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Matched Filters
By: Andy Wang
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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.
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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
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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.
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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
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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 !
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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).
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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
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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.
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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
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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
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The equivalence of correlation andmatched filter receivers (2/3)
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
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The equivalence of correlation andmatched filter receivers (3/3)
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
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Maximum Likelihood Receiver
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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
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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)
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? 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
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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
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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
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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.
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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]
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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
! !
@ !
!
! !
!
@
´´ ´
´ ´
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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
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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
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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
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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)?
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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
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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
a2
Likelihood of s 2p(z|a 2
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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!
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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
® ±¯
" ±°
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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 ! ! !
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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.
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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
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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
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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
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and
maximumlikelihood
Xiaoan WuFeb 16, 2005
Thomas Bayes(1702-1761)
http://www.mrs.umn.edu/~sungurea/introstat/history/w98/Bayes.html
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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
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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)
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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
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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
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1- error = 35, 000
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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%
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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
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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
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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
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X={10, 20, 30, 40, 50}
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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
!
!
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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.
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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
§ §
§
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Test of hypotheses
Likelihood ratioKolgomorov-Smirnov test
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My research work: mass distribution of M87
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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.
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Conclusions
If we know prior information, wemaximize the posterior probability.
If we do not know any prior information, we maximize the
likelihood.
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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)
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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
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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?
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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
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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
Assume that T h < T b
t 8
b
)(t x
A
µ 1¶ bit µ 0¶ bit
8 b
* =)(t y
- A T h
t 8 b
µ 1¶ bit µ 0¶ bit
8 h + 8 b
Inte rsy mbol inte rf e r ence
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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
Assume that T h < T b
* =
t 8
b
)(t x
A
µ 1¶ bit µ 0¶ bit
8 h
+ 8 b
t
)(t y
- A T h
8 b
µ 1¶ bit µ 0¶ bit
8 h
+ 8 b
8 h
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§!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
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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
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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 !
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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
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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
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0 |)()(| |)(|
´
¤
¤
! d f e f f H T g T f j T
M atched Filte r De r ivati on
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
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´
´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
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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
M atched Filte r De r ivati on
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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
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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 ) = ___
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§!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
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)( )()()()(*)()( 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
Digital 2-level PA M Sys tem
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
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) 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 !
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±±±
°
±±±
¯
®
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
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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
bk T k t g at s )()(
)()()( t vt st r !
r (t ) = h (t ) *r (t )
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Bit E rr or P r obabilit y f or 2-PA M
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
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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
Bit E rr or P r obabilit y f or 2-PA M
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)
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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
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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¨!
!!!
Bit E rr or P r obabilit y f or 2-PA M
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
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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
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sym N oise T
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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
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Alte rn ate De r ivati on of Noise P owe rX X X d nT w g nT v
r ´§
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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
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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
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sym symk n sym symk g
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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
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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.
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Optimum Receiver Design
ENSC 428 ± Spring 2007
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Digital Communication System
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Wh at is a Design Problem ?
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yyy
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N oise Model
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Eq uivalent Vector C h annel Model
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cont «
Theorem of Irrelevance:
form sufficient statistics!{ }|1k r k K £ £
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cont «
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yyy yyy
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MAP Optimum Decision Rule
yyy
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ML Decision Ruleyyy
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E valuation of Probabilities
yyy
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yyy yyy
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Optimum Receiver Structure
yyy
yyy
yyy
yyy
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yyy
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yyy
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K ey Observations
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Optimum Correlation Receiver
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Simplified Receiver Cases
yyy
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cont «
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Matc h ed Filter Receiver
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Ex ample : Receiver Design
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K ey Facts
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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?
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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
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³characterizing´ a receiver (or generally, a network)by its ³impulse response function´
Why?
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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.
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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?
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+
+
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 . .
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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 . . .
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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
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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
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. . 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
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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
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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
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. . 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
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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 . .
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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
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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 ´angleof arrivalµ for an echo is indeed a form of (you guessed it) a matched filter, this one in ´angle spaceµ
T hen, we·ll consider (again in angle space) that the ´noiseµ 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)
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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
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