Chapter 9 Random Processesbazuinb/ECE3800SW/SW... · x1 t2 A sin w ... Shift Keyed (PSK) or Quadrature Amplitude Modulation (QAM) communication signals. Notes and figures are based

Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012.

B.J. Bazuin, Fall 2016 1 of 78 ECE 3800

Henry Stark and John W. Woods, Probability, Statistics, and Random Variables for Engineers, 4th ed.,

Pearson Education Inc., 2012. ISBN: 978-0-13-231123-6

Chapter 9 Random Processes Sections 9.1 Basic Definitions 544 9.2 Some Important Random Processes 548 9.3 Continuous-Time Linear Systems with Random Inputs 572 White Noise 577 9.4 Some Useful Classifications of Random Processes 578 Stationarity 579 9.5 Wide-Sense Stationary Processes and LSI Systems 581 Wide-Sense Stationary Case 582 Power Spectral Density 584 An Interpretation of the Power Spectral Density 586 More on White Noise 590 Stationary Processes and Differential Equations 596 9.6 Periodic and Cyclostationary Processes 600 9.7 Vector Processes and State Equations 606 State Equations 608 Summary 611 Problems 611 References 633



9.1 Basic Concepts

A random process is a collection of time functions and an associated probability description.

When a continuous or discrete or mixed process in time/space can be describe mathematically as a function containing one or more random variables.

A sinusoidal waveform with a random amplitude. A sinusoidal waveform with a random phase. A sequence of digital symbols, each taking on a random value for a defined time period

(e.g. amplitude, phase, frequency). A random walk (2-D or 3-D movement of a particle)

The entire collection of possible time functions is an ensemble, designated as tx , where one particular member of the ensemble, designated as tx , is a sample function of the ensemble. In general only one sample function of a random process can be observed!

Think of: 20,sin twAtX

where A and w are known constants.

Note that once a sample has been observed … 111 sin twAtx

the function is known for all time, t.

Note that, 2tx is a second time sample of the same random process and does not provide any “new information” about the value of the random variable.

221 sin twAtx

There are many similar ensembles in engineering, where the sample function, once known, provides a continuing solution. In many cases, an entire system design approach is based on either assuming that randomness remains or is removed once actual measurements are taken!

For example, in communications there is a significant difference between coherent (phase and frequency) demodulation and non-coherent (i.e. unknown starting phase) demodulation.

On the other hand, another measurement in a different environment might measure 21212 sin twAtx

In this “space” the random variables could take on other values within the defined ranges. Thus an entire “ensemble” of possibilities may exist based on the random variables defined in the random process.



For example, assume that there is a known AM signal transmitted:

twtAbts sin1

at an undetermined distance the signal is received as

20,sin1 twtAbty

The received signal is mixed and low pass filtered …

20,cossin1cos twtwtAbthtwtythtx

20,sin2sin5.01cos twtAbthtwtythtx

If the filter removes the 2wt term, we have

20,sin2

1cos tAbtwtythtx

Notice that based on the value of the random variable, the output can change significantly! From producing no output signal, ( ,0 ), to having the output be positive or negative ( 20 toorto ). P.S. This is not how you perform non-coherent AM demodulation.

To perform coherent AM demodulation, all I need to do is measured the value of the random variable and use it to insure that the output is a maximum (i.e. mix with mtw cos , where.

1tm

Note: the phase is a function of frequency, time, and distance from the transmitter.



From our textbook

Random Stochastic Sequence

Definition 8.1-1. Let P,, be a probability space. Let . Let ,nX be a mapping of the sample space into a space of complex-valued sequences on some index set Z. If, for each fixed integer Zn , ,nX is a random variable, then ,nX is a ransom (stochastic) sequence. The index set Z is all integers, n , padded with zeros if necessary,

Definition 9.1-1. Let P,, be a probability space. then define a mapping of X from the sample space to a space of continuous time functions. The elements in this space will be called sample functions. This mapping is called a random process if at each fixed time the mapping is a random variable, that is, ,tX for each fixed t on the real line t .

Example sets of random sequence.

Figure 8.1-1 Illustration of the concept of random sequence X(n,ζ), where the ζ domain (i.e., the sample space Ω) consists of just ten values. (Samples connected only for plot.)

Example sets of random process.

Figure 9.1-1 A random process for a continuous sample space Ω = [0,10].



Example 9.1-2

Separable random process may be constructed by combining a deterministic sequence with one or more random variables.

The classic example already shown is a sinusoid with random amplitude and phase: tfAtX 02sin,

Where the amplitude and phase are R.V. defined based on the probability space selected.

Example9.1‐3

A random process used to model a continuous sequence of random communication symbols. nTtpnAtX

n

In a communication class, Dr. Bazuin would typically use the following TntpAtX

nn , for tp non zero for Tkt 0

Here An is the amplitude and phase of a complex communication symbol and p(t) is the deterministic time function, the simplest of which is a rectangular pulse in time.

This can be used to describe a wide range of digital communication systems, including; Phase-Shift Keyed (PSK) or Quadrature Amplitude Modulation (QAM) communication signals.



TheapplicationoftheExpectedValueOperator

Moments play an important role and, for Ergodic Processes, they can be estimated from a single process in time of the infinite number that may be possible.

Therefore, tXEtX

and the correlation functions (auto- and cross-correlation) *2121, tXtXEttRXX *2121, tYtXEttRXY

and the covariance functions (auto- and cross-correlation) *221121, ttXttXEttK XXXX *221121, ttYttXEttK YXXY

with *212121 ,, ttttRttK XXXXXX

Note that the variance can be computed from the auto-covariance as tttXttXEttK XXXXX 2*,

and the “power” function can be computed from the auto-correlation

2*, tXEtXtXEttRXX For real X(t) tttXEttR XXXX 222,



Example9.1‐5Auto‐correlationofasinusoidwithrandomphase

Think of: ,sin twAtX

where A and w are known constants. And theta is a uniform pdf covering the unit circle.

The mean is computed as

twAEtXEtX sin twEAtXEtX sin

dtwAtXEtX

sin2

1

twAtXEtX cos2

002

coscos2

AtwtwAtXEtX

( What would happen if 0 instead? )

The auto-correlation is computed as

*21*2121 sinsin, twAtwAEtXtXEttRXX 2cos21cos21, 21212*2121 ttwttwEAtXtXEttRXX

2cos2

cos2

, 212

21

2

21 ttwEAttwAttRXX

212

21

2

21 cos20cos

2, ttwAttwAttRXX

( This works if 0 instead. )

Note that if A was a random variable (independent of phase) we would have …

wAERttwAEttR XXXX cos2cos2,2

21

2

21

and we would still have

002

AEtXEtX

Note: this Random Process is Wide-Sense stationary (mean and variance not a function of time)



Definition9.1‐3

All correlation and covariance functions are positive semidefinite.

