# Tag Info

### How/why are the $\mathcal Z$-transform and unit delays related?

Mathematically we can easily show that the z-transform of a unit cycle delay is $z^{-1}$, Just like the s-transform of a time delay $\tau$ is $e^{-\tau s}$. In this post I add some additional insights ...
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### What is the difference between natural response and zero input response?

First it's important to realize that many authors use the terms zero-input response and natural response as synonyms. This convention is used in the corresponding wikipedia article, and for instance ...
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### why is the z transform transfer function 1/(z-1) called an integrator?

There are a couple reasons. One is that $(1-z^{-1})$ represents $x[n]-x[n-1]$ which is a finite difference over a very small period of time. and that is an approximation to a differentiator. The ...
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### Position of poles and Stability in $z$ domain

Short Answer: All the poles of a causal (right-sided) and stable LTI system must be inside the unit circle whereas all the poles of an acausal (left-sided) and stable LTI system must be outside the ...
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### Is the inverse of a causal system also causal?

It's not sufficient to only consider causality, you also need to check whether the inverse system is stable, otherwise it can't be implemented. If $G(z)$ has zeros on the unit circle, it cannot be ...
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### Is $y[k] = y[k-1] + x[k]$ an integrator?

The system $$y[n]=y[n-1]+x[n]\tag{1}$$ is an ideal accumulator, i.e., it computes the cumulative sum of the input samples: $$y[n]=\sum_{k=-\infty}^nx[k]\tag{2}$$ It is in a way analogous to a ...
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### How/why are the $\mathcal Z$-transform and unit delays related?

Given some digital signal $x[n]$, it's z-transform is defined by, $$X(z) = \sum_{n=-\infty}^{\infty}x[n] \,z^{-n}$$ When a system equation is H(z) = z^{-1} ...
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### What is the $\mathcal{Z}$-transform of a constant?

if you define the constant like this: $$x[n] = C ~~~,~~~\text{ for all } n$$ Then its $\mathcal{Z}$ transform $$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} = \sum_{n=-\infty}^{\infty} C z^{-n}$$ ...
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### Stability of system with poles inside unit circle - conflict with differential equation

What you are missing is that this is about a discrete-time system, because we're talking about poles and zeros in the complex $z$-plane and about poles inside or outside the unit circle. So there is ...
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### Poles and zeros of a transfer function

The "poles-inside-unit-circle" stability criterion only applies to causal systems. Your system is not causal because it uses one sample from the future owing to the $z$ term. The general technique to ...
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