3 matched filter

7/31/2019 3 Matched Filter

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Correlated and Uncorrelated Signals

Problem: we have two signals and . How close are they to each other?][nx ][ny

Example: in a radar (or sonar) we transmit a pulse and we expect a return

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0.8

1

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0

0 . 5

1

1 . 5

Receive

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0- 2

- 1 . 5

- 1

- .


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Example: Radar Return

Since we know what we are looking for, we keep comparing what we receivewith what we sent.

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0

0.2

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1

-0.6

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0

0.2

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0.8

1

0 . 5

1

1 . 5

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

- 1 . 5

- 1

- 0 . 5

0

Receive

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0

- 2

Similar? NO! Think so!


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Inner Product between two Signals

We need a measure of how close two signals are to each other.

Inner Product

Correlation Coefficient


4/34

Inner Product

Problem: we have two signals and . How close are they to each other?][nx ][ny

Define: Inner Product between two signals of the same length

1N

=

=

0

* ][][n

xy nynxr

0][][][1

0

21

0

*==

=

=

N

n

N

n

xx nxnxnxr

yyxxxy rrr 2

2

yyxxxy = an on y


5/34

How we measure similarity (correlation coefficient)

Assume: zero mean

xyr ||Compute:

yyxx

xyrr

Check the value:

10 xy

1xy

x,ystrongly correlatedx,yuncorrelated

0xy


6/34

Back to the Example: with no return

1.5

2

2

3

2

3

][nx ][ny ][][ nynx

-

-0.5

0

0.5

-1

0

1

-1

0

1

0 100 200 300 400 500 600 700 800 900-2

-1.5

0 100 200 300 400 500 600 700 800 900-3

-2

0 100 200 300 400 500 600 700 800 900 1000-3

-2

27.2=xyr

003.0

982

=

=yy

xx

r


7/34

Back to the Example: with return

1

1.5

2

2

3

2

2.5][nx ][ny ][][ nynx

-1

-0.5

0

0.5

-1

0

1

0.5

1

1.5

0 100 200 300 400 500 600 700 800 900-2

-1.5

0 100 200 300 400 500 600 700 800 900-3

-2

0 100 200 300 400 500 600 700 800 900 1000

-0.5

0

494=r

754

500

=

=

yy

xx

r

r

8.0= Good Correlation!


8/34

Inner Product in Matlab

Take two signals of the same length. Each one is a vector:

[ ])()2()1( Nyyyy =

Row vector

)2(

)1(

*

*

* y

y

N

Define: Inner Product between two vectors

==

=

)(*1

Ny

n

xy

x

'*yxrxy = 'y

conjugate,transpose


9/34

Example

Take two signals:

2.5 3

0

0.5

1

1.5

2x

0

1

2

0 50 100 150 200 250 300-2.5

-2

-1.5

-1

-0.5

0 50 100 150 200 250 300-3

-2

-1

Compute these:Then:

00856.07.19

==7.19'* == yxr

9.2418.218

x,y are not correlated

8.218'* == xxrxx

'* == .yy


10/34

Example

Take two signals:

Compute these:3

x9.230'* == yxrxy

'* ==01

2

xx

3.234'* == yyryy0 50 100 150 200 250 300

-3

-2

-1

yThen:

19955.09.230

==xy1

2

3

..

x,y are strongly correlated-2

-1

0

0 50 100 150 200 250 300-3


11/34

Example

Take two signals:

33

x

0

1

2

0

1

2

0 50 100 150 200 250 300-3

-2

-1

-3

-2

-1

Compute these:Then:

19955.09.230

==xy

0 50 100 150 200 250 300

'* ==..

x,y are strongly correlated

.xy

6.229'* == xxrxx

' .== yyryy


12/34

Typical Application: Radar

][ns

Send a Pulse

n

N

][ny

and receive it back with noise, distortion

n

0n

Problem: estimate the time delay , ie detect when we receive it.0n


13/34

Use Inner Product

Slide the pulse s[n] over the received signal and see whenthe inner product is maximum:

=

+=1

0

* ][][][N

ys snynr

0if,0][ nnnrys

][y

n

][s0n

N


14/34

Use Inner Product

Slide the pulse x[n] over the received signal and see whenthe inner product is maximum:

if 0nn =MAXsnynrN

ys =+=

=

1

0

* ][][][

][y

0n=

][s

N


15/34

Matched Filter

Take the expression

011...11

][][][***

0

*

nsnsNnNs

snynr nys

+++++=

+=

=

Compare this, with the output of the following FIR Filter

Then

]1[]1[]1[]1[...][]0[][ ++++= NnyNhnyhnyhnr

][ny ][nh ]1[][ += Nnrnr ys

1,...,0],1[][ == NnnNsnh


16/34

Matched Filter

This Filter is called a Matched Filter

][ny ][ nr][nh

]1[][ += Nnrnr ys

1,...,0],1[][ * == NnnNsnh

0

10 += Nnni.e.


