II. Random Processes Review - Facultyfaculty.nps.edu/fargues/teaching/ec4570/EC4570-II-WFY11.pdf ·...

01/14/11 EC4570.WinterFY11/MPF - Section II 1

II. Random Processes Review

- [p. 2] RP Definition- [p. 3] RP stationarity characteristics- [p. 7] Correlation & cross-correlation- [p. 9] Covariance and cross-covariance- [p. 10] WSS property- [p. 13] RP time average and ergodicity- [p. 18] Periodic RP properties- [p. 22] Power Spectral Density- [p. 26] Linear transformations of RPs- [p. 32] Bandpass/lowpass (complex envelope) representations- [p. 46] Linear systems and bandpass/lowpass representations- [p. 58] Noise process: bandpass & lowpass (complex envelope)- [p. 67] Envelope statistics and use in signal detection- [p. 75] Monte Carlo performance evaluation and use in detection


• Consider sequence x(t)= x(t,ξ) ←

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

• x(t) : random signal (→ can be infinite dimensional)

• x(t,ξ) for fixed RV ξ: called realization/trial of the random process

Random Process (RP):

•

x(t, ξ1 )

x(t, ξ3 )

x(t, ξ2 )

ξ1•

••ξ2

ξ3

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

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

t

t

t


• Random processes are characterized by joint distribution (or density) of sample values

•Consider the RP x(t) evaluated at specific points tk ’s, k=1,…,N

• 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:

Stationarity:Definition: a RP is said to be stationary if any joint density or distribution function depends only on the spacing between samples, not where in the sequence these samples occur

fx (x1 , …, xN ; t1 , …, tN ) = fx (x1 , …, xN ; t1+k ,…, tN+k ) for any k and any joint pdf


Stationarity con’t:

Recall: a RP is said to be stationary if any joint density or distribution function depends only on the spacing between samples, not where in the sequence the samples occur

fx (x1 , …, xN ; t1 , …, tN ) = fx (x1 , …, xN ; t1+k ,…, tN+k ) for any k and any joint pdf

• 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 samples


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

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n

n

x(n, ξ1 )

x(n, ξ3 )

x(n, ξ2 )

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x(n, ξ4 )

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... x(n, ξP )

x(n, ξ5 )

Experiment is performed P times

leads to P time sequences

How to computeFx (x1 ; n1 ) = Pr [x(n1 ) ≤

x1 ]

[Probability that the functions x(n,ξ) 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[1]


Stationarity of order N=2 - Physical interpretation for a discrete process

Experiment is performed P timesleads to P time sequences

How to computeFx (x1 , x2 ; n1 , n2 ) = Pr [x(n1 ) ≤

x1 ,

x(n2 ) ≤

x2 ]

[Probability that the functions x(n,ξ) do not exceed x1 at time n1 and x2 at time n2 ]

• Select values for x1 , x2, n1 , n2• Count the number of trials K for which x(n1 ) ≤ x1 and x(n2 ) ≤ x2

Fx (x1 ,x2 ; n1 , n2 ) = K/P

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n

n

x(n, ξ1 )

x(n, ξ3 )

x(n, ξ2 )

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n

n

x(n, ξ4 )

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... x(n, ξP )

x(n, ξ5 )

x1

x1

x1

x1

x1

x1

n1x2

x2

x2

x2

x2

x2

n2 [1]


Random Process autocorrelation function

( ) ( ) ( ) ( ){ }*1 2 1 2 1 2, ,xx xR t t R t t E x t x t= =

• Measures the dependency between values of the process at two different times.

• Allows to evaluate: 1) How quickly a random process changes with respect

to time.2) Whether the process has a periodic component and

what the expected frequency might be, etc…


Random Process cross-correlation function

( ) ( ) ( ){ }*1 2 1 2,xyR t t E x t y t=

• Measures the dependency between values of two process at two different times.

• Allows to evaluate whether/how two processes are related


Random Process (auto)covariance function( ) ( )

( ) ( ){ }1 2 1 2

*1 1 2 2

, ,

( ( ))( ( ))xx x

x x

C t t C t t

E x t m t x t m t

=

= − −

• Similar to correlation function: measures the dependency between values of the process at two different times,

but

• Removes means impacts.

