Dr. Jerrold Prothero

[email protected]

Washington, D.C.

October 30, 2014

Version 1.0

Copyright © 2014 Astrapi Corporation

## Technical Report

**Table of Contents**

Abstract

1. Introduction

2. An Open Field for Innovation

3. Specific Applications of Spiral-Based Signal Modulation

4. The Spectral Efficiency Advantage of Spiral-Based Signal Modulation

4.1 What Shannon Proved (And Did Not Prove)

4.2 LaterResearch

5. Conclusion.

Abstract

Astrapi® is the pioneer of spiral-based signal modulation, which opens an unexplored area for innovation at the core of telecommunications. Based on a generalization of Euler’s formula, the foundational mathematics for telecom, Astrapi

provides fundamentally new ways to design the symbol waveforms used to encode

digital transmissions. Specific potential applications include improved performance

in the presence of phase impairments, for instance due to multipath interference;

greater resistance to interference from competing signals; and the ability to work

well with power-efficient and cost-effective nonlinear power amplifiers. Most

importantly, spiral-based signal modulation for the first time puts non-periodic

signal modulation on a firm theoretical basis. Classical channel capacity theory

implicitly assumes that signals are based on periodic functions. Non-periodic signal

modulation opens a pathway to dramatically higher spectral efficiency, limited by

hardware capabilities rather than solely by available bandwidth and signal-to-noise

power ratio.

** 1. Introduction**

Astrapi is the pioneer of spiral-based signal modulation. Simply put, this is the idea

of basing telecommunication signal modulation on complex spirals, rather than the

complex circles used by standard signal modulation techniques such as Quadrature

Amplitude Modulation (QAM) and Phase-Shift Keying (PSK).

General background on spiral-based signal modulation, including an overview,

engineering FAQ, mathematics, and applicable intellectual property, is available

from the Astrapi website.**¹**

Spiral-based signal modulation requires a fundamentally new approach to radio

design. Why should anyone be interested in a long-term investment in this new

telecommunications technology?

Because the telecommunications industry faces a severe long-term problem, which

requires long-term solutions. The exploding demand for data transfer will be difficult

to meet with existing or proposed solutions, potentially imposing costs in terms of

user needs which are not met, profits which are not realized, and a significant drag

on the world economy as a whole.

Astrapi believes that spiral-based signal modulation technology provides a

fundamental solution to the fundamental problem of the Information Age, which is

helping to ensure rapid, high-volume data transfer to meet exponentially growing

demand.

Why do we believe this? At a high level, for three reasons

- Spirals are a wide-open field for innovation. There has been no prior research

on spiral-based signal modulation. We know that spiral-based signal

modulation is no worse than the current state-of-the-art, because traditional

circle-based signal modulation is a subset of spiral-based signal modulation.

The extra capabilities of spirals open up new possibilities to be exploited for

efficient communication. - . There are specific applications where spiral-based modulation provides

potential benefits. These include situations limited by phase noise, coherent

noise from interfering channels, and where efficient nonlinear power

amplification is required (for reasons of power efficiency or equipment cost). - Spirals are the right way to transition from periodic to non-periodic signal

modulation, which allows for greatly increased spectral efficiency. The

classical theory of channel capacity implicitly assumes that signals are based

on periodic functions. Spirals, and the associated mathematics, opens a path

to exploiting the potentially much higher spectral efficiency of non-periodic

signal modulation.

Each of these points is discussed in more detail below.

2. An Open Field for Innovation

It is surprising but true that the idea of basing signal modulation on complex spirals,

rather than the traditional complex circles, has not previously received research

attention from industry or academics. Consequently, Astrapi holds very broad

foundational patents, with no prior art, on the idea and techniques for spiral-based

signal modulation.

Clearly, a circle is the special case of a spiral whose complex amplitude is constant.

Conversely, spirals can do something circles cannot, which is to change their

complex amplitude continuously over time.

It is common knowledge in any field that extra design parameters open up the

possibility of much better products. Spiral-based signal modulation opens this door

for telecommunications.

3. Specific Applications of Spiral-Based Signal Modulation

What are specific examples of communication applications in which spiral-based

signal modulation may out-perform traditional circle-based signal modulation? The

following three broad areas are intended to be illustrative, rather than

comprehensive.

