z-tranforms?

Question

how to consider ROC in case of z-transforms

Answer 1

In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation.

The Z-transform and advanced Z-transform were introduced (under the Z-transform name) by E. I. Jury in 1958 in Sampled-Data Control Systems (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method".

The (unilateral) Z-transform is to discrete time domain signals what the one-sided Laplace transform is to continuous time domain signals.
Definition

The Z-transform, like many other integral transforms, can be defined as either a one-sided or two-sided transform.

[edit] Bilateral Z-Transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as

X(z) = Z\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \

where n is an integer and z is, in general, a complex number:

z = Aejφ
where A is the magnitude of z, and φ is the angular frequency (in radians per sample).

[edit] Unilateral Z-Transform

Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform is defined as

X(z) = Z\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \

In signal processing, this definition is used when the signal is causal.

An important example of the unilateral Z-transform is the probability-generating function, where the component x[n] is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s), in terms of s = z − 1. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

Answer 2

property 1: the roc of X(z) consists of a ring in the z plane centered about the origin.
property 2: the roc does not contain any pole.
property 3: if x[n] is of finite duration, then the roc is the entire z- plane, except possibly z=0 and/or z = infinity.
property 4: if x[n] is a right handed sequence, and if the circle |z| = r0 is in the roc then all finite values of z for which |z|>r0 will also be in roc.
property5: if x[n] is a left handed sequence, and if the circle |z| = r0 is in the roc then all finite values of z for which 0<|z| property6: if x[n] is a two sided sequence, and if the circle |z| = r0 is in the roc then the roc will consists of a ring in the z- plane that includes the circle |z|=r0
property7: if the z- transform X(z) of X[n] is rational, then its roc is bounded by poles or extends to infinity.

Answer 3

Other than going to a good textbook on them, here's an article that talks a bit about convergence -

http://en.wikipedia.org/wiki/Z_transform#Example_1_.28No_ROC.29

Answer 4

Refer this site: http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Sampled/Sampled1.html