DC Digital Communication PART6

8/7/2019 DC Digital Communication PART6

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Matched Filters

By: Andy Wang



What is a matched filter? (1/1) A matched filter is a filter used in communications to³match´ a particular transit waveform.It passes all the signal frequency components while

suppressing any frequency components where there isonly noise and allows to pass the maximum amount of signal power.The purpose of the matched filter is to maximize thesignal to noise ratio at the sampling point of a bit stream

and to minimize the probability of undetected errorsreceived from a signal.To achieve the maximum SNR, we want to allow throughall the signal frequency components, but to emphasizemore on signal frequency components that are large and

so contribute more to improving the overall SNR.



Deriving the matched filter (1/8) A basic problem that often arises in the study of communication systems isthat of detecting a pulse transmitted over a channel that is corrupted bychannel noise (i.e. AWGN)Let us consider a received model, involving a linear time-invariant (LTI) filter of impulse response h(t).The filter input x(t) consists of a pulse signal g(t) corrupted by additivechannel noise w(t) of zero mean and power spectral density No/2.The resulting output y(t) is composed of go(t) and n(t), the signal and noisecomponents of the input x(t), respectively.

)()()(

0),()()(

t nt g t y

T t t wt g t x

o!ee!

LTI filter of impulseresponse

h(t)

White noisew(t)

Signalg(t)

y(t)x(t) y(T)

Sample attime t = TLinear receiver



Deriving the matched filter (2/8)Goal of the linear receiver

To optimize the design of the filter so as to minimizethe effects of noise at the filter output and improve the

detection of the pulse signal.Signal to noise ratio is:

? A)(

|)(||)(|2

2

2

2

t n E

T g T g SNR o

n

o !!

W where |go(T)|2 is the instantaneous power of the filtered signal, g(t) atpoint t = T, and n

2 is the variance of the white gaussian zero meanfiltered noise.



Deriving the matched filter (4/8)Examine n

2:

? A ? A ? A

? A ? A

´́!!

!!!!

d f f S d f e f S

t nt n

t nt nt n

nn

f j

nn

n

n

n

10

0var 2

2

22

22

X T

X

W

W but this is zero mean soand recall that

autocorrelation is inverseFourier transform of power spectral density

autocorrelation at 0!X



Deriving the matched filter (5/8)Recall:

? A

´´

´ ´!

!!!!

!

df f H N

df e f G f H SNR

sodf f H N df f H N Rt n E

f H N

f S

o

f T j

oonn

on

2

2

2222

2

||

2

|)()(|

||2

||2

0)(

||2

)(

T

W

H (f)

filter

SX(f) SX(f)|H (f)|2 = SY(f)

In this case, SX(f) is PSD of white gaussian noise,Since Sn(f) is our output:

2)( o

X N f S !



Deriving the matched filter (6/8)To maximize, use Schwartz Inequality.

g*

g*

´́ d x x

d x x

22

2

1

||

|| Requirements: In thiscase, they must be finitesignals.

´ ´´

e d x xd x xd x x x 22

21

221 ||||||

This equality holds if 1(x) =k 2*(x).



Deriving the matched filter (7/8)We pick 1(x)=H (f) and 2(x)=G(f)e j2 fTand want to make thenumerator of SNR to be large as possible

oo

oo

fT

fT j fT

N

d f f

N

d f f

S NR

d f f N

d f f d f f

d f f N

d f e f f

d f e f d f f d f e f f

´´´

´´´

´́ ´ ´

!e

e

e

22

2

22

2

2

2222

|)(|2

2

|)(|

|)(|2

|)(||)(|

|)(|2

|)()(|

|)(||)(||)()(|T

T T

maximum SNR

according to Schwarzinequality



Deriving the matched filter (8/8)Inverse transform

Assume g(t) is real. This means g(t)=g*(t)If _ a

_ a f Gt g F

f Gt g F

!!

** )(

)(

then f f f f

!!

*

*

for real signal g(t)through duality

t T kg t h

df e f Gk

df e f Gk

df ee f Gk t h

t T f j

t T f j

f t j f T j

!!

!

!

´´́

T

T

T T

2

2

22)(

Find h(t) (inverse transform of H (f))

h(t) is the time-reversed anddelayed version of the inputsignal g(t).It is ³matched´ to the input signal.



What is a correlation detector? (1/1) A practical realization of theoptimum receiver is thecorrelation detector.The detector part of thereceiver consists of a bank of M product-integrators or correlators, with a set of orthonormal basis functions,that operates on the receivedsignal x(t) to produce theobservation vector x.The signal transmissiondecoder is modeled as amaximum-likelihood decoder that operates on theobservation vector x toproduce an estimate, .mÖ

D etector

Signal TransmissionD ecoder



The equivalence of correlation andmatched filter receivers (1/3)

We can also use a corresponding set of matched filtersto build the detector.

To demonstrate the equivalence of a correlator and amatched filter, consider a LTI filter with impulse responseh j(t).With the received signal x(t) used as the filter output, theresulting filter output, y j(t), is defined by the convolutionintegral:

X X X d t h xt y j j ! ´g

g




From the definition of thematched filter, we can

incorporate the impulseh j(t) and the input signal j(t) so that:

Then, the outputbecomes:Sampling at t = T, we get:

t T t h j j *!

´

*! X X X d t T xt y j j )(

´¡

¡

! X X X d xt y j j




So we can see that thedetector part of thereceiver may beimplemented using either matched filters or correlators. The output of each correlator isequivalent to the output of a corresponding matchedfilter when sampled at t =T.

Matched filters

Correlators



Maximum Likelihood Receiver



The transmitter sends one of M signals

s i(t), for i=1,2,«,MThe M signals forms a constellation in thesignaling space

s 1

s 2

s 3

s 4

s 5

s 6s 7



The received signal x(t) = si(t) + n(t) is

decomposed to its components in thesignal space.

X

] (t)

X

] (t)

X

] 2 (t)

xt)

xi=si1+n1

x2=si2+n2

xN=siN+nN

D ecisionRule

i(t)



? A ? A

? A ? A

¼¼½»

¬¬«!

!!

!!

0

2

0

0

exp1

2var var

N s x

N x f

N n x

sn E s x E

ij X

j j

ij jij j

j T

Since xi are independent Gaussian distributed random variabletheir joint density function is given by:

¼¼½

»

¬¬«

! §!

N

i

jii N j N s X N

s x N s x x x f j

1 0

22

021| exp|,...,, T

T

TT

An ML receiver selectss j that maximizef X|sj .D efine:

¼

¼

½

»

¬

¬«

§!

N

i

jii N

N

s x N

1 0

22

0 expT To be the likelihoodfunction

Maximizing the likelihood function is equivalent to minimizing thequantity

2

1

,minÖ §!

!! N

i jii j s j s x s xd s

j

TTT

T



The maximum Likelihood receiver picks the signal that is closed to thereceived signal in the signal space

s 1

s 2s 3

s 4

s 5

s 6s 7

x dmin



EECE 477/545

Communication Sys tem II(Digital Communication s)

Dr. X. L i

Lectu r e 6: S ection 3.1,3.2(Matched Filte r, ML D etecto r)S eptembe r 11, 2008



Review: we have knownBaseband demodulation ± General procedure, system model, signal model ± Signal space representation of signals & noise

± Matched filter as the optimal demodulation filter Optimize output SNRTo study today: ± Matched filter (MF): signal-space MF design

±O

ptimal detector Note: book¶s approach is limited to binary case.O ur description include more general cases. But the final resultsare identical in binary case.



IV. Matched filter: signal space representation1) Time-domain signal & Signal space representation

2) Use basis as matched filters

1( ) ( )

( ) ( ) ( )

N i ij j j

i

s t a t

r t s t n t

] !!

! § 1

1 1

( , , )

( , , ) ( , , )i i i N

i i i N N

a a

a a n n

!! !

s

r s n

L

L L

)(1t T

)( t T N ]

1 1 1ir a n!

N iN N r a n! ar)(t r

)}({ t j]



Why does the figure give the matched filter output?

thCo nsider th e ma tch ed fil ter ,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

(

= ( ) ( ) (

) (

) ( )

)

i

i i

j j

j j

j j

i i

j i

i j j

j

y t h t r t r h t d

r y T r d r d

s d

h t T t

h T

r a

n

n

d

X X X

X X X X

X ] X X X

]

X ] X

] X X

gg

g gg g

g gg g

! !

@ !

!

! !

!

@

´´ ´

´ ´



3) Convenience of using basis as matched filter

Matched filter & sampler becomes transparent: wheninput is signal space vector, output just equals to thatsignal space vector Matched filters are normalized, so noise componentshave variance (power) N0/2 only.

as if none

Matchedfilter

Samplerr(t) z(T)

r r

Signal spacevector

samplingvalue

Continuous time discrete time

transparent



5) Example 6.1 (Same problem as Example 5.1-5.3,

but do it in signal space): Binary PAM transmission(100 bps, A=1 volt). AWGN with PSD 10-3 W/H z.Find the sampling values after demodulation andthe SNR.

b

t

−At

s1(t)

s2(t)

A

Tb

T



3.2.1 Maximum LikelihoodD etector I. General introduction

1) Where is the detector

2) What is a detector? Detector is a decision-making device that makes decisions of the

transmitted symbols based on the received signal samples (or,map PAM samples z(T) into symbols or binary digits).

3) H ow does a detector work? Detector works based on the statistical property of the received

samples and the transmitted symbols Detector is important because of random noise Objective is to minimize decision error probability

Öi s( ) z T ! r



II. Maximum likelihood detector

± Maximum likelihood detector is a detector thatminimizes decision error probability by choosingthe symbols that most likely produce thereceived samples.

³Most likely ´: quantified by likelihood function ± Likelihood function for si: the probability of the

received samples when the transmitted symbolis si

Example, if tossing a die, what is the likelihood of

obtain ³1´, etc?In our case, if 1 is transmitted, what is the likelihoodthat 0.5 or -.4 is received (due to noise)?



III. Binary PAM transmission case1) Input to the detector (use signal space

representation)

simpli y n ot at io n0

00 0

0

2

( ) ( ) ( )

wh ere ( ), 1, 2, is de termined by symb o l

no ise: Gaussian, zer o mean, varian c e2

i

i b

i

i

z T a T n z a nT

a E i s N

n W

! p

! s !!

!

Be careful: some slight abuse of notations:s i denotes symbol, a i denotes signal space value of s in or n 0 both denote AWGN signal space value,sometimes y sometimes z is used as MF output



2) D istribution of z can be written as a pdf conditionedon the transmitted symbols s

1and s

2The meaning is: if s1 is transmitted, then the probability of having sampling value z is p(z| a1)

Obviously, if s1 is sent, then z more likely lies around a1. If s2is sent, then z more likely lies around a2.

If the actually received value z is near a1, then we wouldbetter say that s1 is sent. O therwise, we say s2 is sent.

2

20

( )

2

0

0

1( ) ,

2

+ , 1, 2,

likeli hood unct io n

i z a

i

i

p z a e

z a n i

W

W T !

! ! ) Likelihood of s 1p(z|a 1)

a10 z

pdf

a2

Likelihood of s 2p(z|a 2



3) Such idea is mathematically defined asWith the received sample z, the decision is made in favor of si such that p(z|a i) is the largestthis gives the maximum likelihood detection rule

Example 6.2. {s1=0,s2=1} produce samples {a1=0.1,a2=-0.1}, N0/2=10-3. If the received sample z=0.05. What isthe best decision?

_ a

Ö arg max ( )

i

ia

s p za!



4) Maximum likelihood decision rule can be further simplified to minimum distance decision rule

Binary minimum distance decision rule

Example 6.3. Same as Example 6.2. For z=.05, then we have(z-a1)2=(.05-.1)2=.0025, while (z-a2)2=(.05+.1)2=1.1025. Sochoose a 1 (or s1).

_ a _ a

2

2

020

1ln ( | ) ln( 2 )2

arg max ( ) is equivalen t to

( )

arg min ( )

ii

i

a

i

aii

p z a

p z

z a

z aa

W T W

!

@

2 21 2 1 1

2 21 2 2 2

i ( ) ( ) , choo se ( o r )

i ( ) ( ) , choo se ( o r )

z a z a a s

z a z a a s

® ±¯

" ±°



5) Explanation of binary minimum-distancedecision rule in signal space

For binary PAM, since

γ=0

S1S2

a1a2

1 2

1 1

2 2

Wi th dec isio n th resho ld ,2

if , choo se (o r )

if , choo se (o r )

a a

z a s

z a s

K

K

K

!

"®¯

1 2 , we h ave 0ba a E K ! ! !



IV. Extend BinaryD ecision Rule to M-ary1) M-ary PAM: there are M different symbols si,

which give M different ai, and hence M likelihoodfunctions p(z|ai).Minimum distance decision rule: find the signal point aithat is closest to the received value z.Exercise 6.4. D etermine the decision rule for M=4

PAM.



2) QPSKWe have 2-dimensional signal space, each samplingvalue z or signal pointa i is a 2-dimensional vector.Minimum distance decision rule

In signal space plane, each symbol has a decisionregion. Samples z falling in this region are decided infavor of this symbol

2 2 21 1 2 2Fin d with min || || ( ) ( )i i i i z a z a ! a z a

1s1 decision

regioin for s



V. Now we know maximum likelihood detector is equivalent to minimum distance detector ± Very simple decision rule in signal space, just

compare the distances from the receivedsample to all the signal points

± It is optimal for equally probable symbols only ± O therwise, it is only sub-optimal ± Some special cases

Binary PAM, M-ary PAM, and QPSK decision rules ± But how optimal is it? We need a metric for

evaluation



ConclusionsMore on Matched filter ± Signal space representation (We like it more!)Maximum likelihood detector ± What is it? ± Can be simplified to ³minimum distance detector´,

especially useful in signal space ± D ecision rule: binary PAM case ± D ecision rule: M-ary PAM or QPSK cases



and

maximumlikelihood

Xiaoan WuFeb 16, 2005

Thomas Bayes(1702-1761)

http://www.mrs.umn.edu/~sungurea/introstat/history/w98/Bayes.html



O utlineAn example: count # of fish in a pondBayes ¶Theorem and maximum likelihoodAnother example: Quasar selection in SDSS

Maximum likelihood estimators: consistentand unbiased?Minimum variance bound: error estimationRelation between MLE and least-squares

fittingMy research work: mass distribution of M87



An example: # of fish in a

pondQ: There are more than 10,000 fish in a pond. How to

estimate the number of a fish if you are themanager of the pond?

A: 6 steps:1) catch 1000 fish2) mark them and put them back to the pond3) catch another 1000 fish in a few days and see

how many have marks on them, say, 10.

4) The fraction of marked fish: 10/1000=1/1005) The number of fish = 1000/(1/100)=100, 000

6) 1- error = 35, 000 (Maximum Likelihood)



Bayes ¶theorem

( ) ( | ) ( ) ( | ) ( )( | ) ( )

( | )( )

: o bserva t io n

: h ypoth eses, e.g. parame ters

: pri o r in o rma t io n

( | , ) ( | ) ( | , ) ( | ) ( | , )

"Wh

at

yo

u kno

w abo

ut

ater

the

da

ta

r

r r r r r

r

P A B P A B P B P B A P A P A B P B

P B A P A

A

B

H

P B A H P B H P A B H P B H L A B H

B A

! !!

w !

I

arrive is wh

at

yo

u knew be o re [ ( | )] an d wh at th e d ata

Po sterio r

to ld

li

yo u [

ke

( | , )

liho (Ma c k od

]

ay)

"

pri o r r r P B H L A B H

w v



Maximum likelihoodB

ayes' Th

eo

em: ( | , ) ( | ) ( | , )Maximum Likeli hood : in c ase we h ave n o prio r

in o rma t io n, w h ere is th e tot al number of h ypoth es

1( | )

es.

r r

r

r P B A H P B H L A B H P B H

N N

w!

Th eo ry ExampleFis h # :

Observa t io n : 10 marke d and 990 unmarke d

Pri o r in fo rma t io n :

r B r

A

H no ne

( | ) co ns tan t

1000 1000

10 990( | , )

1000

r

r

P B H r

L A B H r

¨ ¸ ̈ ¸© ¹ © ¹ª º ª º

¨ ¸© ¹ª º



1- error = 35, 000



Another example: Quasar selection inSDSS

Richards et al., 2002, AJ, 123, 2945Goal: find quasar candidates from SD SS photometric platesRequirements: Efficiency, completeness

Prior information:

1. Quasars are point sources2. Efficiency is poorer in the galactic plane due to contamination

from stars.3. Quasars and stars occupy different regions in color-color space

Result: completeness: 90%, efficiency: 65%



Quasar selection in SDSS

( | , ) ( | ) ( | , )

: h eth er an o bje ct is quasar : all pri o r inf o rma t io n e c an h ave

:o bserved in p

hotome

tri

cpla

tes

r r r

r

H P B H L A B H

B H

A

w



Maximum likelihood estimation

1 2

1 2

1 2

1

1

( ; , , , )

: unk o wn co ns tan t parame ters, { , , , }

: N in d epen d en t o bserva t i

N-d im p. d .f of II

o ns, { , , , }

( | ) ( ; ),

ln ln (

D .

0

; )

s

k

k

N

N

ii

N

ii

f x

x x x

L f x

L f x

U U UU U U

!

!

5

5 ! 5

0 ! ! 5

x0 !x5

§

K

K

K



Maximum likelihood estimators2

2

2

( )

1 2

( )

2

2

2 2

1( )2

: independen t o bserva t io ns, { , , , }

1( | , )2

1ln ln(2 ) ln

2 2

1 ( ) 0

1 1( )

i

x

N

x

i

i

i i

f x e

X x x x

L X e

x N L N

x

x x N N

QW

Q

W

T W

Q T W QT W

W

Q Q W

Q W Q

!

!

¨ ¸! ! © ¹ª ºx ! !x

! !

§

§

§ §

K



X={10, 20, 30, 40, 50}



Mean of estimators: consistency and bias

2 22 2

Co nsis tenc y: , | | 0-

if is co nsis ten t , is als o co nsis ten t-

Unbiased: ( )

st ima to rs:1 1 1

, ( ) , ( )1

1( ) ( ) , co nsis ten t and unbiased

(

i i i

i

N t N a

t t N b

E t

t x t x t x N N N

E t E x N

E

Q W W

Q

Q Q

Q

p g p

!

! ! !

! !

§ § §

§2

2

2

2

( 1)) , co nsis ten t bu t biased

( ) , co nsis ten t and unbiased

N t

N E t

W

W

W

W

!

!



V ariance of estimators: Minimumvariance bound

2

2

2

2 2

1 ln

( )

lnequali ty ho ld s i f is a linear f un ct io n of .

1Fo r example, is a MV B es t ima to r

1( ) is a MV B es t ima to r fo r ,

i

i

V a r t L

E

Lt

t x N

t x N

U

U

¢

W

U

U

W

uxx

xx

!

!

§§

bu t not a MV B es t ima to r fo r In general, maximum likeli hood es t ima to rs are biase d and not

MV B es t ima to rs, th us err o rs s ho uld be o b taine d by b oot strap

es t ima t io n. I f , th e es t ima to rs b N

W

p eco me co nsis ten t ,

unbiase d and minimum varian c e es t ima to rs.



Relation between MLE and least-squares fitting

2

2

2

A fun ct io n ( , )

{ ( ) }

( , )1ln ln(2 ) ln

2 2

( , )maximize maximize

Th e same as leas t -squares pr oc edure.

i i i

i ii

i

i i

i

y f x

y f x y

y f x N L

y f x

U

H

UT W W

U

W

!!

¨ ¸0 ! ! © ¹ª º

¨ ¸0 p ' ! © ¹ª º

§ §

§



Test of hypotheses

Likelihood ratioKolgomorov-Smirnov test



My research work: mass distribution of M87



Summary of MLEsPros

No data binning, all observed informationused.They become unbiased minimum variance

estimators as the sample size increases.They can generate confidence bounds.Likelihood functions can be used to testhypotheses about models and parameters.

ConsMLEs can be heavily biased.Calculating MLEs is often computationally

expensive.



Conclusions

If we know prior information, wemaximize the posterior probability.

If we do not know any prior information, we maximize the

likelihood.



S lide s b y Pr o f. Br ian L. Evan s and Dr. S e r ene Bane rjeeDept . o f Elect r ical and Compute r Enginee r ing

The Unive rs ity o f Te xa s at Au s tin

EE345S Real-Time Digital S ignal Processing Lab Fall 2008

Lecture 1 3

M atched Filte r ing and Di gitalPul se Amplitude Mo dulati on (PA M)



13 - 58

O utli ne

T r ans mitti ng one bit at a time

M atched filte r ing

PA M sys tem

In te rsy mb ol in te r fer ence

C ommu n icati on pe r f or ma nceB it err o r pr o babili ty fo r binary signalsSymb o l err o r pr o babili ty fo r M -ary (mul tilevel) signals

E ye dia gr am



13 - 59

T r ans mitti ng O ne Bit

T r ans mi ss ion on commu n icati on cha nn els is anal ogO ne wa y to tr ans mit di gital i n f or mati on is called2-level di gital pul se amplitude m odulati on (PA M)

8

bt

)(1 t x

A

µ 1¶ bit

Additive NoiseChannel

input output

x (t ) y (t )

8 b

)(0 t x

- A

µ 0¶ bit

t

receive µ 0¶ bit

receive µ 1¶ bit

)(0

t y

8 b

- A

8

bt

)(1 t y

AHow doe s the r eceive r decide which bit wa s

s ent?



