DCSP-20 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 x 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

t6 t5 t4 t3 t2 t1 time

Question: Could you develop a matched filter to detect it?

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.