Signal Processing and Linear Systems-B.P.Lathi copy

# 4 1 graphical procedure for t he convolution sum f k

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Unformatted text preview: e c haracteristic roots a nd characteristic modes. T he z ero-input response is a linear combination o f t he c haracteristic modes. From t he s ystem equation we also determine h[k], t he impulse response, as discussed in Sec. 9.3. Knowing h[kJ a nd t he i nput I [k], we find t he zero-state response as t he convolution o f J[kJ a nd h[kJ. T he a rbitrary c onstants Q ,C2,.·· , Cn in t he zero-input response are determined from t he n auxiliary conditions. For t he s ystem described by t he e quation w ith initial conditions y [-IJ we have determined t he two components o f t he response in Examples 9.3a a nd 9.7 respectively. From t he r esults in these examples, t he t otal response for k 2': 0 is * zk y[kJ = h[kJ Z ero-input c omponent L + O.8{O.8)k - 1.26(4)-k + 0.444{ -O.2)k + 5.81{O.8)k v'" Z ero-input c omponent h[mJz-m F or causal h[k], t he limits on t he s um on t he r ight-hand side would range from 0 t o I n any case, this sum is a function of z. L et us denote it by H[zJ. T hus,t ( x). (9.57a) w here 00 H[zJ = L h[mJz-m (9.57b) m =-oo t This r esult is valid only for t he values o f z for which t he ' (9.59) m =-oo y[kJ = H[zJzk ... Z ero-state c omponent 2:::=-00 h[m]z-m exists (or converges). N atural a nd Forced Response T he c haracteristic modes of this system are {-O.2)k a nd {O.8)k . T he zero-input component is m ade u p of characteristic modes exclusively as expected, b ut t he characteristic modes also a ppear in t he z ero-state response. When all t he c haracteristic mode terms in t he t otal response are lumped together, t he r esulting component is t he n atural r esponse. T he r emaining p art of t he t otal response t hat is made up o f n oncharacteristic modes is t he f orced r esponse. For t he p resent case, Eq. (9.59) yields 598 9 Time-Domain Analysis of Discrete-Time Systems T otal response = 0.644( - 0.2l , + 6.61(0.8)k - l.26(4)-k ''-v---' v N atural r esponse k~O (9.60) Classical Solution of Linear Difference E quations 9.5 599 t he i nputs a nd t he c orresponding forms of forced function w ith u ndetermined coefficients. These coefficients c an b e d etermined b y s ubstituting y.p[k] i n Eq. (9.64) a nd e quating t he coefficients o f s imilar terms. F orced r esponse T able 9 .2 I nput f [k] 9 .5 Classical solution o f Linear Difference Equations l. c lassical method, where t he response is obtained as a s um o f n atural a nd forced c omponents o f t he response. Total response = Yn[k] ' -v-' + modes B ecause t he t otal r esponse Yn[k] we h ave + y.p[k] Q [E] (Yn[k] y.p[k] ' -v-' (9.61) nonmodes is a solution o f t he s ystem e quation (9.9), + y.p[kj) = p [E]f[k] = 1, 2, r 3. 4. . .. , n ) cr k = 'Yi cos (13k + 0) c kr k (f (f ,=0 c cos (13k aiki ) r k + if» Ciki ) r k t=O N ote: By definition, Y4>[k] cannot have any characteristic mode terms. I f any terms shown in the right-hand column for the forced response should also be a characteristic mode of the system, the correct form of the forced response must be modified to k i Y4>[k], where i is the smallest integer t ha...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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