Questions tagged [z-transform]

The Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.

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2answers
4k views

Z-transform of a downsampler

In this paper or multirate filtering, the author establishes the following mathematical relationship. Let $y_D$ be the output of a downsampler such that $$y_D[n] = x[Mn]$$ where $M$ is the ...
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2answers
3k views

Finding the z-transform of $h[n] = a^n\cos(2\pi \frac{n}{F_s}f_0)$ for $n ≥ 0$ and zero for $n < 0$

So I'm trying to decide whether the cosine part is intended to be plugged in for $z$ or whether it is strictly part of $h[n]$. (the number a lies in the open unit disk) I mean I was pretty sure it ...
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3answers
3k views

Intuitive interpretation of Laplace transform

So I am getting to grasps with Fourier transforms. Intuitively now I definately understand what it does and will soon follow some classes on the mathematics (so the actual subject). But then I go on ...
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1answer
9k views

How does the $\mathcal Z$-transform's “region of convergence” work?

I'm a novice in DSP and I have few doubts regarding the $\mathcal Z$-transform and its region of convergence (ROC). I know what a $\mathcal Z$-transform is. But I'm having trouble with understanding ...
9
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2answers
405 views

What is the $\mathcal Z$-transform of Bessel function $J_0(\alpha n)$ sequence

What is the $\mathcal Z$-transform of the sequence $J_0(\alpha n)$ for $n \in \mathbb{Z}$? The Fourier transform of zero$^{\rm th}$ order Bessel function $J_0(\alpha x)$ is known to be $\frac{2}{\...
9
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1answer
285 views

Filter order estimation

Assume some unknown but small and finite number of poles and zeros in the complex Z plane, all with complex conjugates, producing some response. Strictly from the absolute value of a set of equally ...
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3answers
10k views

First order low pass filter

I am trying to better understand the first-order low pass filter: Summary: Per wikipedia, a first order low pass filter yields the following in discrete time: $$ \frac{Y(s)}{U(s)}= \frac{\omega_{c}}...
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2answers
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What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
7
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2answers
15k views

Difference between DFT and Z-Transform

I have searched this question but couldn't find the answer in this network. I know this is very confusing question for DSP beginners. Both DFT and Z-transform work for Discrete signal. I have read ...
6
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1answer
219 views

For a discrete LTI system, does “bounded memory” imply “rational transfer function?”

Every LTI system with a $\mathcal Z$-domain transfer function that is a rational function - aka a quotient of two polynomials in $z$ - can be implemented using a bounded amount of memory. Is the ...
5
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1answer
160 views

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

This is a Question asked in IISC ( Indian Institute of Science,Bangalore,India) interview for MS admission. What is the $\mathcal{Z}$-transform of $\dfrac 1{n^2}$ ?
5
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1answer
13k views

Position of poles and Stability in $z$ domain

We know in Laplace Transform, if the poles lie on the left of $j\omega$ axis, we can say the system is stable. Similarly can we comment on the stability based on poles position in $\mathcal Z$-...
5
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1answer
323 views

Understanding the $\mathcal Z$-transform

I was studying $\mathcal Z$-transforms and found pretty good material on the topic, though I feel I do not have a proper understanding of the concept. Could someone help me clarify this? I know that ...
5
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1answer
3k views

Do Causal Discrete-time systems have proper transfer functions?

In the case of continuous-time systems, if the system is causal, its Laplace transfer function is strictly proper (the degree of the numerator is less than the degree of the denominator). Is this ...
5
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2answers
581 views

Understanding $\mathcal Z$-transforms and pole locations

I am trying to gain a better understanding of pole locations in the $z$-plane of a given discrete transfer function, $H(z)$. I think I have a pretty good understanding of how to use the $\mathcal Z$-...
5
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2answers
261 views

Find the frequency response of a system

I'm trying to find the frequency response $$H(\omega) = Y(\omega)/X(\omega)$$ for this system- the signal equations are given: $$y[n] = v[n - M] - g * v[n]$$ $$v[n] = x[n] + g * v[n - M]$$ I've tried ...
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3answers
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Where can I find a table of $z$-domain coefficients for Butterworth filters?

