ECE310 Problem Set II: Transfer Function Analysis solution




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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 + !
(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 !)
(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.