H (z) is the system function for a stable LTI system and is given by:
H (z) =
(1 − 2z−1)(1 − 0.75z−1)/(z−1(1 − 0.5z−1) )
.
(a) H (z) can be represented as a cascade of a minimum-phase system Hmin1(z) and a
unity-gain all-pass system Hap(z), i. e.,
H (z) = Hmin1(z)Hap(z).
Determine a choice for Hmin1(z) and Hap(z) and specify whether or not they are unique
up to a scale factor.
(b) H (z) can be expressed as a cascade of a minimum-phase system Hmin2(z) and a generalized linear-phase FIR system Hlp(z):
H (z) = Hmin2(z)Hlp(z).
Determine a choice for Hmin2(z) and Hlp(z) and specify whether or not these are
unique up to a scale factor.

Tom is having a problem with his washing machine. he notices that the machine vibrates violently at a frequency of 1500 rpm due to an unknown rotating unbalance. the machine is mounted on 4 springs each having a stiffness of 10 kn/m. tom wishes to add an undamped vibration absorber attached by a spring under the machine the machine working frequency ranges between 800 rpm to 2000 rpm and its total mass while loaded is assumed to be 80 kg a) what should be the mass of the absorber added to the machine so that the natural frequency falls outside the working range? b) after a first trial of an absorber using a mass of 35 kg, the amplitude of the oscillation was found to be 10 cm. what is the value of the rotating unbalance? c) using me-3.5 kg.m, find the optimal absorber (by minimizing its mass). what would be the amplitude of the oscillation of the absorber?