Lesson_10 - EEL 3135: Signals and Systems Dr. Fred J....

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Convolution Lesson Number: 10 (Section 5-3) Background: In Chapter 5 the concept of convolution was introduced in the context of an FIR filter. A finite impulse response (FIR) filter, as the name implies, has an impulse response that persists for only of a finite number of sample values. The impulse response of an N th order FIR is given by (remember the order controversy): { } 1 1 0 ,..., , ] [ - = N h h h k h 1. The coefficients h i are called tap-weights . Observe that if x [ k ]= δ Κ [ k ], the output y [ k ] is first h [0] δ Κ [k]= h [0], followed by h [1] δ Κ [k-1]= h [1], and so forth. The last non-zero output would be h [ N -1] δ Κ [N-1] ]= h [ N -1]. The discrete-time convolution sum shown in Equation 2, defines how an input time-series x [ k ] is modified and manipulated by a N th order FIR to produce an output time-series y [ k ]. [ ] - = - = 1 0 y[k] N m m m k x h 2. where h m is the m th tap weight coefficient. Assuming that the input x [ k ] is infinitely long, Equation 2 states that: ... ... x[N] h .. . + x[2] h x[1] h y[N] 1] - x[N h ... + x[1] h x[0] h 1] - y[N ... ... x[2] h x[1] h x[0] h y[2] x[1] h x[0] h y[1] x[0] h y[0] 0 2 - N 1 - N 0 2 - N 1 - N 0 1 2 0 1 0 = + + = + + = = + + = + = = 3. Suppose that the input time-series consists of L non-zero samples {x[0], x[1], … , x[L-1]}, which is presented to an N th order FIR. Then Equation 2 yields: 1] x[L h 2] L y[ 1] x[L h 2] x[L h 3] L y[ ... .... x[k] h .. . + x[1] h x[0] h y[k] ... ... x[2] h x[1] h x[0] h y[2] x[1] h x[0] h y[1] x[0] h y[0] 1 - N 2 - N 1 - N 0 1 - k k 0 1 2 0 1 0 - = - + - + - = - + = + + = = + + = + = = N N 4. 1
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Example Suppose an FIR filter has an impulse response h={1, 2, 3, 2,1} and input x={1, 2, 3, 4, 5}. Then the filter operation can be studied as follows: n=0 y[0]=1x[0]+2x[-1]+3x[-2]+2x[-3]+1x[-4]= 1*1+2*0+3*0+2*0+1*0=1
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Lesson_10 - EEL 3135: Signals and Systems Dr. Fred J....

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