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Power ofPower ofRandom ProcessesRandom Processes

2/40

Power of a Random Process

Recall : For deterministic signals…instantaneous power is x2(t)

For a random signal, x2(t) is a random variable for each time t. Thus there is no single # to associate with “instantaneous power”. To get the Expected Instantaneous Power (i.e., on average) we compute the statistical (ensemble) average of x2(t):

3/40

Power of a Random Process

Avg. Power of x(t) : PX(t) = E{ x2(t)}

This is also called the “mean square value”of the process

In general, can depend on time.But we’ll see it doesn’t for WSS(& stationary) processes

Often drop the“Average” terminology

4/40

Relationship of Power to ACFRecall: RX(t, t+τ ) = E { x(t) x(t+τ) }

Clearly, setting τ =0 makes this equal to PX(t) ⇒ PX(t) = RX(t, t)

If the Process is WSS (or SS) RX(t,t) = RX(0)

PX = RX(0)

τ

RX(τ)PX (Power of process)

For WSS or SS

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Power vs. Variance

Note: If the WSS process is Zero Mean, then:

Power & Variance are Equal for Zero mean Process

{ }WSSfor)0(

)]()([2

22

xR

txtxE

x

x

−=

−=σRecall Variance:

PX

22 xP xx +=σ“DC” Power

“AC” Power

2xXP σ=

6/40

Power Spectral Density of a Random Process

Recall: PSD for Deterministic Signal x(t) :

∫−

−

∞→

=

=

2/

2/

2

)()(where

)()( lim

T

T

tjT

T

Tx

dtetxX

TX

S

ωω

ωω

7/40

Power Spectral Density of a Random Process

For a random Process: each realization (samplefunction) of process x(t) has different FT and therefore a different PSD.

We again rely on averaging to give the “Expected”PSD or “Average” PSD…

But… Usually just call it “PSD”.

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Define PSD for WSS RPWe define PSD of WSS process x(t) to be :

This definition isn’t very useful for analysis so we seek an alternative form

The Wiener-Khinchine Theorem provides this alternative!!!

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=∞→ T

XES T

Tx

2)()( lim

ωω

<Compare this to PSD for Deterministic Signal>

9/40

Weiner- Khinchine Theorem

Let x(t) be a WSS process w/ ACF RX(τ) and w/PSD SX(ω) as defined in ( )… Then RX(τ) and SX(ω) form a FT pair :

SX(ω) = F{ RX(τ) }

or Equivalently

RX(τ) ↔ SX(ω)

10/40

Proof of WK theoremBy definition : ∫

∞

∞−

−= dtetxX tjTT

ωω )()(

For Real x(t):

21)(

2

2/

2/

2/

2/1

2

2/

2/21

2/

2/1

2

12

21

)()(

)()(

)()()(

dtdtetxtx

dtetxdtetx

XXX

ttjT

T

T

T

tjT

T

tjT

T

TTT

−−

− −

−

−−

∫ ∫

∫∫

=

⋅=

−=

ω

ωω

ωωω

11/40

Proof of WK theorem(cont’d)

Thus :

Move E {.} inside integrals :

21)(

2/

2/

2/

2/12

12)(1lim)( dtdtettRT

S ttjT

T

T

Tx

Tx

−−

− −∞→ ∫ ∫ −= ωω

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= −−

− −∞→ ∫ ∫ 21

)(2

2/

2/

2/

2/1

12)()(1lim)( dtdtetxtxT

ES ttjT

T

T

TT

xωω

12/40

Proof of WK theorem(cont’d)

We were almost there… BUT we have one-too-manyintegrals.

For convenience define: φ(t2 - t1) = RX(t2 - t1) e-jω(t2- t1)

∫ ∫− −∞→

−=2/

2/21

2/

2/12 )(

T1)( lim

T

T

T

TTx dtdtttS φω

13/40

Proof of WK theorem(cont’d)

Change of Variables = Change of AxesChange of Variables = Change of Axes

Instead of integrating “Row-by-Row” as in we integrate “Diagonal-by-Diagonal”.

Let: τ = t2 – t1 t2 = (τ + λ)/2λ = t1 + t2 t1 = (λ – τ)/2

14/40

Proof of WK theorem(cont’d)

t1

t2τ = t2 – t1λ = t2 + t1

15/40

Proof of WK theorem(cont’d)

21

21

21

21

21

J

11

22

=

−

=

∂∂

∂∂

∂∂

∂∂

=

λτ

λτ

tt

tt

Use Jacobian result for 2-D change of variables

From Calculus III

16/40

Proof of WK theorem(cont’d)

∫ ∫∞→

=?

