From Digital Signal Processing using MATLAB,
by Vinay K Ingle, John G Proakis
pg 104
Region of Convergence of z domain functions is defined
as the abs(z) where H(z) exists, z: complex frequency.
Since you have defined in the same question that X is the input signal,
H is the system, both have finite energy,
X(z) ROC is abs(z)>.25 and H(z) ROC is abs(z)>.25
X(z)=1/(1+.5*z^-1) % ROC abs(z)>.5
x(n)=(-.5)^n*u(n)
H(z)=1/(1-.25*z^-1) % ROC abs(z)>.25
h(n)=(.25)^n*u(n)
if X(z) and H(z) had been defined with ROCs abs(z)<.5 and abs(z)<.25 respectively then
x(n)=.5^n*u(-n-1)
h(n)=-.25^n*u(-n-1)
but it's not the case.
Since
Y(z)=H(z)*X(z)
the overlapping ROCs define Y(z) ROC.
Y(z) has same ROC as input signal
abs(z)>.5
If you find this answer of any help solving your question, please click on the thumbs-up vote link,
thanks in advance John
