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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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3answers
61 views

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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3answers
32 views

factoring poles / zeros: off by constant gain compared with textbook

(From Schaum's DSP outline, 2nd edition, problem 5.32) Book says factor it and extract H(z) from the factored product: $$ H(z)H(z^{-1})= \frac{ \frac{5}{4} - \frac{1}{2}z - \frac{1}{2}z^{-1} }{ \...
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2answers
5k 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
256 views

Design discrete controller for zero steady state error

I have the following system where $$G(s)=\frac{0.5}{s+1}+\frac{5}{s+10}$$ How can I design the C(z) controller so that the steady state error for a step input r(t)=1(t) is zero? I know that this ...
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1answer
213 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 ...
2
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1answer
37 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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1answer
22 views

z-transform causality properties: negative coefficents are zero ($x[-1]z^1=0$, $x[-2]z^2=0$, …)

Let's suppose I have a system: $$Y(z)=X(z)H(z)$$ If the system is causal, does that mean that all the negative coefficients (example: x[-1]) of the transform for $Y(z)$, $X(z)$, and $H(z)$ are zero?...
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1answer
50 views

partial fractions expansion inverse Z-transform, help

I have the correct solution from teacher's solution guide, but I was slightly confused by some algebra about the partial fractions expansion evidently difference equation is as follows $ y[n] = \...
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1answer
104 views

Impulse response of a second order LTI

I have a set of measurement which I want to model with 2nd-order difference equations (first order eqs don't model well enough). The equation is $$y[n] = \alpha_1 y[n-1] + \alpha_2 y[n-2] + \beta_0 ...
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1answer
15 views

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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1answer
41 views

ROC of inverse system function

If the region of convergence (ROC) for system function $H(z)$ is $R_h$, what is the ROC of the inverse function $G(z)=\frac{1}{H(z)}$? $$$$
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1answer
60 views

transfer function and 'causal' signal - evaluate transfer function or use z-transform of input?

From my studying difference equations and transfer functions, I understand that when a complex exponential input $x[n]=z_1^n$ is applied to an LTI system with transfer function $H(z)$, determining the ...
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1answer
88 views

How can one know the polynome equation based on the output of a system?

I came across this situation in my textbook: However I have no clue about how you can (starting from the stepresponse on the left), get the polynome equation on the right. Could someone please ...
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1answer
58 views

Taking the inverse $\mathcal Z$- transform with a summation in the denominator

I'm learning about z-transforms, and was going through some practice problems and I've been stuck on this one for a little bit. I'm trying to take the inverse z-transform of the following: $$\frac{1}{\...
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0answers
417 views

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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0answers
75 views

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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0answers
140 views

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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0answers
19 views

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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0answers
38 views

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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0answers
50 views

Sampling and ideal reconstruction of signal

Two time discrete signals $x_1(n)$ and $x_2(n)$ are produced by sampling the continuous signal $$x_a(t) = \cos(2\pi300t) + \cos(2\pi600t) $$ with the sample frequency $F_s = 1000\ \rm Hz$. For the ...
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47 views

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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167 views

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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0answers
364 views

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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0answers
93 views

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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0answers
309 views

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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0answers
20 views

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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0answers
18 views

Is it customary to use the period as coefficient when sampling the impulse response?

I have an analog, continuous impulse response $$h_a(t)=u(t)\cdot\sum_{n=1}^4A_ne^{s_nt}$$, and by sampling it with the usual method I get $$h[n]=\Delta th_a(n\Delta t).$$ Now, that $\Delta t$ is a bit ...
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0answers
160 views

Help needed with finding ROC of transfer function

I've been doing some practice with the $\mathcal Z$-transform for an exam, and I'm not sure if my approach is correct to this problem: My approach: I wrote $y[n]$ as follows: $$ y[n] = 2\cdot\...
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0answers
199 views

Confusion over impulse invariance, matched z-transform, and bilinear transformation methods

In the DSP course that I am taking in my university as an undergraduate student, three methods are presented for mapping analog filters to digital filters - namely, impulse invariance, matched z-...
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64 views

Question regarding ROC of transfer function

I've been trying to understand how to determine the Region of Convergence (ROC) of $H(z)$ given $X(z)$ and $Y(Z)$ for some time, and just can't wrap my head around it. I know that $Y(Z) = X(z)H(z) \...
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0answers
66 views

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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139 views

Inverse Z-Transform of a Complex Filter

What is the inverse z-transform of $$ 1/(1-az) $$ where a is complex and |a| < 1
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71 views

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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143 views

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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213 views

Deterministic autocorrelation sequence from $\mathcal Z$-transform

If we have $$ X(z) =\frac{1}{1-az^{-1}} - \frac{1}{1-bz^{-1}} $$ Can you simply multiply $X(z) \cdot X(z^{-1})$ and take the inverse $\mathcal Z$-transform to find the deterministic autocorrelation ...
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135 views

$\mathcal Z$-transform of auto-correlation

Assume signal $x(n), d(n)$, filter $\mathbf{w}(n)$, define error signal as $$e(n)=d(n)-\mathbf{w}^T(n)\mathbf{x}(n)$$ Auto-power spectrum is defined as $$ S_{ee}(z)= \mathcal{Z}\left[r_{ee}(k)\right]...
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122 views

Solving a linear ODE in the z-Domain

I'm stuck with an excercise or rather unsure how the solution came to be. the problem is the following: The following ODE is given: $$ x[k] + x[k-3] = 0 $$ Calculate the non trivial solution $...
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0answers
188 views

Z transform of $\sum_{k=0}^{n}3^{k}$

My task is to calculate z transform of signal $x[n]=\sum\limits_{k=0}^{n}3^{k}$ ? By definition, $$ \begin{align} X(z) &= \sum\limits_{n=-\infty}^{n=\infty}x[n]z^{-n} \\ &= \sum\limits_{n=-\...
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70 views

Multi-Channel LTI Inverse system problem

A sequence $x[n]$ is the output of a linear time-invariant system whose input is $s[n]$. This system is described by the difference equation $(1.1)$ $$x[n]=s[n]-e^{-8\alpha}s[n-8]$$ $$\alpha>0$$ a)...
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0answers
817 views

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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0answers
396 views

IIR filter SOS and Direct Forms doubt

I have below doubts, so confusing! As I don't want to assume from what I read, I am asking for help here! Are Second Order Sections another name for biQuads ? If I have 2 single pole transfer ...
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221 views

Why the number of poles and zeros for a RHS signal in the Z domain is equal?

I can not understand the reason why the following sentence is true: If we have a Right-Hand signal(RHS) x(n), X(z), the Z-transform of x(n), has the same number of poles and zeros except at z = ...