z transform
I am having confusion regarding ROC of z transform of u(-n+1)
z transform of u(-n+1) is given by
$$X(z)=Z[u(-n+1)] =
\sum_{n=-\infty}^1 z^{-n} =\sum_{n=-1}^\infty z^n = z^{-1} + \sum_{n=0}^\infty z^n $$
ROC given: 0<|z|<1
While I understand that |z| should be less than 1 so that $$\sum_{n=0}^\infty z^n $$ converge, I don't understand why |z| should be greater than 0. Please clarify this doubt.
From definition of inverse z-transforn,
"The inverse Z-transform is, $$x(n) = \frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$$ where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal this means the path C must encircle all of the poles of X(z)."
$$\begin{align} x(n)&= -\frac{1}{3}\times \frac{1}{2\pi j}\oint_{C}^{ } \frac{z^{n-1}}{z-\frac{2}{3}}dz\\ &= -\frac{1}{3}\times \frac{1}{2\pi j}\oint_{C}^{ } \frac{f(z)}{z-\frac{2}{3}}dz\end{align}$$
Where, \$f(z) = z^{n-1}\$. Using Cauchy's Integral formula:
$$x(n) = -\frac{1}{3}\times \left(\dfrac{2}{3}\right)^{n-1}$$
If \$n<1\$, then the function \$f(z) = z^{n-1}\$ will have singularity at \$z=0\$ and hence Cauchy's Integral formula can not be applied. So \$n\$ must be greater than or equal to \$1\$. Or we can write:
$$x(n) = -\frac{1}{3}\times \left(\dfrac{2}{3}\right)^{n-1}u(n-1)$$
PS: Read this page. They have given equations to find inverse z-transform directly (without using integration) using residue method.
Using the Z-transform table on this website
The forward tap, item # 7, has the initial condition, which means for a zero input response, x[k] can be calculated to account for the initial condition in the z domain.
In this case, I was trying to calculate: Z{x[k+1] = (I+TA)x[k]}.
yields-> z(X(Z) - x(0)) = (I+TA)X(Z)
Solving for X(Z) = z*x(0) / [z-(I+TA)]
Solving x[k] = Z^-1{X(Z)} = x(0) * (1+TA)^k
Best Answer
Since
\$|z| = 0 ~~\Leftrightarrow~~ z = 0 \$
for z = 0 you cannot compute the term outside the sum symbol, i.e. \$z^{-1}\$, which is part of your transform. In other words, for \$z = 0\$ the transform does not exist.
Maybe you were put off-tracks because \$|z|\ge 0\$ by definition, but now you have to exclude \$ |z| = 0 \$ from the interval of valid values of \$|z|\$.