September 9, 2004

EE 615 Lecture 2 Review of Stochastic Processes Random Variables DSP, Digital Comm

Dr. Uf TureliDepartment of Electrical and Computer Engineering Stevens Institute of TechnologyHoboken NJ 07030

September 9, 2004

Stochastic Processes Fundamentals Random Variables A mapping between a discrete or a

random event and a real number. (not a variable!)

Ensemble Average Average or Expected value behavior

of a random variable.

September 9, 2004

Continous Random Variables Distribution function FX (a) of RV X is:

Probability density function fX(a)

fX(a) > 0

)Pr()( XFX

)()(

XX Ff

1)()()(

XXX FFf

September 9, 2004

Discrete Random Variables and Probability

Random variable X assumes a value as a function from outcomes of a process which can not be determined in advance.

Sample space S of a random variable is the set of all possible values of the variable X.

: set of all outcomes and divide it into elementary events, or states

1)(}{

x

xp 0)(1 xp

September 9, 2004

Expectation, Variance and Deviation

The moments of a random variable define important

characteristics of random variables:

The first moment is the expectation E[X]=<X>:

Note: The expectation has a misleading name and is not always the value we

expect to see most. In the case of the number on a dice the expectation is 3.5

which is not a value we will see at all!. The expectation is as a weighted average.

The variance is defined by Var[x] = <x2> - <x>2 = M2 - M12.

The standard deviation = Var[x]1/2 evaluates the “spread

factor”or x in relation to the mean.

September 9, 2004

Ensemble Average Mean:

Continuous

Discrete

Variance

)( E XfXX

222 XX

k

kk XX )Pr(

Xx fXXX 222 E

September 9, 2004

Correlation & Covariance Crosscorrelation

Covariance

If <X> or <Y> equal zero, correlation equals covariance

*XYEXYr

YXYYcXY** XYEXX-E

September 9, 2004

Random Process X, Y need not be separate events X,Y can be samples of process

observed at different instants t_1, t_2 )()(tE),( 2

*121 tXXttRX

*2211

*21 )()()(t(E),( tXtXXtXttCX

September 9, 2004

Independence vs. Uncorrelatedness R.V.s X, Y independent if

Uncorrelated (Weaker condition), when

R.V. X, Y uncorrelated if covariance is zero.

Independent R.V. always uncorrelated. Uncorrelated R.V. may not be independent!

)()(),( YXXY fff

*** EEE YXYXXYrXY

*** XYEXX-E YXYYcXY

September 9, 2004

Random Processes Random process is a rule for assigning

every outcome of a probabilistic event to a function

Random process is an indexed sequence of R.V.s

R.P. is stationary in strict sense, if all statistics are time invariant

Wide Sense stationary if first and second order statistics are time invariant.

),( tX

September 9, 2004

WSS Process Properties <X>=constant, For Gaussian process, WSS implies

strict stationarity For WSS:

XX RkkR ),(

2)(E)0(

allfor ),0()(

)()(

tXR

RR

RR

X

XX

XX

September 9, 2004

MOdulation/DEModulation Modulation: Converting digital data into an

analog signal. Demodulation: Converting an analog signal

into digital data

September 9, 2004

DIGITAL SIGNAL DISCRETE WAVEFORM TWO DISCRETE STATES:

1-BIT & 0-BIT ON / OFF PULSE

DATA COMMUNICATION USES MODEM TO TRANSLATE

ANALOG TO DIGITAL, DIGITAL TO ANALOG

September 9, 2004

Digital Comm over Fading Channels Comm Theory 609: Design/

Performance of Digital Comm. In Additive White Gaussian noise

New: Linear Filter Channel with AWGN

Traditional Soln: Equalization Question: How should signals be

designed for complex channels?

September 9, 2004

Statistical Characterization of Channels

Digital Comm. Proakis, 4th Edition, Ch.14 pp.800

Notice channel has time varying impulse response!

September 9, 2004

Propagation Models Channel model provides reliable base

in system design and research For simulations and design, simple

model preferred In transmitter/receiver (transceiver)

design, not accurate but typical and worst case models most relevant

September 9, 2004

Major Channel Effects Propagation Loss is attenuation,

also called path loss Time Dispersion: multiple

reflections due to obstacles leading to multipaths

Doppler Effects: Time variant nature due to mobility of objects in an environment

September 9, 2004

Propogation Loss Free space propagation:

Loss (dB):S(d)=S_0+10a log_10 (d)+b, where a and b depend on operating frequency, environment, obstructed or direct line of sight, around 5 GHz, a=3.75, b=-6.5, such that for distances 10-50m, S=80-100 dB!

