2
$\begingroup$

As we have in Laplace transform that the roots decide the stability of the system i.e. if the roots are complex and lie in the left side of the plane you get a sinusoidal response with decreasing amplitude

similarly is there any significance of the roots , zeros and ROC of the z-transform and the stability criteria . All i read in books is how to find the ROC and the properties of z transform like linearity ,time reversal,time shifting . It has nowhere mentioned why we are even using z-transform

my apologies if this question is too basic or if it shouldn't belong here.

$\endgroup$
  • $\begingroup$ the Z-transform has the same significance to discrete-time signals and systems as does the Laplace transform has on continuous-time signals and systems. the Z-transform is exactly the same as the Laplace transform applied to the ideally sampled signal: $$x_\text{s}(t) = x(t) \ \sum\limits_{n=-\infty}^{\infty} \delta(t-nT) = \sum\limits_{n=-\infty}^{\infty} x(nT) \delta(t-nT)$$ and where $$x[n] \triangleq x(nT)$$. when you compute the Laplace transform of $x_\text{s}(t)$, you will get the Z transform of $x[n]$ with the substitution of $$ z \triangleq e^{sT} $$. that's all $z$ is. $\endgroup$ – robert bristow-johnson Apr 6 '15 at 19:05
6
$\begingroup$

First of all, I think you're reading the wrong books. Almost any basic text on DSP has a chapter on the $\mathcal{Z}$-transform and its significance to describe linear time-invariant (LTI) discrete-time systems. If you're looking for good (and free) books, take a look at this answer.

I will not repeat all the details you can find in those books (and in many other places), but let me just point out a few very basic things to get you started. Each (single) pole $p$ of the transfer function $H(z)$ of a causal LTI discrete-time system contributes a term

$$c\cdot p^nu[n]\tag{1}$$

to the system's impulse response, where $c$ is some constant, $p$ is the (possibly complex) pole, and $u[n]$ is the discrete-time unit step function. From (1) it is clear that this contribution only decays with time if $|p|<1$. So for a causal system to be stable we require that all the poles of the transfer function are inside the unit circle of the complex plane, i.e. they have magnitudes smaller than $1$. So if you're looking for analogies with the Laplace transform, the inside of the unit circle corresponds to the left half plane of the complex variable $s$. Furthermore, the unit circle of the $z$-plane corresponds to the $j\omega$-axis. Knowing these two things, it becomes very easy to carry over everything you know about transfer functions of continuous-time systems (Laplace transform) to the discrete-time domain ($\mathcal{Z}$-transform).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.