Digital signal processing : principles, algorithms, and by J G Proakis; D G Manolakis

By J G Proakis; D G Manolakis

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Manolakis. ISBN 0-13-187374-1. S. 57 y(n) − 4y(n − 1) + 4y(n − 2) The characteristic equation is λ2 − 4λ + 4 λ yh (n) = x(n) − x(n − 1) = 0 = 2, 2. Hence, = c1 2n + c2 n2n The particular solution is yp (n) = k(−1)n u(n). Substituting this solution into the difference equation, we obtain k(−1)n u(n) − 4k(−1)n−1 u(n − 1) + 4k(−1)n−2 u(n − 2) = (−1)n u(n) − (−1)n−1 u(n − 1) For n = 2, k(1 + 4 + 4) = 2 ⇒ k = 29 . The total solution is 2 y(n) = c1 2n + c2 n2n + (−1)n u(n) 9 From the initial condtions, we obtain y(0) = 1, y(1) = 2.

No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G. Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1. , Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G. Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1. 44 (a) s(n) = −a1 s(n − 1) − a2 s(n − 2) − . . − aN s(n − N ) + b0 v(n). 44-1. (b) v(n) = b10 [s(n) + a1 s(n − 1) + a2 s(n − 2) + . . + aN s(n − N )] .

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