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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Approximation of z-transform transfer function
It is common in control theory to approximate a transfer function neglecting the high order terms, in example,
a transfer function with two poles:
$$
P=\frac{1}{(as+1)(bs+1)}=\frac{1}{abs^2+(a+b)s+1}...
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Inverse Z-Transform of $\frac{1}{(1+z)^2}$ with ROC : |z|<1
I'm trying to Find the inverse Z-Transform of $\frac{1}{(1+z)^2}$
My steps are as such :
$\frac{1}{(1+z)^2}$ = $\frac{z^{-2}}{z^{-2}\cdot(1+z)^2}$ = $\frac{z^{-1}\cdot z^{-1}}{(1+z^{-1})^2}$ = $\frac{...
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Inverse Z Transform of $\frac{1}{(z-a)^2}$
I'm Trying to find the Z-transform of $$\frac{1}{(z-a)^2}$$ in discrete, i.e using $u[n]$.
Using the known transformation of:
$$n \cdot \alpha^n \cdot u[n] \Longleftrightarrow \frac{\alpha \cdot z^{-...
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Numerical Inverse Z-Transform - Abate and Whitt
I'm trying to implement an inverse Z-Transform using the Fourier series based technique by Abate and Whitt:
The Fourier-series method for inverting transforms of probability distributions.
Numerical ...
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How helpful is $f^{-1}(z)=\frac1{2\pi i}\oint\ln(1-\frac z{f(w)})dw$, or the method to find it, in deriving integral representations of $f^{-1}(z)$?
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following $\erf^{-1}(z)$ series:
$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via ...
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Integral representation for reverse Bessel polynomials
$\DeclareMathOperator ZZ$Series solutions to partly invert $e^x(x^2+a)$ and $e^xx(x+a)$ exist when more general quadratic-exponential equations are reduced. Using the reverse Bessel polynomials $p_n(x)...
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Why is it valid to use the residue theorem to compute the inverse z-transform, even though the z transform is discrete?
I understand how to use the residue method to compute an inverse z-transform of an elementary function easily enough. I'm confused however, because, from reading on Wikipedia, it seems this method is ...
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Why are the poles of the z-transform of the Fibonacci sequence the golden ratio?
I recently derived the poles of the $z$-transform of the Fibonacci sequence to be the golden ratio by doing the following:
$u(k+2)=u(k+1)+u(k); u(0)=u(1)=1$
$z^2U(z)-z^2u(0)-zu(1) = zU(z)-zu(0)+U(z)$
$...
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Making sense of generating functions in terms of the shift operator
While studying transforms, I stumbled across a neat way to express sequences, and have a couple of questions about its validity and how it connects to other topics. Suppose we are working with ...
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Simplify $\displaystyle \sum_{k=0}^{n}\frac{p^k}{k!}$ [closed]
Note that this is not a infinite sum like $\displaystyle\sum_{k=0}^{\infty}\frac{p^k}{k!}=e^p$, but a finite one.
Or, simplify the infinite sum $\displaystyle\sum_{k=0}^{\infty}\frac{p^k}{(k+n)!}$ may ...
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How do I fix this transfer function with algebra?
Currently, I'm working on a battery analysis system. For now, I have this transfer function:
$$\frac{V_L(z)}{I_L(z)} = \frac{\left(\frac{V_{oc}}{I_L(z)}\right) +b_0 - b_1z^{-1}}{a_0 - a_1z^{-1}}$$
...
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A problem involving a simple discrete-time system
A discrete-time system is described by the state equation
$$ V(k+1) = A V(k) + B u(k) $$
with $V(k) = [x(k), y(k)]^T$ being the state vector, and
$$ A = \begin{bmatrix} 2 && - 3 \\ 0.5 &&...
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Sinc filter and z transfert
I am taking a look to digital filters used on sigma-delta ADCs, my doubt is about sinc filters.
Here an example about sinc filters I've found on datasheet:
Sinc1:
4th order, decimate by 2, 5 taps (1 ...
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Inverse $Z$-transform exercise problem
I'm trying to solve a practice question about inverse $Z$ transform, and my solution is a bit different than the answer key. The question is just a simple polynomial expression
My solution is as ...
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Regarding Region of Convergence in Z-Transform
I am trying to understand Region of Convergence (RoC) of Z-transform.
If $x(n) = a^n u(n)$ then its z-transform is given as
$X(z) = \sum_{-\infty}^{+\infty}x(n) z^{-n} = \sum_{0}^{\infty}a(n) z^{-n}$. ...
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