telecommunications dr. hugh blanton entc 4307/entc 5307

TELECOMMUNICATIONS

Dr. Hugh Blanton

ENTC 4307/ENTC 5307

POWER SPECTRAL DENSITY

Dr. Blanton - ENTC 4307 - Correlation 3

Summary of Random VariablesSummary of Random Variables

• Random variables can be used to form models of a communication system

• Discrete random variables can be described using probability mass functions

• Gaussian random variables play an important role in communications• Distribution of Gaussian random variables is well

tabulated using the Q-function• Central limit theorem implies that many types of noise

can be modeled as Gaussian


Random ProcessesRandom Processes

• A random variable has a single value. However, actual signals change with time.

• Random variables model unknown events.• A random process is just a collection of random

variables.• If X(t) is a random process then X(1), X(1.5),

and X(37.5) are random variables for any specific time t.


TerminologyTerminology

• A stationary random process has statistical properties which do not change at all with time.

• A wide sense stationary (WSS) process has a mean and autocorrelation function which do not change with time.

• Unless specified, we will assume that all random processes are WSS and ergodic.


Spectral DensitySpectral Density

Although Fourier transforms do not exist for random processes (infinite energy), but does exist for the autocorrelation and cross correlation functions which are non-periodic energy signals. The Fourier transforms of the correlation is called power spectrum or spectral density function (SDF).


Review of Fourier TransformsReview of Fourier Transforms

Definition: A deterministic, non-periodic signal x(t) is said to be an energy signal if and only if

dttxE )(2


The Fourier transform of a non-periodic energy signal x(t) is

The original signal can be recovered by taking the inverse Fourier transform

dtetxXtx tj )()()}({

deXtxX tj)()()}({1

)()( Xtx


Remarks and PropertiesRemarks and Properties

The Fourier transform is a complex function in having amplitude and phase, i.e.

jeXX )()(


Example 1Example 1

Let x(t) = eat u(t), then

jae

ja

dtedteetx

tja

tjatjat

11

)(

0

00


AutocorrelationAutocorrelation

• Autocorrelation measures how a random process changes with time.

• Intuitively, X(1) and X(1.1) will be more strongly related than X(1) and X(100000).

• Definition (for WSS random processes):

• Note that Power = RX(0)

)()()( tXtXERX


Power Spectral DensityPower Spectral Density

• P() tells us how much power is at each frequency

• Wiener-Klinchine Theorem:

• Power spectral density and autocorrelation are a Fourier Transform pair!

)()( RP


Properties of Power Spectral DensityProperties of Power Spectral Density

• P() 0

• P() = P(-)

dPPower )(


Gaussian Random ProcessesGaussian Random Processes

• Gaussian Random Processes have several special properties:• If a Gaussian random process is wide-sense

stationary, then it is also stationary.• Any sample point from a Gaussian random process

is a Gaussian random variable• If the input to a linear system is a Gaussian random

process, then the output is also a Gaussian process


Linear SystemLinear System

• Input: x(t)• Impulse Response: h(t)• Output: y(t)

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


Computing the Output of Linear SystemsComputing the Output of Linear Systems

• Deterministic Signals:

• Time Domain: y(t) = h(t)* x(t)• Frequency Domain: Y(f)=F{y(t)}=X(f)H(f)

• For a random process, we still relate the statistical properties of the input and output signal

• Time Domain: RY()= RX()*h() *h(-)

• Frequency Domain: PY()= PX()|H(f)|2


Power Spectrum or Spectral Density Function (PSD)Power Spectrum or Spectral Density Function (PSD)

• For deterministic signals, there are two ways to calculate power spectrum.• Find the Fourier Transform of the signal, find

magnitude squared and this gives the power spectrum, or

• Find the autocorrelation and take its Fourier transform

• The results should be the same.• For random signals, however, the first

approach can not be used.


Let X(t) be a random with an autocorrelation of Rxx() (stationary), then

and

deRS jXXXX

)()(

deSR jXXXX

)(

2

1)(


Properties:

(1)SXX() is real, and SXX(0) 0.

(2)Since RXX(t) is real, SXX(-) = SXX(), i.e., symmetrical.

