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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80 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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97 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
75 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
456 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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914 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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210 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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2answers
126 views

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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313 views

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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611 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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1answer
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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1answer
561 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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112 views

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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1answer
75 views

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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74 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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183 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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3k views

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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576 views

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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337 views

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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2answers
174 views

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
13k views

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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1answer
634 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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228 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 ...
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162 views

$\mathcal{Z}$-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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138 views

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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2answers
220 views

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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1answer
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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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80 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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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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$\mathcal Z$-transform if the output is given

A impulse response for a LTI system is given by: $$h[n]=\left(\frac{2}{3}\right)^n u[n]+2 \left(\frac{1}{5}\right)^n u [n]$$ and if the putput for the system is given by: $$y[n]= \left(\...
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1answer
815 views

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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1answer
222 views

LTI system input upsampling

Let's assume that a linear and time-invariant system is sampled at 2 different frequencies $F_{s}$ and $2F_{s}$ (e.g. 5Hz and 10 Hz). It gives $$Y_{F_{s}}(z) = H_{F_{s}}(z)X_{F_{s}}(z)$$ $$Y_{2F_{s}}(...
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382 views

Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT). CFT can be used in case of continuous ...
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What are the advantages and disadvantages of Laplace transform over Z transform?

Laplace transform for continuous signal $x(t)$ is given by $$ X(s) = \int\limits_{-\infty}^{+\infty} x(t) e^{-s t} dt. \quad (1) $$ Z-transform for discrete signal $x(n)$ is given by $$ X(z) = \...
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0answers
235 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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1answer
170 views

$\mathcal Z$-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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2answers
284 views

Difference equation when transfer function expressed as poles and zeros

The transfer function $H(z)$: $$ H(z) = \frac{Y(z)}{X(z)} = \frac {b_0 + b_1 z^{-1} + b_2 z^{-2}} {1 + a_1 z^{-1} + a_2 z^{-2}} \tag{1} $$ Has difference equation: $$ y[n] = b_0 x[n] + b_1 x[n-1] + ...
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2answers
195 views

Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

I'm currently in the process of going over the $\mathcal Z$-transform and more specifically its derivation. I understand and I am able to follow it up until the final step whereby involving the ...
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1answer
914 views

Applying a time delay with Z-Transform

I am trying to take a signal $F(t)$ that has been sampled at some time DelT, I then wish to pass this signal through a channel $H(s)$. To do this I am sampling my signal $H(s)$ at DelT time intervals ...
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1answer
77 views

Inverse $\mathcal Z$-transform of rational functions

What will be inverse $\mathcal Z$-transform for this function: $$H(z) = \frac{\left(1+\beta z^{-1}\right)\left(1+\beta z\right)}{\left(1+\alpha z^{-1}\right)\left(1+\alpha z\right)}$$
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1answer
45 views

Problems calculating Z-transform

I am trying to solve a class exercise in which I am given the following, in Laplace domain: $$G(s)=\dfrac{e^{-Ts}}{s+3}$$ $$H(s)=\dfrac{1}{s}$$ And I need to calculate $\dfrac{C(z)}{R(z)}$, which is ...
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1answer
129 views

Z-transform of difference equations and stability of a process

According to this paper: $y(t)$ is stationary if all of the roots (of characteristic equation) lie outside the unit circle Here, $y(t)$ is causal. To me it seems the case is exactly the opposite, ...
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3answers
4k views

Is $y[k] = y[k-1] + x[k]$ an integrator?

It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \...
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3answers
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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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225 views

Transfer function of an Exponential system in Z domain

Hi, I am really confused with the system in the diagram. The input-output relation of the system is given by $y[n]=\exp(x^2[n])$ I need to find the transfer function of this system $Y(z)$ in $z$-...
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54 views

Inverse $\mathcal Z$-transform of system with an 8th order pole

Can I find the inverse $\mathcal Z$-transform of this transfer function: $$H(z)=\frac{1}{1-\alpha z^{-8}}$$ in a way other than contour integration and finding the residues of the 8 poles? If so, how?
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1answer
203 views

Fourier Transform of triangle function $x(t)=\Delta\left(\frac{t-1}{2}\right)$

Can you please tell me if my working is right for the Fourier Transform of this function: $$x(t)=\Delta\left(\frac{t-1}{2}\right)$$ My workings are: I have used the fourier transform standard ...
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1answer
75 views

Differentials - Differences: Non causality in the system

I'm still learning DSP and referring to Oppenheim video lectures. In that lectures, differential difference equation is obtained for IIR filter design, in Lecture 14. $$\mathcal{L}[\frac{\mathrm d}{\...
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413 views

Closed form of $\mathcal Z$-transform : decomposition signal $x(n)$

The text of my exercise ask : Determine the closed form of the $\mathcal{Z}$-transform for this $x(n)$ $$ x(n) = \begin{cases} |n-N| & \text{if 0<$n$<2N} \\ 0 & \text{elsewhere} \...

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