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 would one solve this question if no initial conditions are given? Which assumption can I make?

This was a question on our test, I know it can be easily solved by Z-transforms but there are no initial conditions specified. In this case, what would be the right approach? Assume all initial ...
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Z Transform of M-Fold Decimation

I know this is probably a common question, but after some searching I think that my version of the question is slightly different -- apologies if this is a repeat. I have seen that it is inaccurate to ...
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Poles and Zeros of a DiscreteTimeModelFunction with delay in Wolfram Mathematica

help me please, I have a problem with this TransferFunctionModel, When I want to obtain the poles from the Discrete Model Wolfram gives me an error, what am I doing wrong? Could it be the delay of the ...
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Find Fourier Transform of Unit Step using the $z$-Transform [duplicate]

Since the unit step $u[t]$ is not absolutely summable, it has no Fourier Transform. In the DSP book (Proakis), the Fourier Transform of the unit step is formed by evaluating its $z$-Transform on the ...
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Value of $\sum\limits_{n=-\infty}^{\infty}(x*x)[n]$

If $x[n]=(0.5)^nu[n]$ and $y[n]=(x*x)[n]$ then what is the value of $\sum\limits_{n=-\infty}^{\infty}y[n]$ ? I calculated the $\mathcal{Z}$-transform of $x[n]$ and then applied the accumulation ...
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Can I Apply Multiple Properties of the Z Transform Simultaneously?

Using the time shifting, time reversal, and scaling, I want to derive the form of the Z Transform of $$x[n]=-a^n u[-(n+1)]$$ $u[n]$ is the discrete-time unit step function: $$ u[n] \triangleq \begin{...
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ROC of $\mathcal{Z}$-Transform and zeros

Theorem: Let $$f(z) = \sum_{n=0}^{+\infty}a_nz^n$$ where $z\in\mathbb{C}$. If $f(z_0)$ exists for some $z_0\in\mathbb{C}$ then it converges for all $z\in\mathbb{C}$ such that $|z|\lt|z_0|$. Proof: It ...
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How do I proceed to calculate this Z Transform?

I would like to calculate the Z-Transform of following discrete signal: $$x[n] = 3^{-|n|}$$ Plugging it into the known formula, I got: $$X(z) = \sum_{n=-\infty}^\infty x[n]z^{-n} = \sum_{n=-\infty}^\...
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Z-transform of $\cos(\omega_0 n(n+1))u[n]$

I'm doing some research on Zadoff-Chu sequences and as a part of it I wanted to find the Z-transform of: $$\cos(\omega_0 n(n+1))u[n]$$ Wolfram Alpha / Mathematica couldn't help out. I couldn't find a ...
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Z domain Transfer function to magnitude

How would one go about calculating the Magnitude of... $\mathcal{Z}$-transform = $$\frac{1}{1-z^{-2}}$$ I understand that z can be replaced with $${exp}^{j\omega}$$ and I am aware of the identities $${...
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Z transforms doubt -(ROC and its purpose)!

i had this doubt previously queried in another forum, but unfortunately had no answer. Consider a signal 3^n u[n]. Take its Z transform, which is Z/(Z-3). Now i know that in real sense, Z is a delay ...
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Determine transfer function based on a diagram with ai coefficients given

From a diagram with input $x(n)$ a summer and three feedback delay taps I get the difference equation... $$y(n) = x(n)+a_1y(n-1)+a_2y(n-2)+a_3y(n-3)$$ Then I am given values for ai coefficients. Case1:...
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How do I represent any signal in the form x[n]u[n] and x[n]u[-n-1]?

For eg-: If I have the signal x[n]={1,a,a²...}I can represent it as a^n * u[n]. Similarly if I havethe sequence as x[-1]=-a^-1 x[-2]=-a^-2 Then I can represent it as x[n]=-a^n*u[-n-1].
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DSP: newbie not understanding z transform/complex sinusoidal frequency and phase

Im reading Will Pirkles Designing audio effect plugins book and I'm not sure if I'm understanding the z transform correctly. I got up to differential equations in college, but haven't done math in ...
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Polar Fourier Transform and its similarity to the Z-Transform

The DFT is given by $X_{\mathcal{F}}(k) = \sum_{n=0}^{N-1} x_n e^{-j 2 \pi k n / N}$ Knowing that complex numbers can also be represented using a polar form $A \angle\theta$, I was looking for a ...
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Having trouble interpreting z-transform description of a predictor from a codec paper

I've been looking at the opus paper (https://arxiv.org/pdf/1602.04845.pdf); in particular, in section 4.1, they describe the predictor for the current band energy based on energy from both the current ...
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how to calculate $H_{lp} = z(1- H_{hp}) $ , given coefficients for $H_{hp}$?