All auto-correlation functions are diagonal dominate.

Using the Cauchy-Schwartz Inequality

221121 ,,, ttRttRttR XXXXXX

which for a WSS random process becomes

0XXXX RR

AdditionalPropertiesforreal,WSSrandomprocesses.

220 XXXXR

0max XXXX RR

XXXX RR

For signals that are the sum of independent random variable, the autocorrelation is the sum of the individual autocorrelation functions.

tYtXtW

YXYYXXWW RRR 2

If X is ergodic and zero mean and has no periodic component, then

0lim

XXR

InterpretationofWSSautocorrelation…

The statistical (or probabilistic) similarity of future (or past) samples of a random process to other samples of the process for an ergodic random process.

How similar is a time shifted version of a function to itself?

Nominal definition of ergodicity … the time base statistics are equivalent to the probabilistic based statistics of a stationary random sequence or process.



For the autocorrelation defined as:

2121212121 ,, xxfxxdxdxXXEttRXX

For WSS processes:

XXXX RtXtXEttR 21,

If the process is ergodic, the time average is equivalent to the probabilistic expectation, or

txtxdttxtx

T

T

TT

XX 21lim

and

XXXX R



A strange autocorrelation

Arandomprocesshasasamplefunctionoftheform

else

tAtX

,010,

where A is a random variable that is uniformly distributed from 0 to 10.

Find the autocorrelation of the process.

100,101

aforaf

Using

2121, tXtXEttRXX

1,0,, 21221 ttforAEttRXX

1,0,101, 21

10

0

221 ttfordaattRXX

1,0,30

, 2110

0

3

21 ttforattRXX

1,0,3

10030

1000, 2121 ttforttRXX

221121 1,0,1,0,0, ttorttforttRXX

Not WSS as it is a function of time!



Example: tfAtx 2sin for A a uniformly distributed random variable 2,2A

212121 2sin2sin, tfAtfAEtXtXEttRXX

2121

22121 2cos2cos2

1, ttfttfAEtXtXEttRXX

2121221 2cos2cos21, ttfttfAEttRXX

for 12 tt

212

21 2cos2cos1221, ttffttRXX

2121 2cos2cos2416, ttffttRXX

A non-stationary process! It is still a function of both time variables!

The time based formulation:

txtxdttxtx

T

T

TT

XX 21lim

T

TTXX

dttfAtfAT

2sin2sin21lim

T

TTXX

dttffT

A 22cos2cos21

21lim2

T

TTXX

dttfT

AfA 22cos21lim

22cos

2

22

fAfAXX 2cos22cos222

Acceptable, but the R.V. is still present?! To find a value not dependent upon a R.V

ffAEE XX 2cos24

162cos2

2



Example: tfAtx 2sin for a uniformly distributed random variable 2,0

212121 2sin2sin, tfAtfAEtXtXEttRXX

22cos2cos21, 2121

22121 ttfttfEAtXtXEttRXX

22cos2

2cos2

, 212

21

2

2121 ttfEAttfAtXtXEttRXX

Of note is that the phase need only be uniformly distributed over 0 to π in the previous step!

212

2121 2cos2, ttfAtXtXEttRXX

for 12 tt

fARXX 2cos22

but

fARR XXXX 2cos22

Assuming a uniformly distributed random phase “simplifies the problem” !!!

Also of note, if the amplitude is an independent random variable, then

fAERXX 2cos22


txtxdttxtx

T

T

TT

XX 21lim

T

TTXX

dttfAtfAT

2sin2sin21lim

T

TTXX

dttffT

A 222cos2cos21lim

2

2

fAXX 2cos22

This appears to be stationary but not technically ergodic … due to the R.V. in the time AC.



Example:

TttrectBtx 0 for B =+/-A with probability p and (1-p) and t0 a uniformly

distributed random variable

2,

20TTt . Assume B and t0 are independent.

TttrectB

TttrectBEtXtXEttRXX 01012121 ,

Tttrect

TttrectBEtXtXEttRXX 0101

22121 ,

As the RV are independent

Tttrect

TttrectEBEtXtXEttRXX 0201

22121 ,

Tttrect

TttrectEpApAttRXX 0201

2221 1,

2

2

002012

211,

T

TXX dtTT

ttrectT

ttrectAttR

For 21 0 tandt

2

2

002 11,0

T

TXX dtTT

trectAR

The integral can be recognized as being a triangle, extending from –T to T and zero everywhere else.

TtriARXX

2

T

TTT

A

TTT

A

T

RXX

,0

0,1

0,1,0

2

2




txtxdttxtx

T

T

TTXX 2

1lim

C

CCXX

dtT

ttrectBT

ttrectBC

00

21lim

A change in variable for the integral 0ttt . And only integrate over the finite interval T.

2

2

2 11T

TXX dtT

trectT

B

For 20T

T

BTTT

BdtT

BT

TXX

122111 22

2

2

2

For 02 T

T

BTTT

BdtT

BT

TXX

122111 22

2

2

2

And

TttriBXX

2

Not ergodic as taking the expected value of the time autocorrelation … however …

TttriBEE XX

2

TttripApAE XX

122

TttriAE XX

2

This is identical to the probabilistic autocorrelation previously computed!



Some Important Random Processes

AsynchronousBinarySignaling

The pulse values are independent, identically distributed with probability p that amplitude is a and q=1-p that amplitude is –a. The start of the “zeroth” pulse is uniformly distributed from –T/2 to T/2

22

,1 TDTforD

Dpdf

Determine the autocorrelation of the bipolar binary sequence, assuming p=0.5.

kk T

TkDtrectXtX

Note: the rect function is defined as

else

TtT

Ttrect

,022

,1

Determine the Autocorrelation 2121, tXtXEttRXX

kk

nnXX T

TkDtrectXT

TnDtrectXEttR 2121,

n kknXX T

TkDtrectXT

TnDtrectXEttR 2121,

n kkknXX T

TkDtrectXT

TnDtrectXXEttR 2121,



n kkknXX T

TkDtrectXT

TnDtrectEXXEttR 2121,

For samples more than one period apart, Ttt 21 , we must consider apapapapapapapapXXE jk 1111

222 112 ppppaXXE jk 144 22 ppaXXE jk

For p=0.5 0144 22 ppaXXE jk

For samples within one period, Ttt 21 ,

2222 1 aapapXEXXE kkk 0144 221 ppaXXE kk

For samples within one period, Ttt 21 , there are two regions to consider, the sample bit overlapping and the area of the next bit.

kXX T

TkDtrectT

TkDtrectEattR 21221,

But the overlapping area … should be triangular. Therefore

0,112

2

2

2

1

TfordtXXET

dtXXET

RT

Tkk

T

TkkXX

TfordtXXET

dtXXET

RT

Tkk

T

TkkXX

0,112

2

1

2

2

or

0,112

2

2

TfordtT

aRT

TXX

Tfordtt

aRT

TaXX

0,112

2

2



Therefore

TforT

Ta

TforT

TaRXX

0,

0,

2

2

or recognizing the structure

TTforT

aRXX

,12

This is simply a triangular function with maximum of a2, extending for a full bit period in both time directions.

