Random Processes

ECE460Spring, 2012

2

Power Spectral Density

Generalities :

Example: 0, cos 2i iX t A f t

3

Example Given a process Yt that takes the values ±1 with equal probabilities:

Find

2 21 1

1 1 1/ 2

1| 1 1| 1

,21/ 2,

t t

t t t t

P Y P Y

P Y Y P Y Y

TT

T

2 1where t t

1 2,YR t t

4

Ergodic1. A wide-sense stationary (wss) random process is ergodic in

the mean if the time-average of X(t) converges to the ensemble average:

2. A wide-sense stationary (wss) random process is ergodic in the autocorrelation if the time-average of RX(τ) converges to the ensemble average’s autocorrelation

3. Difficult to test. For most communication signals, reasonable to assume that random waveforms are ergodic in the mean and in the autocorrelation.

4. For electrical engineering parameters:

/2

/2

1lim ( ( ; )) [ ( ( ))]

T

iTtg x t dt E g X t

T

X XR x t x t E X t X t R

22

2

22 2

1. is equal to the dc level of the signal

2. equals the normalized dc power

3. 0 is equal to the total avg. normalized power

4. is the avg. normalized ac power

5. is equal to

XX

X

X

x x t

x x t

R x t

x t x t

the rms value of the ac component

5

Multiple Random Processes

Multiple Random Processes• Defined on the same sample space (e.g., see X(t) and Y(t)

above)• For communications, limit to two random processes

Independent Random Processes X(t) and Y(t)

– If random variables X(t1) and Y(t2) are independent for all t1 and t2

Uncorrelated Random Processes X(t) and Y(t)

– If random variables X(t1) and Y(t2) are uncorrelated for all t1 and t2

Jointly wide-sense stationary– If X(t) and Y(t) are both individually wss– The cross-correlation function RXY(t1, t2) depends

only on τ = t2 - t1

Filter( )X t

( )h t

( )Y t

,XY XYR t t E X t Y t R

6

Transfer Through a Linear System

Let X(t) be a wss random process and h(t) be the impulse response of a stable filter.

Find E{Y(t)}

Find the cross-correlation function RXY(t1,t2).

h t X t Y t

7

Transfer Through a Linear System

For a wss X(t) with autocorrelation RX(τ) and a stable (bibo) filter h(t), X(t) and Y(t) are jointly wide sense stationary:

h t X t Y t

8

Example

h t X t Y t

0cos 2

where is a random variable

uniformly distributed on [0,2 ]

X t A f t

differentiatorh t t

:Ym

1 2, :XR t t

:YS f

:X YS f

9

Energy ProcessesRecall that the energy of signal x(t) was calculated by

if Ex < ∞ then this is an energy signal

Define for a random process

Then the energy content of this signal can be given by

2x x t dt

E

2X X t dt

E

X

2

2

X

,

X

X

E E

E t dt

E X t dt

R t t dt

E

10

Power ProcessesRecall that the power of signal x(t) was calculated by

if Px > 0 then this is a power signal

Define for a random process

Then the power of signal x(t) can be given by

2

2

21lim

T

TxT

x t dtT

P

2

2

2

2

2

2

X

2

2

1lim

1lim

1lim ,

T

T

T

T

T

T

X

T

T

XT

P E

E X t dtT

E X t dtT

R t t dtT

P

2

2

21lim

T

TXT

X t dtT

P

11

ExampleShow that SX(f0) ≈ power of X(t) in [f0, f0+Δf]

h t X t Y t

0 01, ,

0, otherwise

f f f fH f

12

Thermal Noise

Because of the wide-band of thermal noise, it is usually modeled as white noise:

R

N t

2 W/HznS f K T R - value of resister in OhmsR - temperature of resister in KelvinT

23 - Boltzmann's constant 1.38 10 J/KK

13

Gaussian ProcessesA random process X(t) is Gaussian if for every t1, t2, …, tn, and every n, the random variables

X(t1), X(t2)…, X(tn)

Are jointly Gaussian.

The Gaussian random process is completely determined by its mean and autocorrelation functions, i. e., by

If a Gaussian process X(t) is passed through a linear filter, the output process is Gaussian

If X(t) is a wss Gaussian process with mean mX(t), autocorrelation RX(τ), and an LTI filter with input response h(t), then Y(t) = X(t)* h(t) is a wss Gaussian process with

1 2 1 2,

X

X

m t E X t

R t t E X t X t

0

*

Y X

Y X

m t m t H

R R h h

14

Zero-Mean White Gaussian NoiseA zero mean white Gaussian noise, W(t), is a random process with

4. For any n and any sequence t1, t2, …, tn the random variables W(t1), W(t2), …, W(tn), are jointly Gaussian with zero mean

and covariances

1. 0

2.2

3. Watt/Hz2

oW

oW

E W t t

NR E W t W t

NS f

cov

2

i j i j

W j i

oj i

W t W t E W t W t

R t t

Nt t

0 for 1,2,...,iE W t i n

15

Bandpass ProcessesSimilar to Chapter 2.5X(t) is a bandpass process

Filter X(t) using a Hilbert Transform:

and define

If X(t) is a zero-mean stationary bandpass process, then Xc(t) and Xs(t) will be zero-mean jointly stationary processes:

0

is a deterministic bandpass signal

and is non-zero about

X

X X

R

S f F R f

1

sgn

h tt

H f j f

0 0

0 0

cos 2 sin 2

cos 2 sin 2c

s

X t X t f t X t f t

X t X t f t X t f t

0

,

,

,

c c

s s

c s c s

c s

X X

X X

X X X X

E X t E X t

R t t R

R t t R

R t t R

16

Bandpass ProcessesThis results in two key formulas for future use:

Note: Xc(t) and Xs(t) are lowpass processes; i.e., their power spectrum vanishes for |f| ≥ W.

Find the power spectrum of the in-phase and quadrature components:

0 0

0 0

cos 2 sin 2

sin 2 cos 2c s

c s

X X X X

X X X X

R R R f R f

R R f R f

17

Bandpass Example 4.6.1The white Gaussian noise process N(t) with power spectrum N0/2 passes through an ideal bandpass filter with frequency response

where W << fc. The output process is denoted by X(t). Find the power spectrum and the cross-spectral density of the in-phase and quadrature components in the following two cases:1. f0 is chosen to be equal to fc.

2. f0 is chosen to be equal to fc-W.

1,

0, Otherwisecf f W

H f