13 - 60

T r ans mitti ng O ne Bit

T wo-level di gital pul se amplitude m odulati on ove r cha nn el that ha s mem ory but d oes not add noise

8 h t

)(t h

18

bt

)(1 t x

A

µ 1¶ bit

8 b

)(0 t x

- A

µ 0¶ bit

Model channel

as LTI systemwith impulseresponse h (t )

Communication

Channel

input output x (t ) y (t )t

)(0 t y

- A T h

receive µ 0¶ bit

t 8 h + 8 b8 h

Assume that T h < T b

t

)(1 t y receive µ 1¶ bit

8 h + 8 b8 h

AT h



13 - 61

T r ans mitti ng T wo Bit s (I n te r fer ence )

T r ans mitti ng tw o bit s (pul ses) back-t o-back will cau se ove r lap (i n te r fer ence ) at the r eceive r

Sample y(t ) at T b, 2 T b, «, a ndth r eshold with th r eshold of ze r o

Ho w d o we p r eve n t i n te rsy mb olin te r fer ence (I SI ) at the r eceive r?

8 h t

)(t h

1


t 8

b

)(t x

A

µ 1¶ bit µ 0¶ bit

8 b

* =)(t y

- A T h

t 8 b


8 h + 8 b

Inte rsy mbol inte rf e r ence



13 - 62

P r eve n ti ng ISI at Receive rO pti on #1: wait T h second s betwee n pul ses in tr ans mitte r (called gua r d pe r iod or gua r d i n te r val )

Disa d van tages?

O pti on #2: u se cha nn el equalize r in r eceive rFIR f ilter d esigne d via training sequen c es sen t by transmi tt er D e sign go al : c asc ad e of ch annel mem o ry an d ch annel

equalizer s ho uld give all-pass f requen c y resp o nse

8 h t

)(t h

1


* =

t 8

b

)(t x

A


8 h

+ 8 b

t

)(t y

- A T h

8 b


8 h

+ 8 b

8 h



13 - 63

§!k

bk T k t g at s )()(

Digital 2-level PA M Sys tem

T r ans mitted sign al

Requi r es syn ch r on izati on of cl ock s betwee n tr ans mitte r and r eceive r

Tr an s mitte r Channel Receive r

b i

ClockT b

PAM g (t ) h (t ) c (t )1

07

a k {- A , A } s(t) x (t) y(t) y(t i )

AWGNw (t )

D ecision

Maker

Threshold P

Sample at

t=iT b

bits

ClockT b

pul s e s hape

r

matched f ilte r

¹¹ º ¸

©©ª¨!

1

00 ln4 p

p AT N

bopt

P



13 - 64

M atched Filte r

Detecti on of pul se in p r esence of additive noiseR ec eiver kn o ws w h at pulse s h ape i t is loo king fo r Ch annel mem o ry ign o red (assume d co mpensa ted by oth er

means, e.g. ch annel equalizer in re c eiver)

Additive white Gau ss ian noi s e (AWGN) with ze r o mean and

va r iance N 0 /2

g (t )

P ul s e s ignal

w (t )

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

t = T

y (T )

Matched f ilte r

)()( )(*)()(*)()(

0 t nt g t ht t ht g t y

!!

T is the sy mbol pe r iod



13 - 65

po wer average po wer o usins tan tane

)}({|)(|

SN R pulse peak iswh ere,max

2

20 !!

t n E T g L

LL

M atched Filte r De r ivati on

De sign of matched filte rMaximize signal p o wer i.e. p o wer of at t = T Minimize n o ise i.e. p o wer of

C ombi ne de sign cr ite r ia

g (t )

P ul s e s ignal

w (t )

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

t = T

y (T )

Matched f ilte r

)(*)()( t ht wt n !)(*)()(0 t ht g t g !



13 - 66

P owe r Spect r aDete r mi n istic sign al x(t )w/ F ou r ier tr ans f or m X ( f )Po wer spe ct rum is square of

abs o lu te value of magni tud e

resp o nse (p h ase is ign o red )

Mul t ipli c atio n in F o urier do main is co nvo lu t io n int ime do main

Co n uga t io n in F o urier do mainis reversal an d co n uga tio nin t ime

Aut ocorr elati on of x(t )

Maximum value a t R x(0) R x(X ) is even symme tric , i.e.

R x(X ) = R x(-X ))()()()( *2

f X f X f X f P x !!

_ a)(*)()()( ** X X ! x x F f X f X

)(*)()( * X X X ! x x R x

t

1 x (t )

0 T s

X

R x (X )

-T s T s

T s



13 - 67

P owe r Spect r a

Powe r spect r um f or sign al x(t ) isA utoco rrela t io n of ran do m signal n(t )

Fo r zer o -mean Gaussian n(t ) wi th varian c e W2

E stimate noise p owe rspect r um i n M atlab

_ a 22* )( )()()()( W X H W X X !!! f P t nt n E R nn

_ a)()( X x x R F f P !

N = 16384; % number of samplesgaussianNoise = randn(N,1);

plot( abs(fft(gaussianNoise)) .^ 2

noise floor

_ a ´g

g!! d t t nt nt nt n E Rn )()()()()( ** X X X

_ a )(*)()()()()()( *** X X X X X !!! ´£

£

nnd t t nt nt nt n E Rn



13 - 68222

0 |)()(| |)(|

´

¤

¤

! d f e f f H T g T f j T


Noise

Sign al

´´g

g

g

g

!! df f H N

df f S t n E N 202 |)(|

2 )(})({

f 2

0 N

Noise power spectrum S W (f )

)()( )(0 f G f H f G !´g

g

! df e f G f H t g t f j )()( )( 20

T

20

|)(|2)()()( f H N

f S f S f S H W N !!

g (t )

P ul s e s ignal w (t )

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

t = T

y (T )

Matched f ilte r

)(*)()(0 t ht g t g !

)(*)()( t ht t n ! AWGN Matched

filter



13 - 69

´

´g

g

g

g!df f H

N

df e f G f H T f j

20

22

|)(|2

|)()(| T

M atched Filte r De r ivati onFi nd h(t ) that maximize s pul se peak SN R L

Schwa r tz¶s inequalit yFo r ve cto rs:

Fo r f un ct io ns:

upper b o und rea ch ed iff

|||| ||||co s |||| |||||| *

baba

babaT

T !e U

Rk xk x ! )()( 21 J J

´´´g

g

g

g

g

ge

-

22

-

21

2

*2

-1 )( )( )()( d x xd x xd x x x J J J J

U

a

b



13 - 70)()( Hen c e,

inequali ty s' Sch war tz by)()(

wh enocc urswh ich , |)(|2

|)(|2

|)(|2

|)()(

|)(||)(||)()(

)()( and )()(Let

*

2*

2

0max

2

020

22

2222

2*21

t T g k t h

k e f k f H

d f f N

d f f N

d f f H N

d f e f f H |

d f f d f f H d f e f f H |

e f f f H f

opt

T f jopt

-

T f j

-

T f j

T f j

!!

