# 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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### How can I plot the frequency response on a bode diagram with Fast Fourier Transform?

Assume that we have an unknow dynamical system and we only want to estimate its parameters. The system can be discribed as: Continous time: $$G(s) = \frac{3s + 5} {5s^2 + 3s + 2}$$ Discrete time ...
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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/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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### z-Transform Methods: Definition vs. Integration Rule

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
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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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### Is Z-transform of $\sin(\omega_0n)$ same as that of $\sin(\omega_0n)u[n]$

The Z-transform tables only mention the transform of $\sin(\omega_0n)u[n]$, e.g. #21 at this link: https://en.wikipedia.org/wiki/Z-transform#Table_of_common_Z-transform_pairs But how can I find of Z-...
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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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### $\mathcal{Z}$-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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### Bilateral $\mathcal Z$-transform of exponential

We all know that $a^nu(n)$ has unilateral $\mathcal Z$-transform. But what is the $\mathcal Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'. But bilateral Laplace ...
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### Why Z-transform is considered as separate transform?

The mathematical formula of the Laplace and Z transforms are same with just one difference. I.e. in the first we use $t$ for continuous-time signal and in the latter uses $n$ for discrete-time signal....
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### Squared magnitude of System Function H(z)

If: $$|\alpha|^2 = \alpha \alpha^*$$ Then, why does: $$|H(z)|^2 = H(z) H(z^{-1})$$ instead of: $$|H(z)|^2 = H(z) H^*(z)$$
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### Why eigen values and poles of a system are equivalent?

In control systems engineering, the stability of a system (modeled in the form of Transfer Function) is determined by the poles of the system in the right or left hand sides. When the model is ...
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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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### 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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### $\mathcal Z$-transform ROC

Let's say I have a $\mathcal Z$-transform that represents some transfer function and its has some ROC. My question is how do I know if this system is causal? I know that if the ROC contains the ...
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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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### Minimum number of Poles and zero of transfer function H(z)?

Suppose $G(z)=H(z)(1-\frac{1}{2}z^{-1})$ now in question its saying ROC of G(Z) is entire Z plane except Z=0,so here we need not to add anything because G(Z) already a right sided signal with ROC ...
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### Conjugate reciprocal pairs of zeros and poles in FIR design

Assuming the impulse response $h[n]$ of an FIR filter is real for all $n$, Why are zeros and poles in FIR design found in reciprocal and conjugate pairs? Is the assumption necessary for this ...
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### Pole/Zero existence at infinity

How can poles and zeros exist at infinity?Can anybody explain using a system function?
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### Z transform stability

What is the causality & stability status for three cases shown (aso in attached photo) ? $$H(z) = \frac{z(z-1)}{(z+1)(z+\frac{1}{3})}$$ for three possible regions of convergence as: a-) |z| > ...
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### Inverse z transform - contour integration

Here is my task: Find inverse z transform of $X(z)=\frac{1}{2-3z}$, if $|z|>\frac{2}{3}$ I need to find it using definition formula, $x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$. How can I ...
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### How do I find the difference equation from a transfer function?

Most of the resources I found online go the other way. If I have the transfer function $H(z) = 1 - cos(\theta) \cdot z^{-1} + z^{-2}$ how do I get the difference equation from it so that I can ...
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### Converting poles/zeros to differential/difference equation solutions

Does anyone have a reference handy on how to convert poles and zeros of a system to differential/difference equations. Here is a quick draft of math, but I am not sure if it's at all correct. First, ...
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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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### Finding Fourier transform of a discrete signal from its Z-transform

Is it possible to find the Fourier transform of a discrete signal if you know its $\mathcal{Z}$-transform of?
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I wish to calculate the Final Value of the following system given initial conditions not at rest, and assuming the X will not change from its initial condition value. The transfer function is $$H(z) = ... 3answers 305 views ### Can I study continuous time Fourier Transform and treat the rest as special cases Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to Lapace transform, Z transform, DTFT, DFT, Fourier sequence ... 1answer 120 views ### Transient response of system with single pole 0 \le p < 1 G(z) = \frac{1-p}{z-p} If the value of p satisfies  0 \leq p < 1 there are no oscillations in the transient response. Question: Why is that \uparrow true? I know roughly what a ... 1answer 4k views ### How to implement a filter in matlab I want to implement the 'filter' function in matlab but I just can't seem to replicate the results I get when using the matlab function. My understanding of the matlab function is that it takes 3 ... 1answer 2k views ### use chirp z transform for spectral resolution let us suppose that we some signal x(t),which consist of sinusoidal components and white noise,i would like to know how to use chirp z transform for improving spectral resolution?i found this ... 3answers 545 views ### Necessary Conditions for stability in z domain? What are the necessary(must) conditions for stability in z domain? I am sure about one(ROC must include unit circle) Is there any other such condition which states that there shouldn't be any poles in ... 1answer 2k views ### Downsampling: Mathematical derivation The problem I am having is related to sample rate conversion and more precise to sample rate reduction. I have been working on the paper Interpolation and Decimation of Digital Signals Tutorial Review ... 1answer 59 views ### Magnitude response of mirrored (with respect to unit circle) poles and zeros I just want to check that my understanding about the following paragraph from Optical Filter Design and Analysis by Christi K. Madsen, Jian H. Zhao is correct: A filter’s magnitude response is equal ... 1answer 916 views ### Calculating filter parameters from system response In trying to answer another question, I got stuck applying what I thought would be a valid way to answer it: using the system function. Here's what I did: First, I put the question ... 2answers 603 views ### Deriving Frequency Response for 2-pole Zero-Delay Feedback State Variable Filter I have an existing zero-delay feedback (ZDF) 2-pole state variable filter implementation (along the lines of the theory presented in VA Filter Design by V. Zavalishin), and I wish to determine the ... 0answers 866 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(-... 1answer 66 views ### Z-transform of x[a -n]…where a is int i try to calculate the \mathcal z-transform of x[a-n] (where n is my variable) i can't find any transform. the best suited transform is x[-n] \longleftrightarrow X(z^{-1}) i took the sum ... 1answer 119 views ### Derive the Forward Euler substitution for transfer function In the book "The control handbook. Volume 1 " by Levine, the author shows that the transfer function: can be aproximated and discretized in the transfer function: using the forwar euler ... 1answer 140 views ### Proof of Forward Euler for discretizing a transfer function In Levine book "The control handbook" it is shown that, for discretizing a transfer function \frac{1}{s} using Forward Euler i simply have to replace s with \frac{z-1}{T}. How can extend the ... 2answers 265 views ### Trouble with inverse Z-transform and calculating of samples I have a little problem. I have to solve this task but I can't. Z-transform of sequence \{x(k)\} describe by the formula:$$X(z) = \frac{2.5 -3.15z^{-1} + 1.2 z^{-2}}{1-2.3z^{-1} + 1.2z^{-2}...
I understand that when you have a Laplace function, you can do a bilinear or forward/backward Euler substitution for $s$ to phrase it in terms of $z^{-1}$. In a typical filter, $z^{-1}$ represents the ...
### 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}{\...