For unequal bit probability

Tforppa

TTforT

ppT

ta

Ra

XX

,144

,144

22

22

As there are more of one bit or the other, there is always a positive correlation between bits (the curve is a minimum for p=0.5), that peaks to a2 at = 0.

Note that if the amplitude is a random variable, the expected value of the bits must be further evaluated. Such as,

22 kk XXE

21 kk XXE

In general, the autocorrelation of communications signal waveforms is important, particularly when we discuss the power spectral density later in the textbook.

If the signal takes on two levels a and b vs. a and –a, the result would be

bpbpapbpbpapapapXXE jk 1111 For p = 1/2

2

22

241

21

41

babbaaXXE jk



And 222 1 bpapXEXXE kkk

For p = 1/2

22

222

2221

bababaXEXXE kkk

Therefore,

Tforba

TTforT

baba

RXX

,2

,122

2

22

For a = 1, b = 0 and T=1, we have

Tfor

TTforTRXX

,41

,141

41

Figure 9.2-2 Autocorrelation function of ABS random process for a = 1, b = 0 and T = 1.



Examples of discrete waveforms used for communications, signal processing, controls, etc.

(a) Unipolar RZ & NRZ, (b) Polar RZ & NRZ , (c) Bipolar NRZ , (d) Split-phase Manchester, (e) Polar quaternary NRZ.

From Cahp. 11: A. Bruce Carlson, P.B. Crilly, Communication Systems, 5th ed., McGraw-Hill, 2010. ISBN: 978-0-07-338040-7

In general, a periodic bipolar “pulse” that is shorter in duration than the pulse period will have the autocorrelation function

wwwp

wXX ttfortt

tAR

,12

for a tw width pulse existing in a tp time period, assuming that positive and negative levels are equally likely.



Digital signal autocorrelation functions give rise to a range of Power Spectral Density results. The following shows some of the expected frequency responses for digital waveforms.



Exercise6‐3.1–CooperandMcGillem

a) An ergodic random process has an autocorrelation function of the form 1610cos164exp9 XXR

Find the mean-square value, the mean value, and the variance of the process.

The mean-square (2nd moment) is 222 41161690 XXRXE

The constant portion of the autocorrelation represents the square of the mean. Therefore 1622 XE and 4

Finally, the variance can be computed as, 2516410 2222 XXRXEXE

b) An ergodic random process has an autocorrelation function of the form

1

642

2

XXR

Find the mean-square value, the mean value, and the variance of the process.

The mean-square (2nd moment) is

222 6160 XXRXE

The constant portion of the autocorrelation represents the square of the mean. Therefore

414

164lim 2

222

t

XE and 2

Finally, the variance can be computed as, 2460 2222 XXRXEXE



PoissonCountingProcess

Applications and properties

arrival times radioactive decay’ memoryless property mean arrival rates

Complicated analysis and derivation left for reading in the textbook.

RandomTelegraphSignal

The random telegraph signal was originally defined based on a telegraph operator or someone manually sending Morse code. The signal may also represent “zero crossings” in an FM modulated signal.

The signal is a binary signal with random transitions in time.

Figure 9.2-4 Sample function of the random telegraph signal.

Let X(0) = +/-a with equal probability. Use the Poisson arrival process from Chap. 8 as the time of transition to the opposite level. The arrival time is now a R.V.

The probability of signal level correlation at two seperate4 times, assuming different symbols

aPaaPaaPaaPaaaPaaPaaaPaaPa

aaPaaaPaaaaPaaaaPa

tXtXEttRXX

||

||,,,,

,

2

2

22

2121



For P(a)=1/2

aaPaaPaaPaaPattRXX ||||21, 221

This becomes the probability of odd or even transitions after the average time interval of the mean of the Poisson arrival time. With this for “the positive time axis” it becomes

0,!

exp!

exp00

2

kodd

k

keven

k

XX kkaR

0,!

1exp0

2

k

kk

XX kaR

The summation can be determined as for tau>0

0,2exp2 aRXX

To include both positive and negative time (the property of positive and negative autocorrelation)

2exp2aRXX

Figure 9.2-5 The symmetric exponential correlation function of an RTS process (a = 2.0, λ = 0.25).



BinaryPhaseShiftKeying

Figure 9.2-6 System for PSK modulation of Bernoulli random sequence B[n].

The communications symbols represent 180 degree phae shifting of a sinusoidal waveform.

ttfts ac 2cos

where

TktTkforkta 1,

and

0,2

1,2nbif

nbifn

Finally

k

c kTktftX 2cos

k

cc kTktfkTktftX sin2sincos2cos

k

c kTktftX sin2sin

Typically T is selected so that the symbols form “complete” cosine waveforms integerTfc

nc

kcXX nTntfkTktfEttR sin2sinsin2sin, 2121

nc

kcXX nTntfkTktfEttR sin2sinsin2sin, 2121

nc

kcXX nkTntfTktfEttR sinsin2sin2sin, 2121



Notice that

nknkE sinsin

The following becomes similar to the previous “rectangular magnitude becoming a triangular autocorrelation. In addition, we typically define a filtering function so to remove frequencies at twice the frequency of interest.

TttTktfTktfttR ck

cXX 212121 ,0,2sin2sin,

This is where the text stops … using some further analysis and assumptions.

