Nonlinearities and Interference—Their Importance for the Study of Information ^†

Abstract

Mixing and interference are respectively the nonlinear and the linear part of combining wave frequencies and it is proposed here that they, together with other processes of wave dynamics, should be considered a lot more in the study of information. From a perspective of physical information—and it is proposed here that basic information is physical—nonlinear mixing processes are important since they first allow the generation of novelty (new frequencies). Second, mixing as in modulation and heterodyning is interesting in that it gives a hint on how similarity and relatedness are represented in wave dynamics. Third, mixing and other processes enabled by nonlinear properties of wave carrying media in balance with the natural dispersive properties of waves can lead to the emergence of solitary waves and localized structures, both of which are genuine wave objects with particle-like properties and interesting behavior, interacting with the matter of their medium and possibly each other. The strongly technical language of the field is translated to arrive at a description as pictorial as the current state of trusted experimental observations allows, based on a thorough literature research.

1. Introduction

Why is heterodyning important for the study of information, especially for information in physical form and what is heterodyning exactly? It is a process of superposition [1] plus nonlinear wave frequency mixing. Spectra of different frequencies can be generated with a stable pattern of amplitude for each frequency in waves. In the study of information, the question how a difference can make a difference in physical systems is important [2], and to approach the answer from an evolutionary perspective [3], recollecting our understanding of basic organizational aspects of energy-matter [4] in science (and engineering) is constructive. Spatio-temporally stable amplitude patterns for frequencies are the result of a linear process of superposing waves and interference of identical frequency (monochromatic) waves. Interference can be observed as the intensity pattern for one frequency at a certain place where the wave hits an obstacle which can map the local amplitude into local intensity on, e.g., a screen or a photographic film, either through absorption or reflection. A screen simply reflects and represents the intensity of incident light while a photographic film chemically reacts in proportion to the incident radiation. If waves with different frequencies are superposed or even if waves with identical frequencies but with a strongly fluctuating local phase relationship are superposed, the incoherence between periods renders impossible a locally stable amplitude profile. In heterodyning mixing is applied to shift a signal in frequency bandwidth.

2. Lawful Generation of Novelties from Nonlinear Mixing in Wave Dynamics

Like heterodyning, nonlinear mixing produces new frequencies (Since in audio engineering the term “mixing” is often used for linear addition of signals as opposite to its common use in radio engineering for nonlinear multiplication of signals in a nonlinear device, with deliberately producing new frequencies, mixing is used here with the adjective “nonlinear” to avoid confusion with the audio engineer´s definition by which generation of new frequencies is not included). Nonlinear wave interactions, which are always facilitated by a nonlinear medium through which the waves travel, can indeed make waves which have new frequencies and thereby lead to new properties (such as different energy or period) of waves when interacting with matter. Nonlinearities in media as “simple” facilitators of nonlinear mixing of waves such as in electromagnetic field waves (comprising e.g., radio waves, visible light waves) as well as in matter waves (including fluids such as water), and sound waves—to name just two types of common media—are essential for evolution, it is claimed. To better visualize the mathematics behind the difference made by the general influence of nonlinear media compared to purely linear interference, the example of audible beating when synchronously sounding two tuning forks slightly different in tune is chosen. It is chosen because the sound waves of tuning forks can be easily modeled as simple sinusoid waves. The transition from the interference-born result of linear superposition and beating in intensity to the occurrence of new frequencies due to a nonlinear process is discussed and a connection to the nonlinear processes in heterodyning and modulation of amplitude is built (Other types of modulation such as angular modulation which comprises phase and frequency modulation are not discussed, given the limitations of an extended abstract). There exists a common mathematical basis for modulation of amplitude and heterodyning, namely the product-to-sum and sum-to-product formulae (also called the prosthaphaeresis formulae [5]) derived from simple trigonometric identities. For mathematical reasons, any observable (real) part of a cyclic process with a certain frequency has to be considered in the domain of complex numbers, i.e., with its conjugate imaginary part. This results in the fact that frequencies always come with added negative frequency partners. When using complex notation, the sum and difference formulas for sine and cosine fall out as a consequence. The derivation is nicely shown, e.g., by Feynman in Lecture 48 about beats [6]. When in our example of two slightly mistuned tuning forks two waves of equal amplitude with different frequencies f₁ = a and f₂ = b are superposed and start without initial phase difference, due to their difference in frequency, one source is shifting its phase relative to the other at a uniform rate leading to alternating constructive and destructive interference and thereby oscillations in amplitude. If the difference between a and b is only marginal, amplitude oscillation will itself happen in a regular pattern, i.e., with a stable frequency. This phenomenon is known as beating. Like logarithms, the prosthaphaeresis formulae relate addition and multiplication. Thus, applied to wave superposition, they show the relation of the linear superposition to multiplication between frequencies. In the presentation, the meaning of these formulae for scaling has been explored. Here, relations between the formulae are explored in applications. Beating is usually described applying the Simpson formulae of prosthaphaeresis (in many sources the trigonometric identities applied in calculating sum-and difference frequencies are not given names hindering differentiation between them; therefore, here, the identifiers from [5] are adopted: Sum-to-product- are called the Simpson formulae and product-to-sum- are called the Werner formulae representation of prosthaphaeresis). For the addition of cosines, the Simpson formula is:

$$\cos \mathit{a}+\cos \mathit{b}=2\cos \frac{\left(a+b\right)}{2}\frac{\left(a-b\right)}{2}$$

(1)

Applied to the example, the left side of the equation describes the linear addition between the two different frequencies; the right side describes processes which occur with frequencies which are the average (half the sum) and half the difference of frequencies a and b. In linear beating (Figure 1), the difference frequency is actually perceivable.

In heterodyning, the dispositional difference frequency and the sum frequency are actually generated by using a nonlinear device (called a beat frequency oscillator (BFO)). As an input, the device is given a carrier signal which had been amplitude-modulated (for details on amplitude modulation of radio frequencies see below), for a simple example by a Morse signal, and received from a radio station. The frequency of the carrier signal is above the human auditory threshold and needs to be shifted into the audible range. The BFO superposes the input with its own local oscillator frequency, which, like in linear beating, is very similar to the other signal. From the two similar added frequencies by the help of the nonlinearity characterizing the device, a signal at the difference between the two input signals is generated which is the desired low frequency signal in audio range (if the device was set up correctly in relation to the input frequency). The sum frequency signal is removed by a filter and not used.

The Simpson formulae thus in both cases (beating and heterodyning) consist of three different types of signals: On the side of the sum are the two signals very similar to each other in their frequency. On the side of the products, there is the sum and the difference between the two inputs multiplied with each other. The product side corresponds to one relatively high (average) frequency being modulated in its amplitude by a relatively low frequency (given the relative similarity between the input frequencies).

Amplitude modulation of radio frequencies on the other hand is usually done based on a formula which is derived from the other representation of prosthaphaeresis formulae, namely the Werner formulae (The correct form of the formulae applied for nonlinear frequency mixing is a bit different from the Werner formulae for practical reasons: the signal in time s(t) = A_a[1 + M cos b]cos a; M is modulation index; A_a is amplitude of the carrier (here a). The mathematical basis is the same). For the multiplication of cosines, the Werner formula is:

cos a cos b = ½ cos (a + b) + ½ cos (a − b)

(2)

The two frequencies which are nonlinearly mixed together are usually strongly different. Very often, a is of relatively high frequency and used as a carrier to send the signal over large distances to another radio station or to radio receivers. This carrier wave is modulated by b, a wave of relatively low frequency (actually in radio communication, signals usually do not consist of a single frequency but of a bunch of waves with frequencies reaching over a large spectrum which covers a certain bandwidth, but for clarity, the simplified picture of a single sinusoid is maintained here) which is representing the message. One possibility for this is that the modulating wave consists of an oscillating voltage, e.g., in case of a Morse code, a voltage which is turned on and off in a pattern. On the other side of the equal sign, we find again sum and difference frequencies of the inputs, which are called the sideband frequencies (upper (a + b) and lower (a − b) sideband) in amplitude modulation. Compared to the carrier and the modulating signal, the sidebands, which are grouped around the carrier frequency in a distance given by the modulating frequency, are similar to each other. Like in the case of the Simpson formulae, in the Werner formulae, three different types of signals can be distinguished: on the side of the products, there are two largely different inputs which are multiplied with each other. On the side of the sum, the sideband frequencies are the signals which are comparatively similar to each other regarding their frequency.

If one compares the Simpson and the Werner formulae of prosthaphaeresis, two things can be seen. First, it is obvious that indeed linear interference in beating and nonlinear generation of sum and difference frequencies in heterodyning are related to amplitude modulation (This is especially true since the two types of prosthaphaeresis formulae can easily be transformed into each other (see, for example, [6])). Second, it seems that nonlinearities in the medium where the wave superposition is occurring decide on the actual distribution of energy from superposed waves into waves which are generated based on the mathematically disposed possibilities. Both aspects might give a hint that different frequencies which do not seem related to each other by a simple harmonic relationship or classical similarity could nevertheless become related by the mutual generation of “sideband”-like mirror pairs around the higher of the two input frequencies in a distance of the lower input frequency. Maybe this is a type of coupling possible for defining relations exclusively in wave dynamics.

The Fourier transform of the beat pattern in the time domain only shows peaks at the two original frequencies f₁ and f₂ in frequency domain, thus the acoustic signal which can be heard is not a combination tone but an aural impression of the mathematically disposed oscillation frequencies. The location of the nonlinear medium, respectively the location where the mixing occurs (it has also been hypothesized as being a purely psychoacoustic phenomenon happening in the neuronal signal processing in the listener’s brain [8]), has been a matter of debate that has lasted centuries long [8]. The mathematical function describing the mixing in the nonlinear medium, the so-called transfer function, needs to be of quadratic order or higher, that is, it must contain exponents that are at least quadratic. It is important to keep in mind, though, that the discussion of sum and difference frequencies only touched on one aspect of mixing products of nonlinear media when frequencies are combined. Other products comprise integer multiples of the input frequencies, called harmonics and sum and differences between different harmonics, as well as mixtures between harmonics, harmonic combination frequencies, and combination frequencies of the fundamental frequencies and harmonics. Higher order mixing products are called intermodulation frequencies and their generation cannot be visualized but only calculated using convolution. Sum and difference frequencies together with higher order harmonics of the fundamental frequency and intermodulation products are the product of nonlinear frequency mixing in more complex mixing processes and are strongly medium-dependent.

3. Other Forms of Important Influences on Dynamics from Nonlinearities in Media

A soliton cannot be viewed separate from its medium; it exists, so to speak, context-dependently. Solitons became known by an observation of John Scott Russell (1808–1882) in water (Union Canal system). The narrative of him observing the generation of a wave by a canal boat pulled by horses stopping suddenly, realizing the special character of the wave and then following its course over more than a mile on horseback is known widely. When nonlinear media enable excitation of localized pulses, or “wavepackets”, they somehow counteract dispersion or diffraction, eliminating phase differences between the components. “Particle-like” behavior gave the phenomenon the name soliton. Localized pulses or “wavepackets” in general maintain their identities even when they undergo collisions with each other. The understanding of the structural basis for the occurrence of solitons took a long time. Korteweg and de Vries realized that the phenomenon described by Russell required an unusually large amplitude and that the medium’s nonlinearity must be grounded in a behavior that is different to waves of different amplitudes [9]. As in the examples of nonlinear mixing and modulation, the nonlinearity seems to play a role as non-biotic selective element on wave dynamics. A balance between dispersion and nonlinear differential acceleration or enhancement (both acceleration and enhancement are meant to be understood in a neutral, analytical way, i.e., positive as well as negative) of vibrational modes gives the solitary wave its long-term stability of form. Similar to in the nonlinear devices of a heterodyning or amplitude modulation process (mainly inside diodes and transistors), nonlinearities also define certain thresholds for changes of states.

4. Conclusions

The interesting question from an evolutionary perspective is how can mechanisms which “handle” waves of different amplitude or frequency in different ways be envisioned? How can mathematically representable nonlinearities be physically acting in structures? A lot has yet to be discovered, preferably in an interdisciplinary scientific research venture. In a way, it can be said that nonlinear mixing between periodic frequencies is among the, if not the, most basic process in physical evolution of organizational aspects of energy-matter in which a dynamical whole can be generated where ‘the whole is more than the sum of its parts’, but with the concession that often, the original “parts” cannot be recovered from the addition as soon as it develops into a nonlinear mixing. The whole process is possible because selective, i.e., differential, enhancement of amplitude into different possible frequencies is re-distributing amplitude from linearly superposed similar (beating) or harmonically related frequencies into otherwise mathematically related frequencies, making them perceivable. Nonlinearity which originates in the medium based on “own” structural dispositions can cause differential amplification of and redistribution between amplitudes.