Given a high pass transfer fn of the form $H_{hp}=a_{1}*z^0 +a_{2}*z^{-1} + ... a_{n}*z^{-n}$ Is it possible to calculate a causal low pass filter using $H_{lp} = z*(1-H_{hp})$ ? attempting $H_{...
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Mixed - Discrete and Continuous system Laplace domain stability - Effect of Sampler and DAC

I have a system whose the plant transfer function is continuous and the compensation is discrete. I have an ADC which allows to measure the output of the system and a DAC which allows to control the ...
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Identify out signal from in signal and impulse response

Hi there I am studying a course in signal processing and systems. I was given an excecise which I am having a great deal of trouble solving. I am allowed to solved using matlab, but for top marks I ...
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Zeros and poles from transfer function

I have a transfer function $$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$ I'm interested in zeros and poles. I know I need to adjust the function to $$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
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Difference equation to FIR filter coefficients

I have a difference equation $$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$ According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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$Z$-transform of a multilinear function/ consecutive multplication of $k$ signals $y_1(n), \ldots, y_k(n)$

How should one go about calculating the $Z$-transform of a signal that is the multiplication of $k$ signals (i.e. a multilinear function with regards to signals $y_1(n) \ldots y_k(n)$ ? Namely, $\...
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inverse z transform performed on 6th order IIR filter

we are told to find coefficients and impulse response of IIR filter of order of 6. There are 6 zeros and 6 poles in the design. Pole and zero pairs are conjugate and poles are within the unit circle ...
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If we take Z-transform of a signal & find its ROC. How to use this ROC? There are ∞ signals in ROC, suppose we choose any signal what to do with it?

If we take Z-transform of a signal & find its ROC. How to use this ROC? There are infinite signals in ROC, suppose we choose any signal from ROC, what to do with it?
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Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically with the signals in ROC?

Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically? Are we supposed to manipulate our O/P by multiplying any signal from ...
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Why does Simulink generate this code for a PID controller?

For the Simulink PID Controller model The Simulink generated code (rewrite for better understanding) is: ...
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I tried two approaches and gained the different conclusions of judging the stability of the transfer function of the system

We want to judge whether the system is stable or not. Given the below transfer function. $$ H\left( z \right) =\frac{\left( 1+2 z^{-1} \right) }{\left( 2+z^{-1} \right) } $$ $$ H\left( z \right) ...
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Filter odd or even harmonics with notch or inverse notch filter

Hi i had the following question. I have a signal containing a 200Hz sine wave and it's odd and even harmonics (no other frequencys or disturbing signals are contained). What i'm looking for is a kind ...
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Let a LTI system be causal and stable with the transfer function being... show that

if the system is an IIR LTI causal and stable one, and the transfer function is \[H(z)=\sum_{n=0}^{\infty}h[n]z{^{n}}= \frac{G}{1 -\sum_{k=1}^{p}a_kz{^{-k}}}\] show that the cepstrum of this system ...
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Where to start with DSP?

I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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Power Spectral Density of a Filter

I need to calculate the output power spectral density of the following digital filter My calculations are as follows: $y\left(n\right)\:=\:x\left(n-1\right)+d\left[x\left(n-1\right)+x\left(n\right)\...
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K-Path sampling overall transfer function

I'm trying to derive the overall Z-transform for the K-path sampling system shown below. I numbered some of the equations that I'll reference. I don't think I understand going from equation 3 to ...
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Z-transform of an affine function

What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship: $$ y[n] = \alpha x[n] + \beta. $$ Using the Z-transform we find: $$ Y[z] = \alpha X[...
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How to find H(z) from just zeros and poles

I have a system with a DC gain of 8, poles at z = +- j/2 and zeroes at e^+-j5, I need to find the H(z). I have tried this but not sure if it is right. $$ H(z) = G_o * z^{-1} \frac{(z-z_0)(z-z_1)}{(z-...
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Digtial FIR Impulse response & transfer function

I am currently working the figure through below. as it is an FIR Filter i have worked out using convolution that the output is 4,2,4,6,0,0. i am trying to obtain the 'z' domain transfer function of ...
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Filter H(z) manipulation frequency response changes

I'm trying to understand how altering the frequency response of a H(z) low pass filter, will visually alter it's frequency response plot. For example: by doing H(z^2), would the frequency response ...
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understanding echo cancellation model

so, here is a very simple model. The CCDE is given, but I was trying to derive it on my own and now I am stuck. first of all, the reverse y[n] when goes through the delayed system, it turns to y[n-M]....
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Order one FIR Filter with complex coefficient

I am trying to learn about the behavior of the FIR filter however with complex coefficients. The filter I am trying to analyze is the following: $$H(z)=a+jbz^{-1}\quad\text{where the variable}\quad j =...
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What is z equal to in z-transform?

In some places, it is said that z is equal to: $$z = e^s \quad where \quad s = \sigma + j \Omega $$ But in some places, it is said that z is equal to: $$z = e^{sT_s} \quad $$ where Ts is a sampling ...
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Is this system causal or not?

My efforts of solving this question are below. I came to a conclusion that this system is causal, since: $$ \begin{cases} w[k]+5w[k-1]+6w[k-2]=x[k] \\ y[k]=w[k]+2w[k-1]+3w[k-2]+4w[k-3] \end{cases} $$...
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Gradient of transfer function (z-transform) with respect to coefficients/parameters?

my problem is perhaps very simple but I just can not find the answer, even though this is used (though not explained) in several books: What is $\frac{\partial G (z, \theta)}{\partial \theta}$ when $G$...
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How do you reduce $H\left(e^{j\frac{\pi}{2}}\right)$ further according to a textbook solution

I want to know how I could get from the first line to the second. I've been trying to figure it out for a while with no luck. Thank you in advance! \begin{align} H\left(e^{j0.5\pi}\right) &= \frac{...
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Why the transfer function is equal to the output in this case

In this description of transfer functions on the z-plane (image linked), I'm confused by equation 1.49, which says that $H(f)=v_{out}(f)$ when $v_{in}(f)= 1 * e^{j 2 \pi (f/f_s)}$. (For another matter ...
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Recovering DTFT from Z-transform

The relationship between the Z-transform and DTFT can be expressed like: $$ H(e^{j \omega}) = H(z)|_{z = e^{j \omega}}$$ Graphically, evaluating the Z-transform on the unit circle is shown as sweeping ...
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Impulse Response for an Anti Causal Linear Shift Invariant System

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I'm asked to determine the impulse response of both an anti-causal and a causal Linear Shift Invariant System ...
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How to break a second-order filter into two first-order filter

Let's assume the transfer function of a continuous-domain filter consists of two poles and one zero: $H(s) = \frac{k_c (s-\omega_{z_1})}{(s-\omega_{p_1})(s-\omega_{p_2})}$. Let's consider we do the bi-...
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How to design IIR digital filters

Practical Infinite-Impulse-Response (IIR) filters are usually based upon analogue equivalents (Butterworth, Chebyshev, etc.), using a transformation known as the bilinear transformation which maps the ...
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transfer function of a sampler in the s domain

I would like to modelize my whole system into the S-domain. This is a mixed system, there a numerical part (corrector, ADC, DAC) and an analogic part (plant transfer function, sensors, etc...). I know ...
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How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?

Let $H(z)$ be the rational system function of an LTI system. How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
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Calculating the output of a pole eigen signal in a difference equation

Let an IAR system be defined by the following difference equation: $$y[n]-\frac{1}{4} y[n-2]=x[n]+3x[n-1]$$ and an input signal $x[n]=(-0.5)^n$. The transfer function is defined as $H^z(z)=\frac{1+3z^{...

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