?

?

?21)(

T1)( lim λττφω ddS

Tx

Q: As τ ranges over -T≤τ≤Thow does λ range?

A: For each τ, λ must be restricted to stay inside original square – see Figure on next Chart

J

17/40

Proof of WK theorem(cont’d)

t1

t2τ = t2 – t1λ = t2 + t1Note: φ(t2 – t1) Doesn’t ChangeIn The DirectionOf t2 + t1 Axis

T/2– T/2

– T/2

T/2τ = 2

τ = 4 τ = 6

τ = T

18/40

Proof of WK theorem(cont’d)

τ = t2 – t1t2 = τ + t1

t1= t2 - τ

λ = (τ + t1) + t1 = 2t1 + τ

λ = t2 + (t2 - τ) = 2t2 - τ

When τ > 0 we need (From Axes Figure):t2 ≤ T/2 λ ≤ T - τt1 ≥ -T/2 λ ≥ -T + τ

λ = t2 + t1

Works out similarly for τ < 0

19/40

Proof of WK theorem(cont’d)

So… τλτφωτ

τ

ddT

ST

T

T

TTx 2

1)(1)( lim ∫ ∫−

−

+−∞→ ⎥⎥⎦

⎤

⎢⎢⎣

⎡=

= T - τ

{ })(

)(

1)(lim

τR

d

dT

x

T

TT

ℑ=

=

⎟⎠⎞

⎜⎝⎛ −=

∫

∫

∞

∞−

−∞→

ττφ

τττφ

<End of Proof>

20/40

Some Properties of PSD & ACF

(1) The PSD is an even function of ω for a real process x(t) proof : since each sample function is real valued then we know that ⏐x(ω)⏐is even.

⇒ ⏐x(ω)⏐2 is even: (even x even = even)⇒ So is SX(w) ( this is clear from )

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Some Prop. of PSD & ACF(2) SX(ω) is real-valued and ≥ 0.

proof : again from ( ) – since ⏐x(w)⏐2 is real-valued & ≥ 0, so is SX(w)

(3) RX(τ) is an even function of τ

Also follows from IFT{ Real & Even } = Real & Even <Property of the FT>

)()}()({

)}()({)(

ττσσ

ττ

x

x

RxxE

txtxER

=+=

−=−

22/40

Graphical View of Prop #3For WSS, ACF does not depend on absolute time only relative time

t- τ t+ττ τ

t

t

t

t

Doesn’t matter if you look forward by τ or look backward by τ

23/40

Computing Power from PSD

From it’s name – Power Spectral Density – we know what to expect :

∫∞

∞−

= ωωπ

dSP xx )(21

So let’s Prove this !!!!

Well … this part is not obvious!!

24/40

Computing Power from PSDWe Know: { }

∫∞

∞−

−

=

ℑ=

ωωπ

ωτ

ωτ deS

SR

jx

xx

)(21

)()( 1

PX = RX(0)

.!Q.E.D)(21

)(21

1

0

∫

∫∞

∞−

∞

∞− =

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

ωωπ

ωωπ

ω

dS

deSP

x

jxX

25/40

Units of PSD Function

∫∞

∞−=

πωω

2)( dSP xx )(ωxS

Watts Hz [ Watts / Hz ]

26/40

Using Symmetry of SX(w)

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∫

∞

0

)(212 ωωπ

dSP xx

Integrate only from 0 → ∞

Double to account for the -∞ → 0 part of integral

27/40

PSD for DT Processes

SX(Ω) = DTFT { RX[m] }

Periodic in Ω with period 2π

ΩΩ= ∫−

dSP xx

π

ππ)(

21

Not much changes – mostly, just use DTFT instead of CTFT!!