(3)Sxx(0) =

(4)

dSR XXXXX )(2

1)0(2

dRXX

)(

)()( XXXX StR


Special CaseSpecial Case

For white noise,

Thus,

)()( 2XXXR

22 )()( Xtj

XXX dteS

RXX()

X

SXX()

X


Example 1Example 1

Random process X(t) is wide sense stationary and has a autocorrelation function given by:

Find SXX.

eR XXX2)(


Example 1Example 1

eR XXX2)(

RXX()

X

deedee

deedeRS

jX

jX

jjXXXX

0

2

0

2

)()(


2

2

2

0

202

0

202

0

0

2

0

0

2

1

2

1

1

1

1

11

11

11

X

XjXjX

jXjX

jjX

jjX

jje

je

j

djej

djej

dededeedee


Example 2Example 2

Let Y(t) = X(t) + N(t) be a stationary random process, where X(t) is the actual signal and N(t) is a zero mean, white gaussian noise with variance N

2 independent of the signal.

Find SYY.

2

2

)()(

)()()(

NXXYY

NXXYY

SS

RR


Correlation in the Continuous DomainCorrelation in the Continuous Domain

• In the continuous time domain

dttxtx

TR )()(

1)( 21

012


• Obtain the cross-correlation R12 () between the waveform v1 (t) and v2 (t) for the following figure.

T 2T 3T

v1(t)

t

1.0

v2(t)

tT 2T 3T

1.0

-1.0


• The definitions of the waveforms are:

and

TtforTttv 0/)(1

TtTfor

Ttfortv

2/1

2/01)(2


• We will look at the waveforms in sections.• The requirement is to obtain an expression

for R12 ()

• That is, v2 (t), the rectangular waveform, is to be shifted right with respect to v1 (t) .


t

The situation for

is shown in the figure. The figure show that there are three regions in the section for which v2(t) has the consecutive values of -1, 1, and -1, respectively. The boundaries of the figure are:

20 Tt

Tt

Tt

t

2

v(t)

T

1.0

-1.0


T

T

T

T

T

T

tdtT

tdtT

tdtT

dtT

t

Tdt

T

t

Tdt

T

t

TR

22

2

20

2

2

2

012

111

11

11

11

)(


42

2

42

2

2

1

2

2

222

2

2

1

2

1

2

1

2

1)(

2222

222

2

22222

2

2

2

2

22

2

0

2

212

TTT

TT

T

TTT

T

t

T

t

T

t

TR

T

T

T


20

4

1

4

2

2

4

2

1

42

2

42

2

2

1

42

2

42

2

2

1)(

22

2

22

2

2

2222

222

212

Tfor

TT

TT

T

TTT

TT

T

TTT

TT

Tr


t

v(t)

T

1.0

-1.0

The situation for

is shown in the figure. The figure show that there are three regions in the section for which v2(t) has the consecutive values of 1, -1, and 1, respectively. The boundaries of the figure are:

TtT 2

Tt

t

Tt

2


T

T

T

T

T

T

tdtT

tdtT

tdtT

dtT

t

Tdt

T

t

Tdt

T

t

Tr

22

2

2

02

2

2

012

111

11

11

11

)(


222

222

22

22222

2

2

2

2

2

2

2

0

2

212

42

2

42

2

2

1

2022

1

2

1

2

1

2

1)(

TTTTT

T

TTTT

t

T

t

T

t

Tr

T

T

T


TT

forT

TTT

Tr

24

3

4

2

2

4

2

1)( 2

2

212


t

T/2 T

0.25

-0.25


• Let X(t) denote a random process. The autocorrelation of X is defined as


Properties of Autocorrelation Functions for Real-Valued, WSS Random Processes

Properties of Autocorrelation Functions for Real-Valued, WSS Random Processes

• 1. Rx(0) = E[X(t)X(t)] = Average Power

• 2. Rx() = Rx(-). The autocorrelation function of a real-valued, WSS process is even.

• 3. |Rx()| Rx(0). The autocorrelation is maximum at the origin.


Autocorrelation ExampleAutocorrelation Example

t 2

2-t

y(

t


t 2

2-t

y(

t

0

dt

Rx 22)(

dtRt

x

2

0

2

4

1)(

dtdRt t

x

2

0

2

0

2

4

1

4

1)(

t

x

tR

2

0

23

234

1)(

2

2

3

2

4

1)(

23 tttRx


2

44

3

6128

4

1)(

2332 ttttttRx

tt

ttttRx 22

2324

3

8

4

1)( 2

332

3

82

64

1)(

3

tt

Rx

12

8

224)(

3

tt

Rx 3

2

224)(

3

tt

Rx


Correlation ExampleCorrelation Example

t

y(t

0 1 2 3 4 5 6 7

1

-1


t=0:.01:2;y=(t.^3./24.-t./2.+2/3);plot(t,y)


