Question 6

Beginning with Eqn. 1, we find the impulse response by taking the inverse Z transform, and isolating y(n). What follows are examples of the computations required to perform the inverse Z transform by:

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Long Division

The terms of this summation will have successively smaller degree. Taking the inverse Z transform, we obtain:

Since the poles of the filter are within the bounds of the unit circle, it can be proven that the coefficients of this infinite summation are steadily decreasing.

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Partial Fraction Expansion (1st Order Factors):

By algebra, A = -0.24, B = 0.2011, C = 0.2011, so

Taking the inverse Z transform yields:

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Partial Fraction Expansion (Quadratic Factors)

using the Z-transform identity pair 12c, on page 359 of Linear Systems and Signals, B.P.Lathi, Berkely-Cambridge Press, 1992: