Questions tagged [z-transform]

The $z$-transform is a discrete analogue to the Laplace transform, in that it maps a time domain signal into a representation in complex frequency plane.

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How would I go along getting the z transform of this?

I have the function: $x(n) = 2^n u[-n] + \frac{1}{4}^n u[-n-1]$ and a chart with some common z-transforms and properties, which say that: $x(n) = a^nu[n]$ transformed is $x(z) = \frac{z}{z-a}$ $-a^nu[...
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Frequency response at $\omega = \frac{4\pi}{9} $ for designing a two-pole BPF with the given specs

$H(\frac{4\pi}{9}) $ following the same steps I did for $H(\frac{\pi}{2})$" /> As shown above, this is an example I found on frequency-domain analysis of LTI Systems and I am unable to achieve the ...
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What is the Z-Transform $u_n =\cos(n+1)\theta$?

I have tried using the putting it into $\cos(n\theta)$ = $\frac{z (z-cos\theta)}{z^2 - 2zcos\theta+1}$ formula, but my answer is wrong. $u_n = \cos(n+1)\theta.$
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Z-Transform of $a^{n^2}$

Is there a expression for the z-transform of the series $f(n)=a^{n^2}$ $$f[z]=\sum_n{a^{n^2}z^{-n}}=?$$
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Z transform using Convolution Integral

Let our transfer function be $G(s) = \frac{10e^{-s}}{5s+1}$. We know that for sampling period of $T = 1$, we have $G(z) = \frac{2}{z-0.8187}$ (You can verify this in MATLAB using c2d function). What I ...
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Using Residues theorem to do inverse Z transform "nearly works".

I have tried to build the habit of using the Residue theorem to do inverse Z transform where ever the table entries are not clearly associated with the function. I have a long example here which "...
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Find the inverse z- transform of $ X(z)=\frac{z^3+2z^2-2z}{(z-2)(z^2+2)}$

Find the inverse transform \begin{equation} X(z)=\frac{z^3+2z^2-2z}{(z-2)(z^2+2)} \end{equation} We solve the partial fractions for: \begin{equation} \frac{X(z)}{z}=\frac{z^2+2z-2}{(z-2)(z^2+2)}...
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Solve $x_{n+1}=2x_n-n^3$

I want to solve this difference equation, using the z-transforms. Using a table I get that: $x_{n+1}\rightarrow zX(z)$ $2x_n\rightarrow 2X(z)$ but then I cannot find any entry for $-n^3$. Using WA ...
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Z transform of $x[k]=\sum_{n=0}^{\infty}c^n * \delta [k-2n] \:; a<0 $

I would want to ask what is the z-transform of this signal x[k]: $$x[k]=\sum_{n=0}^{\infty}c^n * \delta [k-2n] \:; c<0 $$ I know that I have to use the geometric series but I really don't know ...
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Calculate Z transform of a sum

I'm new on the forum and hence not familiar with formal writing of a post so before going in on this post I wanted to make this clear. I'm an second year engineering student with a small knowledge of ...
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Solving a difference equation using z-transform

Solve the difference equation \begin{equation} x_{n+1}-2x_n=n \end{equation} Solution: I use the z-transform table: \begin{equation} \begin{array}{cc} First \ term, x_{n+1}:\\ Table\ ...
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Find the z-transform of $n^2 3^n$

I have $n^2 3^n$, and want to find the z-transform of it. I take $$n^2 3^n= n\cdot n 3^n$$ and we can see from a z-transform table that $$3^n \rightarrow\frac{z}{z-3}$$ $$n3^{n-1} \rightarrow\frac{z}{(...
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Is this proof of s-plane to z-plane mapping correct?

I am trying to prove that the left half-plane of s-plane is mapped into the interior of the unit circle in the z-plane. My proof is the following: We know that $s = σ + jΩ$ , $z = re^{jω}$ and $z = e^{...
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Calculate the value of a series using z-transform

I want to calculate the value of the following series $\sum_{n=1}^\infty\big(\frac{2}{5}\big)^n$ For this I use the z-transform which gives: $X(z)=\sum_{n=1}^\infty\big(\frac{2}{5}\big)^nz^{-n}$ From ...
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Identifying an inverse z-transform using Cauchy's theorem

I have the z-transform of a sequence, and I want to find the sequence. The transform is: \begin{equation} V(z)=\frac{z}{2z^2-7z-4} \end{equation} Instead of using tables of inverse z-transforms, I ...
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Finding a value of a series using z-transform

Having the z-transform formula: \begin{equation} V(x)=\sum_{n=0}^\infty x_nz^{-n} \end{equation} I want to find the value of the series \begin{equation} \sum_{n=0}^\infty \frac{n}{3^n} \end{...
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How can I find the inverse Z-transform of $ -(2z)/(z^2+z+1) $

I'm trying to find the inverse Z-transform of $ -(2z)/(z^2+z+1) $ . The answer from the book is $ 4/\sqrt{3} \cdot \sin(n2\pi/3) $ but I can't understand how to get it.
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Laplace transform of a sampled signal

I have been studying the book Digital Control of Dynamic Systems. In chapter 5 of this book the concept of the ideal sampler is introduced and the Laplace transform of the sampled signal $r^*(t)$ is ...
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Extracting Transient Response from a Rational Transfer Function

I am reading an online book on digital filters and wanted to know how the transient response can be obtained from a rational transfer function. $$\frac{b(1) + b(2)z^{-1} + \cdots + b(n_b+1)z^{-n_b}}{...
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Taking the Z transform and the Fourier transform of the same function

I saw that we could apply two transforms to the propagator of the Continuous-time random walk (CTRW), $$P(x,t)= \sum_{N=0}^\infty [\lambda^{*N}(x)w^{*N}(t)*\int_t^\infty d\tau w(\tau)] $$ where the ...
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Laurent series of a meromorphic function

Let $f(z)$ be a meromorphic function with the poles $\{z_0, z_1, \dots \}$. Suppose that we want to find the Laurent series representation of $f(z)$ centered at $z=0$. Can we claim that the only ...
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Missing link in derivation of z-transfer sum

I'm trying to figure out a derivation of a z-transfer. You should be able to write any z-transfer as a sum of coefficients: $$a + bz^{-1} + cz^{-2}\dots$$ I have this transfer function given, that ...
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Confusion regarding unilateral z-transform and difference equations

Consider the difference equation $$x[k] + \frac{1}{2}[k-1] + \frac{1}{4}x[k-2] = 0 ~\text{and}~ x[-1] = x[-2] = 1$$ It can be solved using unilateral z-transform which yields $$ X(z) = \frac{z(1-z)}{...
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Absolute convergence of the Laurent series

Consider the power series (which is the Laurent series around $z_0 = 0)$ $$\sum_{n=-\infty}^{+\infty}a_nz^{-n} \tag{1}$$ where $z \in \mathbb{C}$. It's known that the ROC (region of convergence) of $(...
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Value of $\sum\limits_{-\infty}^{\infty}x[n]*x[n]$

If $x[n]=(0.5)^nu[n]$ and $y[n]=x[n]*x[n]$ then what is the value of $\sum\limits_{-\infty}^{\infty}y[n]$ ? I calculated the $z-$ transform of $x[n]$ and then applied the accumulation property of $z-$...
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What is the Contour $C$ in Cauchy's Integral Theorem?

In the context of inverse Z Transforms, what does it mean for a pole $z_0$ to be inside the contour $C$ in the definition of Cauchy's Integal Theorem: $$ \frac{1}{2\pi j}\oint_C\frac{f(z)}{z-z_0}dz= \...
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Obtaining transfer function from pole zero plot.

How to determine the transfer function from the plot? By placing the poles in the denominator and zeros in the numerator, I get the following form- $$ H(z) = \frac{(z-8)(z-1)(z-e^{j\frac{2\pi}{20}})(...
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Determine if an equation's roots are inside the unit circle [closed]

I have a polynomial such as, $z^2 - 1.5z + 0.9 = 0$ I need to know if the roots are inside the unit circle and if the transfer function is stable if the equation is the denominator. But I am not ...
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What is the Z-transform of the convolution of a signal with a delayed itself?

By time-shift and convolution property we can infer that: $$ y(t)=x(t)*x(t) $$ $$ y(t-1)=x(t-1)*x(t-1) $$ $$ Z\{y(t-1)\}=z^{-1}Z\{y(t)\}=z^{-1}Z\{x(t)*x(t)\}=z^{-1}X(z)X(z) $$ But, if the signal y is ...
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Can I explicitly find the Taylor series for this function?

I have a following function: $$G(z)=\sqrt{F(z)^2-1}$$ Where: $$F=\frac{\left(1+\alpha^{*2}\right)-\left(1+\alpha^{2}\right)z}{2\left(\alpha^{*}-\alpha z\right)}$$ And $\alpha$ is a complex number. I ...
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Z transform discrepancy at n=0

I was supposed to find the Inverse z transform of $$\dfrac{1}{(z-5)^3}$$. Approach: I first tried to find out the z transform of $\dfrac{z}{(z-5)^3}$.Partial fraction decomposition tells us: $$\dfrac{...
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Determining the inverse Z transform

We are required to find $u_{2}$ and $u_{3}$ of the sequence $u_{n}$ whose z-transform is the function F(z): $$F(z)= \dfrac{2z^2+5z+4}{(z-1) ^{4}}$$ Approach: I managed to split it into partial ...
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A Z-transform for signal processing

I am working on ultrasound scan and the processing of the signal received by the probe. I made the model I wanted and as I do not have an ultrasound scan machine I want to simulate the signal ...
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How can we tell that this is a FIR-filter?

