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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357 views

MATLAB Implementation of Karplus Strong algorithm with filter function?

I want to implement following function: y = ksalgrithm(x, alpha, M, Nout) where x is the input vector with length ...
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295 views

A question regarding z transform and its magnitude response

My teacher of signals and systems gave us a review problem as following: given a DT rightsided LTI system with transfer function $$\frac{1-a^*z}{z-a}, \left | a \right |<1 $$ show that the system'...
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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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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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429 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 ...
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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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54 views

Notation of an LTI system consisting of LTI filters

I would like to find a reference for two notations of an LTI system consisting of LTI filters. In z-domain, the LTI system is given by $$ \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \...
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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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178 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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When can the $\mathcal Z$-transform be inverted? When not?

What are the conditions that must be satisfied to be able to invert the $\mathcal Z$-transform?
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Laplace transform of product of signal and impulse train

I'm reading 'Discrete Time Control Systems' book by Ogata and came across a few statements (specifically, (3-1) and (3-2)) which I have not been able to understand. It is said that any continuous ...
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Help with a $z$-transfrom Problem

I have the function $$(1-e^{-2n})u[n-1]$$ where $u[n]$ is the step input. I want to find the $z$-transform for this function. I know that the transform of $1-e^{-2n}$ will be $$\frac{z}{z-1} - \frac{...
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165 views

What is the rule for manipulating the boundaries of a summation?

When working with DTFTs or $\mathcal Z$-transforms, we sometime get summations that do not go from $n=0$ to $+\infty$. For example, suppose we have the sequence $x(n) = -\alpha^n u(-n-1)$. To find ...
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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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Find the $\mathcal Z$-transform of this function?

I need to find the $\mathcal Z$-transform of $x(n)=a^{-n} u(n)$. Assume, $a$ is a positive constant , but the power of $a$ is negative. Looking at the transform table, I found that $\mathcal Z$-...
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238 views

Finding the minimum phase h[n] and its Z transform

Hello, this is one of my homework questions and i have already solved the first question but im having trouble gettin a relation that helps me solve the second one. From the question i understand that ...
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2answers
229 views

Simplify equation of single pole IIR transfer function

Example - Consider the causal stable IIR transfer function $$ H(z)=\frac{K}{1-\alpha z^{-1}}, \quad 0 < \lvert \alpha\rvert 1 $$ where $K$ and $\alpha$ are real constants Its square-...
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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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Bilinear Transformation Comparison

If I have transfer function coefficients, I can analyze the transfer function in the s-plane and/or the z-plane. If I wanted to demonstrate that the z-plane and s-plane responses are equivalent: ...
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139 views

Does $H(-z)$ produce aliasing? [closed]

Given $H(z)$ is the z-transform of a signal, I know that $H(-z)$ results in shifting of frequencies in DTFT by $\pi$ or $-\pi$. Does it produce aliasing ? How ?
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Poles and zeroes - $\mathcal Z$-transform

I just have a small question, something that I am unsure about. I have a difference equation for a filter: $$y[n] = x[n] - x[n-1]$$ I have worked out the $\mathcal Z$-transform: $$\mathcal Z\left\...
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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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$\mathcal Z$-transform of an equation [Exam question]: Verifying the solution

I'm studying for exams at the moment and I'm trying to reproduce a solution from my professor (I have the solutions). The following signal is given: The excercise says: Calculate the Fourier ...
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429 views

Non invertibility of system $y[n]=x[n]-x[n-1]$ using transform method?

