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[ ]= 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].Block Diagramsx[n]unit delayy[n]=x[n-1]x[n]y[n]=Ax[n]€ Ax[n]y[n]=x[n]+z[n]+z[n]+Block Diagrams: Direct Form€ x n[ ]=δn[ ]= 0 0 1 0 0 0 0{ }€ h[n] = y n[ ]x[n ]=δ[n ]= 0 04828180 0{ }x[n]y[n]€ y n[ ]= bkx n − k[ ]k= 0M∑€ n = −2 −1 0 1 2 3 4€ b0,b1,b2{ }=48,28,18{ }€ = 0 0 b0b1b20 0{ }€ y n[ ]= b0x n[ ]+ b1x n −1[ ]+ b2x n − 2[ ]unit delayx[n-1]unit delayx[n-2]€ b0€ b1€ b2++++€ b0x n[ ]€ b1x n −1[ ]€ b2x n − 2[ ]L=3, M=L-1=2Block Diagrams: TransposeForm€ x n[ ]=δn[ ]= 0 0 1 0 0 0 0{ }€ h[n] = y n[ ]x[n ]=δ[n ]= 0 04828180 0{ }x[n]y[n]€ y n[ ]= bkx n − k[ ]k= 0M∑€ n = −2 −1 0 1 2 3 4€ b0,b1,b2{ }=48,28,18{ }€ = 0 0 b0b1b20 0{ }€ y n[ ]= b0x n[ ]+ b1x n −1[ ]+ b2x n − 2[ ]unit delay unit delay€ b0€ b1€ b2++++€ b2x n[ ]€ b1x n[ ]€ b0x n[ ]€ b2x n −1[ ]€ b2x n − 2[ ]+ b1x n −1[ ]Block Diagrams to Difference Equationsx[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v2n[ ]€ v1n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]Block Diagrams to Difference Equationsx[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v2n[ ]€ v1n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n −1[ ]= b1x n −1[ ]+ v2n − 2[ ]€ v2n − 2[ ]= b2x n − 2[ ]Block Diagrams to Difference Equationsx[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v2n[ ]€ v1n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n −1[ ]= b1x n −1[ ]+ b2x n − 2[ ]Block Diagrams to Difference Equationsx[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v2n[ ]€ v1n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]€ y n[ ]= b0x n[ ]+ b1x n −1[ ]+ b2x n − 2[ ]Block Diagrams to Difference Equationsx[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v2n[ ]€ v1n[ ]€ y n[ ]= b0x n[ ]+ v1n −1[ ]€ v1n[ ]= b1x n[ ]+ v2n −1[ ]€ v2n[ ]= b2x n[ ]€ y n[ ]= b0x n[ ]+ b1x n −1[ ]+ b2x n − 2[ ]€ h n[ ]= b0δn[ ]+ b1δn −1[ ]+ b2δn − 2[ ]difference equationimpulse responseequivalent waysof describing systemblock diagramBlock Diagrams to Difference Equations€ v2n[ ]€ y n[ ]= b0v2n[ ]+ b1v1n[ ]+ b2v1n −1[ ]€ v2n[ ]= x n[ ]x[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v1n[ ]€ v1n[ ]= v2n −1[ ]€ y n[ ]= b0v2n[ ]+ b1v1n[ ]+ b2v1n −1[ ]€ v1n[ ]= v2n −1[ ]€ v2n[ ]= x n[ ]Block Diagrams to Difference Equations€ v2n[ ]€ y n[ ]= b0v2n[ ]+ b1v1n[ ]+ b2v1n −1[ ]€ v2n[ ]= x n[ ]x[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v1n[ ]€ v1n[ ]= v2n −1[ ]€ y n[ ]= b0v2n[ ]+ b1v1n[ ]+ b2v1n −1[ ]€ v1n −1[ ]= v2n − 2[ ]€ v2n − 2[ ]= x n − 2[ ]€ v2n −1[ ]= x n −1[ ]€ v2n[ ]= x n[ ]€ v1n[ ]= v2n −1[ ]Block Diagrams to Difference Equations€ v2n[ ]€ y n[ ]= b0v2n[ ]+ b1v1n[ ]+ b2v1n −1[ ]€ v2n[ ]= x n[ ]x[n]y[n]unit delay unit delay€ b0€ b1€ b2++++€ v1n[ ]€ y n[ ]= b0x n[ ]+ b1x n −1[ ]+ b2x n − 2[ ]€ v1n[ ]= v2n −1[ ]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 response-2 -1 0 1 2 3 400.20.40.60.81x[n]TextEnd-2 -1 0 1 2 3 400.20.40.60.81nh[n]TextEnd€ x n[ ]=δn[ ]= 0 0 1 0 0 0 0{ }€ h[n] = y n[ ]x[n ]=δ[n ]= 0 04828180 0{ }Coefficients from impulse responsex[n]h[n]€ y n[ ]= bkx n − k[ ]k= 0M∑€ n = −2 −1 0 1 2 3 4€ b0,b1,b2{ }=48,28,18{ }€ = 0 0 b0b1b20 0{ }€ x n[ ]=δn[ ]= 0 0 1 0 1 0 0{ }€ y n[ ]=0 0482818+41628116{ }Response from 2 impulsesx[n]y[n]€ n = −2 −1 0 1 2 3 4-2 -1 0 1 2 3 400.20.40.60.81x[n]TextEnd-2 -1 0 1 2 3 400.20.40.60.81€ y n[ ]= bkx n − k[ ]k= 0M∑€ b0,b1,b2{ }=48,28,18{ }€ y n[ ]= 0 0 h 0[ ]x 0[ ]h 1[ ]x 0[ ]h 2[ ]x 0[ ]+ h 0[ ]x 2[ ]h 1[ ]x 2[ ]h 2[ ]x 2[ ]{ }Sum the responses ofeach impulse€ x n[ ]=δn[ ]= 0 0 1 0 1 0 0{ }€ y n[ ]=0 0482818+41628116{ }Response from 2 impulsesx[n]y[n]€ n = −2 −1 0 1 2 3 4-2 -1 0 1 2 3 400.20.40.60.81x[n]TextEnd-2 -1 0 1 2 3 400.20.40.60.81€ y n[ ]= bkx n − k[ ]k= 0M∑€ h[n] = b0,b1,b2{ }=48,28,18{ }€ y n[ ]= 0 0 h 0[ ]x 0[ ]h 1[ ]x 0[ ]h 2[ ]x 0[ ]+ h 0[ ]x 2[ ]h 1[ ]x 2[ ]h 2[ ]x 2[ ]{ }€ y n[ ]= h k[ ]x n − k[ ]k= 03∑€ y 2[ ]= h 0[ ]x 2 − 0[ ]+h 1[ ]x 2 −1[ ]+h 2[ ]x 2 − 2[ ]Convolution sum€ 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]TextEnd€ x n[ ]= 0 ⋅δn[ ]+116⋅δn −1[ ]+416⋅δn − 2[ ]+916⋅δn − 3[ ]+1616⋅δn − 4[ ]€ x n[ ]= 0 0 01164169161616{ }Any discrete signal be thought of a weighted sum of delayed impulses€ n = −2 −1 0 1 2 3 4Convolution € h[n] = y n[ ]x[n ]=δ[n ]=bnn = 0,1K M0 otherwise € y n[ ]= bkx n − k[ ]k= 0M∑€ y n[ ]= h k[ ]x n − k[ ]k= 0M∑convolution sumThe output y[n] is equal to the input x[n] convolved withthe unit impulse response h[n].FIR filter€ y n[ …
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