I am trying to undersant my professors notes. What I don't understand is how we can see that this is a finite impulse response filter by looking at the Z-transform or the graphical representation: I ...
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How to invert a family of matrices with parameter dependent dimensions and position dependent elements?

Let $\mu,\beta,\gamma,\kappa$ be real numbers (parameters) and let $n$ be an integer bigger equal one and let $\left(X_j\right)_{j=0}^{2 n} $ be a vector of real numbers. Consider a following ...
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How are these two sequences equal in discreete time notation?

If we have $h_1(n)=\begin{bmatrix}\underline{3} & 2 & 1\end{bmatrix}$ and $h_2(n)=\begin{bmatrix}\underline{0} & 3 & 2 & 1\end{bmatrix}$. I dont understand why $h_1(n) = h_2(n-1)$. ...
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How should I approach to this recurrence equation with infinite series?

I have an equation with $\{V_{-\infty},...,V_{\infty}\}$, which is determined by the following equation for all $n$. $V_{n}= \pi_n \sum_{i=0}^{\infty} \beta^i \bigg{(} 1- \big{[} 1+ \frac{1-\pi_{n}}{\...
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how to take inverse z transform of 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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Where does variable n gone of trignometric function and sum of infinite series?

I want to compute the Z-transform of the below signal function. $$x[n]=\cos(\omega n)u[n]$$ $$Z[x[n]]=\sum_{n=-\infty}^{+\infty}\cos(\omega n )u[n] \cdot z^{-n}$$ $$=\sum_{n=0}^{+\infty}\cos(\omega n )...
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Is this notation$~X(z)=Z\left[x\left[n\right]\right](z)$ correct of Z-transform?

The textbook which I have wrote the below equation of Z-transform. $$X(z)=Z[x[n]]$$ From the definition of Z-transform, the rightmost term of sigma is held. $$X(z)=Z[x[n]]=\sum_{n=-\infty}^{+\infty} x[...
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Discretization of second order ODE, apply Z transform and inverse

I have the following ODE: $\frac{\mathrm{d^2y(t)} }{\mathrm{d} t} + 2\frac{\mathrm{dy(t)} }{\mathrm{d} t}+4y(t)=e^{-2(t-2)}u(t-2)$ With $u(t)$ being the unit step function. I am than asked to ...
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How did the following factor out to this when doing PFE?

I'm currently applying PFE in getting the Z transform and this was the given: \begin{equation} X(z) = \frac{z+3}{z^4+6z^3+14z^2+16z+8} \end{equation} Dividing both sides by $z$ it resulted to this: \...
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Find the z-transform [closed]

I have these expressions for which I need the z-transformed functions. Please help. The Question Expressions My attempt for the first question My attempt for the second question My attempt for the ...
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What is the Z-transformation of following discrete time signal of this?

I am learning how $Z$-Transforms work, but I have no encountered a situation in which the bound does not account for any signal. Take for example the following discrete time signal: \begin{cases} x(-2)...
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Recurrence relation with Z-Transform

I'm revising the Z-Transform. I am looking at the book which gives an example of how to solve the recurrence relation $$x_{k+2} - 3x_{k+1} +2x_k = 1$$ where $x_0 = 0$ and $x_1 = 1$. The book uses the ...
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Solving difference equation by using z-transform and z-transform confusion

Use the $z$-transform to solve the following difference equation: $$y[n+2]+0.5y[n+1]+0.06y[n]=x[n]$$ where $x[n]=(-3)^{-n}u[n]-(-3)^{-n}u[n-2]$. The first thing to notice is that $x[0]=1$, $x[1]=-\...
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Confusion applying the shifting property of z-transform

I'd like to calculate the z-transform of $Z=(-3)^{-n}u[n-2]=\left(-\dfrac{1}{3}\right)^nu[n-2]$. Also, I'd like to use shifting property, which states: $$x(n-k)\rightarrow z^{-k}\left(X(z)+\sum_{n=1}^...
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Help me with z transform

So the question is basically z transform the given system. $(y[n+2] + 3y[n+1] - 4y[n])=(x[n+2] - 5x[n+1])$ I've to find h[z] first then it's really easy to solve it. So that's what I got so far; $z^2y(...
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z-transformation of difference equation

Let $$x_{n+1}-x_{n}=3x_{n}+2$$ I want to find the z-transformation, in the solution, it is stated that $y(z)=\frac{2}{\frac{1}{z}-3}$ I get a completely different result after multiplying everything ...
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Cannot find the inverse of Z-transform.

Find the invers of $Z$-transform of $X(z)=\dfrac{z}{(z-2)(z^2+6z+9)}$. I try as below. Let \begin{align*} X(z)&=\dfrac{z}{(z-2)(z^2+6z+9)} =\dfrac{z}{(z-2)(z+3)^2}\\ &=\dfrac{A}{z-2}+\...
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