TttTkttfttfttR ck

cXX 21212121 ,0,22cos2cos21,

Setting 21 tt vary over two T. In addition, kT is an integer

TtandTTtfftR ck

cXX 222 0,22cos2cos21,

Averaging across all possible t2, (alternately, if the original equations had a random phase component …)

222cos2cos21, 22 tfEftR c

kcXX

The autocorrelation would become

TTT

triftR cXX

,2cos

21, 2



There is an alternate derivation that focuses on “one” symbol cycle for each of the t1 and t2 sequences, particularly if symbols have zero cross-correlation

nknkE sinsin

only one symbol and in fact the same symbol time shifted is of interest. If one symbol remains fixed and the other varies in time.

tf

Ttrecttf

TtrectEttR ccXX 2cos2cos,

222cos2cos

21, tff

Ttrect

TtrectEttR ccXX

The expected value of the random phase, component goes to zero and

cXX fT

trectTtrectEttR 2cos

21,

If a time average in t is performed the result becomes.

cXX fT

triR 2cos21

If the “symbols” do not have equal probability, there will be a cross-correlation component. Then, the “envelope” of the autocorrelation has a triangular sections (as above) and the rest is a “DC magnitude” that will multiple the cosine “modulated waveform” element. Such as

ccXX fbfT

triaR 2cos2cos21



Note: the following notes are from Cooper and McGillem ….

The Autocorrelation Function

The autocorrelation is defined as:

2121212121 ,, xxfxxdxdxXXEttRXX

The above function is valid for all processes, stationary and non-stationary. For WSS processes:

XXXX RtXtXEttR 21, If the process is ergodic, the time average is equivalent to the probabilistic expectation, or

txtxdttxtx

T

T

TT

XX 21lim

and XXXX R

Properties of Autocorrelation Functions 1) 220 XXERXX or 20 txXX 2) XXXX RR 3) 0XXXX RR 4) If X has a DC component, then Rxx has a constant factor.

tNXtX NNXX RXR 2

5) If X has a periodic component, then Rxx will also have a periodic component of the same period.

20,cos twAtX

wAtXtXERXX cos22

6) If X is ergodic and zero mean and has no periodic component, then 0lim

XXR

7) Autocorrelation functions can not have an arbitrary shape. One way of specifying shapes permissible is in terms of the Fourier transform of the autocorrelation function.

wallforRXX 0



The Crosscorrelation Function The crosscorrelation is defined as:

2121212121 ,, yxfyxdydxYXEttRXY

2121212121 ,, xyfxydxdyXYEttRYX

The above function is valid for all processes, jointly stationary and non-stationary. For jointly WSS processes:

XYXY RtYtXEttR 21, YXYX RtXtYEttR 21,

Note: the order of the subscripts is important for cross-correlation! If the processes are jointly ergodic, the time average is equivalent to the probabilistic expectation, or

tytxdttytx

T

T

TT

XY 21lim

txtydttxty

T

T

TT

YX 21lim

and XYXY R YXYX R

Properties of Crosscorrelation Functions 1) The properties of the zoreth lag have no particular significance and do not represent mean-square values. It is true that the “ordered” crosscorrelations must be equal at 0. .

00 YXXY RR or 00 YXXY 2) Crosscorrelation functions are not generally even functions. However, there is an antisymmetry to the ordered crosscorrelations:

YXXY RR 3) The crosscorrelation does not necessarily have its maximum at the zeroth lag. It can be shown however that 00 YYXXXY RRR As a note, the crosscorrelation may not achieve this maximum anywhere … 4) If X and Y are statistically independent, then the ordering is not important

YXtYEtXEtYtXERXY and



YXXY RYXR 5) If X is a stationary random process and is differentiable with respect to time, the crosscorrelation of the signal and it’s derivative is given by

d

dRR XXXX Similarly,

22

dRdR XXXX



Measurement of The Autocorrelation Function

We love to use time average for everything. For wide-sense stationary, ergodic random processes, time average are equivalent to statistical or probability based values.

txtxdttxtx

T

T

TT

XX 21lim

Using this fact, how can we use short-term time averages to generate auto- or cross-correlation functions?

An estimate of the autocorrelation is defined as:

T

XX dttxtxTR

0

1ˆ

Note that the time average is performed across as much of the signal that is available after the time shift by tau.

Digital Time Sample Correlation

In most practical cases, the operation is performed in terms of digital samples taken at specific time intervals,

t . For tau based on the available time step, k, with N equating to the available

time interval, we have:

kN

iXX ttktixtixtktN

tkR0

11ˆ

kN

iXXXX kixixkN

kRtkR0

11ˆˆ

In computing this autocorrelation, the initial weighting term approaches 1 when k=N. At this point the entire summation consists of one point and is therefore a poor estimate of the autocorrelation. For useful results, k



It can be shown that the estimated autocorrelation is equivalent to the actual autocorrelation; therefore, this is an unbiased estimate.

kN

iXXXX kixixkN

EkREtkRE0

11ˆˆ

tkRkNkN

tkRkN

kixixEkN

tkRE

XX

kN

iXX

kN

iXX

111

11

11ˆ

0

0

tkRtkRE XXXX ˆ As noted, the validity of each of the summed autocorrelation lags can and should be brought into question as k approaches N.

Biased Estimate of Autocorrelation

As a result, a biased estimate of the autocorrelation is commonly used. The biased estimate is defined as:

kN

iXX kixixN

kR0

11~

Here, a constant weight instead of one based on the number of elements summed is used. This estimate has the property that the estimated autocorrelation should decrease as k approaches N.

The expected value of this estimate can be shown to be

tkRN

nkRE XXXX

11~

The variance of this estimate can be shown to be (math not done at this level)

M

MkXXXX tkRN

kRVar 22~

This equation can be used to estimate the number of time samples needed for a useful estimate.



Exercise 6-6.1

Find the cross-correlation of the two functions …

tftX 2cos2 and tftY 2sin10

Using the time average functions

tytxdttytx

T

T

TT

XY 21lim

T

TTXY

dttftfT

2sin102cos221lim

f

XY dttftff

1

0

2sin2cos1120

f

XY dtftff

1

0

2sin222sin2120

ff

XY dtffdttff

1

0

1

0

2sin10222sin10

f

fXY dtfftff

f1

0

1

02sin10222cos

410

fff

fXY 2sin1022cos222cos

410

fffXY 2sin1022cos224cos4

10

fffXY 2sin1022cos22cos4

10

fXY 2sin10



Using the probabilistic functions

tytxERXY

tftfERXY 2sin102cos2

tftfERXY 2sin2cos20

fftfERXY 2sin2222sin10

2222sin102sin10 ftfEfRXY

From prior understanding of the uniform random phase ….

fRXY 2sin10

By the way, it is useful to have basic trig identities handy when dealing with this stuff …

bababa cos21cos

21sinsin

bababa cos21cos

21coscos

bababa sin21sin

21cossin

bababa sin21sin

21sincos

and aaa cossin22sin

1cos2sincos2cos 222 aaaa

as well as

aa 2cos121sin 2

aa 2cos121cos 2



Functions of a random variable and time

Example: tafntX , where a is a wide-sense stationary ergodic process with a known pdf.