- Applications where phase noise is significant, for instance due to

multipath interference or clock recovery problems. Spirals have two

potential advantages: lower dependence on phase information for a

given data rate, since we can also store information in spiral

amplitude variation; and the ability to use spiral amplitude variation

for time alignment. - Applications in which coherent noise from interfering channels is

significant. While it may not be initially apparent, the symbol waveform design space becomes very much larger when symbol

waveforms are constructed from complex spirals, rather than

complex circles. This gives us the ability to make spiral-based signals

look very different from other interfering signals, and therefore more

easily detected in the presence of coherent noise. - Applications benefiting from efficient nonlinear power amplification.

Many current signal modulation techniques require linear power

amplification, whereas the physics of power amplification (and

therefore of efficiency and cost) favors nonlinearity. We can use the

wide complex amplitude variation information that spirals provide on

a per-symbol basis to characterize and correct for nonlinear distortion.

The common theme in these applications is that the extra design space that spirals

provide can be used to compensate for problems that frequently appear with

traditional signal modulation.

4. The Spectral Efficiency Advantage of Spiral-Based Signal Modulation

The most powerful implication of spiral-based signal modulation is that it provides

the most natural transition from periodic to non-periodic signal modulation, and

that making this transition in principle allows spectral efficiency to be dramatically

increased beyond what is usually thought to be possible.

“Periodic signals” are those that return regularly to the same sequence of amplitude

values. This is equivalent to requiring that the frequency domain for the signal does

not vary with time. Which in turn is equivalent to the statement that the frequency

domain is constructed from frequency components (most simply but not necessarily

sinusoidals) which have constant scaling coefficients.

Of course, all real signals are non-periodic, particularly at the boundary between

symbols. However, as we shall see below, Shannon’s foundational proof of the

Shannon-Hartley channel capacity law made an implicit simplifying assumption that

signals are periodic, an assumption carried through into later work in channel

capacity theory.

The assumption of periodicity is a reasonable approximation for traditional signals

whose symbol waveforms are each composed of sinusoidals with constant scaling

coefficients. However, spirals are constructed from sinusoidals with continuously-varying coefficients, and are therefore non-periodic on an instant-by-instant basis.

An assumption of periodicity is therefore a very poor match for spiral-based signal

modulation.

Astrapi has proved that if the assumption of periodicity is removed, then nonperiodic channel capacity can in principle be dramatically higher than the limit

specified by the Shannon-Hartley law.2 However, the proof requires new and

unfamiliar non-periodic mathematical machinery (the generalized Euler’s formula

and spiral-based polynomial decomposition). This can pose a barrier for some

readers.

There is also an existing literature on exceeding the classical Shannon limit in the

presence of nonlinearity, notably with the recent introduction of regenerative

mapping.3 However, nonlinear channel capacity theory is unfamiliar to most

practitioners in the telecommunications industry.4

The remainder of this paper is intended to bring the discussion onto ground more

familiar to a broader technical audience, which is the classical channel capacity

literature. We seek to establish a modest but important point: that, if we examine

what has actually been proven, rather than what is widely believed to have been

proven, there is nothing in classical channel capacity theory inconsistent with the

idea that non-periodic channel capacity can in principle be dramatically higher than

the limit specified by the Shannon-Hartley law.

4.1 What Shannon Proved (And Did Not Prove)

The foundational work for communication theory was provided by Claude Shannon

in two papers: the first, “A mathematical theory of communication”5, in which he introduced the formal framework for describing information transfer; and the

second, “Communication in the presence of noise”6 , in which he proved what we

now call “Shannon’s law”, or the “Shannon-Hartley law”, for the information

capacity of a bandlimited noisy channel. For current purposes, the second of these is

the critical paper, which we will examine below.

Reduced to its basics, Shannon established proofs for two key questions

- What is the maximum rate at which independent amplitude values can be

emitted from a transmitter, given an available bandwidth*B*? Answer: the

Nyquist rate of 𝑓𝑁 = 2𝐵. This fact is known as the “Sampling Theorem”. - Given a sufficiently long sequence, how many bits can be encoded per

amplitude value, assuming signal power S and noise power N? Answer:

log2(√1 + S/N) .

Multiplying these two results together, the maximum rate at which information can

be transmitted is given by the “channel capacity” of

(1)

Shannon chose to express the Nyquist rate as 2𝐵, and to remove the square root

from the logarithm, giving his channel capacity formula in the more elegant (if less

transparent) form

(2)

It is important to understand, however, what mathematical tools Shannon used in

his proof of the Sampling Theorem. Every set of tools makes certain things possible,

while putting other things outside of the scope of the proof.

