20 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 $e^{-\tau s}$. I would like to add some additional ...
user avatar
  • 36.1k
13 votes

Why eigen values and poles of a system are equivalent?

Let's consider a discrete-time state space model (the derivation for a coninuous-time system is completely analogous): $$\begin{align}\mathbf{q}[n+1]&=\mathbf{Aq}[n]+\mathbf{b}x[n]\\ y[n]&=\...
user avatar
  • 79.2k
13 votes
Accepted

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 ...
user avatar
  • 79.2k
10 votes
Accepted

Z-transform of a downsampler

This derivation is a tricky one. The approach suggested before has a flaw. Let me demonstrate this first; then I will give the correct solution. We wish to relate the $\mathcal{Z}$-transform of the ...
user avatar
  • 216
10 votes
Accepted

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 ...
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 ...
user avatar
  • 26.6k
9 votes
Accepted

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 ...
user avatar
  • 79.2k
8 votes

Intuitive interpretation of Laplace transform

Why is the fourier transform a special case of the laplace transform? The Laplace transform produces a 2D surface of complex values, while the Fourier transform produces a 1D line of complex values. ...
user avatar
  • 14.9k
8 votes

Do Causal Discrete-time systems have proper transfer functions?

Note that a stable and causal continuous-time transfer function does not need to be strictly proper but only proper, i.e. the degree of the numerator does not exceed the degree of the denominator, but ...
user avatar
  • 79.2k
8 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 ...
user avatar
  • 79.2k
7 votes
Accepted

What is the significance of Z-transform?

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-...
user avatar
  • 79.2k
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=...
user avatar
  • 79.2k
7 votes
Accepted

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[...
user avatar
  • 4,105
6 votes
Accepted

use chirp z transform for spectral resolution

If you have no prior knowledge about the approximate locations of the frequencies, the Chirp Z-transform is of no immediate use to you. The Chirp Z-transform functions like a magnifying glass, so you ...
user avatar
  • 79.2k
6 votes

Confusion Regarding Bi Linear Transform

The bi linear transform is the transform from the Laplace Transform Domain to the Z Transform. The Laplace Transform Domain is a regular plane. This transform transforms vertical lines in the Laplace ...
user avatar
  • 39.3k
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 ...
user avatar
  • 4,105
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.
user avatar
6 votes
Accepted

$\mathcal Z$-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 ($...
user avatar
  • 79.2k
6 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} ...
user avatar
  • 73
6 votes
Accepted

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 ...
user avatar
  • 4,004
6 votes
Accepted

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 $$ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} = \sum_{n=-\infty}^{\infty} C z^{-n} $$ ...
user avatar
  • 26.6k
6 votes
Accepted

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 ...
user avatar
  • 79.2k
6 votes
Accepted

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 $...
user avatar
  • 79.2k
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 ...
user avatar
  • 79.2k
5 votes
Accepted

How to prove this theorem about the Z transform and final value theorem?

Let me show you a simple way to see this property. Assume $x[k]$ is a causal sequence and let $$x[\infty]=\lim_{k\rightarrow\infty}x[k]$$ be finite. Then the sequence $x[k]$ can be written as $$x[k]...
user avatar
  • 79.2k
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 ...
user avatar
  • 59
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 $...
user avatar
  • 79.2k
5 votes

What is the $\mathcal{Z}$-transform of a constant?

Instead of considering a constant signal, let us consider a rectangular window of length $2 \ell + 1$ $$w (n) = \begin{cases} 1 & \text{if } |n| \leq \ell\\ 0 & \text{otherwise} \end{cases}$$ ...
user avatar
5 votes

Discrete Filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$

You can see by definition that your system is LTI System. Moreover it is casual as it is only depends on input from the past. The question asks for response on the Unit Pulse signal on time 0. In LTI ...
user avatar
  • 39.3k

Only top scored, non community-wiki answers of a minimum length are eligible