DCSP-15: Matched Filter Jianfeng Feng Department of Computer Science Warwick Univ., UK...
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DCSP-15: Matched Filter Jianfeng Feng Department of Computer Science Warwick Univ., UK [email protected] http://www.dcs.warwick.ac.uk/~feng/ dcsp.html
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Transcript of DCSP-15: Matched Filter Jianfeng Feng Department of Computer Science Warwick Univ., UK...
DCSP-15: Matched Filter
Jianfeng Feng
Department of Computer Science Warwick Univ., UK
http://www.dcs.warwick.ac.uk/~feng/dcsp.html
Filters
• Stop or allow to pass for certain signals
There are four main classes of filter in widespread use:
filters. • The name are self-explanatory• This four types are shown in the next figure
Filter Types
• Lowpass
• highpass
• bandpass
• bandstop
Filter Types: TFs
General Filter Design
• Using Chebyshev or Butterworth polynomials to approximate a square wave function in the frequency domain
• Any function can be approximated by a polynomials.
• Optimal filter design aims to minimize these ripples
• We are not discussing them here in details, but the principle is the same as our simple filter
Ideal case
Polynomial approximation
Filters
• Stop or allow to pass for certain signals
as we have talked before
• Detect certain signals such as radar etc
(pattern recognitions)
Ex: Detect it when it comes close
Matched filters: Question
Assume that we can detect
S = (s1, s2, s3, s4, s5 s6)
To find ai
y(0) = a0 x(0)+ a1 x(-1)+a2 x(- 2) + a3 x(-3) + a4 x(-4) +a5 x(-5)
so that we can find S, more precisely Y(0) = a0 s6 + a1 s5 + a2 s4 + a3 s3 + a4 s2 + a5 s1
At time -5 we have x(-5) = S1=1 (length)
detector
Y(-5) = a0 1 + a1 x(-6) + a2 x(-7) + a3 x(-8) + a4 x(-9) + a5 x(-10)
At time -4 we have x(-4) = S2= 2 (length)
detector
Y(-5) = a0 1 + a1 x(-6) + a2 x(-7) + a3 x(-8) + a4 x(-9) + a5 x(-10)
Y(-4) = a0 2 + a1 1 + a2 x(-6) + a3 x(-7) + a4 x(-8) + a5 x(-9)
At time -3 we have x(-3) = S3= 3 (length)
detector
Y(-5) = a0 1 + a1 x(-6) + a2 x(-7) + a3 x(-8) + a4 x(-9) + a5 x(-10)
Y(-4) = a0 2 + a1 1 + a2 x(-6) + a3 x(-7) + a4 x(-8) + a5 x(-9)
Y(-3) = a0 3 + a1 2 + a2 1 + a3 x(-4) + a4 x(-5) + a5 x(-6)
At time -2 we have x(-2) = S4= 4 (length)
detector
At time -1 we have x(-1) = S5= 8 (length)
detector
At time 0 we have x(0) = S6= 6 (length)
detector
Signal
• The input signal is
S = (s1, s2, s3, s4, s5, s6)
= ( 1, 2, 3, 4, 8, 6)
Matched filters: Question
To find ai
y(0) = a0 6 + a1 8 +a2 4
+a3 3 + a4 2 +a5 1
to detect that S = (s1, s2, s3, s4, s5 s6) is here
Matched filters: visulization
a0 x(0) + a1 x(-1) + a2 x(- 2) + a3 x(-3) + a4 x(-4) + a5 x(-5) = y(0)
Matched Matched !!!!
incoming signals
is matched
by the coefficients
a5
Data requirement
Without loss of generality, we can assume that
(normalized signals)
and we can assume that (normalized coefficients)
Remember that
If and only if ai = sN-I for all I which essentially says that the coefficients
a
and the incoming signal
s
are identical (matched, ai = sN-i).
