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149) and y h [ n ] is the homogeneous resolution which satisfies y [ n ] -ay[n - eleven = O think that y,[n] = Abn nrO Substituting Eq. (2. 1521 into Eq. (2. 1491, we receive Abn - a A b n - ' = okay b n from which we receive A = Kb/(b - a), and to procure yh["], we suppose y,[n] = Bzn Substituting this into Eq. (2. 151) provides from which we've z = a and ~ ~ [=Ban n ] Combining yp[n] and yh[n],we get y [ n ] =Ban + okay -b"+l b-a n20 with the intention to ensure B in Eq. (2. a hundred and fifty five) we want the worth of y[O]. environment n and (2. 1501, we've got Y [ O ] -ay[-11 =y[O] - a y - , =x[O] = okay or surroundings n y[O] = ok + a y - , =0 in Eq. (2. 155), we receive y[O] = B +K- b b-a for that reason, equating Eqs. (2. 156) and (2. 1571, we've from which we receive B = a y - , - ok- therefore, Eq. (2. one hundred fifty five) turns into b-a =0 in Eqs. (2. 149) CHAP. 21 LINEAR TIME-INVARIANT structures 103 For n < zero, we now have x [ n ] = zero, and Eq. (2. 149) turns into Eq. (2. 151). as a result. (2. 159) y [ n ] =Ban ,, From the auxiliary situation y[ - eleven = y - we've got y [ - eleven = y - , = BU-' from which we receive B = y - ,a. therefore, Combining Eqs. (2. 158) and (2. 160), y [ n ] might be expressed as notice that as within the continuous-time case (Probs. 2. 21 and a pair of. 221, the method defined through Eq. (2. 149) isn't really linear if y [ - eleven # zero. The process is causal and time-invariant whether it is at first at relaxation, that's, y [ - eleven = zero. be aware additionally that Eq. (2. 149) might be solved recursively (see Prob. 2. 43). 2. forty three. ponder the discrete-time procedure in Prob. 2. forty two. locate the output y [ n ] while x [ n ] = ok S [ n ] and y [ - l ] = y - , = a . we will resolve Eq. (2. 149) for successive values of y [ n ] for n r zero as follows: rearrange Eq. (2. 149) as y[n]=ay[n - eleven +x[n] Then + y [ n ] = a y [ n - 1 1 + x [ n ] =an(acu okay ) =an+'cu + anK equally, we will additionally make sure y [ n ] for n < zero through rearranging Eq. (2. 149) as Then y[-11 = a y[-n] 1 = -{y[-n a + 1 1 - x [ - n + 1 1 ) =a-"+'a Combining Eqs. (2. 163) and (2. 169, we receive y [ n ] = a n + ' a+ okay a n u [ n ] ( 2. sixteen. three) 104 LINEAR TIME-INVARIANT platforms [CHAP. 2 2. forty four. think of the discrete-time approach in Prob. 2. forty three for an in the beginning at relaxation situation. ( a ) locate in impulse reaction h [ n ] of the approach. ( b ) locate the step reaction s [ n ] of the process. (c) locate the impulse reaction h [ n ] from the results of half ( b ) . (a) surroundings ok = 1 and y [ - eleven = a (b) surroundings ok = 1, b = 1 , and y [ - 1] (c) From Eqs. (2. forty-one) and (2. 168) the impulse reaction h [ n ] is given through =0 in Eq. (2. 166), we receive =y - , = zero in Eq. (2. 161), we receive while n = zero. whilst n r 1, therefore, h [ n ]= anu[n] that's just like Eq. (2. 167). 2. forty five. locate the impulse reaction h [ n ] for every of the causal LTI discrete-time structures pleasurable the next distinction equations and point out no matter if each one process is a FIR or an IIR approach. ( a ) y [ n ] = x [ n ] - 2 x [ n - 21 + x [ n - 31 ( b ) y [ n ] + 2 y [ n - eleven = x [ n ] + x [ n - eleven (c) y [ n ] - t y [ n - 21 = 2 x [ n ] - x [ n - 21 (a) via definition (2. fifty six) when you consider that h [ n ] has basically 4 phrases, the method is a FIR approach.

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