LTISystem€ y n[ ]€ x n[ ]Causal FIR filter€ y n[ ]€ x n[ ]Causal FIR filterQ:What is the definition of an FIR filter?A: The output y at each sample n is a weighted sum of the present input, x[n], and past inputs, x[n-1], x[n-2],…, x[n-M].Causal FIR filter€ y n[ ]€ x n[ ]Causal FIR filterQ:What is the definition of an FIR filter?A: The output y at each sample n is a weighted sum of the present input, x[n], and past inputs, x[n-1], x[n-2],…, x[n-M].Causal FIR filter€ y n[ ]= bkx n − k[ ]k= 0M∑ € y n[ ]= b0x n[ ]+ b1x n −1[ ]+ K + bMx n − M[ ]The output y at each sample n is a weighted sum of the present input, x[n], and past inputs, x[n-1], x[n-2],…, x[n-M].3 point average€ y n[ ]= bkx n − k[ ]k= 0M∑€ b0=13€ b1=13€ b2=13M=L-1=2€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]L=32nd orderLength 33 point average€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]-2 -1 0 1 2 3 4 5 6 700.20.40.60.811.21.4nx[n]TextEnd€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }n=-2 -1 0 1 2 3 4 5 6 7€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.51ny[n]TextEnd€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }€ y 0[ ]=13x 0[ ]+13x −1[ ]+13x −2[ ]=131.11+130 +130 = 0.36running onto datasliding windown= -2 -1 0 1 2 3 4 5 6 7b0b1b2b0x[0]€ y 1[ ]=13x 1[ ]+13x 0[ ]+13x −1[ ]=131.16 +131.11+130 = 0.76-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.51ny[n]TextEnd€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }n= -2 -1 0 1 2 3 4 5 6 7b0b1b2b0x[0]b0x[1]+b1x[0]€ y 2[ ]=13x 2[ ]+13x 1[ ]+13x 0[ ]=131.01+131.16 +131.11 = 1.09-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.51ny[n]TextEnd€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }weighted sumn= -2 -1 0 1 2 3 4 5 6 7€ y 3[ ]=13x 3[ ]+13x 2[ ]+13x 1[ ]=131.12 +131.01+131.16 = 1.10€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.51ny[n]TextEndn= -2 -1 0 1 2 3 4 5 6 7-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.511.5ny[n]TextEnd€ y 4[ ]=13x 4[ ]+13x 3[ ]+13x 2[ ]=131.01+131.12 +131.01 = 1.05€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }n= -2 -1 0 1 2 3 4 5 6 7-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.511.5y[n]TextEndn€ y 5[ ]=13x 5[ ]+13x 4[ ]+13x 3[ ]=131.08 +131.01+131.12 = 1.07€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.511.5ny[n]TextEnd€ y 6[ ]=13x 6[ ]+13x 5[ ]+13x 4[ ]=130 +131.08 +131.01 = 0.70€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }running off datan= -2 -1 0 1 2 3 4 5 6 7-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.511.5ny[n]TextEnd€ y 7[ ]=13x 7[ ]+13x 6[ ]+13x 5[ ]=130 +130 +131.08 = 0.36€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0 0{ }running off datan= -2 -1 0 1 2 3 4 5 6 7-2 -1 0 1 2 3 4 5 6 700.511.5nx[n]TextEnd-2 -1 0 1 2 3 4 5 6 700.51bTextEnd-2 -1 0 1 2 3 4 5 6 700.511.5ny[n]TextEnd€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ x n[ ]= 0 0 1.11 1.16 1.01 1.12 1.01 1.08 0.0 0.0{ }running off datarunning onto data weighted sumsliding window€ y n[ ]= 0 0 0.36 0.75 1.09 1.10 1.05 1.07 0.7 0.36{ }n= -2 -1 0 1 2 3 4 5 6 72 point difference€ y n[ ]= x n[ ]− x n −1[ ]€ x n[ ]=n216⋅ u n[ ]€ u n[ ]=1 n ≥ 00 n < 0 -2 -1 0 1 2 3 400.20.40.60.81nTextEnd2TextEnd/16TextEnd-2 -1 0 1 2 3 400.20.40.60.81nu[n]TextEndunit step function2 point difference€ y n[ ]= x n[ ]− x n −1[ ]€ x n[ ]=n216⋅ u n[ ]€ u n[ ]=1 n ≥ 00 n < 0 -2 -1 0 1 2 3 400.10.20.30.40.50.60.70.80.91nx[n]TextEndunit step function€ y n[ ]= x n[ ]− x n −1[ ]€ x n[ ]=n216⋅ u n[ ]€ y n[ ]= 0 0 0116316516716{ }=2n −116u n −1[ ]finite differenceapproximation toa derivative.derivatives enhancenoise (and high frequencies)-2 -1 0 1 2 3 400.20.40.60.81nx[n]TextEnd-2 -1 0 1 2 3 400.10.20.30.40.5ny[n]TextEnd€ x n[ ]= 0 0 01164169161616{ }Impulse response€ y n[ ]= bkx n − k[ ]k= 0M∑FIR filter€ x n[ ]=δ[n] =1 n = 00 otherwise Delta function€ δ[n − k] =1 n = k0 otherwise € y n[ ]= h[n] = bkδn − k[ ]k= 0M∑impulse responseImpulse response€ δ[n − k] =1 n = k0 otherwise € h[n] = y n[ ]= bkδn − k[ ]k= 0M∑€ = bnimpulse responseThe impulse response is just the filter coefficients.Finite length filter, finite impulse response (FIR). € h[n] = b0δn − 0[ ]+ b1δn −1[ ]+ K + bnδn − n[ ]+ K + bMδM −1[ ] € = b00 + b10 + K + bn1+ K + bM0€ δ[z] =1 z = 00 otherwise € = b0δn − 0[ ]+ b1δn −1[ ]+ K + bnδ0[ ]+ K + bMδM −1[ ]€ k = n€ k = 0€ x n[ ]=δ[n] =1 n = 00 otherwise Delta function€ y n[ ]=13x n[ ]+13x n −1[ ]+13x n − 2[ ]€ …
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