The primary source lists Butterworth polynomials in s-domain and provides a link to bilinear transform for digital implementation. But, who needs analog specification in our digital world? Why should ...
5
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2answers
299 views

How to determine if a filter is bandpass/stopband from its pole-zero diagram in z-domain

How can we determine if a filter is bandpass or stopband, just by looking at its pole-zero diagram in z-domain? For exmaple, if we have a system with third-order pole at the origin and a zero on the ...
5
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1answer
308 views

Is there a $z$-transform like for variable sampling rate signals?

I'm working with signals with variable sampling rate (the time space between samples is not constant). I know the delay between samples but I don't wont to interpolate the signal. Is there a way to ...
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2answers
3k views

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

The $\mathcal Z$-transform uses the same notation as the unit delay $z^{-1}$, but in $\mathcal Z$-transform $z$ is a complex number. What's the relation between the $\mathcal Z$-transform and the ...
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3answers
4k views

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

It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \...
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2answers
273 views

Can I determine a system's $z$-domain transfer function from its pole-zero plot?

Is it possible to generate the z-domain transfer function from a pole-zero plot diagram?
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2answers
95 views

Why does this transfer function has a second zero

I'm learning about $\mathcal Z$-transforms in DSP and I have a transfer function of the following form: $$H(z)=\frac{2-3z^{-1}}{1-1.6z^{-1}+0.8z^{-2}}$$ When I calculate zeros and poles of this ...
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3answers
500 views

How to find the difference equation directly from Direct Form II signal flow graph

I am trying to solve for the difference equation of the following signal flow graph: I am aware that Direct Form II can be converted to Direct Form I, which finding the difference equation directly ...
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2answers
2k views

Minimum phase systems with pole at infinity

If a system is given by a transfer function in the $z$ domain that has all poles and zeros inside the unit circle except for a factor of $z^{-1}$ in the denominator (pole at infinity), can it still be ...
4
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1answer
46 views

s-Domin or z-Domain - What to Use for Mixed systems

I recently had to deal with a power electronic system where I had to implement a dynamic model of a power converter in order to design a suitable controller for that converter. The control will be ...
4
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1answer
692 views

Determining the final value of the output of a discrete system

I'm going through an exam question where I've been told that the samples $f(kT)$ of the following function \begin{equation}{F\left(z\right)=\frac{1}{1-0.819z^{-1}}} \end{equation} are applied to a ...
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1answer
11k views

Identifying the magnitude and impulse response from pole zero plot quickly

I have an exam next week and it's verty certain that a task of this kind will be there. Are there some good tips how to match the right pole zero plot to the right responses? No proof is needed in ...
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1answer
11k views

Z-Transform of a^|n|

I am wanting to compute the Z-transform of $f(n) = a^{|n|}$ . 'a' is a positive constant. Looking at the transform table, I found that Z-transform for $a^n u(n)$ is available from the tables and is $...
4
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1answer
149 views

How can the order of a transfer function be derived from its equivalent state space representation?

Suppose I have a discrete state space model: \begin{align} \theta[k+1] &= A \theta[k] + B u[k]\\ y[k] &= C \theta[k] \end{align} I know that the equivalent transfer function can be found by ...
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0answers
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Attempting to use the trapezoidal rule to form a difference equation representing a circuit

I have a differential equation that has been proven to be correct. The transfer function obtained by Laplace domain analysis and Matlab freqs match up and all is well. The problem is somewhere in ...
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2answers
3k views

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

The Fourier transform of a constant exists. Can anyone please tell me what the $\mathcal{Z}$-transform of a constant is? Thanks in advance.
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1answer
2k views

Poles and zeros of a transfer function

What are the poles and zeros of this transfer function (in $z$): $$H(z)=z+2+z^{-1}$$ and how would you approach the resolution of such problem? Personally, I would write $$H(z)=\displaystyle\frac{...
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3answers
2k views

How can I prove stability of a biquad filter with non-zero initial conditions

Ok, so the situation is that I have a DFII biquad with some filter coefficients: \begin{align} w[n] &= x[n] - a_1*w[n-1] - a_2*w[n-2]\\ y[n] &= b_0*w[n] + b_1*w[n-1] + b_2*w[n-2] \end{align}...
3
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1answer
238 views

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

I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity $$xy=\frac{1}{2}x^2 + ...
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2answers
126 views

Finite length truncated exponential sequence $\mathcal Z$-transform, zeros over circle explanation

I'm looking at an example on how to obtain the $\mathcal Z$-transform from a finite length truncated exponential sequence, namely: $$x[n] = \begin{cases} a^N &\text{for} & 0 \leq n \leq N-1\\...
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2answers
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Find output of digital filter given input and transfer function?