!

e!

e!!

¥

¥

¥

¥

¥

¥

¥

¥

¥

¥

¥

¥

¥

¥

´

´´

´

´´´

T

T

T

T

L

L

J J




13 - 71

M atched Filte r

G ive n tr ans mitte r pul se shape g (t ) of du r ati on T ,matched filte r is give n by ho p t(t ) = k g *(T -t ) f or all k Dura tio n an d sh ape of impulse resp o nse of th e o p t imal f ilter is

d etermine d by pulse s h ape g (t )ho p t(t ) is s c ale d , t ime-reverse d , an d sh ift ed versi o n of g (t )

O ptimal filte r maximize s peak pul se SN R

Do es n ot d epen d o n pulse s h ape g (t )Pr o po r t io nal to signal energy (energy per bi t) E bI nversely pr o po r t io nal to po wer spe ct ral d ensi ty of no ise

R 2

|)(|2

|)(|2

0

2

0

2

0

max !!!! ´´g

g

g

g N

E d t t g

N

df f G

N

bL



13 - 72

t=kT T

M atched Filte r f or Recta ng ula r Pul se

M atched filte r f or cau sal r ecta ng ula r pul se ha s an impul se r espons e that i s a cau sal r ecta ng ula r pul se

C onvolve i nput with r ecta ng ula r pul se of du r ati on

T sec a nd sample r esult at T sec i s same a s toFirs t , in tegra te fo r T sec

Seco nd , sample a t symb o l peri od T sec

Th ir d , rese t in tegra t io n fo r nex t t ime peri od

In te gr ate a nd dump ci r cuit

´

S ample and dump

h (t ) = ___



13 - 73

§!k



T r ans mitted sign al

Requi r es syn ch r on izati on of cl ock s betwee n tr ans mitte r and r eceive r

Tr an s mitte r Channel Receive r

b i

ClockT b

PAM g (t ) h (t ) c (t )1

07

a k {- A , A } s(t) x (t) y(t) y(t i )

AWGNw (t )

D ecision

Maker

Threshold P

Sample at

t=iT b

bits

ClockT b

pul s e s hape

r

matched f ilte r

¹¹ º ¸

©©ª¨!

1

00 ln4 p

p AT N

bopt

P



13 - 74

)( )()()()(*)()( w

here)()()(

,i

ik k bk biii

k bk

t nT k i paiT t pat yt ct t nt nk T t pat y

!!!

§§{

Q Q Q

§!k

bk T k t at s )()( H


Wh y is g (t ) a pul se a nd not a n impul se?O th erwise, s(t ) w o uld require in f ini te ban d width

Sin c e we c ann ot sen d an signal of inf ini te ban d width , we limi t its ban d width by using a pulse s h aping f ilter

Neglecti ng noise, w ould like y(t ) = g (t ) * h(t ) * c(t )to be a pul se, i.e. y(t ) = Qp(t ) , t o elimi nate I SI

actual value(note that t i = i

T b )

intersymbol interference (I S I)

noise

p (t ) iscente r ed at

o r igin



13 - 75

) 2

(rect 21

)(

||,0

, 21

)(

W f

W f P

W f

W f W W f P

!

±°

±¯®

"!

E limi nati ng ISI i n PA M

O ne ch oice f or P ( f ) is ar ecta ng ula r pul seW is th e ban d width of th e

sys tem

I nverse F o urier trans fo rmof a re ct angular pulse isis a sin c f un ct io n

T hi s is called the Ideal Ny qui st Cha nn elIt i s not r ealizable becau se pul se shape i s notcau sal a nd i s in fin ite i n du r ati on

)2(sin c)( t W t p T !



13 - 76

±±±

°

±±±

¯

®

ee

e¹¹ º

¸©©ª

¨¹¹ º ¸

©©ª¨

e

!

W f f W

f W f f f W W f

W

f f W

f P

2||20

2|| 22

)|(|sin1

41

||0 21

)(

1

111

1

T

E limi nati ng ISI i n PA M

Anothe r ch oice f or P ( f ) is a r ai sed c osine spect r um

R oll-off fact or give s ba ndwidth i n exce ssof ba ndwidth W f or ideal Ny qui st cha nn elRai sed c osine pul seha s ze r o ISI whe nsampled c orr ectl yL et g (t ) and c(t ) be squa r e r oo t r ai sed c osines

W f 11!E

222 161

2co s

sinc

)( t W

t W

T

t

t p s E

ET ¹¹ º ¸

©©ª¨

!ideal N yquist

channel impulseresponse

dampening adjusted by rolloff factor E



13 - 77

Bit E rr or P r obabilit y f or 2-PA M

T b is bit pe r iod (bit r ate i s f b = 1/T b)

v (t ) is A WGN wi th zer o mean an d varian c e W2

L owpa ss filte r ing a G au ss ia n r andom p r ocess p r oduce s anothe r G au ss ia n r andom p r ocessMean s c ale d by H (0)Varian c e sc ale d by twic e lo wpass f ilter¶s ban d width

M atched filte r ¶s ba ndwidth i s ½ f b

h (t )7s (t )

S ample att =

nT bMatched f ilte r v (t )

r (t ) r (t ) r n §!k


)()()( t vt st r !

r (t ) = h (t ) *r (t )



13 - 78


Bina ry wavef or m ( r ecta ng ula r pul se shape ) is s Aove r nth bit pe r iod nT b < t < (n+1 )T bM atched filte r ing by in te gr ate a nd dump

Set

gainof

match

ed

f il

ter

tobe 1/T b

I ntegra te re c eive d signal o ver peri od , sc ale, sample

n

T n

nT b

T n

nT bn

v A

d t t vT

A

d t t r T

r

b

b

b

b

s!

s!

!

´

´

)(1

)(1

)1(

)1(

0-

Anr

)( nr r P n

AProbability density function (PD F)

S ee s lide 13-16



13 - 79

¹ º ¸©

ª¨ "!"!"!!

W W

Av P Av P v A P AnT s P n

nnb )()0())(|err o r (

0 W / A

W /n

v


P r obabilit y of e rr or give n that the t r ans mittedpul se ha s an amplitude of ± A

Ra ndom va r iableis G au ss ia n with

ze r o mea n andva r ia nce of one

¹ º ¸©

ª¨!!¹

º ¸©

ª¨ "!! g´ W T W W

W

AQd ve Av P AnT s P

v

A

n 21))(|err o r ( 2

2

W

nv

Q f unction on ne xt s lide

PDF f o r N (0, 1)



13 - 80

Q Fu ncti on

Q fu ncti on

C ompleme n ta ry err orfu ncti on er fc

Relati ons hip

´!g

x

y d ye xQ 2/2

21

)( T

´!g

x

t d t e xer f c22

)( T

¹ º ¸©

ª¨!

221

)(x

er f c xQ

Erf c[ x ] in Mathematica

e rf c ( x ) in Matlab



13 - 81

2

2

R h

ere,

21

21

))(|err o r ()())(|err o r ()(err o r)(

W V

V

A

Q

AQ

A

Q

AQ

AnT s P A P AnT s P A P P bb

!!