Or equivalently
a0 SN + a1 SN-1 + …. + aN-1 S1 = 1
if and only if ai = sN-I for all i
a0 SN + a1 SN-1 + …. + aN-1 S1 < 1
otherwise
Ideas
• Define ai = sN-i (reversing the order)
y(n) = a0 x(n) + a1 x(n-1) + … + aN x(n-N)
So when the signal arrives we have
y(n) = SN x(n) + SN-1 x(n-1) + … + S1 x(n-N)
= SN SN + SN-1 SN-1 + … + S1 S1
Matched filter
At time -5 we have y(-5) =0.0877* 0.5262=0.0461
Y(-5) = a0 x(-5)+a1x(-6)+…
0.5262 0.0877 0.7016 0
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) x=( s1 0 0 0 0 0 )
Y(-4) = a0 x(-4)+a1x(-5)+a2x(-6)+…=0.1538
0.5262 0.1754 0.7016 0.0877 0.2631 0
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) s2 s1 0 0 0 0
At time -3 we have Y(-3) = a0 x(-3)+a1x(-4)+a2x(-5)+a3x(-6)…=.2923
0.5262 0.2631 0.7016 0.1754 0.3508 0.0877 0.2631 0
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) s3 s2 s1 0 0 0
At time -2 we have Y(-2) = a0 x(-2) +a1x(-3) +a2x(-4) +a3x(-5)+a4x(-6)…=
0.5262 0.3508 0.7016 0.2631 0.3508 0.1754 0.2631 0.3508
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) s4 s3 s2 s1 0 0
At time -1 we have y(-1) = a0 x(-1)+a1x(-2)+a2x(-3)+a3x(-4)+a4x(-5)…
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) s5 s4 s3 s2 s1 0
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) s6 s5 s4 s3 s2 s1
At time 0 we have Y(0) = a0 x(0) + a1 x(-1) + a2 x(-2) + a3 x(-3) + a4 x(-4) + a5 x(-5) + a6 x(-6) = 1
detector
S= (0.0877 0.1754 0.2631 0.3508 0.7016 0.5262)
a = (0.5262 0.7016 0.3508 0.2631 0.1754 0.0877) 0 s6 s5 s4 s3 s2 s1
At time 1 we have y(1) = a0 x(1) + a1 x(0) + a2 x(-1) + a3 x(-2) + a4 x(-3) + a5 x(-4) + a6 x(-5) = 0.7961
Matched FilterOutcome
Matlab Demo
• Matched-filter-demo
• It is a simple idea, but useful
• A bit theory first
Auto-correlation functionAuto-correlation Function
Detection theory
• A means to quantify the ability to discern between information-bearing patterns and noise
• Patterns: stimulus in human, signal in machines
• Noise: random patterns that distract from the information
Can it be useful?
Any signal here?
Filter output
Filter output s=rand(1,100);s=s/sqrt(s*s');a=s;k=100;N=100*k;sigma=0.2;x=randn(1,N)*sigma;signal=zeros(1,N);for i=3*N/k+1:N*4/k x(i)=a(i-3*N/k)+x(i); signal(i)=a(i-3*N/k);endfor i=1:N*(k-1)/k c(i)=x([i:i+N/k-1])*s';endfigure(1)plot(x);hold onplot(signal,'r');figure(2)plot(c)
Matched filter
• The result is amazing
• It depends on SNR
• We will not go into details, but you might be able to investigate it using Matlab
RGB = imread(‘bush.png');
I = rgb2gray(RGB);
J = imnoise(I,'gaussian',0,0.005);
K = wiener2(J,[5 5]);
figure, imshow(J), figure, imshow(K)
Tomorrow
DCSP-16: Wiener Filter and Kalman Filter*
Jianfeng Feng
Department of Computer Science Warwick Univ., UK
http://www.dcs.warwick.ac.uk/~feng/dcsp.html
Father of cybernetics
Norbert Wiener: Harvard awarded Wiener a PhD, when he was 17.
The RGB format stores three color values, R, G and B, for each pixel.
RGB = imread(‘bush.png');
size(RGB)
ans =
1500 1200 3
imshow(RGB)
Example
Example
SetupRecorded signal x(n,m) = s(n,m) +(n,m)
for example, an image of 1500 X1200
True Signal noise
RGB = imread(‘bush.png');
I = rgb2gray(RGB);
J = imnoise(I,'gaussian',0,0.005);
figure, imshow(I),
figure, imshow(J)
Setup
To find a constant a(m,n) such that
E ( a(m,n) x(m,n) – s(m,n) ) 2
as small as possible
y(m,n) = a(m,n) x(m,n)
Different from the filter before, a(m,n) depends on (m,n) :
Adaptive filter
Setup
2 2 2 2
2 2 2
( ) ( 2 )
2
E ax s E a x axs s
a Ex aExs Es
The quanttity above is minimized if the derivative of it with respect to a is zero
Setup
(optimal Wiener gain)
• Without loss of generality, we assume that
Es=0 E=0
for simplicity of formulation .
