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Lesson_90

# Lesson_90 - Response The output for an input x 1[k at time...

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson 09 FIR Filter Challenge It is claimed that a simple FIR can be used as a signal detector. You are to design an 5 th order FIR (N=5) that can detect the presence of a 5-sample time-series x 1 ={ x 1 [k], x 1 [k-1], x 1 [k-2], x 1 [k- 3], x 1 [k-4]} = {1,-1,1,1,-1} out of a collection of signals: x 1 ={1,-1,1,1,-1} x 2 ={1,1,1,1,-1} x 3 ={1,-1,1,1,1} x 4 ={1,-1,-1,-1,1} x 5 ={-1,1,1,1,-1} x 6 ={-1,-1,1,-1,-1} The FIR coefficients are to have values of +1 or -1. The signal detector’s output is to have the largest possible value in comparison to the filtered value of the other signals. What is the optimum choice of filter coefficients ( i.e ., filter’s impulse response)?
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Unformatted text preview: Response The output for an input x 1 [k], at time k=4 (beginning of steady-state), is given by: y 1 [4]=h x[4]+ h 1 x[3]+ h 2 x[2]+ h 3 x[1]+ h 4 x[5]= h (1)+ h 1 (-1)+ h 2 (1)+ h 3 (1)+ h 4 (-1) The output y 1 [4] is maximized when h =1, h 1 =-1, h 2 =1, h 3 =1, and h 4 =-1 or h[k]={1, -1, 1, 1, -1}. That is, the coefficients are assigned their maximum value (resulting in maximum gain) when the signs of the coefficients are “matched” to the signs of the input (called a matched filter). The outputs are: y 1 [4]=5 ; y 2 [4]=3, y 3 [4]=3, y 4 [4]=1, y 5 [4]=-1, and y 6 [4]=-1. where y 1 [4]=5 is the largest output as required. 1...
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