# Quantifying the value of transient voltage sources

Swati<EMAIL_ADDRESS>Uttam Singh<EMAIL_ADDRESS>Oscar C. O. Dahlsten
<EMAIL_ADDRESS>Department of Computer Science, The University of Hong
Kong, Hong Kong  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, SUSTech, Nanshan District, Shenzhen, China  Center for
Theoretical Physics PAS, Aleja Lotników 32/46, 02-668 Warsaw, Poland

###### Abstract

Some voltage sources are transient, lasting only for a moment of time, such as
the voltage generated by converting a human motion into electricity. Such
sources moreover tend to have a degree of randomness as well as internal
resistance. We investigate how to put a number to how valuable a given
transient source is. We derive several candidate measures via a systematic
approach. We establish an inter-convertibility hierarchy between such sources,
where inter-conversion means adding passive interface circuits to the sources.
Resistors at the ambient temperature are at the bottom of this hierarchy and
sources with low internal resistance and high internal voltages are at the
top. We provide three possible measures for a given source that assign a
number to the source respecting this hierarchy. One measure captures how much
“$\mathrm{UnitDC}$” the source contains, meaning 1V DC with 1$\Omega$ internal
resistance for 1s. Another measure relates to the signal-to-noise ratio of the
voltage time-series whereas a third is based on the relative entropy between
the voltage probability distribution and a thermal noise resistor. We argue
that the $\mathrm{UnitDC}$ measure is particularly useful by virtue of its
operational interpretation in terms of the number of $\mathrm{UnitDC}$ sources
that one needs to combine to create the source or that can be distilled from
the source.

###### keywords:

Voltage sources , Energy harvesters , Thermal noise , Resource theories ,
Energy conversion

##  1 Introduction

As the world seeks renewable and clean sources of electric energy, the
importance of imperfect voltage sources increases. For example, motion-energy-
harvesting devices converting human motion to electricity are expected to play
a crucial role in powering the Internet of Things without wires and batteries
[1, 2]. Larger-scale examples of transient sources range from regenerative
braking and suspension harvesting [3, 4] to wind power [5, 6]. An important
challenge in the usage and design of such transient voltage sources is that
their voltage output often has variability, including randomness, induced by
the motion source behaviour, as well as significant internal resistance [7, 8,
9].

The undeniable rise in importance of such transient voltage sources raises the
question of whether one can capture the value of such a source in a single
number, analogously to how a battery comes with a single number stating how
much energy it can provide, or to the concept of free energy in thermodynamics
[10]. This is important both to fairly compare different source alternatives
and to guide design: what number(s) should be optimised to maximise the value
of a given source? One option in quantifying the value of a source is to focus
on observed averages such as root-mean-squared voltage
($\mathcal{V}_{\mathrm{rms}}$) [1, 2]. This does not distinguish between
random or deterministic sources. Is this justified? Should one and if so by
how much, penalise unpredictability as well? On one hand, thermal noise
voltage should arguably be counted as having no value, but on the other hand,
highly random sources with stronger noise can be used as power sources [7] and
should thus carry value. Moreover, how can one incorporate the internal
resistance into the value assigned to the voltage source?

To tackle this question of how to value transient voltage sources we make use
of a conceptually clear paradigm for quantifying resources in general, known
as the resource theory approach. In our case, the resource is the voltage
source. The resource theory paradigm has proven useful to formalise
thermodynamics [11] and is significantly used in quantum information theory
[12, 13, 14]. A key idea is to define some inter-conversion operations as
free, and consider which resources can be converted to other resources under
these free operations, thus creating a hierarchy that the measure for
quantifying resources must respect. If resource A can be converted into
resource B with free operations, the number we assign to how useful A is, must
be at least as high as that of B.

Here the resources A and B shall correspond to voltage sources and the free
operations as interconversion circuits. We treat a voltage source as a black
box two-terminal electrical device and assume it can be given a (Thevenin)
equivalent circuit of a voltage source $V_{s}(t)$ in series with an ideal
(non-complex) resistor. Examples of such sources are depicted in Fig.1.

Figure 1: Different transient voltage sources and their UnitDC values are
depicted. The sources are modelled as an internal time-dependent source
voltage $V_{s}$ (the circle) in series with an internal resistance $R_{s}$
(zig-zag lines). $V_{s}$ is plotted as a function of time inside the circle.
We assume $V_{s}=S+r$ where $S$ is a deterministic signal and $r$ a Gaussian
random variable. At the foot of each depicted source is the
($\mathrm{UnitDC}$) value we assign to the given source (for a pulse duration
of one cycle):
$\mathrm{UnitDC}=\frac{\tau}{R_{s}}\left(\mathbb{E}|S|^{2}+\left|\mathbb{E}|r|^{2}-\mathbb{E}|r_{th}|^{2}\right|\right)$,
where $\mathbb{E}$ denotes the statistical average of the time-average over
the pulse duration $\tau$, and $\mathbb{E}|r_{th}|^{2}$ is the variance of the
noise corresponding to the internal resistance $R_{s}$ being at the ambient
temperature (see Sec. 6.3, Eq. (9)). We show that if a source A has a higher
$\mathrm{UnitDC}$ value than another source B, conversion from B to A is not
possible with ‘free’ passive operations.

Converting between transient voltage sources here means connecting an
intermediate four-terminal circuit to the harvester, with two terminals
interfacing to the harvester and another two acting as the new output. To
avoid the interface circuit adding value to the original source, we want the
intermediate circuit to be passive (non-powered). More specifically, we allow
the intermediate conversion circuit to implement pre-chosen power-preserving
reversible operations such as voltage sign-flips and phase shifts [9]. At
least some of those conversions are plausible to implement [15, 16, 17, 18,
19]. We moreover take the addition of thermal noise at some ambient
temperature $T$ as free.

Using this approach we derive three quantifiers of the value of transient
voltage sources: (i) one based on the signal-to-noise ratio (SNR) of the
voltage series, (ii) one which quantifies how much $1V$ DC at 1$\Omega$ for
$1s$ one may obtain from the source, (iii) one based on the relative entropy
between the source and a thermal source. We compare the three and conclude
that option (ii) which we term the “UnitDC” value of the source is
particularly attractive due to its clear operational interpretation. Examples
of sources and their UnitDC values are given in Fig. 1. The UnitDC value
moreover turns out to match well with a method used by many energy-harvesting
experimentalists to quantify the performance of a given harvesting setup, in
terms of the energy dissipated into an optimal load resistance. We also show
that certain alternative quantifiers, such as the peak power dissipated into a
matched load, may increase under free inter-conversion operations and are thus
arguably not good quantifiers of the value of a transient voltage source.

We proceed as follows. We begin with describing the resource theory approach
concretely. Then we define the models of transient voltage sources, and the
free interfacing circuits we shall use. We derive which resource quantifiers
respect the inter-conversion hierarchy and which do not. We finally analyse
how the value is increased under composition of sources, and how many unitDC
sources can be ‘distilled out’ from a given source via free operations.

##  2 Resource theory approach

A resource theory is a systematic framework to quantify resources at hand
[14]. Resource theories are primarily developed in the field of quantum
information theory and include the resource theory of quantum entanglement
[12], the resource theory of quantum thermodynamics [11, 20], the resource
theory of asymmetry [21, 22, 23], and the resource theory of coherence [24,
25]. The scope of the resource theory approach is not limited to the quantum
case only as our work here exemplifies.

Let the set of all states (in our case corresponding to voltage sources) be
denoted by $\mathscr{D}$. Let $\mathcal{V}\in\mathscr{D}$ be an arbitrary
state. Let the set of all operations on the states be $\mathscr{M}$. A
resource theory comprises three basic ingredients (not independent of one
another): 1\. A set of free states $\mathcal{F}\subset\mathscr{D}$, 2. A set
of free operations $\Lambda_{f}\subset\mathscr{M}$, 3. Sets of resource states
and operations (complementary sets to $\mathcal{F}$ and $\Lambda_{f}$,
respectively). This is depicted in Figure 2. The set of free operations is
chosen such that it cannot generate resource states from the free states [13].
A key purpose of setting up a resource theory is to determine a resource
quantifier in a systematic manner. Any real valued function $\mathcal{R}$ on
the set of all states $\mathscr{D}$ is said to be a resource monotone if it
satisfies the following set of conditions:
(1) Faithfulness: $\mathcal{R}(\mathcal{V})=0$ if and only if $\mathcal{V}$ is
a free state, i.e., $\mathcal{V}\in\mathcal{F}$.
(2) Monotonicity under free operations.Other conditions, e.g. concerning how
the resource monotone behaves under probabilistic combinations of states, may
also be added but are not needed here as we will not include probabilistic
combinations of sources.

Figure 2: A high-level depiction of a resource theory. The set $\mathscr{D}$
represents the set of all states while the set $\mathcal{F}\subset\mathscr{D}$
represents the set of free states. Let $\mathscr{M}$ denote the set of all
possible operations on $\mathscr{D}$. Then, the free operations
$\Lambda_{f}\subset\mathscr{M}$, map $\mathcal{F}$ into $\mathcal{F}$. The
remaining states $\mathscr{D}\setminus\mathcal{F}$ and the remaining
operations $\mathscr{M}\setminus\Lambda_{f}$ are then called resource states
and operations, respectively.

##  3 Overview of resource theory of transient voltage sources

  * 1.

We identify states as voltage sources, corresponding to a pseudo-random
voltage time series $V_{s}(t)$, placed in series with a source resistance
$R_{s}$. Free states correspond to resistors with thermal noise at the ambient
temperature.

  * 2.

Operations are defined as connecting four-terminal circuits to the two-
terminal source, such that the free two terminals are associated with the new
voltage output. This leads to a new (Thevenin equivalent) source resistance
$R_{s}^{\prime}$ and a new (Thevenin equivalent) voltage source
$V_{s}^{\prime}(t)$. Free operations correspond to certain passive and
thermally noisy four-terminal circuits.

  * 3.

We identify three resource quantifiers (monotones) respecting the
interconversion hierarchy, meaning that the value the quantifier assigns to
the source cannot increase under the free operations.

We now proceed to describe this resource theory concretely, beginning with the
states.

##  4 Model for transient voltage source: The states

We model a two-terminal voltage source as an ideal voltage source in series
with a resistor. An ideal voltage source alone yields, by definition [26], a
voltage across its two terminals that is independent of what they are
connected to. This is an unphysical idealisation and real devices, such as
real batteries [27], are often modelled as a voltage source with voltage
$V_{s}$ in series with a resistor with resistance $R_{s}$. A justification for
this model is that physical circuits are known to be well represented by
networks of resistors, capacitors, and inductors (the lumped element model)
and within that model, it can be shown that a two-terminal circuit acts, at
the black box level, identically to an ideal voltage source in series with an
(in general complex) impedance $Z_{s}$ [26]. Here we choose for technical
convenience $Z_{s}$ to be real 111If $Z_{s}$ is complex, the source will
arguably have the same value as a source with $R_{s}=\mathrm{Re}(Z_{s})$, at
least in the case of a sinusoidal single-frequency voltage $V_{s}$. In that
case, and possibly in general, one can (for idealised circuits) reversibly
convert between $R_{s}=\mathrm{Re}(Z_{s})$ and the complex impedance $Z_{s}$
using an LC circuit (which is passive) connected in series. (An inductor and
capacitor in series have $Z_{LC}(\omega)=i(\omega L)-i\frac{1}{\omega C}$
which can thus be made to equal $-\mathrm{Im}(Z_{s})$ by suitable choice of
constants $L$ and $C$.).

The source voltage $V_{s}$ and resistance $R_{s}$ can be determined
experimentally. If connected to a resistive load of resistance $R_{L}$ then
the voltage across the two terminals equals (up to a sign) that across the
load, namely
$V_{L}=IR_{L}=\frac{V_{s}}{(R_{s}+R_{L})}R_{L}=\frac{V_{s}}{(\frac{R_{s}}{R_{L}}+1)}$,
where $I=\frac{V_{s}}{(R_{s}+R_{L})}$ is the current through the load. One
sees $I$ is impeded by the internal resistance $R_{s}$. That expression for
$V_{L}$ moreover justifies a method for experimentally determining $R_{s}$ and
$V_{s}$: checking the voltage drop when connecting a load. The open-circuit
measured voltage $V_{oc}$ is approximately the load voltage associated with
$R_{L}\rightarrow\infty$ so $V_{oc}=V_{s}$. Moreover, in a few lines, one sees
$R_{s}=\frac{V_{s}}{I}-R_{L}=\frac{V_{s}}{I}-\frac{V_{L}}{I}$ for which all
the terms in the RHS are experimentally accessible with a voltmeter and
ammeter. This procedure is analysed in the context of real batteries e.g. in
[27]. $V_{s}$ moreover has an important operational meaning in terms of the
black-box voltage $V_{L}$ for an optimised resistive load with resistance
$R_{L}$. It is in fact common in experiments in energy harvesting [2] to
compare different harvesters by the energy dissipated in an optimised
resistive load, with $R_{L}$ chosen such that it optimises the power
dissipated in the load: $P_{L}=V_{L}^{2}/R_{L}$. One can see that for all
other parameters being equal, $P_{L}$ is optimised for $R_{L}=R_{s}$, by
noting that $V_{L}=IR_{L}=\frac{V_{s}R_{L}}{(R_{s}+R_{L})}$ and setting
$\partial_{R_{L}}P_{L}=0$. For this choice of $R_{L}$, $V_{L}=V_{s}/2$.

$V_{L}|_{R_{L}=R_{s}}=\mathrm{argmax}_{V_{L}}P_{L}=\frac{V_{s}}{2}.$ (1)

Thus, up to a factor of 2, the internal source voltage $V_{s}$ may be thought
of as the load voltage $V_{L}$ for an optimised resistive load.

We allow for $V_{s}$ to have a possibly complicated structure. Firstly, the
source voltage $V_{s}(t)$ may depend on time and last for some specified time
interval. Experimentally, it will be sampled some $N$ times at fixed time
intervals $\delta t$, yielding a vector
$\vec{\mathcal{V}}:=(V_{1},\cdots,V_{N})^{T}$, where T indicates the
transpose. Secondly, $V_{s}(t)$ may be noisy, a subtle notion. From the
perspective of a given agent, a time-series may be split, in line with the
signal-processing paradigm, into signal plus noise: $V_{i}=S_{i}+r_{i}$, where
$S_{i}$ is deterministic and $r_{i}$ random and $i$ labels the sample. For
example we may have a sinusoidal signal such that $S_{i}=\sin(\omega t_{i})$
and $r_{i}$ being picked randomly from a normal distribution (Gaussian) at
each $i$. We allow for the signal to be a Fourier series in general. For this
class of generating models there are thus well-defined vectors
$\vec{\mathcal{S}}:=(S_{1},\cdots,S_{N})^{T}$ and
$\vec{r}:=(r_{1},\cdots,r_{N})^{T}$ for any run of an experiment. In the
example when the signal is sinusoidal and the noise is Gaussian,
$\vec{\mathcal{S}}$ is distributed as

$\displaystyle
p(\vec{\mathcal{S}})=\prod_{i}p_{i}(\mathcal{S}_{i})=\prod_{i}\delta(\mathcal{S}_{i}-\sin\omega
t_{i}),$ (2)

and the noise $\vec{r}$ as

$\displaystyle
q(\vec{r})=\prod_{i}q(r_{i})=\prod_{i}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[-\frac{r_{i}^{2}}{2\sigma_{i}^{2}}\right].$
(3)

Figure 3: Two ideal resistors with Gaussian voltage time series. Each models a
single physical resistor at some temperature. At each time moment, a voltage
value is selected from a Gaussian distribution centred on 0 and whose standard
deviation is proportional (for some frequency window) to $RT$ where R is the
resistance and T the temperature of the thermal resistor. Thus the right
resistor has a higher temperature than the left.

Such noise is termed additive white Gaussian noise (‘additive’ because
$V_{i}=S_{i}+r_{i}$, ‘white’ because the power spectral density of $r(t)$
turns out to be the same for all frequencies ), abbreviated as AWGN. This
choice of a Gaussian noise model is common, convenient, and in our case
particularly important because we want the resource hierarchy we establish to
be consistent with thermodynamics, such that thermal resistors at the ambient
temperature $T$ are at the bottom of the hierarchy. Thermal resistors are
commonly modelled as a resistance $R_{T}$ in series with a voltage source
$V_{T}$ sampled as additive white Gaussian noise (AWGN) with variance given by
[28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47]

$\sigma^{2}=\Delta f4k_{B}TR_{T},$ (4)

where $\Delta f$ is the frequency bandwidth being sampled, which can be
thought of as a setting in an oscilloscope. (Strictly speaking, Eq. (4) is an
approximation that leads to a diverging variance in $\Delta f$ and more
generally the Planck distribution should be used [28].) Such thermal resistors
are depicted in Fig. 3. There may moreover be Gaussian noise that comes not
from a thermal process, but e.g. due to a variety of factors giving rise to
Gaussian noise via the (Lindeberg) central limit theorem for independent
random variables 222For the Lindeberg central limit theorem see e.g.
https://www.stat.berkeley.edu/users/pitman/s205f02/lecture10.pdf. In this
case, we may interpret the variance of the source voltage as giving rise to an
effective temperature of the noise. For many practical energy harvesting
applications, the genuinely thermal noise is indeed insignificant. A
10k$\Omega$ resistor at room temperature has an open-circuit RMS voltage of
1.3$\mu$V, measured with a bandwidth of 10 kHz [26] whereas the output of
hand-sized harvesters is often of the order of Volts or higher. Thus for many
applications of the results here we expect any noteworthy Gaussian noise not
to be thermal and the variance thus to be associated with an effective
temperature.

If the generating model is a priori not known, one may rather repeat the
experiment many, say $n$ times to build up a set
$\\{\vec{\mathcal{V}}_{k}\\}_{k=1}^{n}$, assume it is generated by a
distribution within a family of models, and match the data to the ‘best’ model
within that family via the maximum likelihood method, as described in E.

In line with the above, we represent states (of the source) as 2-tuples
($p(\vec{\mathcal{V}})$, $R_{s}$) where $\vec{\mathcal{V}}$ is the finite-
dimensional voltage-time series for the internal source voltage $V_{s}$, and
$R_{s}$ the internal resistance. We assume, as described and motivated above,
that $p(\vec{\mathcal{V}})$ corresponds to a Fourier series signal with (AWGN)
Gaussian noise.

We take any resistor with resistance $R$ at ambient temperature $T$ to be a
free state. All other voltage sources, including resistors at other
temperatures, are taken as non-free states.

##  5 Inter-conversion Operations

We now describe the free interconversion operations between sources that we
shall allow.

###  5.1 Reversible energy-preserving operations

Motivated by basic physics and previous work [9] we define one type of free
operations as being those which are reversible and energy preserving, (without
changing the internal resistance $R_{s}$). Here reversible means that the
inverse operation is also allowed (and free). By energy preserving we mean
that the energy delivered to a resistive load,
$\frac{\vec{\mathcal{V}_{L}}\cdot\vec{\mathcal{V}_{L}}}{NR_{L}}$, is
conserved, where $N$ is the number of sampled points. For fixed $R_{L}$ and
$R_{s}$, $V_{L}$ and $V_{S}$ are proportional (recall
$V_{L}=\frac{V_{s}}{1+\frac{R_{s}}{R_{L}}}$). Thus, the demand that
$\vec{\mathcal{V}_{L}}\cdot\vec{\mathcal{V}_{L}}$ is conserved under these
operations is equivalent to demanding that
$\vec{\mathcal{V}}\cdot\vec{\mathcal{V}}$ is conserved.

Matrices which preserve dot products are orthogonal. Modelling the operations
as matrices implies that if sources are added in series, the impact of the
operation is the same as if it had acted on each source individually.
Moreover, multiplication by a factor commutes with applying an orthogonal
matrix, so we may treat the matrix as acting on either $\vec{\mathcal{V}}$ or
$\vec{\mathcal{V}_{L}}$ and choose the former, since we opted to represent the
source as the 2-tuple ($p(\vec{\mathcal{V}})$, $R$). We thus model these
energy preserving reversible operations as orthogonal matrices $O$ acting on
$\vec{\mathcal{V}}$.

There is no measurement during the implementation of the free operations.
There may have been measurements involved in characterising the source’s
probability distribution (a process sometimes called tomography) but we do not
include the one-off cost of those.

A simple example of such operations is the (possibly periodic) bias-flip [48,
9]. This is depicted in Fig. 4. If the aim of the agent is e.g. to outperform
the diode bridge for the task of rectifying the voltage with minimal loss, the
bias flip can in principle succeed provided that the signal-to-noise ratio is
not too high [9]. (Rectification is further discussed in D.) Another simple
example is a phase shift, which shifts the time-label along. As toy examples
of these, consider a voltage time series
$\vec{\mathcal{V}}=(V_{1},V_{2},V_{3},V_{4})^{T}$. An example of the bias flip
transformation denoted by $O_{bf}$ and a phase shift transformation denoted by
$O_{ps}$ is given, respectively, by the following $4\times 4$ orthogonal
matrices.

$\displaystyle O_{bf}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{pmatrix}~{}\mathrm{and}~{}O_{ps}=\begin{pmatrix}0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\end{pmatrix}.$

Figure 4: Example of converting one source to another. There is a non-ideal
random voltage source with associated resistance $R_{S}$. There is a load,
often taken to be purely resistive with resistance $R_{L}$. The source may be
transformed by certain (in principle) energy-preserving reversible operations
such as flipping $V\rightarrow-V$ by switching the circuit architecture
facilitated by switches $S_{1}$ and $S_{2}$ as depicted.

More generally, the voltage time series transform under the energy-preserving
reversible operations as

$\displaystyle\tilde{V}_{i}=\sum_{j}O_{ij}V_{j}=\sum_{j}O_{ij}\mathcal{S}_{j}+\sum_{j}O_{ij}r_{j},$
(5)

where $O$ is an orthogonal matrix.

###  5.2 Adding thermal noise

Adding a thermal resistor $R_{T}$ at the ambient temperature $T$ will also be
treated as a free operation. The variance of such a resistor’s $V_{s}$ is
given by Eq. (4) and is proportional to $R_{T}T$. Thus the addition is
modelled as
$(p(\vec{\mathcal{V}}),R_{s})\rightarrow(p^{\prime}(\vec{\mathcal{V}}),R_{s}+R_{T})$,
where $p^{\prime}$ represents a process where AWGN noise satisfying Eq. (4)
has been added. This free operation is depicted in Fig. 5.

Figure 5: The free operation of adding a thermal source. The schematic on the
left represents an open circuit non-ideal voltage source modelled as an ideal
voltage source $V_{s}$ connected in series with a resistance $R_{s}$. The
schematic on the right represents a closed circuit non-ideal voltage source
which has thermal noise added to it. The non-ideal voltage source is modelled
as an ideal voltage source $V_{s}$ connected in series with a resistance
$R_{s}$ and the thermal noise is modelled as an ideal voltage source $V_{T}$
connected in series with a resistance $R_{T}$. Moreover, from the Johnson-
Nyquist formula (Eq. (4), Refs. [28, 29]), we have
$\mathbb{E}|V_{T}|^{2}\propto R_{T}T$, where $T$ is the ambient temperature
for the thermal noise. We may then measure the power dissipation across the
load resistance $R_{L}$.

##  6 Which resource quantifiers respect conversion hierarchy

We now use the resource theory paradigm to identify quantities that respect
the hierarchy associated with the free operations. If one source can be
converted to another with free operations the latter source must not have a
higher number assigned to it. Quantities respecting the conversion hierarchy
in that sense are then reasonable quantifiers of the value of transient
voltage sources.

The value given by the quantifiers involves taking averages. To explain the
notation we shall use, consider for example a vector
$\vec{u}=\left(u_{1},\cdots,u_{N}\right)^{T}$. Assume it was sampled at $N$
intervals of $\delta t$ over a total time $\tau=N\delta t$. It is natural to
define the time average $\mathbb{E}_{t}(u)=\frac{\sum_{i}u_{i}\delta
t}{\tau}=\frac{\sum_{i}u_{i}}{N}$. Then, for example,
$\mathbb{E}_{t}(u^{2})=\frac{\vec{u}\cdot\vec{u}}{N}$. We shall moreover
employ the statistical average $\mathbb{E}_{p}(u)=\sum_{u}p(u)u$. The
statistical average of a time average will also appear, e.g.
$\mathbb{E}_{p,t}(u^{2})=\mathbb{E}_{p}(\frac{\vec{u}\cdot\vec{u}}{N}):=\mathbb{E}|\vec{u}|^{2}$.

We begin with discussing quantifiers that fail to respect the hierarchy before
giving three that respect it.

###  6.1 Maximal power dissipated does not respect hierarchy

A common measure in the literature on energy harvesting turns out not to be a
monotone. Suppose the experimenter tunes the resistive load $R_{L}$ until
maximum power is dissipated and then records the power for the optimal
resistance and also maximises it over time, i.e. $\max_{R_{L},t}P_{L}$ where
$P_{L}=\frac{V_{L}^{2}}{R_{L}}$. From Eq. (1) one then sees
$\max_{R_{L},t}P_{L}=\max_{t}\frac{{V_{s}^{2}(t)}}{4R_{s}}$. This quantity is
not monotonic under our free operations, however. For example, if $V_{s}=1V$
for 1 second, then we allow for the free interconversion via orthogonal
operations to $V_{s}=\sqrt{2}V$ for half a second, which would increase
$\max_{R_{L},t}P_{L}$.

Moreover, recall that we allow adding thermal noise as a free operation. This
could increase $\max_{R_{L},t}P_{L}$ as well, e.g. if initially $V_{s}=0$.
This addition would even increase the energy delivered to a resistive load,
$\max_{R_{L}}\mathbb{E}_{p}P_{L}\tau$. Thus the energy delivered to a
resistive load is not a monotone either, except for, notably, being an
approximate monotone in the regime where thermal noise is insignificant
relative to the other voltages.

The above argument formalises why we should deduct the thermal noise from the
signal before evaluating the usefulness of the transient voltage source. In
what follows we give three quantities that involve this deduction and are
indeed resource monotones (non-increasing under free operations).

###  6.2 Modified Signal to noise ratio

The standard signal-to-noise ratio (SNR) is defined as
$\frac{\mathbb{E}|\vec{S}|^{2}}{\mathbb{E}|\vec{r}|^{2}}$ for signal vector
$\vec{S}$ and noise vector $\vec{r}$. SNR is widely used in information
processing and information theory so it is arguably natural to consider
whether a voltage source resource quantifier can be built from it.

We firstly propose a modified signal-to-noise-ratio $(\mathrm{MSNR})$, as a
possible resource monotone. Let

$\displaystyle\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s}):=\frac{\mathbb{E}|\vec{S}|^{2}+\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{R_{s}\mathbb{E}|\vec{r}|^{2}}\tau,$
(6)

where $\vec{r}_{th}$ is the thermal noise corresponding to $R_{s}$ at ambient
temperature and the averages ($\mathbb{E}$) are taken as statistical averages
of time averages over the probability distributions of the corresponding
signal and noise sources as defined in the beginning of the section. $\tau$ is
the pulse duration of the source. The reason to include the internal
resistance $R_{s}$ is to ensure that our monotone adds extra penalty on the
sources that have higher internal resistance. The $\vec{r}_{th}$ term ensures
that resistors that are hotter and colder than the ambient temperature
resistors remain nontrivial sources. MSNR, given by Eq. (6), does satisfy the
two requirements we listed for a resource monotone.

  * 1.

Faithfulness: It immediately follows from the definition that $\mathrm{MSNR}$
is zero for the free states ($\vec{\mathcal{S}}=0,~{}\vec{r}=\vec{r}_{th}$),
and is thus, by definition, a faithful quantifier. Resistors at a different
temperature to the ambient one have $\mathrm{MSNR}\neq 0$ since, although
$\vec{\mathcal{S}}=0$, $\vec{r}\neq\vec{r}_{th}$.

  * 2.

Monotonicity under free operations: MSNR is conserved, and thus non-
increasing, under orthogonal transformations for which
$\tilde{\mathcal{S}}_{i}=\sum_{j}O_{ij}\mathcal{S}_{j}$ and
$\tilde{r}_{i}=\sum_{j}O_{ij}r_{j}$ where $O$ is an orthogonal matrix. This
can be shown as follows.

$\displaystyle\sum_{i}\tilde{\mathcal{S}}_{i}^{2}$
$\displaystyle=\sum_{i}\sum_{j}O_{ij}\mathcal{S}_{j}\sum_{k}O_{ik}\mathcal{S}_{k}$
$\displaystyle=\sum_{j,k}\mathcal{S}_{j}\mathcal{S}_{k}\delta_{j,k}=\sum_{j}\mathcal{S}_{j}^{2}.$
(7)

By the same argument, under orthogonal transformations
$\sum_{i}\tilde{r}_{i}^{2}=\sum_{j}r_{j}^{2}$ and
$\sum_{i}\tilde{r}_{th,i}^{2}=\sum_{j}r_{th,j}^{2}$. Moreover, since adding
thermal noise at the ambient temperature, the other free operation, increases
$\mathbb{E}(|\vec{r}|^{2})$ whilst not impacting the signal, it is clear that
$\mathrm{MSNR}$ will not increase under such operations. Under addition of
thermal noise at ambient temperature it is easy to see that the monotone is
non-increasing. When adding a thermal resistor $R$ in series
$\vec{r}\rightarrow\vec{r}+\vec{s}_{th},~{}\vec{r}_{th}\rightarrow\vec{r}_{th}+\vec{s}_{th}~{}\text{and}~{}R_{s}\rightarrow
R_{s}+R$, where $\vec{s}_{th}$ is the thermal noise at ambient temperature
corresponding to resistor with resistance $R$. Then the MSNR value of Eq. (6)
becomes

$\displaystyle\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}+\vec{s}_{th}|^{2}-\mathbb{E}|\vec{r}_{th}+\vec{s}_{th}\right|^{2}|}{(R_{s}+R)~{}\mathbb{E}|\vec{r}+\vec{s}_{th}|^{2}}\tau$
$\displaystyle=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{(R_{s}+R)~{}\mathbb{E}|\vec{r}+\vec{s}_{th}|^{2}}\tau$
$\displaystyle\leq\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{R_{s}~{}\mathbb{E}|\vec{r}|^{2}}\tau.$
(8)

From Eq. (2) and Eq. (2) we see that the resource value assigned by MSNR is
non-increasing under the action of free operations and is hence a valid
monotone.

MSNR may also be conveniently evaluated in several cases. Table 1 gives the
MSNR for different sources $(p(\vec{\mathcal{V}}),R_{s})$, with $R_{s}$ kept
general with AWGN noise (Eq. (3)) with time-dependent variance.

Table 1: MSNR for different sources $(p(\vec{\mathcal{V}}),R_{s})$. It is assumed that signal dominates the noise term, $\mathbb{E}|\vec{\mathcal{S}}|^{2}\gg\max(\mathbb{E}|\vec{r}|^{2},\mathbb{E}|\vec{r}_{th}|^{2}$, such that only the noise in the denominator needs to be included. The AWGN noise standard deviation is allowed to depend on time, labelled as $\sigma_{i}$. The signal duration $\tau=1$. The corresponding proofs are given in A. Quantity |  Value
---|---
$\vec{\mathcal{S}}_{i}$ |  $\sin(\omega t_{i})$
$\vec{r}$ |  Thermal AWGN, $\sigma_{k}$ at time $t_{k}$
MSNR |  $\left(2R_{s}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}\right)^{-1}$ (Eq. (26))
$\vec{\mathcal{S}}_{i}$ |  $\sin(\omega t_{i})+\cos(\omega t_{i})$
$\vec{r}$ |  Thermal AWGN, $\sigma_{k}$ at time $t_{k}$
MSNR |  $\left(R_{s}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}\right)^{-1}$ (Eq. (29))
$\vec{\mathcal{S}}_{i}$ |  $\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{i}+g(\omega^{\prime})\cos\omega^{\prime}t_{i}\right)$
$\vec{r}$ |  Thermal AWGN, $\sigma_{k}$ at time $t_{k}$
MSNR |  $\left(\sum_{k=1}^{N}\left[\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{k}\\!+\\!g(\omega^{\prime})\cos\omega^{\prime}t_{k}\right)\right]^{2})(R_{s}\sum_{k=1}^{N}\sigma_{k}^{2}\right)^{-1}$ (Eq. (A.2))

We have shown that MSNR assigns values to sources in a way that respects the
hierarchy, and that it can be convenient to calculate. However, when there is
no noise ($r=\sigma\approx 0$), MSNR diverges even for finite $R_{s}$,
suggesting the MSNR value itself may not have a direct operational meaning,
such as the number of unit resources that can be distilled. We therefore
propose and investigate a second resource monotone, which turns out to have
such an operational meaning.

###  6.3 UnitDC

There exists a resource monotone whose value, we will show, can be interpreted
as the number of unit DC sources (1V at 1$\Omega$ for 1s) contained within the
source. We accordingly denote this quantifier by $\mathrm{UnitDC}$. We let

$\displaystyle\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s}):=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{R_{s}}\tau,$
(9)

where $\vec{\mathcal{S}}$ is the signal, $\vec{r}$ is the noise part in the
signal, $R_{s}$ the internal resistance of the source, $\vec{r}_{th}$ is the
thermal noise corresponding to $R_{s}$ at ambient temperature and the averages
($\mathbb{E}$) are taken as statistical averages of time averages over the
probability distributions of the corresponding signal and noise sources as
defined in the beginning of the section. $\tau$ is the pulse duration of the
source. The physical unit of $\mathrm{UnitDC}$ is energy (J). It can be seen
easily that $\mathrm{UnitDC}$ is a monotone.

  * 1.

Faithfulness: By inspection $\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})=0$ if
and only if $\vec{\mathcal{V}}$ has no signal and noise is only thermal at
ambient temperature: $\vec{\mathcal{S}}=\vec{0}$ and $\vec{r}=\vec{r}_{th}$.
This is true for our free states. It is also clear that for a hotter or cooler
than ambient temperature resistor, even though $\vec{\mathcal{S}}=0$, the
$\mathrm{UnitDC}\neq 0$ as $|\vec{r}|^{2}\neq|\vec{r}_{th}|^{2}$.

  * 2.

Monotonicity under free operations:
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ is conserved, and thus non-
increasing, under orthogonal transformations since they preserve
$|\vec{\mathcal{S}}|^{2}$, $|\vec{r}|^{2}$, and $|\vec{r}_{th}|^{2}$ (see Eqs.
(2)). The other free operation is to add thermal noise at the ambient
temperature which is equivalent to adding a resistor $R$ in series at the
ambient temperature. Thus, the new internal resistance becomes
$R_{s}^{\prime}=R_{s}+R>R_{s}$. Such a resistor adds an independently
distributed Gaussian noise $\vec{s}_{th}$ whose variance is proportional to
$Rk_{B}T$. Thus, we have

$\displaystyle\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s}+R)$
$\displaystyle=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}+\vec{s}_{th}|^{2}-\mathbb{E}|\vec{r}_{th}+\vec{s}_{th}|^{2}\right|}{(R_{s}+R)}\tau$
$\displaystyle=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{(R_{s}+R)}\tau\leq\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s}).$
(10)

Thus, adding thermal noise at the ambient temperature does not increase
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ and we conclude that
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ is indeed a monotone under free
operations.

The value of $\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ can be evaluated
using similar techniques described above for MSNR.

A motivation for using $\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ over
other monotones is that it can be interpreted as the amount of ‘unit DC’
sources within the source. These are 1V DC sources with internal resistance
$R_{s}=1\Omega$, pulse duration $\tau=1s$, thermal noise corresponding to
$1\Omega$ resistance at the ambient temperature, and are represented as an $N$
length time series

$\displaystyle\vec{\mathcal{V}}_{dc}=\vec{e}+\vec{r}_{th},$ (11)

where $\vec{e}=\\{\overbrace{1,\cdots,1}^{N}\\}$ such that the probability
distribution $p(\vec{e})=\prod_{i}\delta(e_{i}-1)$. For such a source,

$\displaystyle\mathrm{UnitDC}(p(\vec{\mathcal{V}}_{dc}),1)$
$\displaystyle=\mathbb{E}|\vec{e}|^{2}=\frac{1}{N}\int\cdots\int\sum_{i=1}^{N}e_{i}^{2}\prod_{j=1}^{N}\delta(e_{j}-1)de_{j}=1.$
(12)

We show in Sec. 7.3 below that from many copies of the given source
$(p(\vec{\mathcal{V}}),R_{s})$ one can by free operations distill out
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ unit DC sources per original
source. UnitDC further has the desirable property of not diverging for finite
$R_{s}$ whereas MSNR does diverge, as exemplified in Fig. 6.

Figure 6: The figure shows a plot of the logarithm (with base $e$) of
monotones for a sinusoidal voltage time series, denoted by
$\\{\mathcal{S}_{i}\\}_{i=0}^{199}$, vs. the logarithm of the internal
resistance $R_{s}$ which is monotonically related to the thermal noise at the
ambient temperature. We take
$\mathcal{S}_{i}=\sin\left(\frac{\pi}{25}i\right)$ and the noise to be the
thermal noise corresponding to its internal resistance $R_{s}$. The MSNR is
given by
$\mathrm{MSNR}\left((P(\vec{\mathcal{S}}+\vec{r}_{th}),R_{s})\right)=\sum_{i=0}^{199}\sin^{2}\left(\frac{\pi}{25}i\right)/(800k_{B}T\Delta
fR_{s}^{2})$ (see Eq. (24)). The $\mathrm{UnitDC}$ is given by
$\mathrm{UnitDC}\left((P(\vec{\mathcal{S}}+\vec{r}_{th}),R_{s})\right)=\frac{1}{200R_{s}}\sum_{i=0}^{199}\sin^{2}\left(\frac{\pi}{25}i\right)$
(see Eq. (9)). The relative entropy is given by
$\mathcal{R}\left((P(\vec{\mathcal{S}}+\vec{r}_{th}),R_{s})\right)=\frac{1}{400R_{s}}\sum_{i=0}^{199}\frac{\sin^{2}\left(\frac{\pi}{25}i\right)}{4k_{B}T\Delta
fR_{s}}$ (see Eq. (19)). Moreover, we take the pulse duration $\tau=1s$,
$T=300K$, $\Delta f=10^{3}MHz$, and $R_{s}$ is taken to vary in the range of
$1\Omega$ to $10^{12}\Omega$. The plot clearly shows that when $R_{s}$ is
close to zero, the monotones MSNR and relative entropy diverge while UnitDC is
finite.

Finally, we note that unit DC can be written in several forms, including
$\mathrm{UnitDC}=\tau\left(\frac{\mathbb{E}_{t}(S^{2})}{R_{s}}+4\Delta\\!f\mathrm{k_{B}}|T_{s}-T|\right)$,
where $\mathbb{E}_{t}$ denotes a time-average over the pulse duration $\tau$,
$\Delta f$ is the bandwidth under consideration (a setting in an
oscilloscope), $\mathrm{k_{B}}$ Boltzmann’s constant, $T$ the ambient
temperature and $T_{s}$ the effective source noise temperature.

###  6.4 Relative entropy between the noisy and the free source

Relative entropy-based monotones such as relative entropy of entanglement in
the resource theory of entanglement [12] are widely used in resource theories.
Relative entropy measures how different two probability distributions are,
therefore, we now introduce and analyse a relative entropy-based measure in
our resource theory here.

Let $\mathbb{P}$ and $\mathbb{Q}$ be two probability distributions defined on
the same probability space $\xi$ with probability densities $p(x)$ and $q(x)$,
respectively, then the relative entropy or Kullback-Leibler divergence between
$\mathbb{P}$ and $\mathbb{Q}$ is defined as

$\displaystyle
D\left(\mathbb{P}||\mathbb{Q}\right):=\int\mathrm{d}x~{}p(x)\ln\frac{p(x)}{q(x)}.$
(13)

The above expression is finite only when $p(x)=0$ implies $q(x)=0$ [49]. We
let

$\displaystyle\mathcal{R}(p(\vec{\mathcal{V}}),R_{s}):=\frac{\tau}{NR_{s}}D\left(\mathcal{P}||\mathcal{Q}\right),$
(14)

where $\mathcal{P}\equiv p(\vec{\mathcal{V}})$ is the probability distribution
for the noisy signal and $\mathcal{Q}\equiv q(\vec{\mathcal{V}})$ is
probability distribution for the noise corresponding to the internal
resistance $R_{s}$ at the ambient temperature. $\tau$ is again the pulse
duration of the source. $N$ is the number of time steps in the voltage time
series, i.e., $\mathcal{V}=(V_{1},\cdots,V_{N})^{T}$. It is immediate from Eq.
(14) that the value of the monotone for a hotter or cooler than the ambient
temperature resistor is nonnegative. We now show that
$\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})$ is a resource monotone.

  * 1.

Faithfulness: The relative entropy $D\left(\mathcal{P}||\mathcal{Q}\right)$ is
zero if and only if $\mathcal{P}=\mathcal{Q}$ implying that for thermal noise
corresponding to resistance $R_{s}$ at the ambient temperature,
$\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})=0$ and vice-versa. Thus,
$\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})$ is a faithful resource quantifier.

  * 2.

Monotonicity under free operations: The orthogonal transformations reversibly
map different voltage series with the same 2-norm to each other, thus acting
as permutation matrices on the vectors $p(\vec{V})$ whose entries are
probabilities of different voltage distributions. Permutation matrices are
doubly stochastic (rows and columns sum to 1 and entries are in the range 0 to
1). Moreover, the internal resistance $R_{s}$, by definition, remains
invariant under orthogonal transformations and therefore, $\mathcal{Q}$ is
also invariant. Now, the data processing inequality for relative entropy (see
B), which holds for stochastic matrices, applies here.

$\displaystyle\mathcal{R}\left((p(\vec{\mathcal{V}}),R_{s})\right)$
$\displaystyle=\frac{\tau}{NR_{s}}D\left(\mathcal{P}||\mathcal{Q}\right)$
$\displaystyle\geq\frac{\tau}{NR_{s}}D\left(M\mathcal{P}||M\mathcal{Q}\right)$
$\displaystyle=\frac{\tau}{NR_{s}}D\left(M\mathcal{P}||\mathcal{Q}\right)=\mathcal{R}\left((p(\vec{\mathcal{V}^{\prime}}),R_{s})\right),$
(15)

where $M$ is a column stochastic matrix and $\vec{\mathcal{V}^{\prime}}$ is
obtained after the orthogonal transformation on $\vec{\mathcal{V}}$. Thus,
$\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})$ is non-increasing under reversible
energy-preserving operations. Further, for any noisy source, adding thermal
noise corresponding to some resistance $R$ at ambient temperature does not
increase the value of the monotone. The reason is as follows. Note that adding
thermal noise will amount to changing
$\vec{r}\rightarrow\vec{r}+\vec{s}_{th},~{}\vec{r}_{th}\rightarrow\vec{r}_{th}+\vec{s}_{th}~{}\text{and}~{}R_{s}\rightarrow
R_{s}+R$, where $\vec{s}_{th}$ is the thermal noise at ambient temperature
corresponding to a resistor with resistance $R$. Thus, adding thermal noise is
equivalent to a convolution of probability distributions of the signal and
noise by a same probability distribution. From B.2 we see that the relative
entropy is non-increasing under convolution. Therefore, based on the above, we
conclude that the relative entropy of resource is a faithful resource monotone
for our case.

For the case of an arbitrary noisy signal
$\vec{\mathcal{V}}=\vec{\mathcal{S}}+\vec{r}$, where $\vec{\mathcal{S}}$ is a
deterministic signal and is defined by

$\displaystyle p(\vec{\mathcal{S}})$
$\displaystyle=\prod_{i}p_{i}(\vec{\mathcal{S}}_{i})=\prod_{i}\delta(\vec{\mathcal{S}}_{i}-s_{i}),$
(16)

while $\vec{r}$ is satisfying (3), we have

$\displaystyle p(\vec{\mathcal{V}})$
$\displaystyle=\prod_{i}p_{i}(V_{i})=\prod_{i}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right].$
(17)

Also, let the distribution of the thermal noise corresponding to the internal
resistance $R_{s}$ at the ambient temperature be given by

$\displaystyle\mathcal{Q}(\vec{\mathcal{V}})$
$\displaystyle=\prod_{i}\mathcal{Q}(V_{i})=\prod_{i}\frac{1}{\sqrt{2\pi\xi_{i}^{2}}}\exp\left[-\frac{V_{i}^{2}}{2\xi_{i}^{2}}\right].$
(18)

Then, for such a voltage source, we prove in B.3 that

$\displaystyle\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})=\frac{\tau}{2NR_{s}}\sum_{i}\left(\frac{\sigma_{i}^{2}+s_{i}^{2}}{\xi_{i}^{2}}-\ln\left(\frac{\sigma_{i}^{2}}{\xi_{i}^{2}}\right)-1\right).$
(19)

##  7 Composed sources

Composed sources are also sources, and they can be given (Thevenin) equivalent
circuits with one voltage source and one internal resistance. One reason
composed sources are of interest is that we will argue that the
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ corresponds to the fraction of
unit DC sources one can gain from many composed copies of the given voltage
source. Another is that energy harvesters may have different sub-components,
e.g. Piezos with different resonance frequencies, as in Ref. [48]. We first
discuss how to compose sources, go on to derive results on how the resource
quantifiers behave under the composition of sources and then focus on how many
unit DC sources are needed to compose a given source and vice versa how many
unit DC sources one can distill from the source.

###  7.1 How to compose sources

We follow the standard paradigm for composing voltage sources in electronics:
composition in series is allowed for any source and composition in parallel is
allowed for identical sources. We do not consider what will happen if non-
identical sources are connected in parallel 333The characters of each source
would need to alter in such a scenario in order not to violate Kirchoff’s
voltage law which implies e.g. that two sources connected in parallel should
have the same voltage.

We denote composition in series as
$(p(\vec{\mathcal{V}}),R_{s})\circ(p(\vec{\mathcal{V}}),R_{s})$. Under this
composition, the voltages and resistances add. It is natural to define the sum
of two deterministic signals as the signal of the composite system and
similarly the addition of the two noise vectors as the new noise vector.

###  7.2 Resource quantifiers under composition of sources

We see quickly what the MSNR and unitDC are when composing identical sources.
Note that the sum of two independent centered Gaussian variables each with
standard deviation $\sigma$ has standard deviation
$\sigma^{\prime}=\sqrt{2}\sigma$ (since $\sigma^{\prime
2}=\mathbb{E}(r_{1}+r_{2})^{2}=\sigma^{2}+\sigma^{2}$). Thus, letting $\circ$
denote composition in series, we have

$\displaystyle\mathrm{MSNR}((p(\vec{\mathcal{V}}),R_{s})\circ(p(\vec{\mathcal{V}}),R_{s}))$
$\displaystyle:=\frac{1}{2R_{s}}\frac{\mathbb{E}(|2\vec{\mathcal{S}}|^{2})+2\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{2\mathbb{E}(|\vec{r}|^{2})}\tau$
$\displaystyle=\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s})-\frac{1}{2R_{s}}\frac{\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{\mathbb{E}(|\vec{r}|^{2})}\tau.$
(20)

Similarly, for the resource monotone UnitDC, we have

$\displaystyle\mathrm{UnitDC}((p(\vec{\mathcal{V}}),R_{s})\circ(p(\vec{\mathcal{V}}),R_{s}))$
$\displaystyle=\frac{\mathbb{E}|2\vec{\mathcal{S}}|^{2}+2\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|}{2R_{s}}\tau$
$\displaystyle=2\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})-\frac{\tau}{R_{s}}\left|\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}\right|$
(21)

Thus except for the case of $\vec{r}=\vec{r}_{th}$, composing two identical
sources lowers the value per source. This is actually to be expected
intuitively since the composition in series will lead to some of the random
noise cancelling, and is less optimal than firstly rectifying and then
composing, as discussed in D.

For the case when $\vec{r}=\vec{r}_{th}$, we see that MSNR has the strange
feature of not increasing under composition of two identical sources whereas
$\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ doubles. This again speaks in
favour of $\mathrm{UnitDC}(p(\vec{\mathcal{V}}),R_{s})$ as a measure of the
usefulness of a source.

###  7.3 Composing many identical sources

When many identical sources are connected in series, the resulting composed
source can see the Gaussian noise effectively cancelling due to the law of
large numbers. Consider $n$ noisy voltage sources, with the same internal
resistance $R_{s}$, given by $\vec{\mathcal{V}}=\vec{\mathcal{S}}+\vec{r}$
with $p(\vec{\mathcal{V}})=\prod_{i}p_{i}(V_{i})$, where $V_{i}$ are
components of $\vec{\mathcal{V}}$ and $p_{i}(V_{i})$ is given by

$\displaystyle
p_{i}(V_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\int\mathrm{Pr}(\mathcal{S}_{i}=s_{i})\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right]\mathrm{d}s_{i},$

where $\mathrm{Pr}(\mathcal{S}_{i}=s_{i})$ is the distribution of the signal,
which in the deterministic case is just a delta function. These $n$ noisy
sources can be considered as $n$ independent identically distributed vector
valued random variables, namely,
$\vec{\mathcal{V}}_{1},\vec{\mathcal{V}}_{2},\cdots,\vec{\mathcal{V}}_{n}$.
Therefore, given any small number $\epsilon>0$ and $\delta>0$, from weak law
of large numbers [49], we have

$\displaystyle\mathrm{Pr}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\vec{\mathcal{V}}_{i}-\mathbb{E}\left(\vec{\mathcal{V}}\right)\right|\leq\delta\right)\geq
1-\epsilon.$

This means that the source $\frac{1}{n}\sum_{i=1}^{n}\vec{\mathcal{V}}_{i}$ is
very close to the source $\mathbb{E}\left(\vec{\mathcal{V}}\right)$ in
probability, therefore, we can consider
$\frac{1}{n}\sum_{i=1}^{n}\vec{\mathcal{V}}_{i}\approx\mathbb{E}\left(\vec{\mathcal{V}}\right)$
in large $n$ limit. Here
$\mathbb{E}\left(\vec{\mathcal{V}}\right)=\\{\mathbb{E}(V_{1}),\mathbb{E}(V_{2}),\cdots,\mathbb{E}(V_{N})\\}^{T}$
and for all $i=1,\cdots,N$

$\displaystyle\mathbb{E}(V_{i})$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\int
V_{i}\mathrm{d}V_{i}\int\mathrm{Pr}(\mathcal{S}_{i}=s_{i})\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right]\mathrm{d}s_{i}$
$\displaystyle=\int\mathrm{d}s_{i}\mathrm{Pr}(\mathcal{S}_{i}=s_{i})s_{i}=\mathbb{E}\mathcal{S}_{i}.$

Thus, $(\overbrace{\vec{\mathcal{V}}+\cdots+\vec{\mathcal{V}}}^{n})$
correspond to $n$
$(\mathbb{E}\mathcal{S}_{1},\cdots,\mathbb{E}\mathcal{S}_{N})^{T}$ voltage
vectors, each with internal resistance $R_{s}$, for large $n$. In particular,
when $\vec{\mathcal{S}}$ is a deterministic signal then $n$ copies of
$(p(\vec{\mathcal{V}}),R_{s})$ simply corresponds to a source
$(n\vec{\mathcal{S}},nR_{s})$, for large $n$.

###  7.4 How much unit DC in a source

We now consider the question of how much unit DC is contained in a single
source. For example, 2s of 1V at 1$\Omega$ internal resistance should
intuitively be equivalent to 2 unit DCs (recall a unit DC source is 1s of 1V
with 1$\Omega$ internal resistance at ambient temperature). This process is
more generally termed distillation of standard resources (see e.g. [12]). It
is an appealing property of a resource monotone if it can be interpreted as
the number of distillable standard resources. In our case, as discussed above,
the natural standard resource is the unit DC source. Thus in our distillation
protocol, we are interested in adding $n$ noisy voltage sources and hope to
obtain $m$ 1V DC sources, where $m$ is as large as possible. This may be
written as

$\displaystyle(\overbrace{\vec{\mathcal{V}}+\cdots+\vec{\mathcal{V}}}^{n})\xrightarrow[\mathrm{Operations}]{\mathrm{Free}}(\overbrace{\vec{e}+\cdots+\vec{e}}^{m}),$
(22)

where $\vec{e}$ is 1V DC voltage source (or voltage time series) with internal
resistance $R=1\Omega$ (see Eqs. (11) and (12)). It is commonly the case that
such transformations are identified only in the asymptotic limit of
$n\rightarrow\infty$. The law of large number effects can then be employed to
cancel out fluctuations away from the desired state in the asymptotic limit as
described above. The distillation rate is thus commonly defined as
$\lim_{n\rightarrow\infty}\frac{m}{n}$.

We now seek the optimal distillation rate: how many unit DC sources can we
distill from a given source, per copy, via free operations?

Consider firstly, for simplicity, the case of the source having thermal noise
at the ambient temperature, plus some signal. Then the $\mathrm{UnitDC}$ value
of $n$ sources in series has the neat form $nd$ where $d$ is the
$\mathrm{UnitDC}$ value of a single source. It is possible to convert these
$n$ sources in series to $m=nd$ unit DC sources via the free orthogonal
operations provided that $m=n\tau\mathbb{E}|\vec{\mathcal{S}}|^{2}/R_{s}=nd$
(see Eqs. (11) and (12)). This corresponds to the forwards transformation of
Eq. (22). Thus one can distill $\mathrm{UnitDC}$ units per source via free
operations. We give heuristic arguments in D that, in the reverse direction,
free transformations can convert $m$ unit sources into $nd$ copies of the
original source. Thus the unit DC is an achievable and, we believe, optimal
rate for sources where the noise is thermal.

We also believe the unit DC is the optimal rate for the case where the sources
have Gaussian noise that is not at the ambient temperature. In this case
adding the sources in series is not an optimal distillation protocol but
rather one should apply a rectifying operation first. This necessitates
expanding the set of free operations to include a rectifying operation. We
give a possible idealised rectifying operation that can reasonable be called
free in D. It takes free states to free states. Under this operation we show
heuristically that $\mathrm{UnitDC}((p(\vec{\mathcal{V}}),R_{s}))$ should
indeed be the optimal rate for such sources as well.

Table 2: Comparison of the resource theory of transient voltage sources and the well-known resource theory of entanglement. S.No. |  Elements |  Resource theory of entanglement |  Resource theory of transient voltage sources
---|---|---|---
1 |  States |  Positive matrices with trace one [50] |  ($p(\vec{\mathcal{V}})$, $R_{s}$) where $p(\vec{\mathcal{V}})$ is a probability distribution over source voltage time series $\vec{\mathcal{V}}$, and $R_{s}$ the internal resistance
2 |  Operations |  Completely positive trace-preserving maps [50] |  Stochastic transformations of the voltage time series, changes to internal resistance
3 |  Free states |  Separable states [12] |  Gaussian white noise with zero mean at ambient temperature
4 |  Free operations |  Local operations and classical communication (LOCC) [12] |  Reversible energy preserving operations and addition of thermal noise at ambient temperature
5 |  Resource states |  Entangled states |  Voltage sources that are not thermal noise sources at the ambient temperature
6 |  Composition |  Tensor product |  $\mathcal{V}+\mathcal{V}^{\prime}$ (sources in series)
7 |  Typical resource monotones |  Relative entropy of entanglement, distillable entanglement, and entanglement cost |  $\mathrm{MSNR}$, relative entropy and $\mathrm{UnitDC}$

##  8 Conclusion and future directions

We considered transient voltage sources and how to rank them according to
their usefulness. We took a so-called resource theory approach to define an
inter-conversion hierarchy. We defined a resource theory for transient voltage
sources. (Table 2 compares this with the well-known resource theory of
entanglement.) We took power-preserving reversible operations, as well as the
addition of ambient-temperature Gaussian noise as free inter-conversion
operations. We gave three measures of usefulness that respect the inter-
conversion hierarchy induced by the free operations, and show that some
measures currently used do not respect it. One of our proposed measures, the
unitDC, has the interpretation of how many 1V DC over 1s, $1\Omega$ internal
resistance voltage sources can be distilled per source. The results show that
noise that is not from the ambient temperature does carry value and specify
how much so.

We hope this measure may prove useful in comparing different voltage sources
fairly and in guiding the optimisation of energy harvesting. The results
suggest several directions for further development are likely to be fruitful:
(i)the quantifiers can be applied to place value on a range of imperfect
voltage sources, (ii) the approach can be developed further by extending the
set of states (e.g. more rich noise models) and free operations (many more
converting circuits should be possible), (iii) the choice of free operations
to apply for a given conversion is an appealing application of machine
learning techniques, leading to what may be called intelligent energy
harvesting, (iv) the efficiency of energy harvesters can be quantified in
terms of the amount by which they decrease the resource value of the original,
e.g. mechanical source, by converting it into electrical output, (v) the law-
of-large numbers approach taken here should more generally be replaced with
single-shot approaches similar to what is being done in information theory and
statistical mechanics, and the agent’s risk-reward-profile should be taken
into account when assigning value to a resource, and (vi) it will be
interesting to apply the same approach to quantum coherent electronics.

##  9 Acknowledgments

Parts of this work were carried during the visit of U. S. to SUSTech and U. S.
acknowledges the warm hospitality of O. D. at SUSTech. U. S. also acknowledges
the support by Polish National Science Center (NCN) (Grant No.
2019/35/B/ST2/01896). O.D. acknowledges support from the National Natural
Science Foundation of China (Grant No. 12050410246, 12005091) and the Shenzhen
Science Technology and Innovation Commission (Grant No.20200805101139001). We
are grateful for discussions with Guo Hengyu, Dario Egloff and Feiyang Li. We
are particularly grateful to Feiyang Liu for all his inputs and discussions.

In the following appendices we prove the results of the main text.

##  Appendix A Proof of the results in the main text concerning MSNR measure

Everywhere, in this appendix, We assume that the noise in the voltage sources
is given by the thermal noise at the ambient temperature, i.e.,
$\vec{r}=\vec{r}_{th}$ and $\tau=1$.

###  A.1 Modified signal-to-noise ratio for noisy sinusoidal signals

The MSNR in this case is given by (see Eq. (6))

$\displaystyle\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s}):=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}}{R_{s}\mathbb{E}|\vec{r}|^{2}}.$
(23)

First, let us calculate $\mathbb{E}|\mathcal{S}|^{2}$.

$\displaystyle\mathbb{E}|\vec{\mathcal{S}}|^{2}$
$\displaystyle=\frac{1}{N}\int\cdots\int|\vec{\mathcal{S}}|^{2}\prod_{k=1}^{N}\delta(\mathcal{S}_{k}-\sin\omega
t_{k})\mathrm{d}\mathcal{S}_{k}$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}\int\mathcal{S}_{k}^{2}\delta(\mathcal{S}_{k}-\sin\omega
t_{k})\mathrm{d}\mathcal{S}_{k}=\frac{1}{N}\sum_{k=1}^{N}\sin^{2}\omega
t_{k}.$

Similarly,

$\displaystyle\mathbb{E}|\vec{r}|^{2}$
$\displaystyle=\frac{1}{N}\int\cdots\int|\vec{r}|^{2}\prod_{k=1}^{N}\frac{1}{\sqrt{2\pi\sigma_{k}^{2}}}e^{-\frac{r_{k}^{2}}{2\sigma_{k}^{2}}}\mathrm{d}r_{k}$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}\int
r_{k}^{2}\frac{1}{\sqrt{2\pi\sigma_{k}^{2}}}e^{-\frac{r_{k}^{2}}{2\sigma_{k}^{2}}}\mathrm{d}r_{k}=\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}.$

Therefore, from Eq. (23), we have

$\displaystyle\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s})=\frac{1}{R_{s}}\frac{\sum_{k=1}^{N}\sin^{2}\omega
t_{k}}{\sum_{k=1}^{N}\sigma_{k}^{2}}.$ (24)

Let us further assume the voltage time series contains entries at times
$t_{k}=k\frac{T}{N}$. Then, we get

$\displaystyle\frac{1}{N}\sum_{k=1}^{N}\sin^{2}\omega
t_{k}=\frac{1}{N}\sum_{k=1}^{N}\sin^{2}\left(\frac{\omega kT}{N}\right).$

Next, we use the Riemann sum to definite integral formula which looks like the
following.

$\displaystyle\int_{a}^{b}f(x)\mathrm{d}x=\lim_{N\rightarrow\infty}\frac{b-a}{N}\sum_{k=1}^{N}f\left(a+\frac{b-a}{N}k\right).$

Now, by choosing $a=0$, $b=\omega T$ and $f(x)=\sin^{2}x$ in above formula, we
get

$\displaystyle\frac{1}{\omega T}\int_{0}^{\omega
T}\sin^{2}x\mathrm{d}x=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sin^{2}\left(\frac{\omega
kT}{N}\right).$

Solving the left hand side of the above equation, we get

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sin^{2}\left(\frac{\omega
kT}{N}\right)=\frac{1}{\omega T}\int_{0}^{\omega
T}\sin^{2}x\mathrm{d}x=\frac{1}{2}.$ (25)

Combining the above equation with Eq. (24), we have

$\displaystyle\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s})=\frac{1}{2R_{s}}\frac{1}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}}.$
(26)

For $\sigma_{k}=\sigma$ for all $k=1,\cdots,N$, we have

$\displaystyle\mathrm{MSNR}(p(\vec{\mathcal{V}}),R_{s})=\frac{1}{2R_{s}\sigma^{2}}.$
(27)

###  A.2 Modified signal-to-noise-ratio for voltage sources containing more
than one periodic term

In the previous subsection, we have considered probability distribution of
input voltages with only a single frequency. In the following, we consider the
set of all states as before, i.e.,
$\vec{\mathcal{V}}=\vec{\mathcal{S}}+\vec{r}$, except the signal
$\vec{\mathcal{S}}$ contains multiple frequencies and is distributed according
to

$\displaystyle
p(\vec{\mathcal{S}})=\prod_{i}\delta\left(\mathcal{S}_{i}-\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{i}+g(\omega^{\prime})\cos\omega^{\prime}t_{i}\right)\right),$

allowing numerous frequencies. The $\mathrm{MSNR}$ for a such a source is
again given by

$\displaystyle\mathrm{MSNR_{b}}$
$\displaystyle=\frac{1}{R_{s}}\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}}{\mathbb{E}|\vec{r}|^{2}}$
$\displaystyle=\frac{1}{R_{s}}\frac{\sum_{k=1}^{N}\left[\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{k}+g(\omega^{\prime})\cos\omega^{\prime}t_{k}\right)\right]^{2}}{\sum_{k=1}^{N}\sigma_{k}^{2}},$
(28)

where use of a subscript ‘b’ denotes that this $\mathrm{MSNR}$ corresponds to
a broadband source.

Case 1. When
$f(\omega^{\prime})=\delta(\omega^{\prime}-\omega)=g(\omega^{\prime})$. In
this case we have following proposition.

######  Proposition 1.

For a voltage time series $\vec{\mathcal{V}}=\vec{\mathcal{S}}+\vec{r}$ with
$p(\vec{\mathcal{S}})=\prod_{i}\delta\left(\mathcal{S}_{i}-\sin\omega
t_{i}-\cos\omega t_{i}\right)$ and $q(\vec{r})$ satisfying Eq. (3), we have

$\displaystyle\mathrm{MSNR_{b}}$
$\displaystyle=2~{}\mathrm{MSNR}=\frac{1}{R_{s}}\frac{1}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}}.$
(29)

Here $\mathrm{MSNR}$ and $\mathrm{MSNR_{b}}$ are the signal-to-noise ratios
divided by the internal resistance $R_{s}$ when
$p(\vec{\mathcal{S}})=\prod_{i}\delta\left(\mathcal{S}_{i}-\sin\omega
t_{i}\right)$ and
$p(\vec{\mathcal{S}})=\prod_{i}\delta\left(\mathcal{S}_{i}-\sin\omega
t_{i}-\cos\omega t_{i}\right)$, respectively.

###### Proof.

Putting $f(\omega^{\prime})=\delta(\omega^{\prime}-\omega)=g(\omega^{\prime})$
in Eq. (A.2), we have

$\displaystyle\mathrm{MSNR_{b}}$
$\displaystyle=\frac{1}{R_{s}}\frac{\sum_{k=1}^{N}\left(\sin\omega
t_{k}+\cos\omega t_{k}\right)^{2}}{\sum_{k=1}^{N}\sigma_{k}^{2}}$
$\displaystyle=\frac{1}{R_{s}}\frac{N+\sum_{k=1}^{N}\sin(2\omega
t_{k})}{\sum_{k=1}^{N}\sigma_{k}^{2}}.$

Taking $t_{k}=k\frac{T}{N}$, we get

$\displaystyle\frac{1}{N}\sum_{k=1}^{N}\sin(2\omega
t_{k})=\frac{1}{N}\sum_{k=1}^{N}\sin\left(\frac{2\omega kT}{N}\right).$

Now, using Riemann sum formula, we have

$\displaystyle\frac{1}{\omega T}\int_{0}^{\omega T}\sin
2x~{}\mathrm{d}x=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sin\left(\frac{2\omega
kT}{N}\right).$ (30)

Solving the left hand side of the above equation, we get

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sin\left(\frac{2\omega
kT}{N}\right)=\frac{1}{\omega T}\int_{0}^{\omega T}\sin 2x~{}\mathrm{d}x=0.$

Thus, we have

$\displaystyle\mathrm{MSNR_{b}}=\frac{1}{R_{s}}\frac{1}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}}.$
(31)

Using Eq. (26), we complete the proof of the proposition. ∎

Case 2. When
$f(\omega^{\prime})=\delta(\omega^{\prime}-\omega_{1})+\delta(\omega^{\prime}-\omega_{2})$
and $g(\omega^{\prime})=0$. In this case we have following proposition.

######  Proposition 2.

For a voltage time series $\vec{\mathcal{V}}=(1-g)\vec{\mathcal{S}}+g\vec{r}$
with
$p(\vec{\mathcal{S}})=\prod_{i}\delta\left(\mathcal{S}_{i}-\sin\omega_{1}t_{i}-\sin\omega_{2}t_{i}\right)$
and $q(\vec{r})$ satisfying Eq. (3), we have

$\displaystyle\mathrm{MSNR_{b}}$
$\displaystyle=\frac{1}{R_{s}}\frac{1+\frac{\sin\left((\omega_{1}-\omega_{2})\mathcal{T}\right)}{(\omega_{1}-\omega_{2})\mathcal{T}}+\frac{\sin\left((\omega_{1}+\omega_{2})\mathcal{T}\right)}{(\omega_{1}+\omega_{2})\mathcal{T}}}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}}.$
(32)

Here $\mathcal{T}$ can be time period of any of the signal component or any
other constant.

###### Proof.

Putting
$f(\omega^{\prime})=\delta(\omega^{\prime}-\omega_{1})+\delta(\omega^{\prime}-\omega_{2})$
and $g(\omega^{\prime})=0$ in (A.2), we have

$\displaystyle\mathrm{MSNR_{b}}$
$\displaystyle=\frac{1}{R_{s}}\frac{\sum_{k=1}^{N}\left(\sin\omega_{1}t_{k}+\sin\omega_{2}t_{k}\right)^{2}}{\sum_{k=1}^{N}\sigma_{k}^{2}}$
$\displaystyle=\frac{1}{R_{s}}\left[\frac{\sum_{k=1}^{N}\sin^{2}\omega_{1}t_{k}+\sin^{2}\omega_{2}t_{k}}{\sum_{k=1}^{N}\sigma_{k}^{2}}+\frac{\sum_{k=1}^{N}\cos((\omega_{1}-\omega_{2})t_{k})-\cos((\omega_{1}+\omega_{2})t_{k})}{\sum_{k=1}^{N}\sigma_{k}^{2}}\right].$
(33)

Now, taking $t_{k}=k\frac{T}{N}$ and using Riemann sum formula, we have

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\cos\left(\frac{(\omega_{1}-\omega_{2})kT}{N}\right)=\frac{\sin\left((\omega_{1}-\omega_{2})T\right)}{(\omega_{1}-\omega_{2})T}$

and

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\cos\left(\frac{(\omega_{1}+\omega_{2})kT}{N}\right)=\frac{\sin\left((\omega_{1}+\omega_{2})T\right)}{(\omega_{1}+\omega_{2})T}.$

In the above, $T$ can be time period of any of the signal component or any
other constant. Using above in Eq. (A.2), we complete the proof of the
proposition.

∎

###  A.3 Modified signal-to-noise ratio for general broadband sources

From Eq. (A.2), we have

$\displaystyle\mathrm{MSNR_{b}}=\frac{1}{R_{s}}\frac{\sum_{k=1}^{N}\left[\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{k}+g(\omega^{\prime})\cos\omega^{\prime}t_{k}\right)\right]^{2}}{\sum_{k=1}^{N}\sigma_{k}^{2}}.$
(34)

Let us define

$\displaystyle\mathcal{G}$
$\displaystyle:=\sum_{k=1}^{N}\left[\sum_{\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{k}+g(\omega^{\prime})\cos\omega^{\prime}t_{k}\right)\right]^{2}$
$\displaystyle=\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}\left(f(\omega^{\prime})\sin\omega^{\prime}t_{k}+g(\omega^{\prime})\cos\omega^{\prime}t_{k}\right)\left(f(\omega)\sin\omega
t_{k}+g(\omega)\cos\omega t_{k}\right)$
$\displaystyle=\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}\left(f(\omega^{\prime})f(\omega)\sin\omega^{\prime}t_{k}\sin\omega
t_{k}+g(\omega^{\prime})f(\omega)\sin\omega
t_{k}\cos\omega^{\prime}t_{k}\right)$
$\displaystyle~{}~{}~{}+\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}\left(f(\omega^{\prime})g(\omega)\sin\omega^{\prime}t_{k}\cos\omega
t_{k}+g(\omega^{\prime})g(\omega)\cos\omega^{\prime}t_{k}\cos\omega
t_{k}\right)$
$\displaystyle=\mathcal{G}_{1}+\mathcal{G}_{2}+\mathcal{G}_{3}+\mathcal{G}_{4}.$

In the following we will compute each term divided by $N$ in the limit
$N\rightarrow\infty$.

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}_{1}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\sin\omega^{\prime}t_{k}\sin\omega
t_{k}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\frac{1}{2}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\left(\cos\left(\frac{\omega^{\prime}-\omega)kT}{N}\right)-\cos\left(\frac{\omega^{\prime}+\omega)kT}{N}\right)\right)$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\cos(\omega^{\prime}-\omega)x-\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\cos(\omega^{\prime}+\omega)x$
$\displaystyle=\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\frac{1}{2(\omega^{\prime}-\omega)T}\sin(\omega^{\prime}-\omega)T-\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})f(\omega)\frac{1}{2(\omega^{\prime}+\omega)T}\sin(\omega^{\prime}+\omega)T$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{f(\omega^{\prime})f(\omega)}{2}\left(\frac{\sin(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}-\frac{\sin(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}\right).$

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}_{2}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})f(\omega)\sin\omega
t_{k}\cos\omega^{\prime}t_{k}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\frac{1}{2}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})f(\omega)\left(\sin\left(\frac{\omega+\omega^{\prime})kT}{N}\right)+\sin\left(\frac{\omega-\omega^{\prime})kT}{N}\right)\right)$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})f(\omega)\sin(\omega+\omega^{\prime})x-\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})f(\omega)\sin(\omega-\omega^{\prime})x$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{g(\omega^{\prime})f(\omega)}{2}\left(\frac{1-\cos(\omega+\omega^{\prime})T}{(\omega+\omega^{\prime})T}+\frac{1-\cos(\omega-\omega^{\prime})T}{(\omega-\omega^{\prime})T}\right)$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{g(\omega^{\prime})f(\omega)}{2}\left(\frac{1-\cos(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}-\frac{1-\cos(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}\right).$

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}_{3}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})g(\omega)\sin\omega^{\prime}t_{k}\cos\omega
t_{k}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\frac{1}{2}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})g(\omega)\left(\sin\left(\frac{\omega^{\prime}+\omega)kT}{N}\right)+\sin\left(\frac{\omega^{\prime}-\omega)kT}{N}\right)\right)$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}f(\omega^{\prime})g(\omega)\sin(\omega^{\prime}+\omega)x-\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})f(\omega)\sin(\omega^{\prime}-\omega)x$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{f(\omega^{\prime})g(\omega)}{2}\left(\frac{1-\cos(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}+\frac{1-\cos(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}\right).$

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}_{4}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})g(\omega)\cos\omega^{\prime}t_{k}\cos\omega
t_{k}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\frac{1}{2}\sum_{k=1}^{N}\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})g(\omega)\left(\cos\left(\frac{\omega^{\prime}+\omega)kT}{N}\right)+\cos\left(\frac{\omega^{\prime}-\omega)kT}{N}\right)\right)$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})g(\omega)\cos(\omega^{\prime}+\omega)x+\frac{1}{2T}\int_{0}^{T}\mathrm{d}x\sum_{\omega,\omega^{\prime}}g(\omega^{\prime})g(\omega)\cos(\omega^{\prime}-\omega)x$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{g(\omega^{\prime})g(\omega)}{2}\left(\frac{\sin(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}+\frac{\sin(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}\right).$

Combining above equations, we get

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\left(\frac{g(\omega^{\prime})g(\omega)-f(\omega^{\prime})f(\omega)}{2}\right)\frac{\sin(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}+\left(\frac{g(\omega^{\prime})g(\omega)+f(\omega^{\prime})f(\omega)}{2}\right)\frac{\sin(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}$
$\displaystyle+\sum_{\omega,\omega^{\prime}}\left(\frac{g(\omega^{\prime})f(\omega)+f(\omega^{\prime})g(\omega)}{2}\right)\frac{1-\cos(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}+\left(\frac{f(\omega^{\prime})g(\omega)-g(\omega^{\prime})f(\omega)}{2}\right)\frac{1-\cos(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}.$

Let us consider a special case when $\omega$ takes values from the set
$\\{\Omega,2\Omega,\cdots\\}$ and
$\sum_{\omega}\rightarrow\sum_{m=1}^{\infty}$. Further, choose
$f(\omega)\rightarrow\tilde{f}(m)=0$ and
$g(\omega)\rightarrow\tilde{g}(m)=1/m$. This arises in following Fourier
series

$\displaystyle-\ln\left(2\sin\frac{|\Omega
T|}{2}\right)=\sum_{m=1}^{\infty}\frac{\cos m\Omega T}{m}.$ (35)

Then,

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\mathcal{G}$
$\displaystyle=\sum_{\omega,\omega^{\prime}}\frac{g(\omega^{\prime})g(\omega)}{2}\left(\frac{\sin(\omega^{\prime}-\omega)T}{(\omega^{\prime}-\omega)T}+\frac{\sin(\omega^{\prime}+\omega)T}{(\omega^{\prime}+\omega)T}\right)$
$\displaystyle=\sum_{n,m=1}^{\infty}\frac{1}{2mn}\left(\frac{\sin(m-n)\Omega
T}{(m-n)\Omega T}+\frac{\sin(m+n)\Omega T}{(m+n)\Omega T}\right).$

Thus, we have

$\displaystyle\mathrm{MSNR_{b}}=\frac{1}{R_{s}}\frac{\sum_{n,m=1}^{\infty}\frac{1}{2mn}\left(\frac{\sin(m-n)\Omega
T}{(m-n)\Omega T}+\frac{\sin(m+n)\Omega T}{(m+n)\Omega
T}\right)}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\sigma_{k}^{2}}.$

##  Appendix B Data processing inequality for relative entropy and proof of
Eq. (19)

###  B.1 Data processing inequality for relative entropy

Let us consider two probability distributions
$\mathcal{P}=\\{\mathcal{P}_{i}\\}$ and $\mathcal{Q}=\\{\mathcal{Q}_{i}\\}$
and $M$ a column stochastic matrix. Let $\mathcal{P}^{\prime}=M\mathcal{P}$
and $\mathcal{Q}^{\prime}=M\mathcal{Q}$. Now,

$\displaystyle D(\mathcal{P}^{\prime}||\mathcal{Q}^{\prime})$
$\displaystyle=\sum_{i}\mathcal{P}^{\prime}_{i}\ln\frac{\mathcal{P}^{\prime}}{\mathcal{Q}^{\prime}}$
$\displaystyle=\sum_{i}\sum_{j}M_{ij}\mathcal{P}_{j}\ln\frac{\sum_{k}M_{ik}\mathcal{P}_{k}}{\sum_{k}M_{ik}\mathcal{Q}_{k}}$
$\displaystyle\leq\sum_{i}\sum_{j}M_{ij}\mathcal{P}_{j}\ln\frac{\
M_{ij}\mathcal{P}_{j}}{M_{ij}\mathcal{Q}_{j}}$
$\displaystyle=\sum_{j}\left(\sum_{i}M_{ij}\right)\mathcal{P}_{j}\ln\frac{\
\mathcal{P}_{j}}{\mathcal{Q}_{j}}$
$\displaystyle=D(\mathcal{P}||\mathcal{Q}).$ (36)

In above we have used Log sum inequality [49], which states that

$\displaystyle\sum_{i}a_{i}\ln\frac{a_{i}}{b_{i}}\geq\left(\sum_{i}a_{i}\right)\ln\frac{\sum_{j}a_{j}}{\sum_{j}b_{j}}.$
(37)

Therefore, if $M$ is a column stochastic matrix, then

$\displaystyle D(M\mathcal{P}||M\mathcal{Q})\leq D(\mathcal{P}||\mathcal{Q}).$
(38)

###  B.2 Relative entropy under convolution of probability distributions

Let us consider two probability distributions $p$ and $q$ corresponding to
some random variable $X$. Let $Y$ be another random variable with probability
density $r$. Then the probability densities corresponding to $X+Y:=Z$ will be
given by $P:=p*r$ and $Q:=q*r$ for the cases when $X$ is distributed by $p$
and $q$, respectively. In particular,

$\displaystyle P(z)=\int p(x)r(z-x)\mathrm{d}x,~{}~{}~{}~{}Q(z)=\int
q(x)r(z-x)\mathrm{d}x.$

Now,

$\displaystyle D(P||Q)$
$\displaystyle=\int\mathrm{d}zP(z)\ln\frac{P(z)}{Q(z)}$
$\displaystyle=\int\mathrm{d}z\left(\int
p(x)r(z-x)\mathrm{d}x\right)\ln\frac{\left(\int
p(x)r(z-x)\mathrm{d}x\right)}{\left(\int q(x)r(z-x)\mathrm{d}x\right)}$
$\displaystyle\leq\int\mathrm{d}z\int\mathrm{d}x~{}p(x)r(z-x)\ln\frac{p(x)}{q(x)}$
$\displaystyle=\int\mathrm{d}x~{}p(x)\ln\frac{p(x)}{q(x)}$
$\displaystyle=D(p||q),$

where the inequality in line $3$ comes from the continuous analogue of the Log
sum inequality [49]. Thus, the relative entropy is non-increasing under the
convolution of probability distributions.

###  B.3 Proof of Eq. (19)

The relative entropy between the probability distributions of our interest is
given by

$\displaystyle D\left(\mathcal{P}||\mathcal{Q}\right)$
$\displaystyle=\int\cdots\int\mathrm{d}V_{1}\cdots\mathrm{d}V_{N}\prod_{i}p_{i}(V_{i})\ln\left(\frac{\prod_{j}p_{j}(V_{j})}{\prod_{j}\mathcal{Q}(V_{j})}\right)$
$\displaystyle=\int\cdots\int\mathrm{d}V_{1}\cdots\mathrm{d}V_{N}\prod_{i}p_{i}(V_{i})\sum_{j}\ln\left(\frac{p_{j}(V_{j})}{\mathcal{Q}(V_{j})}\right)$
$\displaystyle=\sum_{i}\int\mathrm{d}V_{i}p_{i}(V_{i})\ln\left(\frac{p_{i}(V_{i})}{\mathcal{Q}(V_{i})}\right).$
(39)

Note that, we have

$\displaystyle
p_{i}(V_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right],$
(40)

and

$\displaystyle\mathcal{Q}(V_{i})=\frac{1}{\sqrt{2\pi\xi_{i}^{2}}}\exp\left[-\frac{V_{i}^{2}}{2\xi_{i}^{2}}\right].$
(41)

Then

$\displaystyle\int\mathrm{d}V_{i}p_{i}(V_{i})\ln\left(\frac{p_{i}(V_{i})}{\mathcal{Q}(V_{i})}\right)$
$\displaystyle=\ln\left(\frac{\xi_{i}}{\sigma_{i}}\right)\int\mathrm{d}V_{i}p_{i}(V_{i})+\int\mathrm{d}V_{i}p_{i}(V_{i})\left[\frac{V_{i}^{2}}{2\xi_{i}^{2}}-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right]$
$\displaystyle=\ln\left(\frac{\xi_{i}}{\sigma_{i}}\right)+I_{1}-I_{2},$ (42)

where

$\displaystyle I_{2}$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\int\mathrm{d}V_{i}\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right]=\frac{1}{2},$
(43)

and

$\displaystyle I_{1}$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\int\mathrm{d}V_{i}\frac{V_{i}^{2}}{2\xi_{i}^{2}}\exp\left[-\frac{(V_{i}-s_{i})^{2}}{2\sigma_{i}^{2}}\right]$
$\displaystyle=\frac{1}{\sqrt{\pi}}\int\mathrm{d}y\frac{\left(\sqrt{2\sigma_{i}^{2}}y+s_{i}\right)^{2}}{2\xi_{i}^{2}}e^{-y^{2}}$
$\displaystyle=\frac{\sigma_{i}^{2}}{\sqrt{\pi}\xi_{i}^{2}}\int\mathrm{d}y~{}y^{2}e^{-y^{2}}+\frac{1}{\sqrt{\pi}}\frac{s_{i}^{2}}{2\xi_{i}^{2}}\int\mathrm{d}ye^{-y^{2}}$
$\displaystyle=\frac{\sigma_{i}^{2}}{2\xi_{i}^{2}}+\frac{s_{i}^{2}}{2\xi_{i}^{2}}.$
(44)

Combining Eqs. (B.3), (B.3), (43), and (B.3), we get

$\displaystyle D\left(\mathcal{P}||\mathcal{Q}\right)$
$\displaystyle=\frac{1}{2}\sum_{i}\left(\frac{\sigma_{i}^{2}+s_{i}^{2}}{\xi_{i}^{2}}-\ln\left(\frac{\sigma_{i}^{2}}{\xi_{i}^{2}}\right)-1\right).$
(45)

Therefore, the relative entropy of the resource is given by

$\displaystyle\mathcal{R}(p(\vec{\mathcal{V}}),R_{s})=\frac{\tau}{2NR_{s}}\sum_{i}\left(\frac{\sigma_{i}^{2}+s_{i}^{2}}{\xi_{i}^{2}}-\ln\left(\frac{\sigma_{i}^{2}}{\xi_{i}^{2}}\right)-1\right).$

##  Appendix C Addition of two non-identical noisy sources

We now give an example of adding to two non-identical noisy sources. Let us
consider two voltage sources $(P(\vec{\mathcal{V}})^{(1)},R_{1})$ and
$(P(\vec{\mathcal{V}})^{(2)},R_{2})$ as given by
$\vec{\mathcal{V}}^{(\mu)}=\vec{\mathcal{S}}^{(\mu)}+\vec{r}^{(\mu)}$, where
$\mu=1,2$ and $\vec{\mathcal{S}}^{(\mu)}$ is distributed according to

$\displaystyle
p\left(\vec{\mathcal{S}}^{(\mu)}\right)=\prod_{i}\delta\left(\mathcal{S}^{(\mu)}_{i}-\sin\omega^{(\mu)}t_{i}\right),$
(46)

while $\vec{r}^{(\mu)}$ is distributed according to

$\displaystyle
q\left(\vec{r}^{(\mu)}\right)=\prod_{i}\frac{1}{\sqrt{2\pi\sigma^{(\mu)2}_{i}}}\exp\left[-r^{(\mu)2}_{i}/2\sigma^{(\mu)2}_{i}\right].$
(47)

Here $\vec{r}^{(\mu)}$ is considered as thermal noise at ambient temperature.
If we add these sources in series then the combined source will have voltage
given by $\vec{\mathcal{V}}=\vec{\mathcal{S}}+\vec{r}$, where
$\vec{\mathcal{S}}=\vec{\mathcal{S}}^{(1)}+\vec{\mathcal{S}}^{(2)}$ and
$\vec{r}=\vec{r}^{(1)}+\vec{r}^{(2)}$. Moreover the internal resistance will
become $R_{1}+R_{2}$. From the convolution theorem, the probability
distribution of $\vec{\mathcal{S}}$ is given by
$P(\vec{\mathcal{S}})=\prod_{i}P(\mathcal{S}_{i})$, where

$\displaystyle p(\mathcal{S}_{i})$
$\displaystyle=\int^{\infty}_{-\infty}\mathrm{d}x_{i}~{}\delta\left(\mathcal{S}_{i}-x_{i}-\sin\omega^{(2)}t_{i}\right)\delta\left(x_{i}-\sin\omega^{(1)}t_{i}\right)$
$\displaystyle=\delta\left(\mathcal{S}_{i}-\sin\omega^{(1)}t_{i}-\sin\omega^{(2)}t_{i}\right).$

Similarly, the probability distribution of $\vec{r}$ is given by
$q(\vec{r})=\prod_{i}q(r_{i})$, where

$\displaystyle
q(r_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[-\frac{r_{i}^{2}}{2\sigma_{i}^{2}}\right].$

and $\sigma_{i}^{2}=\sigma^{(1)2}_{i}+\sigma^{(2)2}_{i}$. From Eq. (32) of
previous section, we get

$\displaystyle\mathrm{MSNR}\left((p(\vec{\mathcal{V}})^{(1)},R_{1})\circ(p(\vec{\mathcal{V}})^{(2)},R_{2})\right)=\frac{1}{(R_{1}+R_{2})}\frac{1+\frac{\sin\left((\omega^{(1)}-\omega^{(2)})T\right)}{(\omega^{(1)}-\omega^{(2)})T}+\frac{\sin\left((\omega^{(1)}+\omega^{(2)})T\right)}{(\omega^{(1)}+\omega^{(2)})T}}{\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\left(\sigma^{(1)2}_{k}+\sigma^{(2)2}_{k}\right)}.$
(48)

Similarly, one can compute $\mathrm{UnitDC}$ for such a combined source and we
have

$\displaystyle\mathrm{UnitDC}\left((p(\vec{\mathcal{V}})^{(1)},R_{1})\circ(p(\vec{\mathcal{V}})^{(2)},R_{2})\right)$
$\displaystyle=\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}}{(R_{1}+R_{2})}$
$\displaystyle=\frac{\sum_{k=1}^{N}\left(\sin\omega^{(1)}t_{k}+\sin\omega^{(2)}t_{k}\right)^{2}}{N(R_{1}+R_{2})}.$
(49)

##  Appendix D Distillation rate from noisy voltage sources

In this section we discuss, in a mixture of rigorous and heuristical
arguments, how much $\mathrm{UnitDC}$ one can distill from noisy sources.

###  D.1 Distillation of $\mathrm{UnitDC}$ via rectification of many copies

How can one distill $\mathrm{UnitDC}$ from e.g. a hot resistor, and how much
can be distilled? A hot resistor is a case of the Gaussian noise having higher
variance than that corresponding to the ambient temperature:

$\mathbb{E}|\vec{r}|^{2}>\mathbb{E}|\vec{r}_{T_{amb}}|^{2}=\mathbb{E}|\vec{r}_{th}|^{2}.$
(50)

We can say sources respecting Eq. (50) have ‘hot’ noise. We shall now
initially assume we have a hot noise source rather than a cold noise source.
We let $\vec{r}$ denote Gaussian random variables and $\vec{\mathcal{S}}$
deterministic ones so that a source is written as
$(\vec{\mathcal{S}}+\vec{r},R_{s})$. For example, a resistor with resistance
$R_{s}$ and temperature $T_{H}$ is then written as
$(0+\vec{r}_{T_{H}},R_{s})$.

How can this distillation of unit DCs be achieved? Orthogonal transformations
cannot do the job, since the source voltage is randomly positive and negative
and orthogonal transformations by inspection cannot turn that into a
deterministically positive number. For example in 3-D one can visualise that
if a vector might be pointing in any quadrant-the agent does not know which
one, no orthogonal transformation chosen by the agent is guaranteed make it
point into the positive quadrant.

In practise people use rectifiers (e.g. diode bridges) to achieve guaranteed
positive voltage output. Let us therefore try to define an idealised rectifier
that could in principle be allowed. For a voltage source $V$, it should
satisfy two conditions, (i) rectification, i.e., $V\rightarrow|V|-V_{b}$ where
$V_{b}$ is the penalty associated with potential barriers in the diodes, (ii)
take an ambient thermal source to another ambient thermal source (the 2nd law
of thermodynamics demands this, assuming the diode is not changed by the
operation [9]). Real diodes do have internal resistance but for simplicity let
us firstly see if we can satisfy the two conditions without such resistance,
$R_{s}\rightarrow R_{s}$.

We can consider defining an idealised rectifier operation acting on hot-noise
sources (see Eq. (50)) as

$p(V)\rightarrow\mathcal{N}(|\mathbb{E}|V|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}|^{\frac{1}{2}},\left(\mathbb{E}|\vec{r}_{th}|^{2}\right)^{\frac{1}{2}}),$
(51)

and $R_{s}$ being unmodified. $\mathcal{N}(a,b)$ here means normal
distribution with mean $a$ and standard deviation $b$. Condition (i) of
rectification is then satisfied in the regime where thermal noise is
insignificant ($V^{2}\gg\mathbb{E}|\vec{r}_{th}|^{2}$), since in that regime
the idealised rectification would take $V\rightarrow|V|$. Moreover the output
voltage is generally lower than $|V|$, even though it does not have the form
$|V|$-constant (an expression which anyway should not be taken too seriously
as it could be negative). Condition (ii) is moreover satisfied since

$\mathcal{N}(0,\\!|\mathbb{E}|\vec{r}_{th}|^{2}|^{\frac{1}{2}})\\!\rightarrow\\!\mathcal{N}(|\mathbb{E}|\vec{r}_{th}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}|^{\frac{1}{2}},\\!\left(\mathbb{E}|\vec{r}_{th}|^{2}\right)^{\frac{1}{2}}).$

Let us then accept Eq.51 as a possibly, in principle, realisable rectifier and
go on to use this idealised rectifier operation and see how many unit DCs we
can distill out of a transient source.

Given a source $(\vec{\mathcal{S}}+\vec{r},R_{s})$ we then see that the
rectifier turns the voltage distribution into

$\displaystyle\mathcal{N}(|\mathbb{E}|\vec{\mathcal{S}}|^{2}+\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}|^{\frac{1}{2}}),\mathbb{E}|\vec{r}_{th}|^{2}).$

Now we allow for one more tool to achieve the distillation: composition of
many independent identical copies of the source. Consider many, $n$, such
sources in series. By the law of large numbers, the total voltage is then
approximately deterministic with value

$V_{out}=n|\mathbb{E}|\vec{\mathcal{S}}|^{2}+\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}|^{\frac{1}{2}}.$

How many unit DCs is this equivalent to? A unit DC source can we written as
$(\vec{S}+\vec{r},R_{s})=(1+r_{th},1)$. Here we have instead
$(V_{out}+r_{th},R_{s})$. More generally, suppose we want to create (c,R), a
constant voltage c and internal resistance R from unit DCs meaning (1,1).
Recall that two voltage sources with identical emfs connected in parallel have
a net emf equivalent to one emf source, however, the net internal resistance
is less ($\frac{1}{R_{net}}=\sum_{i}\frac{1}{R_{i}}$). Consider the following
protocol: connect in series c sources each with resistance R/c and voltage 1.
We can create each such elementary source by putting c/R unit DCs in parallel.
Thus we would end up using $\frac{c^{2}}{R}$ unit DCs to mimic the target
source. One reason to believe this is optimal is that it is the same number of
unit DC’s that would give the maximum power for the target source
($\frac{c^{2}}{R}$). We therefore assume (we call it an assumption because we
did not describe a protocol for breaking an arbitrary source into unit DC
sources) that (c,R) can in principle be converted to and from
$\frac{c^{2}}{R}$ unit DC sources.

Accordingly, the source $(V_{out},R_{s})$ composed out of hot-noise sources is
equivalent to
$V_{out}^{2}/(nR_{s})=n\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}}$
unit DC sources. Dividing by the number of sources $n$, the unit DC we can
distill out via the rectifier operation and combination in series is then

$\frac{\mathbb{E}|\vec{\mathcal{S}}|^{2}+\mathbb{E}|\vec{r}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}}$

per source. For the example of a hot resistor we have $\vec{\mathcal{S}}=0$
and $\mathbb{E}|\vec{r}|^{2}=\mathbb{E}|\vec{r}_{T_{H}}|^{2}$, and the number
of unit DCs we can distill out should be
$\frac{\mathbb{E}|\vec{r}_{T_{H}}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}}$.

###  D.2 Formation of noisy sources from unit DCs

In the reverse direction, suppose we want to transform $m$ unit DCs to e.g.
$n$ hot resistors with $T>T_{ambient}:=T_{amb}$, as an important case. This
can in principle be achieved by using the unit DCs to amplify the thermal
fluctuations of (free) thermal resistors with the same resistance as the
target source and at temperature $T_{amb}$. Thermal noise is in fact normally
amplified in order to be detected [28], so amplifying Gaussian noise is
feasible in practise. For this combination of the unit DCs and the thermal
resistors to be a free operation, it is natural to require that it cannot
increase the power that can be delivered to a load. $m$ unit DCs can deliver
$m$ joules into a resistive load at most (for a matched load). The original
$n$ thermal resistors on their own deliver

$P_{o}=n\frac{\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}},$

power and the $n$ target sources should deliver

$P_{T}=n\frac{\mathbb{E}|\vec{r}_{T}|^{2}}{R_{s}}$

power. Thus the missing power, which naively equates to the required $m$ unit
DCs, is
$P_{m}=n\frac{\mathbb{E}|\vec{r}_{T}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}}$,
and

$\frac{m}{n}=\frac{\mathbb{E}|\vec{r}_{T}|^{2}-\mathbb{E}|\vec{r}_{th}|^{2}}{R_{s}}.$

Thus the number of unit DCs needed matches the value of the monotone
$\mathrm{UnitDC}$ of the target source as desired.

##  Appendix E Modelling noisy voltage sources given experimental data

One method of distilling out the signal from the noise is the averaging
method, which works for AWGN by the law of large number arguments used in the
distillation section. The averaging function is built into many oscilloscopes
[26].

In general one can use the maximum likelihood approach. We now show how this
can be used to justify a simple formula for determining the standard deviation
of the AWGN noise from the observed data. Let us consider that we are given
with a data, say some micro voltages $\vec{x}=\\{x_{i}\\}_{i=1}^{N}$. The aim
is to single out a signal and noise from this data. To do so, we will take the
approach of maximum likelihood estimation [49]. Let us assume that the noise
is modeled as additive white Gaussian noise with mean zero and standard
deviation $\sigma$. Further, let us say that the actual signal is
$\vec{S}=\\{S_{i}\\}_{i=1}^{n}$.

######  Proposition 3.

For AWGN with mean zero and standard deviation $\sigma$, the maximum
likelihood estimation of $\sigma$ for the proposed signal $\vec{S}$ and data
$\vec{x}$ is given by

$\displaystyle\sigma=\sqrt{\frac{\vec{r}\cdot\vec{r}}{N}},$ (52)

where $\vec{r}=\vec{x}-\vec{S}$ and
$\vec{r}=\left(r_{1},\cdots,r_{N}\right)^{T}$.

###### Proof.

By assumption, the data $\vec{x}=\vec{S}+\vec{r}$. Then, assuming the noise to
be an additive white Gaussian noise, the probability distribution
$P(\vec{x},\sigma)$ is given by

$\displaystyle
P(\vec{x},\sigma)=\prod_{i=1}^{N}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[\frac{-(x_{i}-S_{i})^{2}}{2\sigma^{2}}\right].$
(53)

The maximum likelihood approach selects the maximally likely distribution,
which can be found by setting $\frac{\partial
P(\vec{x},\sigma)}{\partial\sigma}=0$, which implies

$\displaystyle\sigma=\sqrt{\frac{\sum_{i=1}^{N}(x_{i}-S_{i})^{2}}{N}}=\sqrt{\frac{\vec{r}.\vec{r}}{N}}.$
(54)

This completes the proof of the proposition. ∎

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