Random Process cross-covariance function

( ) ( ) ( ){ }*1 2 1 1 2 2, ( ( ))( ( ))xy x yC t t E x t m t y t m t= − −


Wide-Sense Stationarity:

Definition: a RP x(t) is called wide-sense stationary (WSS) if

(1) the mean is a constant independent of the time

(2) the autocorrelation depends only on the time lag distance τ = t1 −

t2

Consequence: the variance is a constant independent of “t”

( ){ } ( )x xE x t m t m= =

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

*1 2 1 2 1 2

*

,x x

x

R t t E x t x t R t t

R E x t x tτ τ

= = −

= = −

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

2 222

220

x x x

x x x

t E x t m t E x t m t

R m

σ

σ

= − = −

= − =


Correlation Function Properties for wss x(t)

(1) Conjugate symmetry

(3) Rx (t) max at t = 0 and Rx (0)>0

(can we have Rx (0)=0?)

( ) ( )*x xR t R t= −


RP Example: White noiseDefinition: A random process w(n) is called a white noise process with mean mw and variance σ2

w iffE{w(t)}=mwRw (τ)= σ2

ω

δ(τ)=2N0 δ(τ)Notes:

1) all frequencies contribute the same amount (as in the case of white light, therefore the name of “white noise”)

2) if the pdf of w(t) is Gaussian: it is called “white Gaussian noise”

3) White noise is the simplest RP around because it doesn’t have any “structure” and can be used as a building block

4) Physically impossible; in practice restricted to specific bandwidth B leading to power 2N0B

Textbook notation


in many applications only one realization of a RP is available

in general, one single member doesn’t provide information about the statistics of the process

except when process is stationary +ergodic: statistical information cannot be derived from one realization of RP

Def: a RP is called ergodic if:

all ensemble averages = all corresponding time averages

Ergodicity:

RP (time) Average:

( )1( ) lim2

T

TT

x t x t dtT→+∞

−

= ∫


Def: a RP is said to be ergodic in the mean if:

Ergodicity cont’:

Def: a RP is said to be ergodic in correlation at time lag τ if:

( ) ( )1( ) lim * *2

T

x TT

R x t x tT

τ τ→+∞

−

= −∫

Process can be stationary and NOT ergodic


Ergodicity con’t:

• Example 1: Assume RP x(t) which is a dc voltage waveform where the pdf for the voltage is given by U[0, 10].

1) Plot several possible trials for the RP2) Is the process wss ?3) Is the process ergodic in the mean?


Ergodicity cont’:

• Example 2: (Telegraph Signal) Assume the stationary and ergodic RP x(t) takes values ±1 during every time interval Tc with equal probability. Start of the first pulse after t=0, equally likely in the interval [0,Tc ]

1) Plot a possible trial for the RP2) Compute the correlation function



Periodic Random Process

• if x(t) is periodic, ( ) ( )x t x t T= +

• Mean( ) ( ) ( ) ( )x xm t E x t E x t kT m t kT= = + = +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

• Correlation/Covariance for stationary RP( ) ( )1 2 1 2,x xR t t R t t= −

( ) ( )( ) ( )

x x

x x

R t R t kT

C t C t kT

= +

= +


• Example 3: x(t) = A exp (j(ωt + θ)), θ ~ U [0,2π]

Compute Rx (τ) & mx (t)



• Example 4: y(t)=s(t)+w(t), where s(t)=A exp (j(ωt + θ)), θ ~ U [0,2π], w(t) zero-mean white wss noise, w(t) & s(t) are independent.

Compute Ry (τ) and my (t)


Frequency Domain Description of Stationary Processes

Power spectral density (PSD)

( ) ( )( ) ( )

2( ) j fx T x x

x x

S f F R R e d

R IFT S f

π ττ τ τ−= =⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

∫


• Example 5: Compute the PSD for the telegraph signal wave function investigated earlier.




Linear Transformations of RPs

Process stationary random processes using LTI systems

Mean:

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )

*

k k

y t x t h t x h t d

y n k k x n k x k h n k

τ τ τ= = −

= − = −∫

∑ ∑

{ } [ ]( ) ( ) ( ) ( ) ( )

( )x

E y t E h x t d h E x t d

m h d

τ τ τ τ τ τ

τ τ

⎡ ⎤= − = −⎣ ⎦

=

∫ ∫∫

( )x t ( )h t ( )y t

• Output random processes properties


Input-output cross-correlation:

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

* *

*

( ) *( )

( ) *( ) ( ) ( ) ( )

