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# Stub (electronics)

In microwave and radio-frequency engineering, a stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or (especially in the case of waveguides) short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus be considered to be frequency-dependent capacitors and frequency-dependent inductors.

Because stubs take on reactive properties as a function of their electrical length, stubs are most common in UHF or microwave circuits where the line lengths are more manageable. Stubs are commonly used in antenna impedance matching circuits and frequency selective filters.

Smith charts can also be used to determine what length line to use to obtain a desired reactance.

## Short circuited stub

The input impedance of a lossless short circuited line is,

$Z_\mathrm{SC} = j Z_0 \tan(\beta l)\,\!$

where j is the imaginary unit, $Z_0$ is the characteristic impedance of the line, $\beta$ is the phase constant of the line, and $l$ is the physical length of the line.

Thus, depending on whether $\tan(\beta l)$ is positive or negative, the stub will be inductive or capacitive, respectively.

The Length of a stub to act as a capacitor C at an angular frequency of $\omega$ is then given by:

$l = \frac{1}{\beta} \left[(n+1)\pi - \arctan \left(\frac{1}{\omega C Z_0}\right) \right]$

The length of a stub to act as an inductor L at the same frequency is given by:

$l = \frac{1}{\beta} \left[ n \pi + \arctan\left(\frac{\omega L}{Z_0}\right) \right]$

## Open circuited stub

The input impedance of a lossless open circuit stub is given by

$Z_\mathrm{OC} = -j Z_0 \cot (\beta l) \,\!$

It follows that whether $\cot(\beta l)$ is positive or negative, the stub will be capacitive or inductive, respectively.

The length of an open circuit stub to act as an Inductor L at an angular frequency of $\omega$ is:

$l = \frac{1}{\beta} \left[(n+1)\pi - \arccot\left(\frac{\omega L}{Z_0}\right) \right]$

The length of an open circuit stub to act as a capacitor C at the same frequency is:

$l = \frac{1}{\beta} \left[n \pi + \arccot\left(\frac{1}{\omega C Z_0}\right) \right]$

## Resonant stub

Stubs are often used as resonant circuits in distributed element filters. An open circuit stub of length $\scriptstyle l$ will have a capacitive impedance at low frequency when $\scriptstyle \beta l < \pi /2$. Above this frequency the impedance is inductive. At precisely $\scriptstyle \beta l = \pi /2$ the stub presents a short circuit. This is qualitatively the same behaviour as a series resonant circuit. For a lossless line the phase change constant is proportional to frequency,

$\beta = {\omega \over v}$

where v is the velocity of propagation and is constant with frequency for a lossless line. For such a case the resonant frequency is given by,

$\omega_0 = \frac {\pi v}{2 l}$

While this is qualitatively like a lumped element resonant circuit the impedance function is not precisely the same quantitatively. In particular, the impedance will not continue to rise monotonically with frequency after resonance. It will rise until the point where $\scriptstyle \beta l = \pi$ at which point it will be open circuit. After this point (which is actually an anti-resonance point) the impedance will again become capacitive and start to fall. It will continue to fall until at $\scriptstyle \beta l = 3 \pi /2$ it again presents a short circuit. At this point the filtering action of the stub has totally failed. This response of the stub continues to repeat with increasing frequency alternating between resonance and anti-resonance. It is not only a characteristic of stubs, but of all distributed element filters, that there is some frequency beyond which the filter fails and multiple unwanted passbands are produced.[1]

Similarly, a short circuit stub is an anti-resonator at $\scriptstyle \pi /2$, that is, it behaves as a parallel resonant circuit, but again fails as $\scriptstyle 3 \pi /2$ is approached.[1]

## Stub matching

In a stripline circuit, a stub may be placed just before an output connector to compensate for small mismatches due to the device's output load or the connector itself.

Stubs can be used to match a load impedance to the transmission line characteristic impedance. The stub is positioned a distance from the load. This distance is chosen so that at that point the resistive part of the load impedance is made equal to the resistive part of the characteristic impedance by impedance transformer action of the length of the main line. The length of the stub is chosen so that it exactly cancels the reactive part of the presented impedance. That is, the stub is made capacitive or inductive according to whether the main line is presenting an inductive or capacitive impedance respectively. This is not the same as the actual impedance of the load since the reactive part of the load impedance will be subject to impedance transformer action as well as the resistive part. Matching stubs can be made adjustable so that matching can be corrected on test.[2]

A single stub will only achieve a perfect match at one specific frequency. For wideband matching several stubs may be used spaced along the main transmission line. The resulting structure is filter-like and filter design techniques are applied. For instance, the matching network may be designed as a Chebyshev filter but is optimised for impedance matching instead of passband transmission. The resulting transmission function of the network has a passband ripple like the Chebyshev filter, but the ripples never reach 0dB insertion loss at any point in the passband, as they would do for the standard filter.[3]

## References

1. ^ a b Ganesh Prasad Srivastava, Vijay Laxmi Gupta, Microwave Devices and Circuit Design, pp.29-31, PHI Learning, 2006 ISBN 81-203-2195-2.
2. ^ F.R. Connor, Wave Transmission, pp.32-34, Edward Arnold Ltd., 1972 ISBN 0-7131-3278-7.
3. ^ Matthaei, G.; Young, L.; Jones, E. M. T., Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp.681-713, McGraw-Hill 1964.


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# Rolling non-regular tetrahedron. How many orientations?

I've recently studied rolling polyhedra graphs. A polyhedron is rolled over its edges until it has reached all possible orientations. Ideally one more roll puts it back where it started. For example, the snub disphenoid has 72 orientations. If it is rolled through them all and ends back where it started with one more roll then there are 12 possible paths.

I decided to look off-the-grid. Here's the 120 states of the rolling dodecahedron in a closed path.

A bit too hard to simplify. But even the tetrahedron seems challenging. Do all the triangles need angles that are rational fractions of $$\pi$$, or does just the sum of angles around each vertex need to be a rational fraction of $$\pi$$? If the sum of angles around a vertex is not a rational fraction of $$\pi$$ then there are infinite orientations just rolling around that vertex.

A rolling non-regular tetrahedron has $$n$$ orientations. What values can $$n$$ be?

Let $$t^2=1.92756\ldots$$ be the tetranacci constant. Build this tetrahedron with edges having length $$t^0, t^1, t^2, t^3$$.

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# Translation Invariant Wavelet Denoising of Poisson Data

• Published 2001

#### Abstract

In this paper, we propose a novel image denoising algorithm to reduce Poisson noise based on state-of-the-art wavelet domain statistical models. We propose an efficient method to estimate the model parameters from the observations, and solve the optimization problem in orthonormal and translation invariant (TI) wavelet domains. It is observed that TI wavelet transforms produce better estimation performance than orthogonal wavelet transforms. We briefly explain this observation from the viewpoint of approximation theory and minimax estimation theory. I. Reduction of Poisson Noise In many applications, due to the imperfection of imaging systems, the observed image is a degraded version of the original scene. In this paper, we consider the Poisson imaging problem: the observations {yi} are independently Poisson distributed with intensity {fi}. Image denoising aims at obtaining a good estimate of the original image f from the degraded observations y. In image processing literature, Poisson denoising problems have been approached in the spatial domain, see [1, 2, 3, 4, 5] for examples. More recently, methods based on transform-domain models gained in popularity. For instance, wavelet representations are often more suitable to distinguish signal and noise due to energy compaction. Signal singularities such as edges and textures are often modeled based on wavelet representations. Multiscale estimation of Poisson processes has been addressed in [6] and extended in [7]. There, Poisson intensities are estimated in a coarse-to-fine manner using Haar wavelets. Each wavelet coefficient corresponds to a dyadic size square tile in the spatial domain image. The intensity of each tile is related to that at the next finer resolution using a Beta distribution. The estimation methods in [6, 7] are also extended to TI wavelet domain. However, the methods described in [6, 7] are not applicable to other types of wavelets. In this section, we propose a novel and efficient Poisson noise reduction algorithm based on state-of-the-art statistical wavelet models. We consider statistical models for f , and compute the maximum a posteriori (MAP) estimator: f̂ = arg min f [ − log p(y|f) + Φ(f) ] , (1) where Φ(f) = − logπ(f) penalizes unlikely estimates, and π(f) is the prior for f . II. Regularization in wavelet domain Research in image compression has given rise to sophisticated wavelet domain statistical models. Based on these models, we formulate the regularization penalty in the wavelet domain. We use the notation Φ(f̃) to denote the wavelet domain regularization penalty, where f̃ is the wavelet domain representation of the original image. A Choice of the penalty function The models considered for Φ(f̃) are listed below. The first two models are classical ones and are used for comparison purposes. The third one is a more accurate model that accounts for the spatial inhomogeneity of wavelet coefficients [8, 9]. 1. Gaussian model. Assume the wavelet coefficients within a subband j are iid N(0, σ f̃ ,j ). The corresponding penalty takes the form Φ(f̃) = 1 2 ∑ j,i f̃ 2 j (i)/σ 2 f̃ ,j , where f̃j(i) denotes the signal coefficient in subband j at location i. This quadratic penalty is used for instance in [10]. 2. Laplacian model. The statistics of wavelet coefficients within a subband j are modeled as iid Laplacian [11]. The penalty function Φ(f̃) is thus the l norm. 3. EQ model [8]. It is assumed that wavelet coefficients are independent Gaussian with zero mean and slowly varying variance. Indeed it has been verified [9] that wavelet coefficients normalized by their local standard deviation estimated from a small neighborhood approximately follows the N(0, 1) distribution. Under this model, the regularization penalty takes the form Φ(f̃) = 1 2 ∑ j,i f̃ 2 j (i)/σ 2 f̃ ,j (i), where σ f̃ ,j (i) is the local signal variance in subband j, at location i. The three models described above make different assumptions about the wavelet coefficients in the detail subbands. For the approximation subband (coarse scale), coefficients are modeled as stationary Gaussian with positive mean. The difference between each coefficient and the average of its four neighbors is iid Gaussian. As a benchmark, we also consider a spatial domain Gaussian Markov random field (MRF) model which assumes that the difference between each pixel and the average of its four neighbors is iid Gaussian. B Gradient descent algorithm Gradient descent algorithms can be applied to solve the optimization problem (1). Under the Poisson imaging model yi ∼ Poisson(αfi), where α is a positive constant controlling signal-to-noise ratio (SNR), the cost function to be minimized in (1) takes the form

### Cite this paper

@inproceedings{Liu2001TranslationIW, title={Translation Invariant Wavelet Denoising of Poisson Data}, author={Juan Liu and Pierre Moulin}, year={2001} }


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Semirings where solving linear systems is in P

Solving linear systems appears hard in semirings.