XXERRttR XXXXXX 0,0, 21

daapdftXtXRttRttR XXXXXX ,, 21

Exponential

tutAtX exp where A is a uniformly distributed random variable BA ,0 .

B

XX dABtutAtutAttR

0221121

1expexp,

B

XX dAABttuttttR

0

2212121

1,maxexp,

3

,maxexp,2

212121BttuttttRXX

For 21 0 tandt

0,exp3

,02BRXX

For 21 0 tandt

0,exp3

,02

BRXX

Therefore

exp3

2BRXX



MATLAB Signal Processing Examples

Fig_6_2: Cross Correlation Rxy and Ryx

Run the various y waveforms. Chirp and Sinc are popular and interesting.

Fig_6_3: Auto Correlation random Gaussian noise

Fig_6_4: Auto Correlation smoothers random Gaussian noise

Fig_6_9v2: Sin wave correlation in noise



Section 9.4 Classifications of Random Processes

Definition9.4‐1.:LetXandYberandomprocesses.

(a) They are Uncorrelated if 21*21*2121 ,, tandtallfortttYtXEttR YXXY

(b) They are Orthogonal if 21*2121 ,0, tandtallfortYtXEttRXY

(c) They are Independent if for all positive integers n, the nth-order CDF of X and Y factors. That is

nnYnnX

nnnXY

tttyyyFtttxxxFtttyxyxyxF

,,,;,,,,,,;,,,,,,;,,,,,,

21212121

212211

Note that if two processes are uncorrelated and one of the means is zero, they are orthogonal as well!

Stationarity

A random process is stationary when its statistics do not change with the continuous time parameter.

TtTtTtxxxF

tttxxxF

nnX

nnX

,,,;,,,,,,;,,,

2121

2121

Overall, the CDF and pdf do not change with absolute time. They may have time characteristics, as long as the elements are based on time differences and not absolute time.

0,;,,;, 21212121 ttxxFttxxF XX

0,;,,;, 21212121 ttxxfttxxf XX

This implies that 0,0,, 21*2121 XXXXXX RttRtXtXEttR

Definition9.4‐3.:WideSenseStationary

A random process is wide-sense stationary (WSS) when its mean and variance statistics do not change with the continuous time parameter. We also include the autocorrelation being a function of one variable …

tofntindependedforRtXtXE XX ,,*



Power Spectral Density

Definition9.1‐1:PSD

Let Rxx(t) be an autocorrelation function for a WSS random process. The power spectral density is defined as the Fourier transform of the autocorrealtion function.

diwRRwS XXXXXX exp

The inverse exists in the form of the inverse transform

dwiwtwStR XXXX exp21

Properties:

1. Sxx(w) is purely real as Rxx(t) is conjugate symmetric

2. If X(t) is a real-valued WSS process, then Sxx(w) is an even function, as Rxx(t) is real and even.

3. Sxx(w)>= 0 for all w.

Wiener–Khinchin Theorem For WSS random processes, the autocorrelation function is time based and has a spectral decomposition given by the power spectral density.

Also see:

http://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem

Why this is very important … the Fourier Transform of a “single instantiation” of a random process may be meaningless or even impossible to generate. But if the random process can be described in terms of the autocorrelation function (all ergodic, WSS processes), then the power spectral density can be defined.

I can then know what the expected frequency spectrum output looks like and I can design a system to keep the required frequencies and filters out the unneeded frequencies (e.g. noise and interference).



Relation of Spectral Density to the Autocorrelation Function

For “the right” random processes, power spectral density is the Fourier Transform of the autocorrelation:

diwtXtXERwS XXXX exp

For an ergodic process, we can use time-based processing to arrive at an equivalent result …

txtxdttxtx

T

T

TT

XX 21lim

T

TT

XX dttxtxTtXtXE

21lim

diwdttxtx

TtXtXE

T

TT

XX exp21lim

dtdiwtxtxT

T

TT

XX

exp21lim

dtdiwttiwtxtxT

T

TT

XX

exp21lim

dtdtiwtxiwttxT

T

TT

XX

expexp21lim

dtdtiwtxiwttxT

T

TT

XX

expexp21lim

If there exists wXX

dtwXiwttxT

T

TTXX

exp

21lim

dttwitxT

wXT

TTXX

exp21lim

2wXwXwXXX



Properties of the Fourier Transform:

diwxxwX exp

For x(t) purely real

dwiwxxwX sincos

dwxidwxxwX sincos

dwxidwxwOiwEwX XX sincos

dwxwEX cos and

dwxwOX sin

Notice that:

wEdwxdwxwE XX

coscos

wOdwxdwxwO XX

sinsin

Therefore, the real part is symmetric and the imaginary part is anti-symmetric!

Note also, for real signals *wXwXconjwX

X(w) is conjugate symmetric about the zero axis.



Relatingthistoarealautocorrelationfunctionwhere XXXX RR wOiwER XXXX

dwiwRR XXXX sincos

dtwtiwttRR XXXX sincos

dtwttRidtwttRR XXXXXX sincos

wOiwER XXXX

Since Rxx is symmetric, we must have that

XXXX RR and wOiwEwOiwE XXXX

For this to be true, wOiwOi XX , which can only occur if the odd portion of the Fourier transform is zero! 0wOX .

This provides information about the power spectral density,

wERwS XXXXX

wEwS XXX

0 wS XX

The power spectral density necessarily contains no phase information!



Example9.5‐3

Find the psd of the following autocorrelation function … of the random telegraph.

0,exp forRXX

Find a good Fourier Transform Table … otherwise

dwjRwS XXXX exp

dwjwS XX expexp

0

0

expexpexpexp dwjdwjwS XX

0

0

expexp dwjdwjwS XX



0

0

expexp

wjwj

wjwjwS XX

wjwj

wjwj

wjwj

wjwjwS XX

exp0exp

0expexp

wjwjwjwj

wjwjwS XX

11

222222

ww

wS XX

For a=3

Figure 9.5-2 Plot of psd for exponential autocorrelation function.



Example9.5‐4

Find the psd of the triangle autocorrelation function … autocorrelation of rect.

