Sep 22, 2005 CS477: Analog and Digital Communications

1

Random Processes and PSD

Analog and Digital Communications

Autumn 2005-2006

Sep 22, 2005 CS477: Analog and Digital Communications 2

Random Processes Cross-correlation

(Processes are orthogonal if )

Cross-covariance

RX;Y(t1;t2) = E[X(t1)Y(t2)]

CX;Y(t1;t2) = E[X(t1)Y(t2)]à mX(t1)mY(t2)

CX;Y(t1;t2) = RX;Y(t1;t2) à mX(t1)mY(t2)

RX;Y(t1;t2) = 0

Sep 22, 2005 CS477: Analog and Digital Communications 3

ExampleX(t) = A cos2ùt

mX(t) = E[A]cos2ùt

RX(t1;t2) = E[A2]cos2ùt1cos2ùt2

CX(t1;t2) = Var(A)cos2ùt1cos2ùt2

A is random variable

Sep 22, 2005 CS477: Analog and Digital Communications 4

Example

mX(t) = 0

RX(t1;t2) = 21cos(2ùf c(t1à t2))

CX(t1;t2) = RX(t1;t2)

X(t) = cos(2ùf ct + Ê); Ê ø U(0;2ù)

Mean is constant and autocorrelation is dependent on t1à t2

Sep 22, 2005 CS477: Analog and Digital Communications 5

Example

RX;Y(t1;t2) = RX(t1;t2) + mX(t1)mN(t2)

Y(t) = X(t) + N(t)

X(t) and N(t) independent

Sep 22, 2005 CS477: Analog and Digital Communications 6

Stationary and WSS RP Stationary Random Process (RP)

Wide sense stationary (WSS) RP Mean constant in time Autocorrelation depends only on

Stationary WSS (Converse not true!)

pX(t)(x) = pX(t+T)(x) 8T

t1à t2

RX(t1;t2) = RX(t2à t1)= RX(ü)

Sep 22, 2005 CS477: Analog and Digital Communications 7

Power Spectral Density (PSD) Defined for WSS processes Provides power distribution as a

function of frequency Wiener-Khinchine theorem

PSD is Fourier transform of ACFSX(f ) = R

à1

1RX(ü)eà j2ùfüdü

Sep 22, 2005 CS477: Analog and Digital Communications 8

PSD: ExampleX(t) = K cos(2ùf ct + Ê); Ê ø U(0;2ù)

RX(t1;t2) = 2K 2cos(2ùf c(t1à t2))

RX(ü) = 2K 2cos(2ùf cü)

SX(f ) = 4K 2âî (f à f c) + î (f + f c)

ã

Sep 22, 2005 CS477: Analog and Digital Communications 9

Deterministic Signals and PSD

Pv = hjv(t)j2i = T1 R

à T=2

T=2jv(t)j2dtlim

T! 1

Pv = hv(t)vã(t)iFor energy signals, multiply above expression with time

Rv(ü) = hv(t)vã(t à ü)i

= T1 R

àT=2

T=2v(t)vã(t à ü)dtlim

T! 1

Rv(0) = PvACF is a more generic function than average power

Sep 22, 2005 CS477: Analog and Digital Communications 10

Deterministic Signals

Rvw(ü) = hv(t)wã(t à ü)i = hv(t + ü)wã(t)i

= T1 R

à T=2

T=2v(t)wã(t à ü)dtlim

T! 1

Cross-correlation function

Cross power spectral densitySvw(f ) = R

à 1

1Rvw(ü)eà j2ùfüdü

(Also applicable to jointly stationary random signals)

Sep 22, 2005 CS477: Analog and Digital Communications 11

LTI Systems RevisitedFor random and deterministic signals:

Ryx(ü) = h(ü) ? Rx(ü) (Prove it at home!)

Ry(ü) = h(à ü) ? Ryx(ü) (For real channels!)

Ry(ü) = h(à ü) ? h(ü) ? Rx(ü)

Sy(f ) = Hã(f )H(f )Sx(f )

Sy(f ) = jH(f )j2Sx(f )

x yh(t)

Sep 22, 2005 CS477: Analog and Digital Communications 12

Example: Random SignalConsider White Noise input to an LTI filter

Sy(f ) = jH(f )j2 2N0

Sx(f ) = 2N0

LTIx y