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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Find moments of a discrete process using $\mathcal{Z}$-Transform method

For a linear process relating a variable $Y_i$ to random, independent variables $X_i$ using the equation: \begin{equation} Y_i = aY_{i-1} + (1-a)X_i \end{equation} which has the solution: \begin{...
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Understanding convolution of Chirp z algorithm

I dont´t understand how works the convolution part of the Chirp z. I understand how the DFT is transformed \begin{align*} x(k) = \sum_{n=0}^{N-1} x(n) W_N^{kn} \end{align*} to this expresion: \...
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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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Inverse z-transform. Where is mistake?

I've already wrote about that trouble (link here), but I don't understand where I've made a mistake. Full description of the task is as follows: Z-transform of sequence {x(k)} describe by the ...
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How to transform a Fractional Order Laplace Transfer Function into a digital filter?

I'm working with loudspeaker impedance analysis. Electrical behavior of loudspeakers can be modeled with RLC networks. But real loudspeakers have components, that exhibit some non-linear and frequency ...
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Why does Simulink generate this code for a PID controller?

For the Simulink PID Controller model The Simulink generated code (rewrite for better understanding) is: ...
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chirp z-transform for different output sizes

I am attempting to use the chirp z-transform for an application that requires arbitrary FFT output sizes less than or equal to the length of the input signal. However, I've encountered an issue where ...
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How does the ROC (Region of Convergence) related to a real world application?

In class, we are often given exercises to find the impulse response, output, and Z-transform of a system. In addition, we are often asked to define the Region of Convergence (ROC) depending on where ...
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Z Transform of M-Fold Decimation

I know this is probably a common question, but after some searching I think that my version of the question is slightly different -- apologies if this is a repeat. I have seen that it is inaccurate to ...
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Z-transform of $\cos(\omega_0 n(n+1))u[n]$

I'm doing some research on Zadoff-Chu sequences and as a part of it I wanted to find the Z-transform of: $$\cos(\omega_0 n(n+1))u[n]$$ Wolfram Alpha / Mathematica couldn't help out. I couldn't find a ...
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Alternative function for MatLab iztrans to Octave?

I am trying to compute the reverse Z transform on Ocatve and I get the following error: error: 'iztrans' undefined near line 1, column 1 The code I am running is ...
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Summations in Z-Transform

I'm currently working on a problem that involves a Z-Transform. Basically, the essence of the problem is that if: \begin{equation} H\left(z\right) = \sum_{n=0}^{N-1}h\left(n\right)z^{-n} \end{equation}...
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What are the missing steps in the derivation of this equation?

I am trying to understand the model of a piano hammer described here. The relevant excerpt is: As I understand it, they are saying in (12a) that if the hammer spring's uncompressed end's y position ...
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Powers of $z$ in a discrete transfer function

I have been working with discrete-time systems and I am not yet able to understand the reason behind using powers of $z^{-1}$ in transfer functions. I get that $z^{-1}$ means a delay, but when we ...
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Region of convergence for discrete system

I am confused by two questions on the region of convergence (ROC) when applying the $Z$-transform. Consider a discrete-time linear system $$x[n+1] = Ax[n]+Bu[n], \qquad y[n]=Cx[n]+D$$ whose transfer ...
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Z-transform of not quite an upsampler

I know the z-transform of an upsampler is: $$ y[n] = \begin{cases} x(n/L) &n=0,\pm L, \pm 2L, ...\\ 0&otherwise \end{cases} \longrightarrow Y(z)= X(z^{L}) $$ if $x[n]_L$ is defined to zero ...
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Finding Z transform of a signal: Intermediate steps

Find the Z transform of $y(n)=x(n+2)u(n)$ I have solved the problem. I have doubt whether it is correct or not. It would be very helpful if someone could check whether the steps that I have ...
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How do I convert a two-pole two-zero transfer function from the s-domain to the z-domain?

I'm trying to convert the following IIR transfer function from the s-domain to the z-domain: $$ H(s) = \displaystyle\frac{\frac{s^2}{\omega_z^2} + 2\zeta_z\frac s\omega_z + 1}{\frac{s^2}{\omega_p^2} + ...
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periodicity, minimum phase, maximum phase, interpretation

I have a finite linear difference equation $$y(n)=ax(n-1)+bx(n-2)+cx(n-3)+\ldots+fx(n-m)\text,$$ relating an input $x(n)$ to an output $y(n)$. If I assume periodicity of type $x(n-2)=x(n)$, the ...
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Efficient computation of Chirp Z Transform

Chirp Z Transform (1, 2, 3) is more powerful than zooming techniques (I use it to actually trace non-stationary chirp signals) and very usable in signal processing, but it's flexibility comes at price ...
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Inverse z transform - Pair of complex conjugate poles

How can I perform the inverse z-transform on the following $H(z)$ to be able to calculate a real-valued impulse response $h[n]$? $$ H(z)=\frac{z^2}{z^2+0.8\sqrt{2}z+0.64} $$ My idea was to find an ...
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Deriving finite impulse response for polylogarithm

As a part of my research i have to use the following z-transform in matlab 'filter' function so as to derive the convoluted signal from the original one. $$\frac{1}{{\rm Li}_{k}(z^{-1}e^{-b})}$$ ...
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Z-transform and binomial series

I am reading a paper on frequency warping and I need to do a little manipulation of the Z-transform. Can somebody help me on how can I go about deriving equations $(3)$ and $(4)$ from equations $(1)$ ...
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How to compute the inverse Z-transform

How to compute the inverse Z-transform of the form $$ G(z)=\frac{z^{2n}}{a(z^{2n})+b(z^n)+c} $$ I started by taking $$ F(z^n)=G(z) $$ so $$ \frac{F(z)}{z}=\frac{z}{az^2+bz+c}$$ This can be solved ...
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Convolution between bit stream sampled by N number of samples per bit with a channel modeled as low pass filter

I generated a bit stream in MATLAB, from the transmitter to be convoluted by the channel I use MATLAB function "randi" between 0 & 1, so it generates random vectors of 0s & 1s and ...
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Poles and Zeros of a DiscreteTimeModelFunction with delay in Wolfram Mathematica

help me please, I have a problem with this TransferFunctionModel, When I want to obtain the poles from the Discrete Model Wolfram gives me an error, what am I doing wrong? Could it be the delay of the ...
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How do I represent any signal in the form x[n]u[n] and x[n]u[-n-1]?

For eg-: If I have the signal x[n]={1,a,a²...}I can represent it as a^n * u[n]. Similarly if I havethe sequence as x[-1]=-a^-1 x[-2]=-a^-2 Then I can represent it as x[n]=-a^n*u[-n-1].
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$Z$-transform of a multilinear function/ consecutive multplication of $k$ signals $y_1(n), \ldots, y_k(n)$

How should one go about calculating the $Z$-transform of a signal that is the multiplication of $k$ signals (i.e. a multilinear function with regards to signals $y_1(n) \ldots y_k(n)$ ? Namely, $\...
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Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically with the signals in ROC?

Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically? Are we supposed to manipulate our O/P by multiplying any signal from ...
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Let a LTI system be causal and stable with the transfer function being... show that

if the system is an IIR LTI causal and stable one, and the transfer function is \[H(z)=\sum_{n=0}^{\infty}h[n]z{^{n}}= \frac{G}{1 -\sum_{k=1}^{p}a_kz{^{-k}}}\] show that the cepstrum of this system ...
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How to find H(z) from just zeros and poles

I have a system with a DC gain of 8, poles at z = +- j/2 and zeroes at e^+-j5, I need to find the H(z). I have tried this but not sure if it is right. $$ H(z) = G_o * z^{-1} \frac{(z-z_0)(z-z_1)}{(z-...
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Order one FIR Filter with complex coefficient

I am trying to learn about the behavior of the FIR filter however with complex coefficients. The filter I am trying to analyze is the following: $$H(z)=a+jbz^{-1}\quad\text{where the variable}\quad j =...
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Gradient of transfer function (z-transform) with respect to coefficients/parameters?

my problem is perhaps very simple but I just can not find the answer, even though this is used (though not explained) in several books: What is $\frac{\partial G (z, \theta)}{\partial \theta}$ when $G$...
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Why the transfer function is equal to the output in this case