2

4

dGG

P

PRT

T

R

September 9, 2004

Noise White Noise Interference: Narrowband Interference

KelvinJxkkN /1038.1, WT 230

Microwave Emission

Frequency Hopping

September 9, 2004

Multipath Propagation Natural

obstacles, buildings, furnitures, etc.

Each path:delay, attenuation, phase shift

September 9, 2004

Terminology Static Channel Impulse response k:path index, a_k:path gain,

theta_k:path phase shift, tau_k:path delay

1

0

)()(N

kk

jk teath k

September 9, 2004

Power Delay Profile (PDP) PDP

RMS Delay Spread

)()(1

0

2

N

kkk tatP

1

0

2

1

0

2

1

0

2

1

0

22

2

,

)(

)(

,

)(

)(

N

kk

N

kkk

N

kk

N

kkk

RMS

RMS

a

a

a

a

dttP

dtttP

dttP

dttPt

September 9, 2004

Coherence Bandwidth Autocorrelation of channel frequency response For class of channels with exponential delay profile, autocorrelation can be computed as a statistical expectation

Coherence BW:

dfffHfHfR )()()( *

fj

ffHfHfRRMS

21

1)()(E *

2

1

)0(

)(

cohBf

A

R

fR

September 9, 2004

Flat vs Frequency Selective Fading For channel with exponential delay

spread

If BW > B_coh: Frequency selective fading

If BW < B_coh: Flat fading

)2/(1 RMScohB

September 9, 2004

Effect of channel Transmitted signal fc:carrier frequency, j=sqrt(-1) Received signal

tfj

l

cetsts2

)( Re)(

dtsttx

ttsttx nn

n

)(),()(

)()()(

September 9, 2004

Time Variant ChannelsCorrelation:

tR

dttththtR )()()( *

Coherence Time:

2

1

)0(

)(

cohTt

A

R

tR

September 9, 2004

Doppler Spectra Doppler

Spectrum:

T_coh ~ 1/ f_d

dtetRfP ftjh

2)()(

September 9, 2004

Example: OFDM Modulation

September 9, 2004

Multicarrier Modulation DFT/FFT to generate subcarriers Real representation:

Tttttts

Ttttttfjdts

ss

sssTi

c

N

NNi

s

s

i

,,0)(

,)))((2exp(Re)( 5.012/

12/2/

Tttttts

Ttttttfjdts

ss

sssTi

c

N

NNi

s

s

i

,,0)(

,)))((2exp()(12/

12/2/

Complex:

September 9, 2004

Demodulation

Tddtttjd

Ttttttfjdttj

s

s

s

s

s

i

s

s

i

s

s

Njs

Tt

t

Tji

N

NNi

sssTi

c

N

NNis

Tt

t

Ti

2/

12/

12/2/

12/

12/2/

))(2exp(

,)))((2exp())(2exp(

September 9, 2004

IFFT for modulation N point transform N^2 operations

(complexity grows quadratically) NlogN complexity in the FFT/IFFT

(slightly faster than linear) Radix-4 butterfly

September 9, 2004

FFT Implementation Decimation in Time

September 9, 2004

Multicarrier System -Wireless Complex Transmission

September 9, 2004

Wireline, Baseband Transmission

September 9, 2004

Decimation Decimation in Time, vs Frequency

September 9, 2004

Scalability-repetetive structure Partial FFT, if you use a subset of

transmitted carriers

September 9, 2004

Cyclic Extension Transmission in frequency domain

(FFT) DFT properties

Signal and channel linearly convolved

Prefix and postfix extension

}{}{}{

}{}{}*{

nnnn

nnnn

hxDFTdDFThdDFT

hFTdFThdFT

September 9, 2004

Cyclic Prefix Make the convolution linear Filtering:

Cylic Prefix and Removal makes linear convolution into Circular convolution

)()()(*)()( knxkhnhnxny

September 9, 2004

Time/ Frequency Domain -Processing Why not equalize in frequency

domain? Stu Schwartz (Princeton) Hikmet Sari (France Telekom)

(w/cylic prefix) Falconer (Carleton)