E x t y t E x t h u x t u du

E y t x t E h u x t u x t

τ τ

τ τ

⎡ ⎤− = − −⎣ ⎦⎡ ⎤− = − −⎣ ⎦

=

∫∑


Output correlation:

{ } ( ) *

*

( ) *( ) ( ) ( ) ( )

( ) ( ) ( )

E y t y t E h u x t u du y t

h u E x t u y t du

τ τ

τ

⎡ ⎤− = − −⎣ ⎦

⎡ ⎤= − −⎣ ⎦

∫∫


Output covariance: same properties as for correlation

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

*

*

( ) ( )

*

*

* *

y xy

xy x

y xy

y x

R h t R t dt

R h R

R R h

R R h h

τ τ

τ τ τ

τ τ τ

τ τ τ τ

= −

= −

=

= −

∫ ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )

*

*

( )

*

*

* *

y xy

xy x

y xy

y x

C h t C t dt

C h C

C C h

C C h h

τ τ

τ τ τ

τ τ τ

τ τ τ τ

= −

= −

=

= −

∫

( ) ( ) 2| |y y yC R mτ τ= −


• Example 6: Given a wss zero mean white noise RP x(t) with covariance Cx (l)=σ0

2 δ(l)

Compute the mean, correlation function, covariance function, and PSD of the output RP y(t) to the LTI system with impulse response h(t) = e-atu(t) , a>0.



Bandpass & complex envelope signal representations

• Communication and radar signals are usually concentrated in a narrow bandwidth around a center frequency• Complex lowpass equivalent (i.e., complex envelope) signals derived from the bandpass signal usually simplify analysis

• Need to define lowpass/baseband and bandpass signals

Baseband/lowpass signal

Signal s(t) with frequency information is restricted for f B≤

Bandpass signal

Signal s(t) with frequency information restricted around cf±


Bandpass & complex envelope signal representations

( ) c o s ( 2 ( ) ) { c o s ( ( ) ) c o s ( 2 )

s in ( ( ) ) s in ( 2 )}c o s (( (

( )(

2 ) s in) ( )

)

2)

c

c

c

c cI Q

s t f t tt f t

t f tf t ts t

a t

t

a t

fs

π θθ π

θ ππ π

= +

=−

= −

Assume

( ) c o s ( ( ))a t tθ( ) s in ( ( ))a t tθ

Information signal

In-phase component(lowpass)

Quadrature component(lowpass)

Note: representation shows that 2 signals may be transmitted within the same bandwidth


I and Q signal contributions are orthogonal

( ) ( ) co s ( ( ))Is t a t tθ=

( ) ( ) s in ( ( ))Qs t a t tθ=

Correlation between two signals ?

[ ]0

( ) cos(2 ) ( )sin(2 )T

I c Q cs t f t s t f t dtπ π⎡ ⎤− =⎣ ⎦∫


Bandpass & complex envelope signal representations, cont’

{ }

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

Real ( )(cos

(

(2 ) sin

( )

( ) ( ) (2 )

)

)I

Q

c

c c

cQ c

I

s t a t f t tf t f ts t

f ts t js j

s t

t f t

π θπ π

π π

= +

= −

= ++

( ) ( ) cos( ( ))Is t a t tθ=

( ) ( ) s in ( ( ))Qs t a t tθ=

Defined as the complex envelope (complex baseband, i.e., lowpass, equivalent signal)

( )u t

{ }2( )( ) R e e x p cj f tus t t π=

Note: There is a relationship between bandpass signal s(t) and complex envelope u(t)



Note: There is a relationship between S(f) and U(f), when s(t) is deterministic

{ }2( ) R e ( ) exp cj f ts t u t π=

2 2 *1( ) ( ) exp ( ( ) exp )2

=

c cj f t j f tS f FT u t u tπ π−⎡ ⎤⇒ = +⎣ ⎦



Note: There is a relationship between PSD expressions Ss (f) and Su (f) when u(t) is random



Note: The energy of s(t) may be expressed as

2 ( )s t dtε∞

−∞

= =∫


• Communication signal applicationAmplitude shift keying (ASK) – On-Off Keying (OOK) signal

• Send either a sinusoid for 1 or nothing for 0• Assume T is duration of one bit symbol

0

1 0

( ) 0( ) sin( )

s ts t A tω

==

0 t T≤ ≤


{ }

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

Real ( )(cos

(

(2 ) sin

( )

( ) ( ) (2 )