!¹ º ¸©

ª¨!¹

º ¸©

ª¨¹

º ¸©

ª¨!

!!!


P r obabilit y of e rr or give n that the t r ans mitted pul seha s an amplitude of A

Assume that 0 a nd 1 a r e equall y likel y bit s

P r obablit y of e rr ordec r ea ses exp onen tiall y with SN R

)/())(|err o r ( W AQ AnT s P b !!

VT V

T

V

21)( )(

22

ee eQ x

e xer f c x

V, po si t ivelargef o r x



13 - 82

PA M Sy mb ol E rr or P r obabilit y

Ave r age sign al p owe r

G T ([ ) is square r oot of th e

raise d co sine spe ct rum Normalization by T sym will

be rem o ved in le ct ure 15 slid esM -level PA M amplitude s

Assumi ng each symb ol is equall y likel y

sym

nT

sym

nSignal T

a E d G

T a E

P }{

|)(|21}{ 2

22

!v! ´¦

¦

[[T

sym

M

i

M

ii

symSignal T

d M id

M T l

M T P

3)1()12(

21

11 22

2

1

2

1

2 !¹¹¹

º

¸

©©©

ª

¨!¹

º ¸©

ª¨! §§

!!

2,,0,,12 ),12(M M

iid l i ......!!

2-PAM

d

-d

4-PAM

Constellations withdecision boundaries

d

-d

3 d

-3 d



13 - 83

sym N oise T

N d N P

sym

sym2

2

21

0

2/

2/

0 !! ´ [T

[

[

)()( sym Rn sym nT vanT x !


Noise p owe r and SN R

Assume ideal cha nn el,i.e. one with out I SI

C ons ide r M -2 inn er level s in cons tellati onErr o r if and o nly i f

wh erePr o babli ty of err o r is

C ons ide r tw o oute r level s in cons tellati on

d nT v sym R "|)(|

¹ º ¸©

ª¨!"W d

Qd nT v P sym R 2)|)((|

2/02 N !W

¹ º ¸©

ª¨!"W d

Qd nT v P sym R ))((

two-sided power spectral density of AWGN

channel noisefiltered by receiver

and sampled

0

22

3

)1(2 R

N d M

P

P

Nois e

Signa l v!!

Op tional



13 - 84

Alte rn ate De r ivati on of Noise P owe rX X X d nT w g nT v

r ´§

§

! )()()(

_ aÀ¿¾

°¯®

¼½»

¬«! ´

g

g

22 )()()( X X X d nT w g E nT v E r

})()()()({ 212211´ ´!g

g

g

gX X X X X X d d nT w g nT w g E T T

´ ´

g

g

g

g!

212121)}()({)()( X X X X X X d d nT wnT w E g g

T T

T d Gd g

sym

sym

r r

22/

2/

2222 )(21

)(W [[

T W X X W

[

[

!!! ´´g

g

Noi s e

powe r

T = T symFilte r ed noi s e

W 2 H (X 1 ±X 2 )



13 - 85

¹ º ¸©

ª¨!¹

º ¸©

ª¨

¹¹

º

¸©©

ª

¨ ¹ º ¸©

ª¨!

W W W d

QM

M d Q

M d

QM

M P e

)1(2

2 2

2


Assumi ng that each symb ol is equall y likel y,symb ol e rr or p r obabilit y f or M -level PA M

Symb ol e rr or p r obabilit y in te r m s of SN R

13

R sin c e R 1

3 12 22

22

1

2 !!¹¹¹ º

¸

©©©ª

¨¹ º ¸©ª̈! M d

P P

M Q

M M P

Nois e

Signa l e W

M- 2 interior points 2 e x terior points



13 - 86

V isualizi ng ISI

E ye dia gr am i s empi r ical mea su r e of sign al qualit y

In te rsy mb ol in te r fer ence (I SI ):

Rai sed c osine filte r ha s ze r oISI whe n corr ectl y sampled

See slide s 13-31 a nd 13-32

§ §¨

¨ !

¨

{¨ ! ¹

¹¹

º

¸

©©©

ª

¨!!

k nk

k

sym symk n sym symk g

k T nT g aa g T k T n g an x

)0(

)( )0()()(

§§{!{!

!enk k

sym

nk k

sym sym

g

k T g d M

g

k T nT g d M D

,, )0(

)( )1(

)0(

)( )1(



13 - 87

E ye Dia gr am f or 2-PA M

Useful f or PA M tr ans mitte r and r eceive r anal ys is and t r ouble shoo ting

T he m or e ope n the e ye, the bette r the r ecepti on

M =2

t - T sym

Sampling instant

Interval over which it can be sampled

Slopeindicates

sensitivity totiming error

D istortion over zero crossing

Margin over noise

t + T symt



13 - 88

E ye Dia gr am f or 4-PA M

3 d

d

-d

-3 d

D ue to startuptransients.Fix is todiscard first

few symbolsequal tonumber of symbol periodsin pulse shape.



Optimum Receiver Design

ENSC 428 ± Spring 2007



Digital Communication System



Wh at is a Design Problem ?

yyy

yyy



N oise Model



Eq uivalent Vector C h annel Model



cont «

Theorem of Irrelevance:

form sufficient statistics!{ }|1k r k K £ £



cont «

yyy

yyy yyy

yyy



MAP Optimum Decision Rule

yyy

yyy



ML Decision Ruleyyy



E valuation of Probabilities

yyy

yyy



cont «



cont «

yyy yyy



cont «



Optimum Receiver Structure

yyy

yyy

yyy

yyy



cont «

yyy

yyy



cont «

yyy



K ey Observations



Optimum Correlation Receiver



Simplified Receiver Cases

yyy

yyy

yyy



cont «

yyy



cont «



Matc h ed Filter Receiver



Ex ample : Receiver Design



cont «



K ey Facts



119119

The UbiquitousThe UbiquitousMATCH ED FILTERMATCH ED FILTER. . . it¶s everywhere!!. . . it¶s everywhere!!

an evening with a v ery im portant principle that¶s finding excitingnew applications in modern radar

R. T. H ill AES Societyan IEEE Lecturer D allas Chapter

25 September 2007

How do receivers work?



120120

How do receivers work? A brief review, then, of

network theory

characterizing a receiver by its³impulse response function´

representing radio signals

reminding ourselves of ³convolution´in linear systems . .

Wow! . . all in twenty minutes or so!

Well, first we¶ll need a receiver block diagram



121

³characterizing´ a receiver (or generally, a network)by its ³impulse response function´

Why?



122

D o you see how we can represent any signal s(t) as acollection of im pulses . .?

. . the im pulse is a won d er f ul function, so useful ±I¶ll make some comments about it.



123

Also, our signal (certainly in radio/radar work)is surely bipolar-cyclic . . a modulated carrier

. . that is, we can think of it, in radio work, as a sine waveof v oltage f iel d intensity an d polarity , leaving all the relationshipsto its accompanyingmagnetic f iel d and the medium at hand³to Maxwell´, so to speak!

D o you see nowhow im pulsescould still be usedto represent eventhis com ple x radiosignal?



124

+

+

Now, we see here that the output is indeed the sumof the many impulse response functions, weighted and

translated by the input signal . . that the action of thislinear network is, bysu per position , a con v olution !