In this description of transfer functions on the z-plane (image linked), I'm confused by equation 1.49, which says that $H(f)=v_{out}(f)$ when $v_{in}(f)= 1 * e^{j 2 \pi (f/f_s)}$. (For another matter ...
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How to break a second-order filter into two first-order filter

Let's assume the transfer function of a continuous-domain filter consists of two poles and one zero: $H(s) = \frac{k_c (s-\omega_{z_1})}{(s-\omega_{p_1})(s-\omega_{p_2})}$. Let's consider we do the bi-...
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How to design IIR digital filters

Practical Infinite-Impulse-Response (IIR) filters are usually based upon analogue equivalents (Butterworth, Chebyshev, etc.), using a transformation known as the bilinear transformation which maps the ...
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Calculating the output of a pole eigen signal in a difference equation

Let an IAR system be defined by the following difference equation: $$y[n]-\frac{1}{4} y[n-2]=x[n]+3x[n-1]$$ and an input signal $x[n]=(-0.5)^n$. The transfer function is defined as $H^z(z)=\frac{1+3z^{...
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finding power spectral density from a vector

I have been given a vector: \begin{equation} v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix} \end{equation} my job is to find the power spectral density from this vector \...
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Stability of $x(n) = A x(n-1)+b$

I am looking at the following system: $x(n) = A x(n-1) + b$ where x and b are vectors and A is a matrix. How can I derive the stability and causality conditions for such a system using Z transform? If ...
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Are discrete time functions expected to be this inaccurate?

I am attempting to test some discrete time functions against their continuous counterparts and I am surprised by how inaccurate the discrete versions are coming out. I am modeling the displacement, ...
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Does separation of variables of a transfer function followed by a z-domain transform work?

I tried taking a low pass filter transfer function in the form $A/(s^2+Bs+C)$ and separating it into multiple fractions over each root $(D/(s+r1))+(E/(s+r2))$. Then I substituted $s =((2/T)*((z-1)/(z+...
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non-uniform antenna array design

There are tons of papers discussed this topics and most of them are related to fancy optimization techniques (convex/non-convex). I am wondering if we can have a simpler way to find the antenna ...
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What is prewhitening filter mode?

In this paper, the following prewhitening filter is described: $$ C(z) = \sum_{k=0}^n c_{k}z^{-k} $$ where $n$ and $c_k$ are known. The paper also describes the values $C(\lambda_{k})$, with $\...
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Where to start in the design of my filter to remove 50 Hz

pick any 3 random files from this database, : https://physionet.org/pn3/ecgiddb/ They are subject to 50 Hz powerline interference. We wish to convert from time domain to frequency domain and remove ...
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The range of r can be r<1 and r>1

I have to find the range of r which makes H(z) stable. There is no restriction of left sided or right sided. Then, both r<1 (z>r) and r > 1 can be the answer?
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Please explain Multiplication property in Z transform?

I am having problem visualizing contour any example will be great help. As far as as where I need it I was trying to find Z transform of a contracted signal by first multiplication by impulse train.
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Determine filter type using recurrence relation

Given the recurrence relation: $y[n] = x[n] + 0.5y[n-1]$ I want to determine the filter type (i.e. LPF, HPF etc.) I try to use Z transform, and get that the the transfer function is $H(z) = \frac{2z}{...
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DTFT of $ f[k] = 3^k u(-k-1)$

Find the Discrete-time Fourier transform of $ f[k] = 3^k u(-k-1)$ (then sketch it and find its magnitude & angle). It doesn't fit any templates on the Fourier table, and I don't see how one ...
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z-transform of $2^k$

It seems that you can decompose it as such: $f(n) = a^n u(n) + a^{-n} u(-n-1)$ But I already have issue here, is it basically saying that $ u(n) + u(-n-1) = 1$? this is the plot of u(n) and u(-...
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Inverse Chirp Z Transform

I am working to understand and use the Chirp Z-Transform. I want to use the algorithm for simple signal processing on data sets that are not a power of two. I need to be able to inverse transform as ...
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