## Description

1. Given the following digital transfer function:

H (z) = (2z 1) (z + 4)

3 (3z 1) (2z + 5)2

(a) Specify the ROC that corresponds to a stable system.

(b) Write the general form of h [n] for that ROC: do NOT Önd the exact signal (e.g.,

via partial fractions). My intention is for you to list the modes with unspeciÖed

constant coe¢ cients.

(c) Repeat the above for the case of a causal system.

(d) Find an allpass factor A (z) and a minimum-phase Hmin (z) such that Hmin = HA.

You can leave your answer in factored form. Also, superimpose graphs of the three

phase responses (all unwrapped, in degree) in MATLAB.

2. Find minimum-phase H (s) whose magnitude squared response is:

jH (!)j

2 =

9 + !

2

(9 + 4!2

) (16 + 9!2

)

3. The power spectral density of a discrete-time WSS random signal is given by:

S (!) = 13 + 12 cos !

(5 + 4 cos !)

2

(a) Express the PSD in the z-domain, S (z). You do not have to multiply out the

factors. In fact, your formula can have both positive and negative powers of z.

(b) Determine whether S (z) has poles and/or zeros (and their multiplicity) at 0

and/or 1.

(c) Use the method of spectral factorization to Önd the innovations Ölter H (z) and

the whitening Ölter G (z), assuming the innovations signal is normalized to have

unit variance. Again, you do not have to multiply out the factors. Note:

You can use MATLAB or a calculator to help you avoid doing messy algebra

by hand; e.g.,Önd roots of polynomials. Iím NOT asking you to write generalpurpose spectral factorization code. As a hint, the (Önite) poles and zeros are

fairly simple rational numbers.