TtriRXX

or TT

RXX

,1

T

TXX dwjT

wS

exp1

T

TXX dwjT

dwjT

wS0

0

exp1exp1

TTTT

XX

dwjT

dwj

dwjT

dwjwS

00

00

expexp

expexp

TT

T

TXX

wjwj

wjwj

T

wjwj

wjwj

T

wjwj

wjwjwS

02

0

2

0

0

expexp1

expexp1

expexp

22

22

1expexp1

expexp11

1expexp1

wwTwj

wjTwjT

T

wTwj

wjTwjT

wT

wjwjTwj

wjTwj

wjwS XX



222expexp121

expexp1

expexp

wTwj

wTwj

TwT

wjTwjT

wjTwjT

T

wjTwj

wjTwjwS XX

22cos212sin2sin2

wTw

TwTwTw

wTwwS XX

TwwT

wS XX cos112

2

2

2

2

2

2

2sin

2sin212

Tw

Tw

TTwjwT

wS XX

Don’t you love the math ?!

Using a table is much faster and easier ….



Deriving the Mean-Square Values from the Power Spectral Density

Using the Fourier transform relation between the Autocorrelation and PSD

diwRwS XXXX exp


The mean squared value of a random process is equal to the 0th lag of the autocorrelation

dwwSdwiwwSRXE XXXXXX 210exp

2102

dffSdwfifSRXE XXXXXX 02exp02

Therefore, to find the second moment, integrate the PSD over all frequencies.

As a note, since the PSD is real and symmetric, the integral can be performed as

0

22120 dwwSRXE XXXX

0

2 20 dffSRXE XXXX



Converting between Autocorrelation and Power Spectral Density

Using the properties of the functions we can actually different variations of Transforms!

The power spectral density as a function is always real, positive, and an even function in w/f.

You can convert between the domains using any of the following …

The Fourier Transform in w

diwRwS XXXX exp


The Fourier Transform in f

dfiRfS XXXX 2exp

dfftifStR XXXX 2exp

The 2-sided Laplace Transform (the jw axis of the s-plane)

dsRsS XXXX exp

j

jXXXX dsstsSj

tR exp21



NotesonusingtheLaplaceTransform

ECE 3100 and ECE 3710 stuff …

(1) When converting from the s-domain to the frequency domain use:

jws or jsw

(2) As an even function, the PSD may be expected to have a polynomial form as: (Hint: no odd powers of w in the numerator or denominator!)

0

22

4242

2222

20

22

4242

2222

20

bwbwbwbwawawawaw

SwS mm

mm

m

nn

nn

nXX

This can be factored and expressed as:

sTsTsdsdscscwS XX

To compute the autocorrelation function for 0 use a partial fraction expansion such that

sd

sgsdsgwS XX

and solve for 0 using the LHP poles and zeros as

0,exp21

tfordsstsdsgtR

j

jXX

for determining 0 , use the RHP expansion, replace –s with s, perform the Laplace transform and replace t with –t.

Another hint, once you have 0 , make the image and skip the math tRtR XXXX .

Final Note … for sTsTsdsdscscwS XX

If you “define” T(s) as all the LHP poles and zeros and T(-s) as all the RHP poles and zeros, then T(s) will represent a (1) causal, (2) minimum phase, and (3) stable filter (if there are no poles on the jw axis).

You give me a power spectral density and I can design a filter that passes “signal energy” and filters out as much of the rest as possible!



Example:InverseLaplaceTransform.

222

22

22

1

2

wA

w

AwS

X

XXX

Substitute s for w

ssA

sAsS XX

2

22

2 22

Partial fraction expansion

ss

Ass

sksks

ks

ksS XX

21010 2

20210

1010

222

0

AkAkk

kkskk

sA

sAsS XX

22

Taking the LHP Laplace Transform

Taking the RHP with –s and then –t.

0expexpexp222

2

tfortAtAtA

sAL

Combining we have

tARXX exp2

0exp2

tfortA

sAL



7-6.3 A stationary random process has a spectral density of.

else

wwS XX ,0

2010,5

(a) Find the mean-square value of the process.

02

22

10 dwwSdwwSR XXXXXX

20

10

10

20

20

10

52

1252

152

10 dwdwdwRXX

501020

210

2100 20

10

wRXX

(b) Find the auto-correlation function the process.

dwtwjwStR XXXX exp21

10

20

20

10

expexp2

5 dwtwjdwtwjtRXX

10

20

20

10

expexp2

5tj

twjtj

twjtRXX

tj

tjtj

tjtj

tjtj

tjtRXX20exp10exp10exp20exp

25

tj

tjtj

tjtj

tjtj

tjtRXX10exp10exp20exp20exp

25



ttttj

tjtj

tjtRXX

10sin20sin510sin220sin22

5

ttt

ttt

tRXX

15cos5sin10

21020cos

21020sin25

tttt

ttRXX

15cos5sinc5015cos5

5sin50

(c) Find the value of the auto-correlation function at t=0..

015cos05sinc50015cos05

05sin500

XX

R

115011500 XX

R

500 XXR

It must produce the same result!



White Noise

Noise is inherently defined as a random process. You may be familiar with “thermal” noise, based on the energy of an atom and the mean-free path that it can travel.

As a random process, whenever “white noise” is measured, the values are uncorrelated with each other, not matter how close together the samples are taken in time.

Further, we envision “white noise” as containing all spectral content, with no explicit peaks or valleys in the power spectral density.

As a result, we define “White Noise” as tSRXX 0

20

0N

SwS XX

This is an approximation or simplification because the area of the power spectral density is infinite!

Nominally, noise is defined within a bandwidth to describe the power. For example,

Thermal noise at the input of a receiver is defined in terms of kT, Boltzmann’s constant times absolute temperature, in terms of Watts/Hz. Thus there is kT Watts of noise power in every Hz of bandwidth. For communications, this is equivalent to –174 dBm/Hz or –204 dBW/Hz.

For typical applications, we are interested in Band-Limited White Noise where

fW

WfN

SwS XX

020

0

The equivalent noise power is then:

WNNWSWdwSRXEW

WXX

00

002

2220

For communications, we use kTB where W=B and N0=kT.

How much noise power, in dBm, would I say that there is in a 1 MHz bandwidth?

dBmBdBkTdBkTBdB 11460174



Receiver Sensitivity

What does it mean when you buy a radio receiver?

For a great receiver (spectrum analyzer grade), assume a 200 kHz FM radio bandwidth.

Noise Power kT -174. dBm/Hz

Equivalent Noise Bandwidth B 53. dB Hz

Receiver Noise Figure NF 10. dB

Signal Detection Threshold D 8. dB

Minimum Detectable Signal MDS -103. dBm

FM radio stations can transmit up to 1 Megawatt +90 dBm

Why doesn’t your receiver get blasted? Path loss, distance, higher noise figure, receiving antenna inefficiency, etc.

https://en.wikipedia.org/wiki/Path_loss

https://en.wikipedia.org/wiki/Friis_transmission_equation

But notice that your commercial; receiver is in microvolts, where 2.0 uV is very good. Power into 50 ohms is V^2/R or 8e-14 W = -130.97 dBW -101 dBm.

