p. 75] Periodic Random process Definition [p. 82] Uncorrelated Random process Definition [p. 83] Cyclostationary Random process Definition

[p. 85] Multiple Random Processes Joint Properties [p. 90] Application to data analysis

How to assess signal stationarity [p. 94] Application to data analysis -

How to check IID assumption--

Autocorrelation --

Lag plot [p. 98] Application to data analysis -

How to extract an IID sequence out of an non IID sequence [p. 105] Application: target range detection [p. 107] Introduction to the spectrogram [p. 111] Application: Gas furnace reaction time [p. 114] Application: Detection of the periodicity of stationary signals in noisy environments [p. 116] How to estimate correlation lags; biased/unbiased estimator issues [p. 125] Frequency domain description for a stationary process

Power spectral density (PSD), definition & properties

Cross Power spectral density , definition & properties [p. 146] Appendices [p. 170] References

11/10/12 EC3500.FallFY13/MPF - Section II 2

Examples

[p. 5] Example 1

[p. 7] Example 2

[p. 9] Example 3

[p. 24] Example 4

[p. 29] Example 5

[p. 32] Example 6

[p. 34] Example 7

[p. 43] Example 8

[p. 50] Example 9

[p. 64] Example 10

[p. 67] Example 11

[p. 70] Example 12

[p. 76] Example 13

[p. 78] Example 14

[p. 80] Example 15

[p. 87] Example 16

[p. 93] Example 17

[p. 97] Example 18

[p. 104] Example 19

[p. 110] Example 20

[p. 114] Example 21

[p. 115] Example 22

[p. 121] Example 23

[p. 128] Example 24

[p. 129] Example 25

[p. 130] Example 26

[p. 132] Example 27

11/10/12 EC3500.FallFY13/MPF - Section II 3

Consider sequence x(t) =x(t,)

for a fixed t, x(t) is a Random Variable (RV)

x(t): random signal ( can be infinite dimensional) x(t,) for fixed RV : called realization/trial of the random process

if x(t)is

discrete for a fixed , the RP x(n)= x(nT, )is called a discrete

random process

Random Process Signal/Sequence:

A RP is a mapping function that attributes a function x(t) = x(t,) to each outcome of the random experiment

x(t, 1

)

x(t, 3

)

x(t, 2

)

1

2

3

t

t

t

11/10/12 EC3500.FallFY13/MPF - Section II 4

4 Types of Random Processes

Discrete time / discrete valued

Discrete time/ continuous valued

Continuous time/ discrete valued

Continuous time / continuous valued

11/10/12 EC3500.FallFY13/MPF - Section II 5

Example 1: x(t)=x(t,) = cos(t/10), where = U[0,1].

x(n)=x(nT, ) = cos(t/10+), where = U[0,2].

11/10/12 EC3500.FallFY13/MPF - Section II 6

Signal mean value (ensemble average):

Signal variance: ( ) ( ){ }( ){ } ( )

22

2 2

( )x x

x

t E x t m t

E x t m t

=

=

n1n

x[n]

n2

2 1n n= Lag

(dimensionless)

( ) ( ){ }xm t E x t=

Discrete process case Continuous process case

t1t

x(t) 2 1t t = Time lag (sec)

t2

11/10/12 EC3500.FallFY13/MPF - Section II 7

x(t,) = cos(t/10), where = U[0,1].Example 2

Compute process mean and variance

11/10/12 EC3500.FallFY13/MPF - Section II 8

Signal autocorrelation function:

( ) ( ) ( ) ( ){ }*1 2 1 2 1 2, ,xx xR t t R t t E x t x t= =measures the dependency between 2 values of the process at two different times.

Allows to evaluate: 1) How quickly a random signal changes with respect to time.2) The amount of memory

a signal may have.

2) Whether the process has a periodic component and what the expected frequency might be, etc

t1t

x(t) 2 1t t = lag

t2

11/10/12 EC3500.FallFY13/MPF - Section II 9

Let x(t) be a real valued process defined as

Compute mx (t)Compute: Rx (t1 ,t2 )

Example 3

x(n, 1

)

x(n, 3

)

x(n, 2

)

1

2

3

x(t,) = cos(t/10+), where = U[0,2].

