28 votes

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 ...
Dan Boschen's user avatar
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13 votes
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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 ...
Matt L.'s user avatar
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13 votes
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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 ...
robert bristow-johnson's user avatar
10 votes
Accepted

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 ...
Fat32's user avatar
  • 28.2k
10 votes

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 ...
Matt L.'s user avatar
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9 votes
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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 ...
Matt L.'s user avatar
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8 votes

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

Given some digital signal $x[n]$, it's z-transform is defined by, \begin{equation} X(z) = \sum_{n=-\infty}^{\infty}x[n] \,z^{-n} \end{equation} When a system equation is \begin{equation} H(z) = z^{-1} ...
nm15's user avatar
  • 93
7 votes
Accepted

$\mathcal{Z}$-transform of $\frac{1}{n^2}$

The problem is not sufficiently specified, because the range of admissible values of $n$ is missing. Here I make the assumption that we consider $n>0$. With this assumption we have $$X(z)=\sum_{n=...
Matt L.'s user avatar
  • 90k
7 votes
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Understanding the $\mathcal Z$-transform

Consider a liner discrete-time system. Assume we can define it in terms of an input-output relation as follows (you can assume a more general model but it is enough for our purpose): $$a_0y[n]+a_{1}y[...
msm's user avatar
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7 votes
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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 $$ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} = \sum_{n=-\infty}^{\infty} C z^{-n} $$ ...
Fat32's user avatar
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7 votes
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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 ...
Matt L.'s user avatar
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7 votes
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z-Transform Methods: Definition vs. Integration Rule

Just to avoid a misunderstanding: the $\mathcal{Z}$-transform is a transform defined for sequences, comparable to the Laplace transform for continuous functions. What you are talking about is not the $...
Matt L.'s user avatar
  • 90k
7 votes

Filter odd or even harmonics with notch or inverse notch filter

What you are looking for are what we, in the audio space, call comb filters. Comb filters may or may not have a feedback path, just like FIR and IIR filters. In fact, there is a generalized theory ...
robert bristow-johnson's user avatar
6 votes

Is $y[k] = y[k-1] + x[k]$ an integrator?

Your simple integrator is called a "Rectangular Rule" integrator. There are more complicated (and more accurate) integrators called "Trapazoidal Rule", "Simpson's Rule", and "Tick's Rule" integrators.
Richard Lyons's user avatar
6 votes

DSP interview question: use of the identity in development of a significant transform

This is related to Chirp Z-transform (CZT) (refer to the Bluestein's algorithm). Using this identity, the CZT can be expressed in terms of a convolution. Hence, it can be efficiently implemented using ...
msm's user avatar
  • 4,285
6 votes
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$\mathcal Z$-transform ROC

If the ROC is outside a circle in the complex $z$-plane ($|z|>a$), then the corresponding system is causal. If it is inside a circle ($|z|<a$), the system is anti-causal. If the ROC is a ring ($...
Matt L.'s user avatar
  • 90k
6 votes
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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 ...
Atul Ingle's user avatar
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6 votes
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Why must the Region Of Convergence (ROC) contain infinity and the system function be a right-sided sequence for it to be causal?

A causal impulse response is zero for negative argument: $$h[n]=0,\qquad n<0\tag{1}$$ Hence its $\mathcal{Z}$-transform is given by $$H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n}=\sum_{n=0}^{\infty}...
Matt L.'s user avatar
  • 90k
6 votes
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How to find the difference equation directly from Direct Form II signal flow graph

Darkling, these things are quite clearly explained in standard signals & systems textbooks. But I assume you have little time left to read (as most undergraduate courseware are full of homeworks, ...
Fat32's user avatar
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6 votes
Accepted

Help with my first (simple) Z-transform

First of all, it's important to understand that there is no single best way to transform a continuous-time system to a discrete-time system. The method you're using is called backward Euler method, ...
Matt L.'s user avatar
  • 90k
6 votes
Accepted

Is this system causal or not?

Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial ...
Matt L.'s user avatar
  • 90k
6 votes

Filter odd or even harmonics with notch or inverse notch filter

If the OP is actually interested in selecting only one individual frequency from the even or odd harmonics, then a moving average filter (MAF) would be ideal since this can provide a null at every ...
Dan Boschen's user avatar
  • 51.1k
6 votes

How can I solve such an inverse Z-transform?

The correct answer to the question What is the inverse $\mathcal{Z}$-transform of $F(z)=z^{-\frac12}?$ is $F(z)=z^{-\frac12}$ is not a valid $\mathcal{Z}$-transform, hence its inverse transform ...
Matt L.'s user avatar
  • 90k
6 votes
Accepted

For unit step $g(t) = u(t)$, why does $G(z) = \frac{z}{1-z}$, whereas $G(s) = \frac{1}{s}$?

Answer to the updated question: You wonder why there are several mappings from the continuous domain to the discrete domain, and why we don't just use the optimal mapping $z=e^{sT}$. Let's see how we ...
Matt L.'s user avatar
  • 90k
5 votes

Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

The short answer is yes, if you have the Laplace or Z-transform of a function you do not need the Fourier transform. This is because the CFT is a special case of the Laplace transform and the DTFT ...
Matt's user avatar
  • 59
5 votes
Accepted

How to identify causality, stability and ROC from the pole-zero plot?

For systems with more than one pole (with different radii), there are $3$ types of ROCs: inside the circle with a radius corresponding to the smallest pole radius; the corresponding time-domain ...
Matt L.'s user avatar
  • 90k
5 votes
Accepted

Is Z-transform of $\sin(\omega_0n)$ same as that of $\sin(\omega_0n)u[n]$

The two sequences $\sin(\omega_0n)u[n]$ and $\sin(\omega_0n)$ are very different, so their transforms can't be identical. The $\mathcal{Z}$-transform of $x[n]=\sin(\omega_0n)u[n]$ can be computed as $...
Matt L.'s user avatar
  • 90k
5 votes
Accepted

Z-domain transfer function to difference equation

$$\begin{align*}\dfrac{Y(z)}{X(z)} &= \dfrac{1+z^{-1}}{2(1-z^{-1})}\\ \\ 2(1-z^{-1})Y(z)&=(1+z^{-1})X(z)\\ \\ Y(z) -Y(z)z^{-1}&= \frac{1}{2}X(z) +\frac{1}{2}X(z)z^{-1}\\ \\ y[n]-y[n-1]&...
Andy Walls's user avatar
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