Need only look at -π ≤ Ω < π

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Big Picture of PSD & ACFi.e. High Frequencies have Largepower contentProcess exhibits

Rapid fluctuations

Narrow ACF Broad PSD

Less CorrelatedSample-to-sample

i.e. High Frequencieshave Smallpower contentProcess exhibits

Slow fluctuations

Broad ACF Narrow PSD

More CorrelatedSample-to-sample

<<See “Big Picture: Filtered RP” Charts in “V-3 RP Examples” >>

29/40

White NoiseThe term “White Noise” refers to a WSS process whose PSD is flat over all frequencies

C-T White Noise

ω

SX(ω)

Convention to use this form (i.e. w/ division by 2)

White Noise Has Broadest Possible PSD

N /2

ωω ∀=2

)( NXS

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C-T White Noise

NOTE : C-T white noise has infinite Power :

Can’t really exist in practice but still a veryuseful Model for Analysis of Practical Scenarios

∫∞

∞−

∞→ωd2/N

31/40

C-T White NoiseQ : what is the ACF of C-T white Noise ?A: Take the IFT of the flat PSD :

{ }

)(

2/)( 1

τδ

τ

2N

N

=

ℑ= −xR

x(t1) & x(t2) are uncorrelatedfor any t1 ≠ t2

Delta function !Narrowest ACFRX(τ)

Area = N /2τ

PX = RX(0) = N /2δ(0) → ∞Infinite Power.. It Checks!

Also….

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D-T White NoisePSD is:SX(Ω) = N /2 ∀ Ω

…but focus on Ω∈[-π,π]

ACF is:RX[m] = IDTFT {N /2 }

= N /2 δ [m]

Delta sequence

x[k1] & x[k2] are uncorrelated for any k1 ≠ k2

Ω

SX(Ω)

-π π

N /2 Broadest Possible PSD

Narrowest ACF

m

RX[m]

-3 1-2 -1 2 3

N /2

33/40

D-T White Noise

Note:

D-T White Noise has Finite Power(unlike C-T White Noise)

2221

2]0[

NN

N

=Ω=

==

∫−

π

ππ

dP

RP

x

xx watts

watts

34/40

Examples of PSDExample 1: “BANDLIMITED WHITE NOISE”This looks like white noise within some bandwidth but its PSD is zero outside that bandwidth – hence the name.

Thus:

⎟⎠⎞

⎜⎝⎛=

BSx π

ωω4

rect2/)( N

B

BBdPB

Bx

N

NN

=

+⋅⋅== ∫−

)22(2/212/

21 2

2

πππ

ωπ

π

π

And…

SX(ω)

ω-2πB 2πB

N /2

35/40

Example: BL White NoiseThe ACF (using FT pair for rect and sinc) is:

RX(τ) = N B sinc (2πBτ)

Again we see PX = N B , since RX(0) = N B

Note: For τ > 1/2B …x(t) & x(t + τ) are Approximately Uncorrelated

RX(τ)

-32B

-22B

-12B

1 2B

22B

32B

N B

τ

36/40

Example #2 of PSDExample 2 : “SINUSOID WITH RANDOM PHASE”

x(t) = A cos (ωCt + θ)

We examined this RP before: RX(τ) = A2 cos (ωCτ)2

So using FT Pair for a Cosine gives the PSD:

SX(ω) = (πA2)/2 [δ (ω + ωC) + δ (ω – ωC)]

ω

SX(ω)

- ωc ωc

Area = (πA2)/2Area = (πA2)/2

37/40

Example #2 of PSDNote : Can get PX in two ways:

221

2

2

2

)(.2

)0(.1

Ax

Ax

dS

R

=

=

∫∞

∞−ωωπ

⇒ PX = A2

2

<Sifting Property>

38/40

Example #3 of PSDExample 3: “FILTERED D-T RANDOM PROCESS”

< See Also: Class Notes on “Filtered RPs” >

D-T Filterx [k]

Zero mean ⇒ RX[m] = σ2δ[m] (Input ACF) White noise

⇒ SX(Ω) = σ² ∀Ω (Input PSD)

y[k] = x[k] + x [k +1]

SX(Ω)

Ω

σ²

39/40

Example #3 of PSDFor this case we showed earlier that for this filter output the ACF is :

RY[m] = σ² { 2δ[m] + δ[m-1] + δ[m+1] }

So the Output PSD is:

SY(Ω) = σ² [2 + e-jΩ + e-jΩ]

= 2σ² [cos (Ω) + 1]

Use the result for DTFT of δ[m] and also time-shift property

= 2 cos (Ω) By Euler

40/40

Example #3 of PSD

General Idea…Filter Shapes Input PSD:Here it suppresses High Frequency power

-2π -π π 2π

Sy(Ω)

Ω

4σ²Replicas Replicas

SY(Ω) = 2σ² [cos (Ω) + 1]

Remember: Limit View to [-π,π]