)

)I

Q

c

c c

cQ c

I


f ts t js j

s t

t f t

π θπ π

π π

= += −

= ++

Recall:

( )u t

0

1 0

( ) 0( ) sin( )

s ts t A tω

=

=

Compute:• In phase component sI (t)• Quadrature component sQ (t)• Complex envelope u(t)


• Communication signal application

Phase shift keying (PSK) signal• Send either a sinusoid with one specific phase for 1 or with different phase for 0• Assume T is duration of one bit symbol

0 0

1 0

( ) sin( )( ) sin( )

s t A ts t A t

ωω

== −

0 t T≤ ≤

Phase changes


{ }

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

Real ( )(cos

(

(2 ) sin

( )

( ) ( ) (2 )

)

)I

Q

c

c c

cQ c

I


f ts t js j

s t

t f t

π θπ π

π π

= += −

= ++

Recall:

( )u t

0 0

1 0

( ) sin( )( ) sin( )

s t A ts t A t

ωω

=

= −

Compute:• In phase component sI (t)• Quadrature component sQ (t)• Complex envelope u(t)


• Example 7:Consider the real signal s(t) defined as

( ) cos(2 ), 04cs t A f t t Tππ= + ≤ ≤

Compute 1) the complex lowpass equivalent (complex envelope) signal u(t), Fourier transform U(f), and S(f)

2) the signal energy




Linear systems & bandpass signals/complex envelopes• LTI filter output of a bandpass signal may be computed using signal and impulse response complex envelope expressions much simpler than by using original bandpass expressions

hB (t)s(t) s0 (t)

BandpassHas a complex

envelope expression

Can be modeled as a bandpass filter

Has a complex envelope expression{ }

[

2

*

( ) Re ( )

1( ) ( )2

( )]

cjf t

c

c

s t u t e

S f U f f

U f f

π=

= −

+ − −

{ }[

2

*

( ) 2 Re ( )

( ) ( )

( )]

cjf tB

B c

c

h t h t e

H f H f f

H f f

π=

= −

+ − −

h(t)u(t) v(t)


Linear systems & bandpass signals/complex envelopes, cont’

hB (t)s(t) s0 (t)

{ }[

2

*

( ) Re ( )

1( ) ( )2

( )]

cjf t

c

c

s t u t e

S f U f f

U f f

π=

= −

+ − −

{ }[

2

*

( ) 2 Re ( )

( ) ( )

( )]

cjf tB

B c

c

h t h t e

H f H f f

H f f

π=

= −

+ − −

[ [

0

0

**

( )*( )

1 = ( )

( )

( )]2

( )

( ) ( )]

( )( )

B

B

c cc c

s tS f

U f f

s h tH f

H f

tS f

U ff ff H f

==

− + − + − −− −

h(t)u(t) v(t)



hB (t)s(t) s0 (t)

[ [*

*

*

*

*

0

*

( )

( ) ( )]

( ) ( )

( )1 = ( ) ( )]21 [

( ) + ( )]

( ) ( )2

( ) (

( )

)

B

c c c c

c c

c

c

c

c

c c

S f

U f f U f f

U

H f

H f f H f

f f U f f

U f f U

f

H

S f

f f H f f

f HfH ff f f

− + − −

−

=

− + − −

= − − −

− − −

+ − −

+ − − −

h(t)u(t) v(t)


hB (t)s(t) s0 (t)

*0

0

**

*

*

*

( ) (1 [2

1 [ ]

)

( ) ( )

( ) ( )

(

) ( )

2

( )

(

)

( ) ( )]

( ) ( )

c c

c

c c

c c

c

c

c c c

S f

S

U f f f f

f

H f f U H f f

HU f f Uf f H f f

H f f H f f

f f

U f f U f f

= − −− − −

− − −

−

− −

+

+ +

− − −

−

−= − + −

h(t)u(t) v(t)



hB (t)s(t) s0 (t)

0

*

* *1( ) [ ]21 = [ ( ) ( ) ]2

( )) ) ( )( (

c c

c cc cU f f U f fS f

V f f V

H f f H f f

f f

= +− − −−

+ − −

− −

−

h(t)u(t) v(t)

0

is the complex envelope expression of

(

(

)

)

v t

s t

( )

( )

(

( )

)

( ))

(V f

v

U f

u t

H f

ht t

=

⇓= ∗

Complex envelope of output signal

Complex envelope of input signal

Complex envelope of filter response



hB (t)s(t) s0 (t) h(t)u(t) v(t)

( )

( )

(

( )

)

( ))

(V f

v

U f

u t

H f

ht t

=

⇓= ∗


Complex envelope of output signal

Complex envelope of input signal

Complex envelope of filter response∗=


( ) ( ( ) ( ))

( ) ( ( )

) (

))( I Q I Qu t s t js th t h t h tv t j+= += ∗ ∗

=



Conclusion:• Complex baseband (complex envelope) representation of bandpass signals allows for accurate representation and analysis of signals independent of the signal carrier frequency.

