DCSP-17: Matched Filter Jianfeng Feng Department of Computer Science Warwick Univ., UK

Filters Stop or allow to pass for certain signals as we have talked before Detect certain signals such as radar etc

Matched filters 0,0,0.0,0,0, 0,0,0,1,1,-1,1,-1, 0,0,0,0,0, 0,0,0,0,0, 0 At time 1

Matched filters 0,0,0.0,0,0, 0,0,0,1,1,-1,1,-1, 0,0,0,0,0, 0,0,0,0,0, 0 At time 2

Matched filters 0,0,0.0,0,0, 0,0,0,1,1,-1,1,-1, 0,0,0,0,0, 0,0,0,0,0, 0 At time 3

Matched filters X=( 0,0,0.0,0,0, 0,0,0,1,1,-1,1,-1, 0,0,0,0,0, 0,0,0,0,0, 0 …) To develop a filter to detect the arriving of the signal (1,1,-1,1,-1).

y(n)= a(0) x(n)+ a(1) x(n-1)+…+ a(N) x(n-N) find a = ( a(0),a(1),a(2),a(3),a(4) ) ???? X=( 0,0,0.0, 0, 0, 0, 0, 0, 1, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0,0,0, 0 …) at time zero, a tank or a flight is appeared and detected by a radar.

y(-9) = a(0) x(-9)+ a(1) x(-10)+ a(2) x(-11) + a(3) x(-12) +a(4) x(-13) = 0 y(-8) = a(0) x(-8)+ a(1) x(-9) + a(2) x(-10) +a(3) x(-11) +a(4) x(-12) = 0 … y(-4) = a(0) x(-4)+ a(1) x(-5) + a(2) x(-6) +a(3) x(-7) +a(4) x(-8) = a(0) y(-3) = a(0) x(-3)+ a(1) x(-4) + a(2) x(-5) +a(3) x(- 6) +a(4) x(-7) = a(0)+a(1) y(-2) = a(0) x(-2)+ a(1) x(-3) + a(2) x(-4) +a(3) x(-5) +a(4) x(-6) = -a(0)+a(1)+a(2) y(-1) = a(0) x(-1)+ a(1) x(-2) + a(2) x(-3) +a(3) x(-4) +a(4) x(-5) = a(0)-a(1)+a(2)+a(3) y(0) = a(0) x(0)+ a(1) x(-1) + a(2) x(-2) +a(3) x(-3) +a(4) x(-4) = -a(0)+a(1)-a(2)+a(3)+a(4) y(1 ) = a(0) x(1)+ a(1) x(0) + a(2) x(-1) +a(3) x(-2) +a(4) x(-3) = -a(1)+a(2)-a(3)+a(4) …..

We have The equality is true if and only if a (i) = x (-i)

Matched Filter A filter is called a matched filter for sequence s if Advantage: easy to implement and efficient Disadvantage: we know the exact signal we want to detect before hand. Also known as Linear correlation detector

time

Question: Could you develop a matched filter to detect it? y (0) = a(0) x(0) + a(1) x(-1) + a(2) x(-2) + a(3) x(-3) +a(4) x(-4) + a(5)x(-5) = a(0) a(0) + a(1) a(1) +a(2)a(2) +a(3)a(3) + a(4)a(4)+a(5)a(5) 2ab <= a^2 + b^2 and the equality holds iff a=b

Correlation

r xx (0) r xx (1) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2)

Correlation between two sequences

r xy (0) r xy (1) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) y(1)y(8)y(9)y(10)y(11)y(12)y(3)y(4)y(5)y(6)y(7)y(2) X(1)X(8)X(9)X(10)X(11)X(12)X(3)X(4)X(5)X(6)X(7)X(2) y(1)y(8)y(9)y(10)y(11)y(12)y(3)y(4)y(5)y(6)y(7)y(2)

Correlation measures the difference between two objects

D(X,Y)= 2 – 2 r XY (0) if we assume that ||X|| = || Y || =1.

Image Enhancement Contrast enhancement

Ideas: Look at the following two images, generated from the following Matlab code x1=abs(1*randn(205,232)); figure, imshow(x1); X2=(rand(205,232)); figure,imshow(x2);

grey level

grey level

Property For a random variable X, let F be its distribution function i.e. F(x) = P(X<= x) Then the distribution of F(X) is uniform P(F(X) <=x)=x

Histogram Equalization Note how the image is extremely grey; it lacks detail since the