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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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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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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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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 ...
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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}{\...
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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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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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$\mathcal{Z}$-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 $\dfrac 1{n^2}$ ?
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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 $\mathcal Z$-...
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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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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 ...
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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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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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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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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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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 $\mathcal Z$-...
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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 ...
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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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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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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 ...
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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 ...
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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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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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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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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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Finding the $\mathcal Z$-transform of $((-\frac{1}{3})^n + \frac{1}{2})^n \mu[n-2]$?

What is the $\mathcal Z$-transform of $\left(\left(-\frac{1}{3}\right)^n + \frac{1}{2}\right)^n \mu[n-2]$? Doing raw computations with large $n$ gives a sum of $0.6030$ which doesn't seem right, and ...
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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 ...
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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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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}...
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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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Downsampling and Then Upsampling

Given this system: I need to show the $\mathcal Z$-transform of $y[n]$ as a function of the $\mathcal Z$-transform of $x[n]$. Now I know that for downsampling alone: $$Y(z) = \frac1M\sum_{m=0}^{M-1}...
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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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When inverting a transfer function, solving for the input using the output does the causality status change

suppose $y(n)=ax(n-1)+bx(n-2)+\dots$ ($y$ is the output and $x$ the input). What happens if I want to solve $x(n)$ from $y(n)$? Z transform: $$Y(z)=G(z)X(z)\tag{1}$$ then $$X(z)=\frac{1}{G(z)...
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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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Find the transfer function of the difference between IIR and FIR filter

I am using a filter following equations from papers. It is basically the difference between exponential moving average and simple moving average. $$L[n]=\frac{1}{\alpha_L}f[n]+\left(1-\frac{1}{\...
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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 ...
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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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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 ...
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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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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: .....
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174 views

Impulse response of a continuous system sampled with zero-order hold

I've a continuous system $$F(s) = \frac{K}{Ts+1}.$$ I sample it with zero-order hold with sampling period $T_s$. The discrete system transfer function is $$ \begin{aligned} G(z) &= % \frac{z-1}{z}...
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Problem designing a specific FIR filter

Consider an LTI system whose impulse response is $$h[n]=\frac{1}{2^n}u[n]+\frac{1}{3^n}u[n]$$ The input signal to this system is $x[n]$ and is null for $n<0$ but may or may not be null for $n=0$. ...
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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) =...
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Control systems and convolution

I think i am not understanding the concept of convolution well. Lets say we are given a system impulse response in the S-domain, and we have implemented a controller $G_c(s)$ that will adjust the ...
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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 ...