For the system to be invertible, we should have different outputs for different inputs. In terms of constant functions say, $$X_1[n]=3 \quad \forall n \in \mathbb{Z}$$ and $$X_2[n]=4 \quad \forall ...
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848 views

Inverse $z$-transform of a transfer function in MATLAB

I have designed a Butterworth highpass filter (HPF) of 4th order with cutoff frequency high enough to give a gain of $3$ at high frequencies. I want to find the inverse $z$-transform using MATLAB. <...
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204 views

Gain at given frequency from $z$-plane zero-pole plot. Two methods gives different results

I have two zeros at $z=-1$ and two complex conjugate poles at $z=A\cos\theta\pm jA\sin\theta$ This gives me the next transfer function $$H(z)=\frac{1+2z^{-1}+z^{-2}}{1-2A\cos\theta z^{-1}+A^2z^{-2}}$...
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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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What additional information do we get from z-transform that we don't get from DTFT? [duplicate]

As an engineer analyzing a system (whether it be a circuit or an audio sample), you should know when to apply the analysis tools you've been given--such as Discrete Time Fourier Transform and Z-...
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467 views

Is the following system stable and causal?

Suppose the following function describes the unit step response of a system, where $u[n]$ is the unit step function. $$ y[n]=\left(\frac{1}{2}\right)^{n-1}u[n+1] $$ I want to find out the system ...
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2k views

Magnitude of function in $z$ domain

I am newbie to $\mathcal Z$-transform, I searched to find the magnitude of a function in $z$-domain, but I couldn't find anything, for example when we have $$ H(z) = \frac{z-3}{z-0.5} $$ How do you ...
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526 views

Phase Contribution of Zeros in Z-plane

I have been studying Proakis, and stumbled upon minimum/maximum/mixed phase concept. He said that all zeros inside unit circle in Z-plane contributes zero phase change, and all zeros outside unit ...
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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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Finding the inverse Inverse $\mathcal Z$-transform

I'm working through and learning about Inverse $\mathcal Z$-transforms right now, and I'm getting caught up on trying to find the Inverse $\mathcal Z$-transform of the following: $$\sum_{k=1}^P\frac{1}...
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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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166 views

Getting the $\mathcal Z$-transform of the moving average summation

I'm getting started with signal processing, and I'm stuck on a problem that asks me to take the $\mathcal Z$-transform of the following causal, DT-LTI system: $$\sum_{k=0}^M b_k x[n-k] $$ I'm not ...
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How to find ROC of system when input is two sided and output is one-sided

Given the $\mathcal Z$-transform of input $x[n]$ and output $y[n]$, how can I find the ROC of the system function $H(z) = Y(z)/X(z)$? I have $$X(z) = \frac{2z\left(z-\frac{10}{3}\right)}{\left(z-\frac ...
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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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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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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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1answer
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How to identify causality, stability and ROC from the pole-zero plot?

To preface, this is not a homework related question but purely for self-study purposes. If I am given the following Pole-Zero Plot: (Source: Berkeley Exam1) How would I go about trying to ...
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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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576 views

Find the difference equation and draw the simulation diagram

Calculate the difference equation and then draw the simulation diagram of the below transfer function. $$ H(z) = \frac{Y(z)}{X(z)} = \frac{0.4142 + 0.4142z^{-1}}{1.4142 - 0.5858z^{-1}} $$ I ...
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Why correlation property of $\mathcal Z$-transform contains a time reversal operation

I'm reading through Digital Signal Processing, Proakis and Manolakis, third edition. I've reached section 3.2: Properties of $\mathcal Z$-transform. One property is the convolution: $$x(n) = x_1(n)\...
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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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$\mathcal{Z}$-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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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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Learning about inverse-z-transform and how to apply it to a rational transfer function

I have been studying IIR filters and know that a rational transfer function: $$ H(z) = \frac {b_0 + b_1 z^{-1} + ... + b_N z^{-N}}{1 + a_1 z^{-1} + ... + a_N z^{-N}} $$ has a finite difference ...
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Inverse $\mathcal Z$-transform problem

$B(z)+B(-z) = 2c$, explain the structure of $b[n]$ and find the constraint of its length given that $c$ cannot be $0$. This is a homework problem. "Explain the structure" means that $b[n]$ is zero ...
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$\mathcal Z$-transform, ROC of a system from dividing to others

TRUE / FALSE Given three systems with rational $\mathcal Z$-transform. Systems A and B are not stable with $\mathcal Z$-transform $H(z)$, $G(z)$ respectively. A and B have no common poles. System's C ...
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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 ...