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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1answer
15k views

Z-Transform of a^|n|

I am wanting to compute the Z-transform of $f(n) = a^{|n|}$ . 'a' is a positive constant. Looking at the transform table, I found that Z-transform for $a^n u(n)$ is available from the tables and is $...
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3answers
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How to compute magnitude and phase response from transfer function in Z-domain?

I have a transfer function $$H(z)=\frac{1+1.2z^{-1}+0.8z^{z^-2}}{1-0.9z^{-1}}$$ from which I'm supposed to sketch the magnitude and phase response. I know that you can transform $z=e^{j\omega}$ to get ...
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1answer
171 views

Z-transform of alternating sequence

I'm having some difficulty in going through the z-transform of a sequence that is "on" every other sample. The sequence is $$x(n) = na^{|n|/2},$$ when $n$ is an even integer, and 0 otherwise. I have ...
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1answer
55 views

$\mathcal Z$-Transformation in Discrete Time [closed]

I want to find the inverse $\mathcal Z$-transform of this, in discrete time: $$X(z) = \frac{1}{1+3z^{-1}+2z^{-2}}$$
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3answers
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How to compute the Laplace transform of a discrete signal?

Assume I have a discrete random signal, $f(t)$ for which I want to calculate the laplace transform. How can I do it in matlab without using sym variables, for ...
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1answer
752 views

Bilinear transformation confusion

Wikipedia says in bilinear transformation from \$s\$ domain to $z$ domain relation is $$\boxed{s \longleftarrow \frac{2}{T}\frac{z-1}{z+1}}$$ But here this relation is given like this $$\boxed{w=\...
2
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1answer
259 views

Can a unit delay in discrete time be represented by exponential functions?

If we have a signal $y[n]$ and its unit delayed version $y[n-1]$, can we write $y[n-1]$ in terms of $y[n]$ times some exponential? The reason I want to do this is to then take $y[n]$ common and ...
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1answer
7k views

Determine whether the system is a FIR or IIR by looking the transfer function

I have the following system: $$ y[n]=\frac{1}{3}(x[n+1]+x[n]+x[n-1]) $$ After the Z-Transform we get $$ \frac{y[z]}{x[z]}=\frac{z^2+z+1}{3z} $$ which is of course the transfer function of the system. ...
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1answer
1k views

Z-transform of an impulse signal in discrete time

am trying to compute the Z-transform of the following signal \begin{equation*} x\left[n\right]\:=\:\sum_{k=-\infty \:}^{\infty \:}\:\delta \:\left[n-k\right] \end{equation*} so I thought it would be \...
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1answer
1k views

ROC of transfer function

given: $$ H(z) = \frac{4z(z-1)}{z-0.5} $$ I would say, when all poles are in the unit circles, the impulse response is right sided and causal.
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875 views

Z transform of a function with delay?

i have this open loop system , and i've been asked to find out the response $C(kT)$ due to a unit step input. I am able to find the transfer function without the delay unit i.e $$\frac{C(z)}{R(z)}=\...
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1answer
667 views

How to find the inverse Z transform of this function in z domain?

i have a function $$F(z)=\frac{z-0.4}{z^2+z+2}$$ i need to find the inverse z transform of it , i have tried it with residues but the roots are too much ugly and it involves lots of messy ...
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3answers
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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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1answer
55 views

which of the analog sinusoidal frequency can not pass through the filter?

First time I am encountering this type of question so i just tried but not getting whether my logic correct or not. First let the sinusoidal signal be $X(t)=\cos(2\pi ft)$. After sampling this ...
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1answer
132 views

Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$

I'm considering the following problem from some course notes. Suppose the following is known about a discrete-time LTI system: Given the input $x[n]=(1/3)^n$ for all $n$, the system ...
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1answer
2k views

Transfer function and difference equations: why does $H(z)$ numerator polynomial not correspond to $Y(z)$?

For a discrete time LTI system, I understand that from a difference equation description of the system in the form $$ \sum\limits_{k=0}^N{a_k y[n-k]}=\sum\limits_{k=0}^M{b_k x[n-k]} $$ I can ...
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1answer
472 views

How does a digital filter $H(z)=1/(1-z^{-1})$ change a continuous rectangular pulse? [closed]

I inputted the following signal in this discrete time filter and got this output Could anyone explain how ? Also how does a discrete filter process a continuous time output ?
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2answers
152 views

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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2answers
76 views

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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1answer
149 views

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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2answers
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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 ...
2
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1answer
116 views

What do I with the sampling period when I inverse $\mathcal Z$-transform?

Given the input $$e(kT_s) = {0.4, 0.8,1.2, - 0.9}, \quad k = 0,1,2,3$$ and transfer function $$T\left(z^{-1}\right)=\frac {z^{-1}-0.8z^{-2}}{1-1.1z^{-1}+0.3}$$ Find the output $Y(kT_s),\...
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1answer
122 views

Find transfer function given impulse

I am given the following discrete time transfer function : $$G_d(z^{-1})=z^{-d}\frac{b_0+b_1z^{-1}}{1+a_1z^{-1}}$$ which has the following impulse response $$g_d[n]=\{0,1,-0.1,-0.05,...\}$$ How can I ...
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1answer
2k views

Determine impulse response given input and output: which ROC?

Let's suppose I have to find the impulse response of a discrete time LTI system given a specified input and its output through the system. I think I'm going to get the $\mathcal Z$-transform of input-...
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1answer
372 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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1answer
403 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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1answer
2k views

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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1answer
265 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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1answer
776 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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2answers
535 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 ...
2
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1answer
59 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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0answers
103 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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1answer
81 views

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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2answers
2k views

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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0answers
385 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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1answer
295 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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1answer
158 views

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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1answer
306 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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1answer
244 views

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 ...
3
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1answer
678 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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2answers
245 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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3answers
298 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 ...
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1answer
496 views

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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1answer
81 views

$\mathcal Z$-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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1answer
163 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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1answer
81 views

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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1answer
77 views

$\mathcal Z$-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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1answer
517 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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1answer
1k 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. <...
0
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1answer
218 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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