11/10/12 EC3500.FallFY13/MPF - Section II 10

11/10/12 EC3500.FallFY13/MPF - Section II 11

11/10/12 EC3500.FallFY13/MPF - Section II 12

( ) ( )( ) ( ) ( )( ){ }( ) ( )

*

1 2 1 1 2 2

*

1 2 1 2

( , )

( , )

xx x x

x x x

C t t E x m x m

R t t m m

t t t t

t t

=

=

Signal autocovariance function (remove impact due to process mean):

Signal normalized correlation function (remove impact due to process mean and normalizes max value to 1):

( ) ( )( ) ( )1 2

1 21 2

,, xx

x x

C t tt t

t t

= ( )1 2| , | 1 !!x t t

11/10/12 EC3500.FallFY13/MPF - Section II 13

Signal cross-correlation function:

Signal cross-covariance function:

( ) ( ) ( ) ( )*1 2 1 2 1 2, ,xy xy x yC t t R t t m t m t=

( )1 2,xyR t t =

Measures the dependency between values of two processes at two different times.

Allows to evaluate whether/how two processes are related

Similar to cross-correlation function: measures the dependency between values of two processes at two different times,

but also

Removes

impact of mean values.

Note: unless there is a good reason to keep the signal

means, remove or use covariance based

expressions!

11/10/12 EC3500.FallFY13/MPF - Section II 14

Normalized cross-correlation function:

( ) ( )( ) ( )1 2

1 21 2

,, xyxy

x y

C t tt t

t t

= ( )1 2| , | 1 !!xy t t

11/10/12 EC3500.FallFY13/MPF - Section II 15

Random processes are characterized by joint distribution (or density) of samples

Fx (x1

, x2

, , xk , t1

,, tk ) = Pr [x(t1

)

x1

,

x(tk )

xk ]

F(.) is highly complex to compute -

difficult or impossible to obtain in practice

Statistical Characterization of Random Processes:

11/10/12 EC3500.FallFY13/MPF - Section II 16

Independent, Identically Distributed (I.I.D.) Random Process:

A Random Process is said to be:

An independent process (i.e., independent of itself at earlier and/or later times) if for any ti :

Fx (x1

, x2

,,xk ;t1 ,,tk ) = F1 (x1 ;t1 )Fk (x2 ;tk )

An I.I.D. process if all RVs have the same pdf fx (x)Note: I.I.D. processes have no memory, where a future value would depend on past values they can be viewed as building blocks for more realistic random processes

Mean of I.I.D. Process:

mx (t) = E{x(t)} =

11/10/12 EC3500.FallFY13/MPF - Section II 17

Independent, Identically Distributed (I.I.D.) RP, cont

Autocovariance of IID RP:

{ }{ } { }

{ }

*1 2 1 1 2 2

*1 1 2 2 1 2

21 1 1 2

( , ) ( ( ) ( ))( ( ) ( ))

( ( ) ( )) ( ( ) ( )) ,

| ( ( ) ( )) | ,

=

x x x

x x

x

C t t E x t m t x t m t

E x t m t E x t m t t t

E x t m t t t

=

= =

Autocorrelation of IID RP:

Rx (t1

, t2

) =

11/10/12 EC3500.FallFY13/MPF - Section II 18

( , ) 0.05* , ~ (0,1)x t t

N

= +

I.I.D. process ?

[ ( , )]E x t

11/10/12 EC3500.FallFY13/MPF - Section II 19

Stationarity Concept:

If x(t) is stationary for all orders N = 1, 2, x(t) is said to be strict-sense stationary.

If x(t) is stationary for order N = 1,

Stationary up to order 2 wide-sense stationary (WSS).

( ; ) ( ; ) x xf x t f x t T = +

Pdf is identical for all times instants t

Definition: a RP is said to be stationary if any joint density or distribution function depends only on the spacing between time instants, not

where on the timeline the time instants occur.

fx (x1

, , xN ; t1

, , tN ) = fx (x1

, , xN ; t1

+T ,, tN +T) for any ti , T & any joint pdf

11/10/12 EC3500.FallFY13/MPF - Section II 20

Stationarity of order N=1 - Physical interpretation for a discrete process

...

...

...

n

n

n

x(n, 1

)

x(n, 3

)

x(n, 2

)

...

...

n

n

n

x(n, 4

)

... x(n, P

)

x(n, 5

)

Experiment is performed P timesleads to P time sequences

How to computeFx (x1

; n1

) = Pr [x(n1

)

x1

][Probability that the functions x(n,) do not exceed x1 at time n1 ]

Select values for x1 and n1

Count the number of trials K for which x(n1 ) x1

Fx (x1 ; n1 ) = Pr [x(n1 ) x1 ]= K/P

x1

x1

x1

x1

x1

x1

n1[11]

11/10/12 EC3500.FallFY13/MPF - Section II 21

Stationarity of order N=2 - Physical interpretati

Recommended