how can determine the region of convergence?

hi all. I am a student, I should solve this question in matlab. can you help me please?

Answer 1

[John BG]

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

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thanks in advance John

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[sara]

thanks dear John. it is so helpful. but I am confused. z is variable and changes. for x: abs(z)>.5 and for h: abs(z)>1/4... how can I define this? and is there any way to show this in a plot?

[John BG]

it's defined by the exercise, not you: signal X(z) gets in H(z), Y(z)=H(z)*X(z) out it goes.

z is complex, you usually plot the magnitude with abs(z) and phase with angle(z).