The essential idea behind Shannon’s proof of the Sampling Theorem is that sampling

at the Nyquist rate in the time domain is sufficient to fully specify the Fourier series

coefficients for the signal’s frequency domain, and therefore to fully determine the

signal. Sampling above the Nyquist rate therefore provides no additional

information.

However, in order for this proof to work, the frequency domain has to be stationary

(non-time varying), which is equivalent to assuming that the signal is periodic.

If the assumption of periodicity is removed, Shannon’s proof of the Sampling

Theorem simply fails, as does the Shannon-Hartley law which depends upon it. This

implies that Shannon’s proof of the Shannon-Hartley law does not necessarily apply

to signals based on non-periodic functions such as spirals.

The fact that using Fourier series coefficients, as Shannon did, assumes periodicity is

not controversial: it is mentioned in every introduction to the subject.

Shannon was no doubt aware that his channel capacity law was limited to periodic

signals, and probably thought this point too obvious to be worth mentioning.

Certainly, in the context of what was possible or useful for 1940’s communication

engineering, there was little point to discussing non-periodic signals.

Unfortunately, since it was published Shannon’s proof has been more admired than

studied, and the fact that it is limited to periodic signals has been lost from the

common understanding. This fact is not mentioned even in modern textbooks, at a

time when non-periodic signals are certainly realizable.7

It is perfectly reasonable to ask whether Shannon’s assumption of periodicity was

very limiting. While non-periodic signals are not covered by the Shannon-Hartley

law, does this leave much room in practice to increase channel capacity through

non-periodicity? Or is non-periodicity simply a messy version of periodicity which

has little new to offer?

A formal answer to this question is available, proving that non-periodicity in

principle allows channel capacity to be dramatically increased beyond the periodic

Shannon-Hartley limit.8 However, in keeping with the theme of this paper, which is

to stick with familiar mathematics, let us instead make two intuitive arguments.

The first intuitive argument that non-periodicity has something new to offer is as

follows. If sampling at the Nyquist rate is required to fully reconstruct a signal with a

stationary (periodic) frequency domain, perhaps a higher sampling rate will be

necessary to reconstruct a signal with a frequency domain which can in principle

vary at each instant in time. If so, the Shannon-Hartley law will be a poor

approximation to non-periodic channel capacity.

A second and more detailed intuitive argument arises from considering why

periodicity matters when using the Fourier series, as Shannon did in his proof of the

Sampling Theorem.

“Fourier’s trick” for moving from the time domain to the frequency domain is to

assume that the time domain of a function 𝑓(𝑡) can be represented by sinusoidals

whose frequencies are integer multiples of each other, then to use the fact that such

sinusoidals are orthogonal to each other to isolate the coefficient for each sinusoidal.

Clearly, this works fine in the simple case where 𝑓(𝑡) = 1. But what if 𝑓(𝑡) is a more

general function? For the integral of 𝑓(𝑡) cos(𝑛𝑡) sin(𝑛𝑡) to equal zero, as it must

for Fourier’s trick to work, 𝑓(𝑡) must provide exactly the same weight for values of 𝑡

where cos(𝑛𝑡) sin(𝑛𝑡) is positive as it does for values of 𝑡 where cos(𝑛𝑡) sin(𝑛𝑡) is

negative; and similarly for all other orthogonal pairs.

This constraint on the allowable functions 𝑓(𝑡) (which is equivalent to assuming the

periodicity of 𝑓(𝑡)) is obviously very restrictive. It raises a strong suspicion that more

general non-periodic functions 𝑓(𝑡) might allow a wider scope for symbol waveform

design, greater noise resistance, and higher spectral efficiency. In principle while

using the same amount of bandwidth.

Viewed from this light, Shannon’s Sampling Theorem could be seen as proving that

periodic functions are so constrained and self-correlated that a periodic analog

waveform is determined by a remarkably small number of amplitude values

(provided by sampling at the Nyquist rate).

As shown above, Shannon’s proof of the Sampling Theorem, on which the ShannonHartley law depends, does not necessarily apply to non-periodic signals. However, it is certainly possible in principle that later researchers might have resolved this limitation.

To the best of my knowledge, no researcher since Shannon has both recognized that

Shannon’s proof of the Sampling Theorem was limited to periodic signals, and

claimed to have provided a more general proof of the Sampling Theorem showing

that sampling at the Nyquist rate is sufficient to reconstruct non-periodic signals. Such an extension would be necessary to prove that the Shannon-Hartley channel

capacity law applies to non-periodic signals.