-103 dBm 1.6 uV



FMRadioDesignDiagram–Somethingyoumayencounterinthefuture

Assume input to be digitized by a 12-bit ADC with 60 dB SNR



Band Limited White Noise

fW

WfNSwS XX

02

00

The equivalent noise power is then:

002 20 SWdwSRXEW

WXX

Butwhatabouttheautocorrelation?

W

WXX dfftiStR 2exp0

tiWti

tiWtiS

tiftiStR

W

WXX

22exp

22exp

22exp

00

ti

WtiiStRXX

22sin2

0

For xtxtxt

sinc

WtSWtRXX 2sinc2 0

Using the concept of correlation, for what values will the autocorrelation be zero? (At these delays in time, sampled data would be uncorrelated with previous samples!)

,2,12

2

kforWkt

kWt

Sampling at 1/2W seems to be a good idea, but isn’t that the Nyquist rate!!

Also note, noise passed through a filter becomes band-limited, and the narrower the filter the smaller the noise power … but the wider is the sinc autocorrelation function.



Noise and Filtered Noise Matlab Simulation

Based on Cooper and McGillem HW Problem 6-4.6.

x=randn(N,1); % zero mean, unit power random signal [b,a] = butter(4,20/500); y=filter(b,a,x); % applying a digital filter y=y/std(y); % normalizing the output power Rxx=xcorr(x)/(N+1); Ryy=xcorr(y)/(N+1); DFTx = fftshift(fft(x))/N; DFTy = fftshift(fft(y))/N; DFTRxx = fftshift(fft(Rxx,2*N))/N; DFTRyy = fftshift(fft(Ryy,2*N))/N;



Pink_Noise … If we call constant at all frequencies white noise, then noise in a limited low frequency band is sometimes called pink noise.

OK. I’m getting ahead …. we just did this.



The Cross-Spectral Density

Why not form the power spectral response of the cross-correlation function?

The Fourier Transform in w

diwRwS XYXY exp and

diwRwS YXYX exp

dwiwtwStR XYXY exp21

and

dwiwtwStR YXYX exp21

Properties of the functions

wSconjwS YXXY

Since the cross-correlation is real, the real portion of the spectrum is even the imaginary portion of the spectrum is odd

There are no other important (assumed) properties to describe

Note: the trick using the Laplace transform to form the positive and negative portions of the “time-based” cross-correlation is required to determine the correct “inverse transform” of the “Cross” Power Spectral Density.

OK … Time to talk about linear transfer functions … filters!.



Section 9.3 Continuous-Time Linear Systems with Random Inputs

Linear system requirements:

Definition 9.3-1 Let x1(t) and x2(t) be two deterministic time functions and let a1 and a2 be two scalar constants. Let the linear system be described by the operator equation

txLty then the system is linear if “linear super-position holds”

txLatxLatxatxaL 22112211 for all admissible functions x1 and x2 and all scalars a1 and a2.

For x(t), a random process, y(t) will also be a random process.

Linear transformation of signals: convolution in the time domain txthty

th ty tx

Linear transformation of signals: multiplication in the Laplace domain

sXsHsY

sX sH sY

The convolution Integrals (applying a causal filter)

0

dhtxty

or

t

dxthty

Where for physical realize-ability, causality, and stability constraints we require

00 tforth and

dtth



Example: Applying a linear filter to a random process 03exp5 tfortth

tMtX 2cos4

where M and are independent random variables, uniformly distributed [0,2].

We can perform the filter function since an explicit formula for the random process is known.

t

dxthty

t

dMtty 2cos43exp5

tt

dtdtMty 2cos3exp203exp5

t

t

diiiit

tMty

2exp2exp3exp10

33exp5

t

iiit

iiitMty

232exp3exp

232exp3exp10

35

23

2exp23

2exp103

5i

itii

itiMty

49

2exp232exp23103

5 itiiitiiMty

ttMty 2sin22cos31320

35

Linear filtering will change the magnitude and phase of sinusoidal signals (DC too!).

tMtX 2cos4

69.33,2cos4135

35

tMty



Expectedvalueoperatorwithlinearsystems

For a causal linear system we would have

0

dhtxty

and taking the expected value

0

dhtxEtyE

0

dhtxEtyE

0

dhttyE

For x(t) WSS

00

dhdhtyE

Notice the condition for a physically realizable system!

The coherent gain of a filter is defined as:

00

Hdtthhgain

Therefore, 0HXEhXEtYE gain

Note that:

dttfithfH 2exp

For a causal filter

0

2exp dttfithfH

At f=0

0

0 dtthH

And 0HtyE



Whataboutacross‐correlation?(Convertinganauto‐correlationtocross‐correlation)

For a linear system we would have

dhtxty

And performing a cross-correlation (assuming real R.V. and processing)

dhtxtxEtytxE 2121

dhtxtxEtytxE 2121

dhtxtxEtytxE 2121

dhttRtytxE XX 2121 ,

For x(t) WSS

dhRRtytxE XXXY

hRRtytxE XXXY

What about the other way … YX instead of XY

And performing a cross-correlation (assuming real R.V. and processing)

2121 txdhtxEtxtyE

dhtxtxEtxtyE 2121

dhtxtxEtxtyE 2121

dhttRtxtyE XX 2121 ,

For x(t) WSS … see the next page



For x(t) WSS

dhttRRtxtyE XXYX

dhRRtxtyE XXYX

Perform a change of variable for lamba to “-kappa” (assuming h(t) is real, see text for complex0

dhRRtxtyE XXYX

Therefore

dhRRtxtyE XXYX

hRRtxtyE XXYX

Whatabouttheauto‐correlationofy(t)?

And performing an auto-correlation (assuming real R.V. and processing)

222211112121 , dhtxdhtxEttRtytyE YY

112222112121 , dhdhtxtxEttRtytyE YY

112222112121 , dhdhtxtxEttRtytyE YY

112222112121 ,, dhdhttRttRtytyE XXYY

For x(t) WSS

122112 ddhhRRtytyE XXYY

112221 dhdhRRtytyE XXYY



The output autocorrelation can also be defined in terms of the cross-correlation as

111 dhRRtytyE XYYY

hRRtytyE XYYY

The cross-correlation can be used to determine the output auto-correlation!

Continue in this concept, the cross correlation is also a convolution. Therefore,

hhRRtytyE XXYY

If h(t) is complex, the term in h(-t) must be a conjugate.