17/34

Example

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1

1,...,0],[ = NnnsWe transmit the pulse shown below, withlength 20=N

0 2 4 6 8 10 12 14 1 6 18 2 0-1

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0

0.2ns

Max at n=119

10

121.5

][ny

Received signal:1001201190 =+= n

0

2

4

6

8

-1

-0.5

0

0.5

1

][ny ][ nr

][nh

0 20 40 60 80 100 120 140 160 180 200-6

-4

-2

0 20 40 60 80 100 120 140 160 180-2

-1.5

1,...,0],1[][ * == NnnNsnh


18/34

How do we choose a good pulse

1,...,0],[ = NnnsWe transmit the pulse and we receive(ignore the noise for the time being)

][][ 0nnAsny=

][nr][nh

]1[][ += Nnrnr ys

1N

1,...,0],1[][ * == NnnNsnh

][

][][][

0

0

0

nnrA

snnsnr

ss

n

ys

=

+=

=

The term

is called the autocorrelation of s[n]. This characterizes

=

+=1

0

* ][][][N

ss snsnr

the pulse.


19/34

Example: a square pulse

][nrss][ns

1

N

n1N0 N N n

?][][][1

0

*=+=

=

N

ss snsnr

NsssrNN

===

1

2

1

* ][][][]0[ == 00

11][]1[]1[2

0

2

0

*==+=

=

=

NssrNN

ss

See a few values:

kNskskrkNkN

ss ==+=

=

=

1

0

1

0

* 1][][][

NkN 11*

ssrk

ss ===

==0


20/34

Compute it in Matlab

][ns

118

20

n1N012

14

16

N=20; % data length

s=ones(1,N); % square pulse 46

8

rss=xcorr(s); % autocorr

n=-N+1:N-1; % indices for plot

-20 -15 -10 -5 0 5 10 15 200

2

stem(n,rss) % plot


21/34

Example: Sinusoid

15

20

25

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5

10

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0

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-10

-5

0 5 10 15 20 25 30 35 40 45 50-1

-0.8

49,...,0],[ =nns- - - - -

49,...,49],[ =nnrss


22/34

Example: Chirp

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30

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0 5 10 15 20 25 30 35 40 45 50-1

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-5

49,...,49],[ =nnrss49,...,0],[ =nns

s=chirp(0:49,0,49,0.1)


23/34

Example: Pseudo Noise

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2

2.5

40

50

-2

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-1

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1

0

10

20

0 5 10 15 20 25 30 35 40 45 50-2.5

-50 -40 -30 -20 -10 0 10 20 30 40 50-20

-10

49,...,49],[ =nnrsss=randn(1,50)

,...,, =nns


24/34

Compare them

2

2.5

0.6

0.8

1

0.8

1

ns

cos chirp pseudonoise

-1.5

-1

-0.5

0

0.5

1

.

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0

0.2

0.4

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0

0.2

0.4

0.6

50

25

30

0 5 10 15 20 25 30 35 40 45 50-2.5

-2

0 5 10 15 20 25 30 35 40 45 50-1

.

0 5 10 15 20 25 30 35 40 45 50-1

25

0

10

20

30

40

0

5

10

15

20

-10

-5

0

5

10

15

20

ss

-50 -40 -30 -20 -10 0 10 20 30 40 50-20

-10

-50 -40 -30 -20 -10 0 10 20 30 40 50-10

-5

-50 -40 -30 -20 -10 0 10 20 30 40 50-20

-15

Two best!


25/34

Detection with Noise

Now see with added noise

][][][ 0 nwnnAsny += ][]1[][ 0 nrNnnrnr ywys ++=

1,...,0],1[][ * == NnnNsnh


26/34

White Noise

A first approximation of a disturbance is by White Noise.

White noise is such that any two different samples areuncorre a e w eac o er:

2

3

nw

0

1

-3

-2

-

0 100 200 300 400 500 600 700 800 900 1000-4


27/34

White Noise

The autocorrelation of a white noise signal tends to be adelta function, ie it is always zero, apart from when n=0.

][nrss

n


28/34

White Noise and Filters

The output of a Filter

=

1N

nw

][nh=0

=

1 1 1

2121

12

][][][][

1

][

1 M N NM

nwnwhhnw

=

= = ==

1 1 1

2121

0 0 00

][][1

][][

1 2

N N M

nn

nwnwhh

=

= = =

12

12

0 0 0

1

1 2

MN

n

nwh

== 00 n


29/34

White Noise

The output of a Filter

=

N

nw

][nh=0

related to the Power of the Noise at the input as

w

n

W PnhP

=

=0

2][


30/34

Back to the Match Filter

][][][ 0 nwnnAsny +=

][nh

0 nwnnrnr ss ++=

1,...,0],1[][ * == NnnNsnh

]1[]0[]1[ 00 ++=+ NnwArNnr ss


31/34

Match Filter and SNR

At the peak:

]1[]0[]1[ 00++=+

NnrArNnr swss

=

=

=

1

0

21

0

22 |][||][|]0[N

n

N

n

ss nsnAsAr

W

n

WPnsP

=

=0

2|][|

N 1 2

SNRN

Pns

ns

SNRN

nS

peak =

=

=

12

0

][n=0


32/34

Example

Transmit a Chirp of length N=50 samples, with SNR=0dB

2 30

0

0.5

1

1.5

10

15

20

25

-2

-1.5

-1

-0.5

-15

-10

-5

0

0 50 100 150 200 250 300

Transmitted Detected withMatched Filter


33/34

Example


1

1.5

2

30

40

50

-1

-0.5

0

0.5

0

10

20

0 50 100 150 200 250 300-2

-1.5

0 200 400 600 800 1000 1200-20

-10

Transmitted Detected witha c e er


34/34

Example


2 140

160

0

0.5

1

1.5

60

80

100

120

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

-20

0

20

40

0 200 400 600 800 1000 1200 1400

-40

TransmittedDetected with

i r

3 matched filter

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