t=0:.01:2;y=(-t.^3./24.+t./2.+2/3);plot(t,y)


tint=0; tfinal=10; tstep=.01; t=tint:tstep:tfinal; x=5*((t>=0)&(t<=4));subplot(3,1,1), plot(t,x)axis([0 10 0 10])h=3*((t>=0)&(t<=2));subplot(3,1,2),plot(t,h)

axis([0 10 0 10])axis([0 10 0 5])t2=2*tint:tstep:2*tfinal;y=conv(x,h)*tstep;subplot(3,1,3),plot(t2,y)axis([0 10 0 40])


Matched FilterMatched Filter


Matched FilterMatched Filter

• A matched filter is a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform.• Consider that a known signal s(t) plus a

AWGN n(t) is the input to a linear time-invariant (receiving) filter followed by a sampler.


• At time t = T, the sampler output z(t) consists of a signal component ai and

noise component n0. The variance of the

output noise (average noise power) is denoted by 0

2, so that the ratio of the

instantaneous signal power to average noise power, (S/N)T, at time t = T is

20

2

i

T

a

N

S


Random Processes and Linear SystemsRandom Processes and Linear Systems

• If a random process forms the input to a time-invariant linear system, the output will also be a random process.

• The input power spectral density GX(f)

and the output spectral density GY(f) are

related as follows:

2)()()( fHfGfG XY


• We wish to find the filter transfer function H0(f) that maximizes

• We can express the signal ai(t) at the filter output in terms of the filter transfer function H(f) and the Fourier transform of the input signal, as

20

2

i

T

a

N

S

dfefSfHa ftj

i2)()(


• If the two-sided power spectral density of the input noise is N0/2 watts/hertz, then we can express the output noise power as

• Thus, (S/N)T is

dffH

N 2020 )(

2

dffHN

dfefSfH

N

Sftj

T 20

2

2

)(2

)()(


• Using Schwarz’s inequality,

• and

dffSdffHdfefSfH ftj 22

2

2 )()()()(

0

2)(2

N

dffS

N

S

T


• Or

• where0

2max

N

E

N

S

T

dffSE

2

)(


• The maximum output signal-to-noise ratio depends on the input signal energy and the power spectral density of the noise.

• The maximum output signal-to-noise ratio only holds if the optimum filter transfer function H0(f) is employed, such that

tfj

tfj

efkSth

or

efkSfHfH

21

20

)(*)(

)(*)()(


tfj

tfj

efkSth

or

efkSfHfH

21

20

)(*)(

)(*)()(


• Since s(t) is a real-valued signal, we can use the fact that

• and

)(*)( fXfX

tfjefXttx 20 )()(


• to show that

• Thus, the impulse response of a filter that produces the maximum output signal-to-noise ratio is the mirror image of the message signal s(t), delayed by the symbol time duration T.

elsewhere

TttTksth

0

0)()(


Tt

s(t)

-Tt

s(-t) h(t)=s(T-t)

tT

Signal waveform Mirror image of signal waveform

Impulse response of matched filter


• The impulse response of the filter is a delayed version of the mirror image (rotated on the t = 0 axis) of the signal waveform.• If the signal waveform is s(t), its mirror

image is s(-t), and the mirror image delayed by T seconds is s(T-t).


• The output of the matched filter z(t) can be described in the time domain as the convolution of a received input wavefrom r(t) with the impulse response of the filter.

t

dthrthtrtz0

)()()()()(


t

t

dtTsr

dtTsrtz

0

0

)()(

)(()()(

Substituting ks(T-t) with k chosen to be unity for h(t) yields.

When T = t

T

dsrtz0

)()()(


• The integration of the product of the received signal r(t) with a replica of the transmitted signal s(t) over one symbol interval is known as the correlation of r(t) with s(t).


• The mathematical operation of a matched filter (MF) is convolution; a signal is convolved with the impulse response of a filter.

• The mathematical operation of a correlator is correlation; a signal is correlated with a replica of itself.


• The term matched filter is often used synonymously with correlator.

• How is that possible when their mathematical operations are different?


s0(t) s1(t)

Tb

Tb

A A

-A


h0=s0(Tb -t) h0=s1(Tb -t)

TbTb

A A

-A


y0(t)

Tb

A2Tb

2Tb

y0(t)

Tb 2Tb

telecommunications dr. hugh blanton entc 4307/entc 5307

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