TheMeanSquareValueataSystemOutput

Based on the output autocorrelation formula

1221122 0 ddhhRRtyE XXYY

211122

2 0 ddhRhRtyE XXYY

dhRhRtyE XXYY 02

Based on the input to output cross-correlation formula

dhRRtytyE XYYY



Example:WhiteNoiseInputstoacausalfilter

Let tNtRXX 20

0122

0211

2 0 ddhRhRtYE XXYY

0122

021

01

2

20 ddhNhRtYE YY

0

11102

20 dhhNRtYE YY

0

12

102

20 dhNRtYE YY

For a white noise process, the mean squared (or 2nd moment) is proportional to the filter power.



Example:RCfilter

The RC low-pass filter

CRs

CRCRs

CsR

CssH

1

1

11

1

1

Inverse Laplace Transform

tuCRt

CRth

exp1

Coherent Gain of the RC Filter

00

Hdtthhgain

0

exp1 dtCRt

CRhgain

CR

CRt

CRCR

CRt

CRhgain

1

exp1

1

exp1 0

10expexp1

CRCR

hgain

If driven by a white noise process, what is the output power?

0

202

2 dhNtYE

0

202 exp1

2 d

CRCRNtYE



0

202 2exp1

2 d

CRCRNtYE

CR

CRCR

NtYE

2

2exp1

20

202

CR

NCR

NtYE

411

21

2 002

ComparingNoisePowerinthefilterbandwidth

Power in band-limited noise

B

B

W

WW df

NdwNNE 10102 1

21

21

2

BNWNWNNE W 0002 222

2 where W is in rad/sec and B in Hz

The noise power in an RC RC

NYE RC 41

02

For an equivalent band-limited noise process to have the same power (assume a brick wall filter)

2002 41

2 RCWYE

RCN

WNNE

RCNWN

41

2 00

Therefore RC

W2

or RC

BW4

12

where B is in Hz

Note that the nominal -3dB band (½ power) of an RC network is

RCW dB

13 or RC

B dB 2

13



Comparing these two, the equivalent noise bandwidth is greater than the –3dB bandwidth by

dBWW 32

or dBBB 342

Note: B in Hz and W in rad/sec.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.2

0

0.2

0.4

0.6

0.8

1



The power spectral density output of linear systems

The first cross-spectral density

hRR XXXY

diwRwS XYXY exp

diwhRwS XXXY exp

Using convolution identities of the Fourier Transform (if you want the proof it isn’t bad, just tedious)

wHwSwS XXXY

The second cross-spectral density

hRR XXYX

diwRwS YXYX exp

diwhRwS XXYX exp*

Using convolution identities of the Fourier Transform (if you want the proof it isn’t bad, just tedious)

*wHwSwS XXYX



The output power spectral density becomes

hhRR XXYY

diwRwS YYYY exp

diwhhRwS XXYY exp

Using convolution identities of the Fourier Transform

*wHwHwSwS XXYY

2wHwSwS XXYY

This is a very significant result that provides a similar advantage for the power spectral density computation as the Fourier transform does for the convolution.

This leads to the following table.



AdditionalTopics

System analysis with a noise input …

tx tn

th ty tr

Where the signal of interest is x(t), n(t) is a noise or interfering process. The signal plus noise is r(t) and the received system output is y(t) which has been filtered.

We have tntxtr

Assuming WSS with x and n independent and n zero mean

tntxtntxEtrtrERRR

tntntxtntntxtxtxERRR

NNXXRR RtxtnEtntxERR

NNNXXXRR RRR 2

NNXXRR RRR

And then

* hhRRR NNXXYY

hhRhhRR NNXXYY



Signal‐to‐Noise‐RatioSNR(alwaysdoneforpowers)

The signal-to-noise ratio is the power ratio of the signal power to the noise power.

The input SNR is defined as

00

2

2

NN

XX

Noise

Signal

RR

tNEtXE

PP

The output SNR is defined as

0

02

2

hhRhhR

thtNEthtXE

PP

NN

XX

Noise

Signal

For a white noise process and assuming the “filter” does not change the input signal (unity gain), but strictly reduces the noise power by the equivalent noise bandwidth of the filter.

We have

dhN

dwwHwSR XXYY202

2210

With appropriate filtering with unity gain where the signal exists and bandwidth reduction for the noise

EQXXYY BNdwwSR

02

10

or EQXXYY BNRR 000

The output SNR is defined as EQ

XX

Noise

Signal

BNR

PP

0

0

The narrower the filter applied prior to signal processing, the greater the SNR of the signal! Therefore, always apply an analog filter prior to processing the signal of interest!

From the definition of band-limited noise power, the equation for the equivalent noise bandwidth (performed as a previous example)

12

10

12

12

20 dhNdhRthtNE NN

dffHdtthBEQ22

21

21

Under the unity gain condition

dtth1



Otherwise, the equivalent noise bandwidth can be defined as

dffHfH

BEQ2

2max12

For a real, low pass filter this simplifies to

dffHH

BEQ2

2012

Using Parseval’s Theorem

dffHdwwHdtth 22221

2

2

2

2

02

dtth

dtth

H

dtthBEQ



ExamplesofLinearSystemFrequency‐DomainAnalysis

Noise in a linear feedback system loop.

sX sY

1

1 ssA

sN

Linear superposition of X to Y and N to Y.

sNsYsXssAsY

1

sNsXssA

ssAsY

11

1

sNsXssA

ssAsssY

11

2

sNAss

sssXAss

AsY

2

2

2

There are effectively two filters, one applied to X and a second apply to N.

Ass

AsH X 2 and Ass

sssH N

22

sNsHsXsHsY NX

Generic definition of output Power Spectral Density:

wSwHwSwHwS NNNXXXYY 22



Change in the input to output signal to noise ratio.

dwwS

dwwSSNR

NN

XX

In

dwwHN

wSH

dwwSwH

dwwSwHSNR

N

XXX

NNN

XXX

Out20

2

2

2

2

0

EQ

XX

X

N

XX

Out BN

wS

dwH

wHN

wSSNR

0

2

20

02

Where

dwH

wHB

X

NEQ 2

2

0

If the noise is added at the x signal input, the expected definition of noise equivalent bandwidth results.



Systems that Maximize Signal-to-Noise Ratio

SNR is defined as EQNoise

Sign

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Chapter 9 Random Processesbazuinb/ECE3800SW/SW... · x1 t2 A sin w ... Shift Keyed (PSK) or Quadrature Amplitude Modulation (QAM) communication signals. Notes and figures are based