**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:

Long DivisionThe 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.

Partial Fraction Expansion (1st Order Factors):By algebra, A = -0.24, B = 0.2011, C = 0.2011, so

Taking the inverse Z transform yields:

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: