Einstein’s special relativity theory postulates that, as long as we are dealing with uniform motion, there is no frame of reference preferable over another. Also, all the laws of physics (and thus electromagnetism) must be valid irrespective of the chosen reference frame. One immediate conclusion then must be that we need not differentiate between a moving source (with stationary receiver) and a moving receiver (with stationary source) as in the classical case with a medium.

Again, we start with the plane wave, represented by its phasor $A(t)$, as in the classical case.

$$A(\mathbf{r},t)=A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr) \label{plane-wave}$$

where $A_0$ describes the (constant) amplitude and phase of the emitted signal, $\omega$ is its frequency at the source and $\mathbf{k}$ is the wave vector, describing the spatial frequency of the wave and its direction.

In the reference frame of the source, a moving receiver can be described similarly to the classical case, with the wave field at the location of the receiver moving with velocity $\mathbf{v}$ (in the source frame of reference) given by

$$A(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}\cdot\bigl(\mathbf{r}_{0}+\mathbf{v} t\bigr)\Bigr] \label{field-R}$$

In the classical case, the received frequency was determined by differentiation of the phase of the field with respect to time,

$$\begin{aligned}
\omega^\prime_{R} &= \partial_t \arg\bigl[A(\mathbf{r},t)\bigr] \vphantom{\biggl(\biggr)}\\
& =\omega-\mathbf{k}\cdot\mathbf{v}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}}{c}\biggr)
\end{aligned}\label{phase-differentiation}$$

In relativistic physics, however, in order to obtain the frequency observed by the receiver we have to do the phase differentiation in the reference frame of the receiver. Einstein postulated that we need to apply the Lorentz transformation to convert the measurements of time and space in a reference frame which is defined to be at rest (source) into the measurements of the same quantities that would be obtained in another reference frame (receiver) which is moving with velocity $\mathbf{v}$ relative to the frame at rest. The Lorentz transforms for time and space are

$$\begin{gathered}
t^\prime = \gamma \biggl(t - \frac{\mathbf{v}\cdot \mathbf{r}}{c^2} \biggr)\\
\mathbf{r}^\prime = \mathbf{r} + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}}{\mathbf{v} \cdot \mathbf{v}} - \gamma \, \mathbf{v} \, t
\end{gathered}\label{Lorentz-transformation}$$

where the primed variables denote the quantities in the moving reference frame and

$$\gamma = \biggl(1 - \frac{\mathbf{v}\cdot\mathbf{v}}{c^2}\biggr)^{-\frac{1}{2}}$$

is the Lorentz factor. The inverse transforms are

$$\begin{gathered}
t = \gamma \biggl(t^\prime + \frac{\mathbf{v}\cdot \mathbf{r}^\prime}{c^2} \biggr)\\
\mathbf{r} = \mathbf{r}^\prime + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}^\prime}{\mathbf{v} \cdot \mathbf{v}} + \gamma \, \mathbf{v} \, t^\prime
\end{gathered}\label{inverse-Lorentz-transformation}$$

The difference in the forward and inverse transforms is merely in the sign of $\mathbf{v}$ in agreement with the postulated equivalence of all reference frames. It is equally valid to define the receiver reference frame to be at rest with the source frame moving in the opposite direction $-\mathbf{v}$ (assuming the coordinate systems in both cases to have the same orientation). Hence the inverse transform can only differ in the sign of the velocity vector.

We can write $\eqref{plane-wave}$ in terms of the receiver coordinates $t^\prime$ and $\mathbf{r}^\prime$ by inserting $\eqref{inverse-Lorentz-transformation}$ and regrouping the arguments

$$\begin{align}
A(\mathbf{r},t)&=A_{0}\exp\biggl(i \omega \gamma \biggl[t^\prime + \frac{\mathbf{v}\cdot \mathbf{r}^\prime}{c^2} \biggr] - i\mathbf{k}\cdot \Bigl[ \mathbf{r}^\prime + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}^\prime}{\mathbf{v} \cdot \mathbf{v}} + \gamma \, \mathbf{v} \, t^\prime \Bigr] \biggr)\notag\\
&= A_{0}\exp\bigl( i \omega^\prime t^\prime - i \mathbf{k}^\prime \cdot \mathbf{r}^\prime \bigr)\vphantom{\biggl(\biggr)}
\end{align}$$

with

$$\omega^\prime = \gamma \bigl(\omega - \mathbf{k}\cdot\mathbf{v}\bigr) \quad \text{and} \quad \mathbf{k}^\prime = \mathbf{k} - \frac{\omega \gamma}{c^2} \mathbf{v} + \frac{\mathbf{k} \cdot \mathbf{v}}{\mathbf{v}\cdot\mathbf{v}} \bigl(\gamma - 1\bigr) \mathbf{\hat{v}}\label{frequency-R}$$

We see that the frequency detected by a receiver at a fixed location $\mathbf{r}^\prime$ within its own reference frame, $\omega^\prime$, is just the frequency $\omega^\prime_R$ from $\eqref{phase-differentiation}$, describing the receiver motion through the wave field in the source reference frame, multiplied by the Lorentz factor $\gamma$. Intuitively, this is sensible since the Lorentz factor also describes the relation between the rates of passage of time in both reference frames, cf. $\eqref{Lorentz-transformation}$. Unlike the classical Doppler effect, time dilation causes a “true” frequency shift in the receiver reference frame since the very rate of passage of time is now different.

To determine which of the two is the dominating effect at some velocity, we plot the relative frequency shift due to time dilation, which is given by $\gamma - 1$, and the relative detected frequency shift due to $\eqref{phase-differentiation}$, which is given by $\mathbf{\hat{k}}\cdot\mathbf{v}/c$, versus $v$ and $\mathbf{\hat{k}}\cdot\mathbf{v}$, respectively. For the case that $\mathbf{k}$ and $\mathbf{v}$ are (anti-) parallel, $\bigl|\mathbf{\hat{k}}\cdot\mathbf{v}\bigr| = v$.

Figure 1: Contribution to frequency shifts due to receiver motion in the source wave field and relativistic time dilation.

For $\mathbf{\hat{k}}\cdot\mathbf{v} = \pm 1$, the effect of time dilation is less than the geometric phase $\eqref{phase-differentiation}$ for all velocities not very close to light speed. However, for some applications like GPS the relativistic correction significantly increases accuracy. For applications like vibration measurement via the Doppler effect for light the relativistic contribution can usually be neglected.

The geometric frequency shift in $\eqref{phase-differentiation}$ depends on the angle between wave and receiver motion while the time dilation does not. When the receiver moves nearly orthogonally to the wave, Doppler shifte due to time dilation can dominate at comparatively low speeds, as is shown in Figure 2.

Figure 2: Ratio of the contributions due to time dilation and spatial phase accumulation for the receiver in motion. Colors denote different angles between receiver motion and wave direction.

Depending on the angle between the wave and the receiver motion, both effects can offset each other. In particular, for any $v$ so that $0 \lt v \lt c$ there exists a direction with $\mathbf{\hat{k}} \cdot \mathbf{v} \lt 0$ so that $\omega^\prime = \omega$. This is shown exemplary for $v = 10^6$m/s – and thus an angle of $\cos^{-1} \bigl(\mathbf{\hat{k}} \cdot \mathbf{\hat{v}}\bigr) = 89.90^\circ$ – in Figure 3 which plots the compound Doppler shift due to both effects versus velocity.

Figure 3: Relative frequency shift for a receiver moving at various angles to the plane wave emitted by the source.

We also see in $\eqref{frequency-R}$ that the direction of the wave in the receiver reference frame, described by $\mathbf{k}^\prime$, changes when $\mathbf{k}$ and $\mathbf{v}$ are not (anti-) parallel. This effect is referred to as aberration and is significant e.g. in astronomy when determining the location of stars.

Equation $\eqref{frequency-R}$ is written in very general terms and valid for arbitrary directions of $\mathbf{k}$ and $\mathbf{v}$. In the “standard” case of both being (anti-) parallel, it simplifies to

$$\omega^\prime = \omega\biggl(1-\frac{v_{R}}{c}\biggr) \bigg/ \sqrt{1 - \frac{v_R^2}{c^2}} = \omega \sqrt{\frac{c-v_R \vphantom{v_R^2}}{c+v_R}}$$

which may look more familiar from physics books.

The Doppler effect is described in various levels of detail in a multitude of places [1–3]. What I couldn’t find on the ‘net, though, is a clear mathematical derivation of it by means of plane wave functions in the form

$$\begin{aligned}
E(\mathbf{r},t) & =\frac{1}{2}\Bigl(A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)+\mathrm{c.c.}\Bigr)\\
& =\Re\Bigl\lbrace A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)\Bigr\rbrace\\
& =\Bigl|A_{0}\Bigr|\cos\Bigl(\omega t-\mathbf{k}\cdot\mathbf{r}+\mathrm{arg}\bigl[A_{0}\bigr]\Bigr)\label{plane-wave}
\end{aligned}$$

as I regularly use them here (e.g. in this article) and wherever else they might come in handy. To be as general as possible without being bound to any particular coordinate system, these plane waves should be given in terms of the position vector $\mathbf{r}$ and the reciprocal wave vector $\mathbf{k}$, describing a spatial frequency with direction.

Plane Waves

Just about any field distribution can be decomposed into plane waves in order to mathematically propagate them from here to there, as has been done e.g. in the article on the angular spectrum. So if we can describe a linear effect in terms of the plane wave we can describe it for any other wave by decomposing that wave into plane waves, calculating the effect on each of the plane waves and superposing the plane wave solutions to obtain the desired result. Note that this superposition will only work for linear effects / systems.
Now, $E$ in $\eqref{plane-wave}$ can represent the real-valued electric field in the case of light propagation or air pressure in the case of sound waves, $A_{0}$ is the (constant) complex amplitude which contains both magnitude and phase information of the source wave, $\omega$ is the oscillation frequency of the source, $\mathbf{k}$ is the aforementioned wave vector, and “c.c.” stands for the complex conjugate of everything that precedes it. The term

$$A(t)=A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)\label{plane-wave-complex}$$

is called the phasor and fully describes the wave. We will use this phasor a lot for linear systems, because it allows us to work with the relatively simple exponential functions instead of the more complicated cosine. Both are equally valid, though.

The frequency $\omega^\prime$ that the receiver at position $\mathbf{r}$ detects is simply the time derivative of the phase of the phasor $A(t)$:

$$\begin{aligned}
\omega^\prime & =\partial_{t}\arg\bigl[A(\mathbf{r},t)\bigr] \label{frequency}\\
& =\omega
\end{aligned}$$

when the source amplitude $A_{0}$ is constant. As long as nothing moves, the receiver sees (or hears) the same frequency that was sent. When things get moving, however, this changes as a result of the Doppler effect.

There are actually two slightly different Doppler effects in the classical domain. When we have a transporting medium like we have with sound waves, it makes a difference whether the wave source is stationary with respect to the medium (and only the receiver moves) or whether the source is moving, because the velocity $c$ of the waves is defined relative to the medium. Both cases can also be combined in various ways.

Moving Receiver

The mathematically most straightforward case is the moving receiver with a stationary source: The vector $\mathbf{r}$ in $\eqref{plane-wave}$ describes the position of the receiver in space (the source is assumed to be at the origin of the coordinate system). The motion of the receiver is described by

$$\mathbf{r}=\mathbf{r}_{0}+\mathbf{v}_{R}t=\mathbf{r}_{0}+v_{R}\mathbf{\hat{v}}_{R}t$$

in which $v_{R}=\bigl|\mathbf{v}_{R}\bigr|$ is its velocity, $\mathbf{\hat{v}}_{R}$ is a unit vector in the direction of motion and $\mathbf{r}_{0}$ is its initial position. For the moving receiver, $\eqref{plane-wave}$ becomes

$$A_{R}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}\cdot\bigl(\mathbf{r}_{0}+\mathbf{v}_{R}t\bigr)\Bigr]\label{plane-wave-R}$$

The received frequency is determined analogous to $\eqref{frequency}$:

$$\begin{aligned}
\omega_{R}^\prime & =\partial_{t}\arg\bigl[A_{R}(\mathbf{r},t)\bigr]\\
& =\omega-\mathbf{k}\cdot\mathbf{v}_{R}\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)
\end{aligned}\label{frequency-R}$$

where we used

$$\mathbf{k}=\mathbf{\hat{k}}\frac{\omega}{c}\label{normalized-wave-vector}$$

Hence, there is a frequency shift which depends on the direction of the receiver motion relative to the direction of the wave coming from the source. If the receiver approaches the source directly ($\mathbf{k}$ is antiparallel to $\mathbf{v}$), the frequency increase is maximum. The observed frequency shift due solely to the change of the received phase as a result of the receiver motion within the field. Neither the spatial distribution nor the oscillation frequency of the field has changed.

Also note that the case $\omega\rightarrow\infty$ (describing a sonic boom) is not achievable since the receiver velocity cannot reach infinity.

Moving Source

Things change a bit with a source on the move. We could try and switch our frame of reference to the moving source to re-create the situation with a moving receiver and a stationary source in the new coordinates, apply $\eqref{plane-wave-R}$ and $\eqref{frequency-R}$, and be done with it. In this case we would disregard, however, that the medium is now also moving with respect to our coordinate system. And since the wave velocity $c$ is defined relative to the medium, our result would be wrong.

To correct this error, we need to figure out the new wave vector $\mathbf{k}_{S}$ in the new frame of reference in which the medium is moving. We’ll start by decomposing the medium motion into a motion parallel to the initial wave vector $\mathbf{k}$ and one orthogonal to it. The magnitude of the former is

$$\Delta c=-\mathbf{\hat{k}}\cdot\mathbf{v}_{S}=-\frac{c}{\omega}\mathbf{k}\cdot\mathbf{v}_{S}$$

in which $\mathbf{v}_{S}$ is the velocity vector of the source in the reference frame in which the medium is at rest and hence $-\mathbf{v}_{S}$ is the velocity vector of the medium in the moving frame. This component of the medium velocity directly alters the phase velocity $c$ of the wave since the phase velocity is defined relative to it. The other component, orthogonal to $\mathbf{k}$ and parallel to the wave fronts, will shift the wave sideways, but since the plane wave is infinitely extended in that direction, such a shift causes no observable change and can thus be neglected. Since the only change is in the direction $\mathbf{\hat{k}}$ of the wave vector, that direction does not change, merely its magnitude is affected.

Then, with the modified phase velocity

$$c^\prime=c+\Delta c=c\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)\label{velocity-S}$$

we have the new wave vector

$$\begin{aligned}
\mathbf{k}_{S} & =\frac{\omega}{c^\prime}\mathbf{\hat{k}}\\
& =\mathbf{k}\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}\\
& =\mathbf{k}\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}
\end{aligned}\label{wave-vector-S}$$

As our frame of reference transformation is purely translational, the same wave vector $\mathbf{k}_{S}$ is observed in the moving as well as the fixed frame. Note that a simple addition of velocities as in $\eqref{velocity-S}$ is only possible in classical physics. To determine the received frequency as before, we’ll remain in the moving frame, noting that in this frame the receiver also moves with $-\mathbf{v}_{S}$ as does the medium. We insert $\eqref{wave-vector-S}$ into $\eqref{plane-wave-complex}$,

$$A_{S}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}_{S}\cdot\bigl(\mathbf{r}_{0}-\mathbf{v}_{S}t\bigr)\Bigr]$$

and determine $\omega_{S}^\prime$ analogous to $\eqref{frequency}$:

$$\begin{aligned}
\omega_{S}^\prime & =\partial_{t}\arg\bigl[A_{S}(\mathbf{r},t)\bigr]\\
& =\omega+\mathbf{k}\cdot\mathbf{v}_{S}\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1+\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega-\mathbf{k}\cdot\mathbf{v}_{S}}\biggr)\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}\label{frequency-S}
\end{aligned}$$

Note that if the component of the source velocity parallel to the wave vector equals the wave velocity $c$ in the medium, both $\mathbf{k}$ and $\omega_{S}^\prime$ become infinite, resulting in a sonic boom in the case of sound waves. Also, the apparent frequency shift is now due the receiver motion within the wave field as above, as well as the change of the wavelength within the wave field. The frequency of the wave again remains the same.

Both Moving

When both source and receiver are moving, we only need to alter $\eqref{plane-wave-S}$ to additionally account for the receiver motion:

$$A_{RS}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}_{S}\cdot\bigl(\mathbf{r}_{0}-\mathbf{v}_{S}t+\mathbf{v}_{R}t\bigr)\Bigr]$$

resulting in a received frequency of

$$\begin{aligned}
\omega_{RS}^\prime & =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}
\end{aligned}$$

The correction factor to the frequency $\omega$ in this case is the product of the correction factors in $\eqref{frequency-R}$ and $\eqref{frequency-S}$. Also, if both source and receiver move with the same velocity relative to the medium, the received frequency is again equal to the source frequency $\omega$.

Note that the frequency $\omega_{RS}^\prime$ depends on the absolute velocities of source and receiver, not just how they move relative to each other. This is due to the crucial role of the medium in classical physics.

Echo from a Moving Reflector

A stationary source emits a wave which is reflected back towards the source by a moving receiver. A Doppler shift as above occurs in both directions. The frequency observed by the source can again be found by a combination of the first two cases. The outgoing wave is received by the reflector with an observed frequency of $\eqref{frequency-R}$,

$$\omega_{SRS}^\prime=\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)$$

The reflector in turn becomes a moving source which emits waves with the modified frequency $\omega_{SRS}^\prime$ which is observed by it and also a modified wave vector corresponding to that frequency

$$\mathbf{k}^\prime=-\mathbf{k}\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)$$

cf. also $\eqref{normalized-wave-vector}$. The minus sign results from the change of direction upon reflection. The frequency observed by the source upon reception of the waves emitted by the reflector is given by $\eqref{frequency-S}$ in which $\mathbf{v}_{R}=\mathbf{v}_{S}$ (both describing the motion of the reflector) and $\omega$ and $\mathbf{k}$ have been replaced by $\omega_{SRS}^\prime$ and $\mathbf{k}^\prime$, respectively:

$$\begin{aligned}
\omega_{SRS}^{\prime\prime} & =\omega_{SRS}^\prime\biggl(1-\frac{\mathbf{k}^\prime\cdot\mathbf{v}_{R}}{\omega_{SRS}^\prime}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\biggl(1+\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)\biggl(1+\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)^{-1}\\
& =\omega\frac{c-\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c+\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}
\end{aligned}
\label{frequency-SRS}$$

Decomposition

Note that the perceived Doppler shift in $\eqref{frequency-SRS}$ depends solely on the component of the reflector motion which is parallel to the propagation velocity of the plane wave. If the source is not a source of plane waves, which it cannot be for plane waves are infinitely extended, we can decompose whatever waves the source does emit into an angular spectrum of plane waves, as mentioned above. See also the article on the angular spectrum in the context of Fourier diffraction. Each of these plane waves has a different direction $\mathbf{\hat{k}}$ and thus will generally cause a different detected frequency shift upon reflection and subsequent reception. The source will thus see (or hear) a whole spectrum of reflected frequencies.

This is compounded by the finite extent of the receiver and thus a finite cross-section for reflection. Each incoming plane wave is spatially filtered upon reflection so that the reflected portion of it must again be decomposed into its angular spectrum, each component of which is received with a different Doppler shift back at the source. The reflection spectrum can possibly be deconvoluted and analyzed to reveal information about the reflection cross-section, but such an analysis exceeds the scope of this article.

[1] Doppler effect [Wikipedia]
[2] Doppler effect [James B. Calvert]
[3] Doppler Shift for Sound and Light [Reflections on Relativity]

There’s not much to that, though; the plugin simply removes the ## section links indicator and all text following it prior to comparing each link to the title of the current tiddler.

If you’d like to always have the list of incoming links displayed beneath the list of tags for the current tiddler, edit the ViewTemplate tiddler and change the line

<div class='tagged' macro='tags'></div>

to

<div class='tagged'>
  <div macro='tags'></div>
  <div macro='whatLinksHere "<br />links: <br /> "'></div>
</div>

To install the plugin, copy and paste everything below the divider into a new tiddler in your TiddlyWiki, tag it with systemConfig and reload.

/***
|Name|WhatLinksHerePlugin|
|Source|http://rumkin.com/tools/tiddlywiki/#WhatLinksHerePlugin|
|Version|1.1.0|
|Author|Tyler Akins, Marcus Winter|
|License|Public Domain|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|Displays a list of tiddlers that link to the current tiddler.|

!Usage
{{{
<<whatLinksHere txtIfList txtIfNoList>>
}}}
* txtIfList: String to print at the top of the list if there is at least one result.
* txtIfNoList:  String to print at the top if there are no tiddlers that link to the current tiddler.

!Configuration
Do not list these tiddlers:
{{wideInput{<<option txtWhatLinksHereExclude 40>>}}}

!Examples
This is a live example of what links to this tiddler:
|<<whatLinksHere "List of tiddlers that link to me: <br />" "Sorry, nobody links to me.">>|

!Installation
# Copy/paste the WhatLinksHerePlugin tiddler into your TiddlyWiki and tag it with systemConfig.
# Modify a tiddler or template to use the whatLinksHere macro.

!Revision History
* 1.1.0 (2012-02-24)
** Modified the code to also recognize section links like tiddler##section.
* 1.0.0 (2007-09-30)
** Initial version by Tyler Akins (http://rumkin.com/tools/tiddlywiki/#WhatLinksHerePlugin).

!Code
***/
//{{{
version.extensions.WhatLinksHerePlugin={major: 1, minor: 1, revision: 0, date: new Date(2012,2,24)};

// initialize 'autozoom' and 'exclude' tree options (defaults are not to zoom, and to follow all links)
if (config.options.txtWhatLinksHereExclude===undefined)
	config.options.txtWhatLinksHereExclude='GettingStarted DefaultTiddlers tabTimeline';
if (config.optionsDesc)
	config.optionsDesc.txtWhatLinksHereExclude = "List of tiddlers to not show."

config.macros.whatLinksHere={
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var span = createTiddlyElement(place, "span");
		var excludes = config.options.txtWhatLinksHereExclude.readBracketedList();
		var results = [];
		var tid = story.findContainingTiddler(place);
		var root = "";
		var out = ""
		if (tid) {
			root=tid.getAttribute("tiddler");
		}

		store.forEachTiddler(function(title,tiddler) {
			if (! excludes || ! excludes.contains(title)) {
				var links = tiddler.getLinks();
				for (var i=0; i < links.length; i++) {
					if (links[i].split("##")[0] == root) {
						results.pushUnique(title);
					}
				}
			}
		});
		results.sort();
		out = "[[" + results.join("]]\n[[") + "]]";
		if (results.length && params[0])
			out = params[0] + out;
		if (! results.length && params[1])
			out = params[1] + out;
		out = "{{whatLinksHere{" + out + "}}}";
		wikify(out, place);
	}
}

//}}}

If you really wanted to know details about what Polytec does, this is the wrong place to look anyway – there is infinitely more information in books like Optical Inspection of Microsystems, co-authored by Polytec engineers, or in conference proceedings such as those sponsored by A.I.VE.LA, to which Polytec contributes regularly.

Distribution of focal lengths used during three weeks in Hawai’i. Blue bars include shots that were stitched into panoramas, green bars represent only single-frame images.

Only the shots that I kept out of the 3,180 total were used to make the figure.

Interestingly, the short focal lengths see the most use, with the minimum possible length dominating the statistics, even more so when including the shots comprising the 205 panoramas. I guess that’s not unusual, since most of these are pictures of landscape (after all, it’s Hawai’i). However, I was obviously to lazy (or too hurried) to even once switch to the 14mm pancake lens that I lugged around with me.
There were a couple of situations (<1.5% of the pictures) where I likely could have used a longer lens, but had to settle for the 55mm long end of my 18-55mm zoom.

OTDM Recap

Some basics of OTDM / Nyquist WDM are explained in The Art of Nyquist WDM which I recommend reading if you’re new to the concept. Recently, I find the name OTDM much more fitting due to its duality with OFDM. However, what’s commonly termed OTDM in optics (multiplexing really short pulse trains) I assume stands for optical TDM, since it doesn’t make any use of an orthogonality relation (though technically, the tributaries are orthogonal).

Anyway, the basic premise of OTDM is that $K$ pulses are overlapping in time. Only by their special shape (the truncated sinc function) can they be detected without crosstalk. The sinc function has the fortunate property of having zeros at regular intervals. By making it so that these zeros occur at the centers of all neighboring symbol slots, there will only be one symbol not equal to zero at each slot center (this being the symbol that is centered on that particular slot). At all other times within the symbol slot, all symbols can have values other than zero and interfere. Hence, we immediately expect quite different signal statistics depending on the position within the symbol slot we are interested in.

The spectrum of the sinc function is rectangular (or almost rectangular for the truncated sinc) so that many such channels could be put very close to each other without spectral crosstalk.

Importance Sampling

In contrast to OFDM which was (in a special case) the superposition of randomly chosen letters from the same alphabet (the QAM constellation diagram) with equal weights, the weights of the different superposed symbols in OTDM are defined by the sinc function. At the symbol slot center, for instance, the weight for the symbol centered in that slot is unity while for all other symbols it is zero. In this case the PMF for a quadrature, the signal or its power can be simply determined from the constellation diagram of the modulation format.

At all other positions the weights are different and non-zero. We cannot therefore use combinatorical means to tackle this problem. Due to the low probabilities involved (which are expected to be similar to OFDM), an exhaustive numerical simulation is not feasible and a Monte-Carlo simulation will not yield the relevant low-probability details. We can, however, apply the concept of importance sampling to our Monte Carlo simulations [1]. Here, we bias the random drawing (of points from the constellation diagram for each symbol) to favor the occurrence of signals with high powers, i.e. we prefer to stay in the corners of the constellation diagram and switch to the opposite corner whenever the sign of the particular sinc-weight for a symbol becomes negative. In this way we get samples for which all the constituents are quite large and in phase. We have to retain some randomness, however, since we are interested in the PMF/CDF and not just the maximum possible value (more on that below). The biased probability for the symbols is compensated when later calculating the PMF.

Biasing toward higher sample powers results in a smaller number of samples with low power, which in turn leads to larger variance in that part of the PMF. We would like to have an accurate representation of the whole PMF, though. Hence, we will generally simulate the same modulation format with different bias values and later combine their results using the balance heuristic which is basically just a pretty optimal way of combining such data [2].

You will find the MATLAB code (not optimized for speed or anything) at the end of the article.

Signal Power

Since the code is available to play around with, we will look at just a few results to highlight the differences between OFDM and OTDM. In particular, the symbol length will be $16T$, i.e. 16 times the time between consecutive symbols (or the inverse symbol rate), and thus $K=16$. The alphabets used will be 16-QAM ($N=4$) and 64-QAM ($N=8$). As can be seen in Fig. 5 in the OFDM article, this results in probabilities in the vicinity of $10^{-28}$ for the largest powers in OFDM signals, something not feasible to be figured out by “regular” Monte Carlo techniques.

Figure 1: Cumulative distribution functions of sample power for OTDM signals with K = 16 overlapping symbols. Also shown are comparable distributions of OFDM signals (K = 16) and the χ-squared distribution corresponding to complex Gaussian signals. Samples are taken at the symbol slot boundaries.

The figure shows the worst case regarding high signal powers, with the sampling instants chosen at the symbol slot boundaries. See also the next section for more details. While the probabilities of the maximum values are the same (they depend only on $K$ and $N$ as shown in the OFDM post), the maximum observable power is less than half in OTDM. At high probabilities like $10^{-6}$ the difference is smaller. This is a result of the heterogeneous weights of the contributions from the different symbols to each sample, as opposed to equal (magnitudes of) weigths in OFDM.

Sometimes there will be “kinks” in the graphs so obtained by importance sampling. These are a result of outliers whose effect is amplified by the algorithm (in the code, data with very low dataweights contribute significantly more). This is a known problem of importance sampling. As these occur randomly, they might or might not be there. Their position is also random, so that they can be identified by multiple runs of the code.

Other Sample Times

In OFDM, the sample power distribution varied only little with sample timing within the symbol slot (cf. Figs. 6 and 7 in the OFDM post). As the constellation diagrams of consecutive OTDM symbols do not rotate relative to each other, it is quite straightforward to determine the maximum possible sample power at various instants within the symbol slot by simply adding the weight magnitudes (plus squaring and scaling). They can even be given analytically.$^1$

Figure 2: Maximum possible sample power versus sampling instant within the symbol slot. Colored markers correspond to curves in Fig. 3.

Unlike OFDM, the maximum power does not have multiple local maximums within the symbol slot, but varies by about an order of magnitude between symbol slot center and boundaries. The sample power CDFs vary accordingly:

Figure 3: Cumulative distribution functions of sample power for OTDM signals with K = 16 overlapping symbols and 16-QAM modulation for various sampling instances within the symbol slot.

For $t_S=0.5$ (the symbol slot center), the CDF corresponds to the constellation diagram, as at this instant only one of the overlapping symbols is non-zero.

Oversampling

For (always necessary) oversampling, the distributions for the appropriate sampling instances can simply be combined. Interestingly, for simple two-fold oversampling, half the samples will be taken in the symbol slot center and the other half at its boundaries. Thus, the worst case regarding symbol power is included and the high-power samples resulting thereof need to be accounted for by leaving appropriate headroom in the digital-analog converters, amplifiers and whatnot.

MATLAB Code

clear variables

LoS = 16; % length of symbol in units of symbol slots
QAM = 64; % number of possible values in each quadrature for 16-QAM - only powers of 2 allowed
NoS = 2^20; % number of Monte Carlo samples

offset = 0.5; % relative sampling position within symbol slot - 0.5 is center

% list of biasing coefficients used; need not be integer
explist = [0,1,2,3]; % works well for 16-QAM, LoS = 16
% explist = [0,2,4,6,8]; % works well for 64-QAM, LoS = 16

bins = linspace(0,16,1001); % set histogram bins for graphing

%%% set-up of some variables

normalization = 1/sqrt(2/3*(QAM-1)); % alphabet normalization
normalization = normalization / 0.987345; % unit energy per symbol for LoS = 16
% normalization = normalization / 0.993669; % unit energy per symbol for LoS = 32
normweight = (1/QAM)^LoS; % probability of any particular symbol w/o biasing

% sinc coefficients of each sample within the symbol
coeffs = sinc((-LoS/2:LoS/2-1) + offset);

% constellation diagram
NoV = sqrt(QAM); % number of possible values in I and Q
values = 0:2:2*(NoV-1); values = values - NoV + 1; % possible values in I and Q for each symbol
values = values * normalization;
symbols = repmat(values, [NoV,1]) + 1i * repmat(values.', [1,NoV]);
symbols = symbols(:); % complex QAM alphabet

bins = [bins, Inf];
PMF = zeros(length(bins), length(explist));
bincounts = PMF;

%%% importance sampling Monte Carlo simulation

for jj = 1:length(explist)

    % biasing is based on distance to opposite corner symbols (depending on coeff sign)
    distplus = abs(symbols - symbols(1));
    wplus = 1 ./ (1 + distplus).^explist(jj); % weights avoid division by zero
    wplus = wplus / sum(wplus); % normalization

    distminus = abs(symbols - symbols(QAM)); % opposite corner
    wminus = 1 ./ (1 + distminus).^explist(jj);
    wminus = wminus / sum(wminus);

    origdata = rand(NoS,LoS); % generate random data (symbol plus neighbors)

    data = zeros(NoS,LoS);
    dataweights = zeros(NoS,LoS);

    for kk = 1:LoS
        if coeffs(kk) >= 0
            for ii = (QAM):-1:1; % draw from biased distribution
                data(origdata(:,kk) < sum(wplus(1:ii)),kk) = symbols(ii);
                dataweights(origdata(:,kk) < sum(wplus(1:ii)),kk) = wplus(ii);
            end
        else % for negative coeffs bias towards opposite corner
            for ii = (QAM):-1:1;
                data(origdata(:,kk) < sum(wminus(1:ii)),kk) = symbols(ii);
                dataweights(origdata(:,kk) < sum(wminus(1:ii)),kk) = wminus(ii);
            end
        end
    end

    data = data .* repmat(coeffs, [NoS, 1]); % each row corresponds to one sample

    powers = abs(sum(data, 2)).^2;
    dataweights = prod(dataweights, 2);

    if explist(jj) ~= 0 % remove IS outliers
        index = find(dataweights >= normweight);
        powers = powers(index);
        dataweights = dataweights(index) * NoS / 4;
            % division by 4 is necessary to compensate for biasing towards only 1 corner
    else
        dataweights = dataweights * NoS;
    end

    % build PMF histograms for each biasing value
    for ii = 1:length(bins)-1
        index = find(powers >= bins(ii) & powers < bins(ii+1));
        PMF(ii,jj) = sum(normweight ./ dataweights(index));
        bincounts(ii,jj) = length(index);
    end

end

% average histograms using balance heuristic
weights = PMF .* bincounts ./ repmat(sum(PMF .* bincounts, 2), [1, length(explist)]);
weights(isnan(weights)) = 0; % bincount of zero causes weights to become NaN

avgPMF = sum(PMF .* weights, 2);

% complementary CDF
for ii=1:length(avgPMF)
    CDF(ii) = sum(avgPMF(ii:end));
end

Note the normalization of the symbol amplitude in order to obtain unit mean energy per symbol. This accounts for the distribution of the constellation points for the particular QAM format as well as the truncated sinc pulse shape. Since

$$\intop_{-K/2}^{K/2} \mathrm{sinc}^2 x \, dx < 1 \quad \text{for} \quad K < \infty$$

the energy reduction resulting from truncation is compensated by appropriate scaling.

1 The main problem is to find

$$S_\mathrm{max}^2 = \left[\sum_{k=0}^{K-1} \left| \mathrm{sinc} \left(k - K/2 + t_S\right) \right| \right]^2 = \left[\sum_{k=0}^{K-1} \left| \frac{\sin \bigl[\pi \left(k - K/2 + t_S\right)\bigr]}{\pi \left(k - K/2 + t_S\right)} \right| \right]^2$$

which is then multiplied by the highest power occuring in the constellation diagram of the particular modulation format (which also depends on $K$ when properly normalized). For simplicity, we’ll assume $K$ to be even and we will only try and find $S_\mathrm{max}$, and leave the squaring to the interested reader. The sine in the numerator will then have a constant value, determined by $t_S$ that simply alternates its sign with $k$ and thus can be written

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi}\sum_{k=0}^{K-1} \left| \frac{c_S \left(-1\right)^k}{k - K/2 + t_S} \right|$$

with

$$c_S = \begin{cases} 1 & K/2 \text{ even}\\ -1 & K/2 \text{ odd}\end{cases}$$

We split the sum to get rid of the negative denominator and with it the absolute value function and the sign coefficients:

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi}\sum_{k=0}^{K/2-1} \frac{1}{k + t_S} - \frac{1}{k - K/2 + t_S}$$

These are then just two general harmonic series. Their solution can be given by generalized harmonic numbers or in terms of the Digamma function $\psi(\cdot)$ [3]

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi} \left[ \psi\left(t_S - \frac{K}{2}\right) + \psi\left(t_S + \frac{K}{2}\right) - 2 \psi\left(t_S \right) \right]$$

[1] Importance Sampling [Wikipedia]
[2] E. Veach, Robust Monte Carlo methods for light transport simulation, dissertation, Stanford University, 1998.
[3] T. M. Rassias and H. M. Srivastava, “Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers,” Applied Mathematics and Computation, vol. 131, pp. 593–605, 2002. ]]> http://www.marcuswinter.de/archives/1612/feed 0 http://www.marcuswinter.de/archives/1610 http://www.marcuswinter.de/archives/1610#comments Mon, 05 Dec 2011 13:19:46 +0000 Marcus http://www.marcuswinter.de/?p=1610 I am still catching up to getting some order into the pictures I took during our last vacation on Hawai’i. And that was back in September! Anyway, here are some of the better panoramas from our first stop, Big Island. Click through for the large version at flickr, with a short description.

scenic point on Hwy 19, Big Island, Hawai’i (panorama)

Pololu Valley lookout, Big Island, Hawai’i (panorama)

halfway down Pololu Valley, Big Island, Hawai’i (panorama)

Pololu Valley, Big Island, Hawai’i (panorama)

Pololu Valley, Big Island, Hawai’i (vertorama)

Rainbow Falls, Big Island, Hawai’i (panorama)

cove in Onomea Bay, Big Island, Hawai’i (panorama)

off Hawai’i Belt Road, Kau, Big Island, Hawai’i (panorama)

Halema’uma’u Crater in Kilauea Caldera, Volcanoes National Park, Big Island, Hawai’i (panorama)

cinder cone off Saddle Road, Big Island, Hawai’i (panorama)

clouds rolling in, Mauna Kea Access Road, Big Island, Hawai’i (panorama)

central Big Island seen from Mauna Kea Visitor Center, Hawai’i (panorama)

seeing double at Mauna Kea Visitor Center, Big Island, Hawai’i (panorama)

W.M. Keck Observatory, Mauna Kea summit, Big Island, Hawai’i

cinder cones and observatories, Mauna Kea summit, Big Island, Hawai’i

Lake Waiau, Mauna Kea summit, Big Island, Hawai’i (panorama)

Kealakekua Bay, Big Island, Hawai’i (panorama)

Kealakekua Bay, Big Island, Hawai’i (panorama)