About designing our receiver . .



125125

out designing our receiver . .

What impulse response function do I want??

Well, what do we want our receiver to do . . produce areplica of our signal? NO!! . . . discuss . . .



126

For maximumsensiti v ity to our own signal¶s being atthe in put , we don¶t need to see a copy of it at the output . .

. . we simply need, at the output, the greatest possibleind ication of its presence at the input.

In other words, we need to have an impulse responsefunction that, whencon v ol v e d with our signal, wouldgive the greatest possible ³signal to noise ratio´ at theoutput.

Oh, yes . . did I neglect to mention ´noiseµ?

Yes noise always present in radio wor



127

Admitting, then, that our total input is, say, ³gi(t)´,made up of our desired signal AND accompanying(but wholly independent) noise, we see that we cantreat these two inputs separately as they pass throughour receiver:

Y es, noise . . always present in radio wor

Oh, yes , , looks complicated,but nothing new here . . jremember the wordshere! Let the math ´talkµ

and our output is just the sum of the two convolutions ± one due

to the signal being present and one from the always- present noise.

So we might ask



128

. . WH AT Impulse Response Function

would produce the greatest possible indication, at theoutput, of our signal¶s arrival at the input?

Answer . .. . various methods (differential calculus; the ³Schwartz

inequality´) lead to the conclusion that maximum sensitivityis achieved when the IRF is thecom ple x conjugate of thesubject signal:

Concepts:

Signal to Noise Ratio (SNR) as a measure of sensitivityRepresenting our cyclic signal in this ³A e j ³ formand, note that some details of necessary timedisplacement notation are overlooked here

S o, we might ask . .

Thi h i



129129

T his, then, is

T he Matched Filter A receiver the impulse response function of whthe complex conjugate of a particular signal wthe greatest possi ble signal-to-noise ratio at the outpuwhen that signal is at the input and in the presindependent and completely random noise . . .receiver is the most ´sensitiveµ to the particula . . it is ´matchedµ to it.

i i lb i



130

R epresenting our signal b y a rotating vector . .

. . a great convenience

. . a vector rotating at the carrier frequency, amplitude modulatedby s(t) with possible phase modulation (a binary phase coded signal,for example) shown as (t) . . do you know, or remember, thisconvention? Sure helps in diagramming a lot of things in today¶ssignal processing.

S ome discussion . . to improve our understanding



131

. . consider a four-segment amplitude- and phase-modulated signal,and for the moment, wit h out noise . .

. . something similar to a child¶sscattering the blocks with which he hadmade a tower . . what if we wanted tosee that maximum height (rebuild thetower) again? I would re-align the blocksby multi plying each by its conjugate(remember: angles a dd when vectorsare multiplied) and ±v oila! ± the tower appears again, maximum possible height!Is ´angleµ then enough? T he conjugate involves the amplitude . .why?

In conjugating the phase modulation of our signal, why multiply



132

j g g p g y p yin the convolutionb y the amplitude as well?

Ah . . we cannot ignore the noise!

The circles here show an e x pectation of the noise contribution.O ur input signal gi(t) is our own signal abcdan d this noise . . butnotice, the noise is (of course) of the same strength regar d less of the amplitude of abcd at that time. Also note, the noise is completelyran d om. This utter ran d omness and ind e pen d ence of abcd are propertiesof ³Gaussian´ noise, ³white´ noise, as from natural thermal phenomena.Now, to convolve with, say, aunit level phase-only conjugate woulde x aggerate some of the noise effects in the angle-corrected vector addition ± unwise. The be s t thing f o r u s to do , in s uch noi s e , isindeed to ignore it! ³Match´ to our signal alone!

OK . .but just one further thought . .



133

Remember the child¶s block tower? Consider:

The child¶s playroom is subject to a mild earthquake(good grief!) as the blocks are tumbled in the waywe expected.

³White´ noise . . we¶reO K . . use the matched filter

O n the other hand, what if a ³wind´ had been blowing

distinctly from, say, thewest as the blocks were tumbledin addition to the earthquake¶s vibratory behavior?

Not random!! Biased! We¶d better com pensatefor that, assuming we can sense it. That is,we ma y wi s h to u s e a ³ whitening´ f ilte r to

r andomi ze the di s tu r bance be f o r e the matched f ilte r !T hat idea is indeed ´keyµ to much of the adaptive signal processing so strong in today·s radar literature . . . more about that to come



Illustration # 1 . . Pulse Compression



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Illustration # 1 . . Pulse Compression

First, some remarks about pulse compressionin modern radar . . fine range resolution desired,but still with long pulse for lots of energy

Achieved b y modulating the transmitted pulse,then ´compressingµ the pulse on receivewith (of course) a matched filter

T echniques ²binary phase coding widely used;typical lengths of hundreds to one ² our example here? A mere four to one!

Binary phase codingith ³t d d l li ´

X X



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X

first bit out

with a ³tapped delay line

Showing a 180o

phase shift on one tap,giving the binary code sequence ³ + - + + ³ , one of the Barker codes.

7

On receive (after down conversion), the signal is sent throughits Matched Filter . . in this case, the same circuit with the taps reversed:

7

+ + - +

X Clearly, pulse compression is a ³convolution´ process, and we see the ³time´ or ³range´ sidelobesin the output which, for all the Barker binary codes, are never more than unity value, while the narrowmain peak is full value, the number of bits in the code. In this matched situation, the output is the³autocorrelation function´, and a low sidelobe level is a very desirable attribute of a candidate code.

A peculiar thing about binary phase coding



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The idea that the ³tapped delay line, backwards´

is indeed a conjugating matched filter is not so clear in binary phase coding . . adding or subtracting 180°results in the same zero phase for that bit (all bitsthen being phase aligned).

Just to illustrate conjugation more clearly, imagine that our

four-segment sequence had been 0°, -30°, 0°, 0° ± terribleautocorrelation function, but it makes our point about phase³realignment´ byconjugation in this convolution process.

Pulse expander on transmit

Pulse compressor on receive

Compressed pulse output (hereshowing rather poor range sidelobes)

The Barker codes: sidelobe level



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Length 2 + - and + + - 6.0 dB3 + + - - 9.54 + + - + and + + + - - 12.05 + + + - + - 14.07 + + + - - + - - 16.9

11 + + + - - - + - - + - - 20.813 + + + + + - - + + - + - + - 22.3 dB

Modulo 2 adder

Seven-stage shift register

This seven-stage shift register is used to generate a 127-bit binary sequence

that can in turn be used to control a (0,180o

) phase shifter through which our IF signal is passed in the pulse modulator of our waveform generator. Suchshift-register generators produce sequences of length 2N ± 1 (before repeating;N is the number of stages). Today, computer programs generate the modulation,storing sequences known to have good autocorrelation functions, for many lengthsot h er than just 2Int- 1.