In $\mathbb{N}_0 (+,\times)$ it is NP-complete via reduction to 1-in-3 SAT.

In the min-plus semiring the complexity is $NP \cap coNP$ according to this

Are there nontrivial semirings where solving linear systems is in P?

Is they exist I would be interested what constraints they can encode.

Added clarification

By "linear system" mean a system of linear equations of the form: $$c_1 \otimes x_1 \oplus c_2 \otimes x_2 \oplus \cdots \oplus c_n \otimes x_n = c_{n+1}$$ where $c_i$ are constants and $x_i$ are variables. If the standard definition is other, say $A x = B x$, would accept an answer about the standard definition.

By "solving a linear system" system mean finding at least one solution or claim that no solution exist. Though finding some kind of "basis" for all solutions would be interesting too.

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What do you mean exactly by "solving" a linear system? Deciding whether it has a solution? Interesting semirings are $\mathbb{R}_{\geq 0}$ and $\mathbb{Q}_{\geq 0}$, over which deciding whether a solution exists is linear programming and hence in $P$. On the other hand, finding a minimal set of generating solutions such that every other solution is a linear combination of the given ones is very difficult (vertex enumeration problem, e.g. for polytopes). –  Tobias Fritz Feb 10 at 21:51
Please state your question more precisely. $\mathbb{N}_0$ carries numerous semiring structures, like $(+,\times)$, $(\max,+)$, $(\min,+)$, $(\min,\times)$, $(\min,\max)$ etc. State that you mean linear systems of the form $Ax=Bx$ for matrices $A,B$ and an unknown vector $x$. (If this is the case) Is something known about the $(\min,\max)$ semiring, over some linearly ordered domain? –  Günter Rote Feb 10 at 22:09
Tobias Fritz, Günter Rote thank you for the comments. Edited the question. –  joro Feb 11 at 7:33
@Tobias your answer will be interesting, especially if you can explain why some semirings are "easy" –  joro Feb 11 at 16:20
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# Understanding proof that if $c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$ then $d_1$ and $d_2$ are topologically equivalent metrics

Theorem. If there are strictly positive constants $c_1$ and $c_2$ such that

$$c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$$

for all $x,y \in X$, then $d_1$ and $d_2$ are topologically equivalent metrics on $X$.

Proof. (for one direction) Let $U$ be open in $(X, d_1)$. Then for $a \in X$ there > exists $\epsilon > 0$ such that

$$\{x \in X; d_1(x,a) < \epsilon\} \subset U.$$

But then

$$\{x \in X; d_2(x,a) < c_1\epsilon\} \subset \{x \in X; d_1(x,a) < \epsilon\} \subset U$$

and $U$ is open in $(X, d_2)$.

The line I don't understand is the one that says $\{x \in X; d_2(x,a) < c_1\epsilon\} \subset \{x \in X; d_1(x,a) < \epsilon\}$. Why is the former a subset of the latter? I don't see the justification of this.

• Hint: the inequality $c_1d_1(x,y)\le d_2(x,y)$ can be written $d_1(x,y)\le\frac{1}{c_1}d_2(x,y)$, and then apply $d_2(x,y)<c_1\epsilon$. – Jesus RS Mar 7 '15 at 16:54

We know that $c_{1}d_{1}(x,y) \leq d_{2}(x,y)$ for all $x,y \in X$.
For membership in $\{x \in X: d_{2}(x,a) < c_{1} \epsilon \}$, we have that $d_{2}(x,a) < c_{1} \epsilon$.
Therefore, we conclude that $c_{1}d_{1}(x,a) \leq d_{2}(x,a) < c_{1} \epsilon$, thus $c_{1}d_{1}(x,a) < c_{1} \epsilon$, and thus we can say $d_{1}(x,a) < \epsilon$ since $c_{1}$ is positive.
• I can figure out why $d_1(x,a) < \epsilon$, but I don't see how $d_2(x,a) < c_1\epsilon$ is $< d_1(x,a) < \epsilon$, since it's supposed to be a subset – mr eyeglasses Mar 7 '15 at 17:04
• If you can conclude that $d_{1}(x,a) < \epsilon$ from $d_{2}(x,a) < c_{1} \epsilon$, then you are done. You are showing that the restriction on the first set guarantees the restriction on the second, so the former must be a subset. – Jonathan Hebert Mar 7 '15 at 17:12