Hey guys so I have an input sequence that are real values that represent intensity of a column of an image. I also have the transfer function in the Z domain of a 2nd order high pass filter. I have ...
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2answers
2k views

Determining which Filter from a Z-Plane Plots?

How do i determine which FIR filter (LP, HP, BS, BP) it is from looking at it's z-plane plot?
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2answers
3k views

Are Z-transform time shifting and differentiation properties always compatible?

I'm currently studying the Z-transform, and I'm having issues in understanding the time shift and differentiation properties, to be precise: calculating a Z-transform explicitly, and obtaining it by ...
3
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1answer
287 views

Generating a time series given a transfer function

I'm trying to work my way through a paper I found online, titled "Three Models of Wind-Gust Disturbances for the Analysis of Antenna Pointing Accuracy" by W. Gawronski, 2002. http://ipnpr.jpl.nasa....
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1answer
142 views

Is this a valid z-transform of $ \frac{1}{n^2}u[n] $?

I was looking over some DSP and came across the following signal: $x[n] = 1/n$. So I wondered whether it had a z-transform but I soon realized that it fails to meet two conditions described here: .....
3
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1answer
434 views

ROC of this LTI system given $x[n]$ and $y[n]$

So I have a system with the following inputs and outputs: \begin{align} x[n]&=\left( \frac12 \right)^{n}u[n] + 2^{n}u[-n-1]\\ y[n]&=6\left( \frac12 \right)^{n}u[n] - 6\left( \frac34 \right)^{...
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3answers
539 views

Question about z transform

After studying z transform from different books and literature on internet I want to ask few which makes me confuse. a) From the Discrete Time Fourier Transform we have drive equation for z ...
3
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1answer
123 views

Why is a feedback loop not represented by the least order transfer function?

I have a feedback loop with transfer $L(z)= \frac{H(z)C(z)}{1+H(z)C(z)}$. $$H(z) = h\quad \text{and} \quad C(z) = \frac{K}{z-\alpha}.$$ If I manually calculate the transfer function, I get: $$L(z) =...
3
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1answer
130 views

Converting transfer function that is a sum of unusual rational polynomials to finite difference equation

I have the following rather exotic transfer function: $$ H(z) = cz^{-m} + \frac{b_0 z^{-1} + b_1z^{-2} + \dots + b_{2m}z^{-2m}}{1 + a z^{-1}} + \frac{q_0 z^{-1} + q_1z^{-2} + \dots + q_{2m}z^{-2m}}{1 ...
3
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1answer
200 views

Can the Z-Transform be used to create smoothed 3D surfaces from point clouds?

According to Dr. Math the Z-transform can create closed-form solutions for 1D series defined by difference equations (e.g. the Fibonacci series). My 3D surface $z=LC(x,y)$ is defined by difference ...
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2answers
598 views

How to intuitively understand the state space formulation of discrete time system?

The SS formulation of DT system is given by $$x[(k+1)T] = Ax(kT) + Bu(kT)$$ $$y(kT) = Cx(kT) + Du(kT)$$ Note: T is the sampling period and often omited Can someone explain to me why the state ...
3
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1answer
69 views

Potential issues arising from too stable discretization

When numerically simulating a system, usually some kind of discretization is necessary, obtained by some kind of z-transform, such as, for instance, the bilinear transform $s\mapsto \frac{2}{\triangle ...
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3answers
353 views

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

Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT). CFT can be used in case of continuous ...
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2answers
190 views

Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

I'm currently in the process of going over the $\mathcal Z$-transform and more specifically its derivation. I understand and I am able to follow it up until the final step whereby involving the ...