Illustration # 2 . .Antennas; the ´ A daptiveµ Array



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First, some remarks about how antennas form receivebeams, phased arrays the simplest and very pertinent illustration

Next, we·ll ob serve that compensating for the ángleof arrivalµ for an echo is indeed a form of (you guessed it) a matched filter, this one in ángle spaceµ

T hen, we·ll consider (again in angle space) that the ńoiseµ may NO T be utterly random, not ´whiteµ (statisticallyuniform) in angle . . we may need a ´whiteningµ filter before our matched filter

First, consider a few discrete elements of a phased array,



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, p y,

These simple sketcheswill remind us of howantennas, phased arraysspecifically, perform beamsteering . . a matter of compensating for the

element-to-element phasedifference resulting fromthe path-length differencesassociated with the desiredbeam-steering angle (theangle-of-arrival ³under test´,

so to speak)



T he ´ A daptiveAntennaµ . .



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Bottom line ± the coherent sidelobe cancellers (CSLC) ± the more elaborate ³adaptive phased arrays´

are forms of spatial ³whitening filters´, here to ³whiten´the heterogeneous disturbance ± noise ± in angle. Why? So thata straightforwardangle matched filter can be used most effectively.

Array signal processing . . first, spatial analysis,then compensation to ³whiten´

D iscussion

Spatial analysis analogousto ³spectral analysis´

Finding compensating weightsfor each element involvessolving as many simultaneousalgebraic equations . . inverting

the covariance matrix NO T EASYAdapted pattern will be ³inverse´ to the

angular distribution of noise, ³whitening´ it

Today¶s art state . . 16 DO F rather standard . .

Illustration # 3 . .S pace-T imeA daptive Processing,ST AP



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A daptive antenna processing is S pace-A daptive. What is meant b yS pace-T imeA daptive Processing?

A few remarks about Doppler processing in radar, itself an application of the Matched Filter . . Doppler filters are indeed filters matched to a particular Doppler shift

Many radars, air borne ones particularly, need to do Doppler processing when theb ackground (noise, continuous ground clutter) is certainly NO T spectrally uniform . . once again,we·ll need a ´whiteningµ filter

Doppler filtering



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Theory View a singleD oppler ³filter´ as a classic ³Matched Filter´,

that is, we multiply (convolve) the input signalwit h t h e conjugate o f t h e signal being soug h t .

sample #1 2 3 4

signal

x

reference

Recall, phase angles add when complex numbers (vectors) are multiplied ± that is,the signal is ³rotated back´ in phase by the amount it might have been progressingin phase . . To the extent that such a component was in the input signal will we getan output in this particular filter. We¶ve built a Matched Filter for t h at component(that f requency component) alone. HO WEVER , this is best O NLY IF backgroundnoise is utterly random inD oppler frequencies . .

=

product

The airborne radar situation for discussion



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T he air borne radar situation . . for discussion

D o you see the need for ³whitening´ the background

in BO TH the angle dimension (the broadband interferenceis purelyangle dependent) and in D oppler shift (the groundreflectivity may NO T be utterly random, uniformly distributed in angle)?

Broadband interference

(jamming) suggests needfor adaptive antenna

Terrain featurescontribute tonon-uniform spectrum

of the side-lobe coupledground clutter

An airborne radar

ST AP ² tobe adaptive inboth the antenna·s pattern (as before discussed) and also in the weights to put on each pulse return to shape theD l filt i t ( ti g f th if



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Doppler filters in spectrum (compensating for the non-uniform spectrum of theb ackground clutter)

The ³data field´available to us

To adapt to the background¶s sensed heterogeneity in both angle and spectrum,we must solve (to be ³fully adaptive´) MN simultaneous equations (size of the covariancematrix to invert: MN x MN. No wonder, then, today¶s literature is full of STAP papersaddressing ways to ³reduce the dimensionality´ of the processing, find the best thatwe can do in ³partial adaptivity´! Very exciting work!

Space

Time

Illustration # 4 . .T he Polarimetric Matched Filter



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First, a short general review of polarimetry in radar,its uses, its value

T hen, an example of the Polarimetric Whitening Filter and how a polarimetric radar image ( b yS AR )is improved just from PWF application to the area clutter

Of course, the ´whiteningµ to randomize the polarization stateof the surrounding area (local clutter in a scene) permits us then to search for targets ( building, vehicles) the´polarimetric signatureµ of which may havebeenestimated in advance.

R adar Polarimetry . . a little review



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Polarization of an Electro-Magnetic wave is taken as thespatial orientation of the E-field . . most, but certainlynot all, radars are designed to operate, for variousreasons, in either horizontal or vertical (linear)polarization, fixed by the antenna design ± that is,they are not ³polarimetric´

A ³Fully Polarimetric Radar´ (FPR) can, first, transmit onepolarization and separately measure the receivedsignal in each of two orthogonal polarizations, thendo the same, transmitting the orthogonal polarization

(e.g., transmit H , receive H and V;then transmit V, receiveH and V)

We learn a lot about a target by sensing its polarimetric scattering

D eveloped well by the meteorological radar community, someother specialty radars

P ola r imet ry u s ed f o r image enhancement

³Whit i ´ d ³M t hi ´ filt



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³Whitening and ³Matching filters

The work under D r. Les Novak (MIT/Lincoln Laboratories) in the 1990s is extremely

valuable in establishing these approaches to image enhancement by polarimetry. A number of papers in our conferences (to be cited here) and other teaching materialhe has provided me contribute to this instruction. An airborne SAR at 33 GH z,fully polarimetric, was used in many valuable experiments there.

Review . . D etection o f thing s o f inte r e s t (ta r get s) in the p r e s ence o f r etu r n not o f inte r e s t (noi s e , clutte r) r equi r e s contrast between the two in someob s e r vable dimen s ion s pace (he r e , ou r image ).

The idea of ³whitening´ and ³matching´ is universal, formsmatc h

e d f ilter theory.The whitening f ilte r: attempts to minimize the ³speckle´ of the background,

± that is, the standard deviation among the pixels of the clutter ± inimages formed by combining the complex images inHH , H V and VVusing complex weights among them, weights that minimize thecorrelation in cluttered regions among the three images. The weightsare based on knowledge of the clutter covariance matrix, a priori in theNovak work reviewed here, and involved its inversion, not difficult withorder three. With such PWF, cells that do not ³belong´ to the clutter willhave increased contrast with the background and are more easily seen.

The matched f ilte r: attempts to ma x imize the target intensity toclutter intensityin the combined image, by using weights based on knowledge (estimates)of the polarimetric covariance of target AND clutter returns.

(Polarimetry and image enhancement, cont.)



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. . from the Novak, Lincoln Laboratory work onPWF . . a dual-power-line scene, images bythe 33 GH z fully polarimetric airborne SAR.

The histograms show the increased contrast(separation of the clutter and towers compilations)

afforded by PWF processing compared to anon-polarimetric image, here theHH .

O ne can see the visible effect in the two imagesabove, HH on the left, PWF-processed at right.

All here with 1 foot x 1 foot resolution.

Well . . did we make it to this concluding slide??

The Matched Filter



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T he Matched Filter

The conjugate impulse response function ±max sensitivity to a signalin t h e presence o f w h ite noise

Normally taught in the context of just temporal signalprocessing . . functions of time, etc

Should be no less seen by students of radar as theunderlying principle to many advances, in³other´ dimension spaces: angle (antennapatterns), spectral analysis (D oppler filtering),polarimetric analysis (as in synthetic apertureradar image enhancement)

Today¶s ³adaptive´ processes are generally the MF-related³whitening´ required in non-random environments

DC Digital Communication PART6

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