• Leads to simpler evaluation of filter outputs and system performance analysis


• Example 8: Consider the bandpass signal

0

0 0

( ) 2cos(2 )cos(2 ) sin(2 )sin(2 ),

c

c c

s t f t f tf t f t f f

π ππ π

=− <

0 0 0

0 0 0 0

0 0

Consider the bandpass filter with/ ,

2, 2 2 , 2( ) , ( )

0, ow , 20, ow

I Q

jf f f f ff f f j f f f

H f H fj f f f

− ≤ ≤⎧⎪− ≤ ≤ ≤ ≤⎧ ⎪= =⎨ ⎨ − − ≤ ≤ −⎩ ⎪⎪⎩

1) Compute in-phase and quadrature components of s(t)2) Compute the complex envelope of s(t)3) Compute the complex envelope of the filter output4) Compute the bandpass filter output





Bandpass noise process

• Defined as a process which is 1) centered at a non zero frequency2) does not extend to zero frequency (i.e., has no DC term)

f

Sn (f)


{ }

{ }2

Noise process may be represented as:( ) ( ) cos(2 ( ))

cos(2 ) sin(2 )

Real ( )(cos(2 ) sin(2 ))

Rea

( ) (

(

(

(

)

)

)

)

l c

n c n

c c

c c

j f

Q

I

I

Q

t

n tn t a t f t t

f t f t

f t j f

n

n t jn t

t

t

z t e π

π θπ π

π π

= += −

+= +

=( ) complex envelopez t →

( ) co s ( ( ))na t tθ( ) s in ( ( ))na t tθ

Bandpass noise process, cont’


Correlation properties between bandpass noise components

( ) ( ) co s ( ( ))( ) ( ) s in ( ( ))

I n

Q n

n t a t tn t a t t

θθ

==

( )nR τ =

Bandpass noise process, cont’

Trig identitiescos cos (1/ 2)(cos( ) cos( )), sin sin (1/ 2)(cos( ) cos( ))sin cos (1/ 2)(sin( ) sin( )), cos sin (1/ 2)(sin( ) sin( ))

a b a b a b a b a b a ba b a b a b a b a b a b

= + + − = + − −= + + − = + − −



Bandpass noise process, cont’*( ) ( ) ( ) 2 ( ) 2 ( )

( ) ( )

( ) 2 ( ) 2 ( )

I Q I

I Q I

z N N N

I Q

z N N N

R E z t z t R jR

n t jn t

S f S f jR f

τ τ τ τ⎡ ⎤= − = +⎣ ⎦+

⇒ = +

[ ]*

1( ) Re ( )exp( 2 )2

( ) (1/ 4) ( ) (1/ 4) ( )

n z c

n z c z c

R R j f

S f S f f S f f

τ τ π τ=

⇒ = − + − −

Conclusion: PSD of a random bandpass process can be derived from the PSD of the complex envelope and vice-versa



Example: bandpass white noise process n(t) with PSD shown belowCompute the PSD and the correlation of the complex envelope

fc-fc

BN0 /2

Sn (f)




Envelope statistics and their use in signal detection

Noise-only case:Bandpass noise n(t) is defined as:

{ }2( ) Re ( )

with ( ) ( ) ( )

cj f t

I Q

n t w t e

w t w t jw t

π=

= +

• Commonly used in non-coherent signal detection where unknown signal parameters are treated as random variables

• Look at - noise only- signal + noise

wI (t) & wQ (t) are zero-mean, statistically independent Gaussian processes, with variance σ2


Noise-only case cont’

{ }2( ) Re ( )

with ( ) ( ) ( )

cj f t

I Q

n t w t e

w t w t jw t

π=

= +

( )w tρ = =

Pdf of the envelope is _____________

Noise envelope is defined as:


Noise-only case cont’