• S, are independent
Setup
The filtered output is given by
Note that the coefficient a depends on the position
is the local variance, E2 is the global variance (noise)
Algorithm
where summation is over an area of N and M
1 2
1 2
1 2,
2 2 21 2
,
1( , )
1( , )
n n
n n
x n nNM
x n nNM
3X 3 area
(N=3, M=3)
Algorithm
Algorithm in Matlab
• wiener2 lowpass - filters an intensity image that has been degraded by constant power additive noise.
• wiener2 uses a pixelwise adaptive Wiener method based on statistics estimated from a local neighborhood of each pixel.
RGB = imread(‘bush.png');
I = rgb2gray(RGB);
J = imnoise(I,'gaussian',0,0.005);
K = wiener2(J,[5 5]);
figure, imshow(J), figure, imshow(K)
Example
• Large M and N reduce noise, but blur the photo• small M and N will not affect noise • optimal size should be used
Further issues
N=M=50 N=M=2
•A more general version of Wiener: Kalman filter
•Kaman filter is widely used in many areas
(still a very hot research topic)
• As an introductory course, we only give you a taste of its flavour
Further issues
Kalman Filter: find out xk
• Have two variables state variable x which is not observable
xk = Fk xk-1 + Bkuk + wk
• Observable variable z which we can observe, but is contaminated by noise
zk = Hk xk +vk
where wk ~ N(0,Qk) and vk ~ N(0,Rk) are noise, Fk and Hk are constants.
Xk-1Input uk
Bk
Output xk
Kalman Filter For given xk-1|k-1 and pK-1|K-1
Predict•Predicted (a priori) state estimate xk|k-1 = Fk-1 xk-1|k-1+Bkuk
•Predicted (a priori) estimate covariance pk|k-1 = Fk-1 pk-1 |k-1 FK-1 +QK
Update•measurement residual yk = zk –Hk X k|k-1
•measurement residual covariance Sk = Hk Pk|k-1HK+Rk
•optimal Kalman gain Kk = P k|k-1 Hk (SK)-1
•updated (a posteriori) state estimate X k|k = x k|k-1 + Kk y k
•updated (a posteriori) estimate covariance pK|K = (I-K kH k) P k|k-1
Kalman Filter
• It is one of the most useful filters in the literature, as shown below
• We do not have time to go further, but do read around
clear allclose allN=1000;q=1;r=1;f=0.1;h=0.2;x(1)=0.5;xsignal(1)=0.5;xhat=zeros(1,N+1);phat=zeros(1,N+1);xhat(1)=x(1);phat(1)=x(1);p(1)=1;z(1)=0.1;y(1)=1;for i=1:Nxsignal(i+1)=f*xsignal(i)+randn(1,1)*sqrt(q)+1*cos(2*pi*0.01*i);z(i+1)=h*xsignal(i)+randn(1,1)*sqrt(r);endfor i=1:Nx(i+1)=f*xhat(i)+1*cos(2*pi*0.01*i);p(i+1)=f*phat(i)*f+q;y(i+1)=z(i+1)-h*x(i+1);s(i+1)=h*p(i+1)*h+r;k(i+1)=p(i+1)*h/s(i+1);atemp=x(i+1)+k*y(i+1);xhat(1,i+1)=atemp(1,i+1);atemp=(1-k*h)*p(i+1);phat(i+1)=atemp(1,i+1);clear atempendplot(xsignal);hold onplot(xhat,'r')plot(z,'g')plot(xhat,'r')
Applications• Attitude and Heading Reference Systems
• Autopilot
• Battery state of charge (SoC) estimation [1][2]
• Brain-computer interface
• Chaotic signals
• Tracking and Vertex Fitting of charged particles in Particle Detectors [35]
• Tracking of objects in computer vision
• Dynamic positioning
• Economics, in particular macroeconomics, time series, and econometrics
• Inertial guidance system
• Orbit Determination
• Power system state estimation
• Radar tracker
• Satellite navigation systems
• Seismology [3]
• Sensorless control of AC motor variable-frequency drives
• Simultaneous localization and mapping
• Speech enhancement
• Weather forecasting
• Navigation system
• 3D modeling
• Structural health monitoring
• Human sensorimotor processing[36]
Flight control unit of an Airbus A340
modern weather forecasting.
Final slide
• Information theory
• FT (continuous, discrete, discrete +finite)
• Filters (band related filters, matched, Wiener + Kalman)
No lectures next week, but drop me a line if you need help for your assignment