Nonetheless, more sophisticated mathematical tools have appeared since Shannon’s

foundational work, notably prolate-spheroidal functions and the Karhunen-Loeve

transform. It is reasonable to inquire whether these techniques implicitly extended

the scope of the Shannon-Hartley law to non-periodic signals. This topic is addressed

below.

4.2 Later Research

In classical channel capacity theory after Shannon, the most familiar names are

Landau, Pollak, Slepian and Wyner. This section briefly reviews their work from the

perspective of whether they extended the Sampling Theorem, and thus the

Shannon-Hartley law, to cover non-periodic signals.

It is important to observe that none of them made the claim of having done so.

Wyner, for instance, writes of Shannon’s proof that “there is no question as to the meaning and validity of the capacity formula”9, and makes no attempt to extend its

scope to non-periodic signals.10

The question at hand is therefore whether the more sophisticated mathematical

techniques they employed implicitly did extended the scope of the Shannon-Hartley

law.

In a series of papers,11 12 13 Landau, Pollak and Slepian introduced the prolate

spheroidal wave functions as a means to precisely describe time-limited and bandlimited functions, an area in which Shannon relied on the intuition of perfect bandlimiting. 14 The third paper in this sequence can be seen as a more sophisticated

approach to the Sampling Theorem than Shannon presented.

However, as the titles of their papers would suggest, Landau, Pollak and Slepian’s

techniques rely on Fourier analysis, and specifically on the ability to describe the

time domain with a stationary frequency domain, which carries forward Shannon’s

assumption of signal periodicity

In the first of these papers, Slepian and Pollak15 identified a set of functions (the prolate spheroidal wave functions) as the eigenfunctions of the Fourier transform over finite intervals. As eigenfunctions simplify many problems, this provides a very useful tool for describing functions that are limited in both time domain and frequency domain. It allows the idea of band-limiting to be discussed with considerable rigor. However, the underlying assumption of the paper is that functions possess Fourier transforms (Section II) in the sense that the time domain

can be described by a frequency domain that does not vary over time. This assumes

periodicity, and therefore anything built upon this paper also assumes periodicity.

In the second paper in the sequence, Landau and Pollak16 apply the theory of the

first paper to the study of time limited and bandlimited signals. They show that the

common understanding that a function cannot be simultaneously confined tightly in

the time and frequency domain can be described precisely in terms of the amount of

energy contained in a frequency band, and discuss applications.

In the third paper in the sequence, Landau and Pollak 17 provide a more sophisticated proof of Shannon’s Sampling Theorem, making use of the prolate spheroidal wave functions. They show that a signal can be accurately approximated by a number of basis functions equal to twice the signal’s bandwidth times the signal’s duration, which implies that sampling at the Nyquist rate is sufficient to reconstruct the signal.

However, this proof inherits the periodicity assumption of the first paper in the

sequence as described above, and therefore does not extend the Sampling Theorem

to cover non-periodic signals.

Wyner, 18 who as mentioned above makes no claim to have extended Shannon’s proof of the Sampling Theorem to non-periodic signals, inherits periodicity assumptions from both Shannon’s Sampling Theorem and Slepian and Pollak’s prolate spheroidal wave functions. His application of the Karhunen-Loeve expansion also contains an implicit periodicity assumption.

The Karhunen-Loeve Transform (KLT)19 is in a sense a generalization of the Fourier

transform. Fourier analysis seeks to resolve a signal into sinusoidals; the KLT instead

identifies a set of weighted orthogonal functions which ideally represent the signal + noise input sequence in terms of the signal + noise self-correlation matrix. However,

the KLT assumes time-invariant coefficients for its orthogonal functions, which repeats in a more general context the Fourier assumption of periodicity.20

Nor, to briefly cover two other important authors, do either Van Trees21 22 or Harmuth23 claim to extend Shannon’s Sampling Theorem to cover non-periodic

signals.

In short, to the best of our knowledge there is no researcher after Shannon who

either deliberately or implicitly extended Shannon’s proof of the Sampling Theorem

to cover non-periodic signals. In the absence of such a proof, the Shannon-Hartley

law cannot be considered to apply to non-periodic signal modulation.

The fact that periodic signal modulation has dominated both theoretical and applied

attention in telecommunications, to the extent that the potential benefits of nonperiodic signal modulation are generally unrecognized, probably largely reflects the available mathematical tools.

Certainly, the practical engineering difficulties associated with non-periodic signal

modulation are sufficient to explain why it is not already the basis for everyday

telecommunications. But that its very possibility is not familiar to otherwise wellinformed engineers and researchers reflects a deeper issue.

The Whorfian hypothesis in linguistics states that one’s language determines one’s

conception of the world. The underlying mathematical language for the telecommunications industry arises from Euler’s formula, introduced in the mid 18th

Century. Euler’s formula generates the sinusoidals used in standard signal modulation, and Fourier analysis which is used to analyze these signals. This is the

mathematical language for thinking about periodic functions.

It is true that the windowed Fourier transform, wavelets and sparse representations24 open an approach to non-periodicity. But they do not present

non-periodicity natively in its own language, as something with an inherent structure, elegance and power of its own.

Such a native mathematical language for non-periodicity arises from generalizing

Euler’s formula.25 And, consistent with the Whorfian hypothesis, it is from this new

non-periodic mathematical language that Astrapi’s spiral-based signal modulation

arose.

5. Conclusion

Astrapi’s introduction of non-periodic spiral-based signal modulation opens a new

frontier in telecommunications innovation. Fundamental problems require

fundamental solutions. The telecommunications industry faces a fundamental

problem in the exponential growth of data transmission demand. Astrapi’s

technology provides a fundamental solution through new tools for symbol

waveform design, making possible dramatic improvements in spectral efficiency.

- http://www.astrapi-corp.com/technology/
- J. Prothero (2012). The Shannon law for non-periodic channels. Astrapi Technical Report R-2012-1. Available from http://www.astrapi-corp.com/technology/white-papers/
- M. Sorokina, S. Turitsyn (2014). Regeneration limit of classical Shannon capacity. Nature Communications 5:3861 doi: 10.1038/ncomms4861. Available from http://www.nature.com/ncomms/2014/140520/ncomms4861/full/ncomms4861.html
- Nonlinearity and non-periodicity are of course distinct but related ideas. The spirals introduced by Astrapi are both. We believe that it is more useful to focus on the non-periodicity than the nonlinearity of spiral-based signal modulation, since it is natural to think of the spectral efficiency advantage that spiral-based signal modulation provides as arising from continuous non-periodic variations in the frequency domain.
- C. Shannon (1948). A mathematical theory of communication. Bell System Technical Journal 27, 623-656
- C. Shannon (1949). Communication in the presence of noise. Proceedings of the IRE 37(1), 10-21.
- See, for instance, J. Proakis & M. Salehi (2008). Digital communications. 5th Edition, McGraw-Hill 74, 365-367.
*Ibid*., 2.- A. Wyner (1966). The capacity of the band-limited Gaussian channel. Bell System Technical Journal 659-395. Available from http://www.alcatel-lucent.com/
- See also A. Wyner (1976). A note on the capacity of the band-limited Gaussian channel. Bell System Technical Journal 343-346. Available from http://www.alcatel-lucent.com/
- 1 D. Slepian & H. Pollak (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty – I. The Bell System Technical Journal 43-63. Available from http://www.alcatel-lucent.com/
- H. Landau & H. Pollak (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty – II. The Bell System Technical Journal 65-84. Available from http://www.alcatel-lucent.com/
- H. Landau & H. Pollak (1962). Prolate spheroidal wave functions, Fourier analysis and uncertainty – III. The Bell System Technical Journal 1295-1335. Available from http://www.alcatel-lucent.com/
- Slepian provided interesting comments in his later John von Neumann Lecture. D. Slepian (1983). Some comments on Fourier analysis, uncertainty and modeling. SIAM Review, 25(3) 379-393.
*Ibid*., 11*Ibid*., 12*Ibid*., 13*Ibid*., 9., 9- A very readable introduction to the KLT is provided in C. Maccone (2012). Mathematical SETI: statistics, signal processing, space missions. Springer. Maccone devoted 15 years to applying the KLT to special relativity
- And the assumption of orthogonality is itself less general that the spirals-based signals introduced by Astrapi.
- 1 H. Van Trees (2013). Detection estimation and modulation theory Part 1 – detection, estimation and filtering theory. 2nd Edition, John Wiley & Sons
- H. Van Trees (1971). Detection estimation and modulation theory part 2 – nonlinear modulation theory. John Wiley & Sons.
- H. Harmuth (1972). Transmission of information by orthogonal functions. 2nd Edition, SpringerVerlag.
- S. Mallet (2009). A wavelet tour of signal processing: the sparse way. 3rd Edition, Elsevier
- See http://www.astrapi-corp.com/technology/white-papers/