{ }2( ) Re ( )

with ( ) ( ) ( )

cj f t

I Q

n t w t e

w t w t jw t

π=

= +

Noise phase is defined as

Pdf of the noise phase:

1 ( )( ) t a n

( )Q

nI

w tt

w tθ − ⎛ ⎞

= ⎜ ⎟⎝ ⎠


Signal + Noise case:Assume the received signal

{ }{ } { }{ }{ }

2

2 2

2

2

( ) R e ( ) ( )

R e ( ) R e ( )

R e ( ( ) ( ))

R e ( ( ) ( ) ( ))

c

c c

c

c

j f t

j f t j f t

j f t

j f tI Q

y t v t e n t

v t e w t e

v t w t e

v t w t jw t e

π

π π

π

π

= +

= +

= +

= + +

Complex envelope ρ for y(t) is given by

(Assume v(t)=A)


Turns out the pdf for the envelope is Ricean and given by:

2 2 2( | | ) / 202 2

| | , 0( )

0, otherwise

AAI ef

ρ σ

ρ

ρ ρ ρρ σ σ

− +⎧ ⎛ ⎞ ≥⎪ ⎜ ⎟= ⎝ ⎠⎨⎪⎩


ExampleConsider the radar application where we want to decide whether a signal x(t) is present or not using envelope statistics only

Noise envelope is given by:

Signal +noise envelope is given by:


Define the probability of detectionPD =

Define the probability of false alarmPfa =

More details later on….


[Sch]

2

2

Aγσ

=


Monte Carlo performance evaluation

• Deals with computer evaluation of a probability

• Useful in cases where one cannot determine analytically or numerically expressions of the form

• Can be found in detection problems where we may wish to evaluate probability that a given statistic exceeds/falls below a threshold

1 2( ), ( ), ( )P x K P x K P K x K≥ ≤ ≤ ≤


( ) ,

assume ( ) : (1) is a valid pdf, (2) 0 over integra

( )( )

tion range

b

a

I f x dx

f x

h xf x

=

≠

∫

Expressions involve computations of an integral expression

( )b

a

I h x dx= ∫Expressions can be rewritten as

[ ]1

1 ( )( )N

xi

iI E wN

xw x=

= = ∑xi with pdf f(x)


Example:

Compute where exponential RV with =1E x x λ⎡ ⎤⎣ ⎦ ∼

0

1

exp( )

1 = ,

where has exponential pdf

N

ii

i

E x x x dx

xN

x

∞

=

⎡ ⎤ = −⎣ ⎦ ∫

∑


MATLAB Implementation% generates data x with exp. pdfu=rand(1000,1);x=-log(u);h= x.^(0.5);%Value obtained using the Monte Carlo methodmean(h)

% Value obtained using numerical integrationF = @ (x) sqrt(x).*exp(-x);quad(F,0,10) % evaluate integration from 0 to 50

0.8747

0.8861


Check exponential data fit for x


Evaluate ( ), where (0,1)P x S x N≤ ∼Example:

( ) ( )

= ( ) , with

( )

1,

( )0,ow

S

h x

x S

P x S f x dx

f d

x

x

h

x

−∞

+∞

−∞

≤

≤= ⎨

=

⎧

⎩

∫

∫


1

( ) ( ) , with

( )

1,( )

0,ow

=

# 1 (N

)i

Ni

i

h x

x Sh

P x S f x dx

x

h x x SN

+∞

−∞

=

≤⎧⎨

≤

=

=

=

⎩

≤ ∫

∑

xi ~ N(0,1)

Example: ( 2) (2) 0.9772 MC with N=10,000 0.9751 MC with N=100,000 0.9771

P x Q≤ = =→

→

Problem with MC method: Need

high number of samples !!


• MC is an estimation process, estimated value will have a certain variance.

• MC performance evaluation works but….large number of trials needed to insure accurate estimates of PD and PFA estimates in radar/coms applications

Evaluating a small PFA requires ~ 100/PFA samplesin radar applications PFA =10-5 not unusual……

• Above discussion emphasizes need for reduction of the sampling size and insure probability estimates are still accurate.

• Reduction may be obtained via importance sampling.

•Reduction obtained by reframing the problem into one where events of interests are not rare, so fewer evaluations are needed, i.e., use samples where the value of the function to integrate is NOT small.



References


[1] W. Chan, Foundation Course on Probability, Random Variable and Random Processes

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