Similar to the LTE modulation in Chapter 12, modulation of a periodic impulse can lead to complex waveforms.
Find behaviors that are periodic impulse trains that are modulated by another periodic process, leading to another waveform with strong subbands.
ChatGPT: The description you provided suggests a phenomenon where a periodic impulse train, a series of regular, sharp signals, is modulated by another periodic process, leading to a composite waveform characterized by pronounced subbands. This can be observed in various fields, including signal processing, telecommunications, and even in natural systems. Here are a few examples where such behaviors might be observed:
Frequency Modulation (FM) in Telecommunications: In FM, the frequency of a carrier wave is varied in accordance with the amplitude of a modulating signal. If the modulating signal is a periodic impulse train, the resulting FM signal will exhibit a spectrum with sidebands at multiples of the modulating frequency. These sidebands can be viewed as subbands in the spectrum, resulting from the modulation process.
Amplitude Modulation (AM) in Radio Broadcasting: In AM, the amplitude of a carrier signal is varied in proportion to that of a modulating signal, which could be a periodic impulse train. This modulation results in a waveform where the carrier frequency is accompanied by sidebands (subbands) at frequencies equal to the sum and difference of the carrier frequency and the modulating frequencies.
Biological Rhythms: In biological systems, certain rhythms or signals can modulate others. For example, the circadian rhythm (a roughly 24-hour cycle) can modulate other physiological processes, leading to patterns that have periodic qualities modulated by the circadian rhythm, resulting in complex waveform behaviors with distinguishable sub-patterns or subbands.
Vibration Analysis in Mechanical Systems: In mechanical systems, a periodic impulse (such as that from a rotating imbalance) can modulate the vibration signatures of other components of the system. This can lead to complex vibration signatures that contain subbands corresponding to the modulating frequencies.
Electroencephalography (EEG) in Neuroscience: In EEG signals, certain brain wave patterns can be modulated by underlying rhythmic processes, such as the sleep-wake cycle. This can result in EEG waveforms where the primary rhythms are modulated, leading to complex patterns with distinct frequency bands.
Optical Communications: In optical fiber communications, a technique known as Wavelength Division Multiplexing (WDM) can be seen as modulating multiple light signals (each a periodic process) onto a single fiber. The combined signal exhibits a spectrum with distinct subbands, each corresponding to one of the modulated signals.
These examples illustrate the diversity of systems and contexts in which periodic impulse trains modulated by another periodic process can lead to complex waveforms with pronounced subbands, each reflecting the interplay between the modulating signals and the base periodic impulses.
The illustration above showcases vibration analysis in mechanical systems, depicting a complex waveform resulting from the modulation of a periodic impulse train by another periodic process. It includes a spectral analysis graph, highlighting the pronounced subbands within the waveform, and a representation of a mechanical system, such as a rotating shaft with an imbalance, along with the corresponding vibration signal being analyzed. This image is designed to be detailed and technical, suitable for educational purposes in the field of engineering.
More ChatGPT: The spectral domain representation below illustrates the result of a periodic impulse train, such as that from a rotating imbalance, being modulated by another vibrating periodic process. In this simulation, the periodic impulse train has a frequency of 50 Hz, and it is modulated by a lower frequency signal at 2 Hz.
The Fourier Transform of the modulated signal reveals pronounced subbands around the base frequency of the impulse train and its harmonics. These subbands are a direct consequence of the modulation process, where the amplitude of the impulse train is modulated by the lower frequency signal, leading to the creation of sidebands in the frequency domain. This graphical representation captures the complex interplay between the periodic impulse train and the modulating signal, resulting in a composite waveform characterized by these pronounced subbands.
This is a good recent citation : He, G., Ding, K., Li, W. and Li, Y. (2017), Frequency response model and mechanism for wind turbine planetary gear train vibration analysis. IET Renewable Power Generation, 11: 425-432. https://doi.org/10.1049/iet-rpg.2016.0236
Frequency components of complicated asymmetric modulation sidebands, existing in the vibration of the healthy planetary gear train, are prone to be erratically diagnosed as fault characteristics, which leads to difficulties in fault diagnosis of wind turbine planetary gearbox. The factors affecting the modulation sideband, i.e. the periodical time-varying transmission path and meshing force direction, are analysed. Considering both the meshing vibrations of the planet–ring and planet–sun gear pairs, a mathematical model was developed to analyse the planetary gear train’s vibration response. Simulation and experiments were conducted, and the mechanism of vibration modulation sidebands was revealed. The modulation sideband is not caused by the meshing vibration itself, but by the testing method that sensors are fixed on the ring gear or gearbox casing. The frequency components and amplitudes of the sidebands are determined by the tooth number of the ring gear and sun gear, the number of planet gears and their initial assembling phases. The asymmetric modulation sideband is mainly caused by the phase difference of the initial planets’ assembling phase.
Abstract: Frequency response model and mechanism for wind turbine planetary gear train vibration analysis
Something I learned early on in my research career is that complicated frequency spectra can be generated from simple repeating structures. Consider the spatial frequency spectra produced as a diffraction pattern produced from a crystal lattice. Below is a reflected electron diffraction pattern of a reconstructed hexagonally reconstructed surface of a silicon (Si) single crystal with a lead (Pb) adlayer ( (a) and (b) are different alignments of the beam direction with respect to the lattice). Suffice to say, there is enough information in the patterns to be able to reverse engineer the structure of the surface as (c).
Now consider the ENSO pattern. At first glance, neither the time-series signal nor the Fourier series power spectra appear to be produced by anything periodically regular. Even so, let’s assume that the underlying pattern is tidally regular, being comprised of the expected fortnightly 13.66 day tropical/synodic cycle and the monthly 27.55 day anomalistic cycle synchronized by an annual impulse. Then the forcing power spectrum of f(t) looks like the RED trace on the left-side of the figure below, F(ω). Clearly that is not enough of a frequency spectra (a few delta spikes) necessary to make up the empirically calculated Fourier series for the ENSO data comprising ~40 intricately placed peaks between 0 and 1 cycles/year in BLUE.
click to expand
Yet, if we modulate that with an Laplace’s Tidal Equation solution functional g(f(t)) that has a G(ω) as in the yellow inset above — a cyclic modulation of amplitudes where g(x) is described by two distinct sine-waves — then the complete ENSO spectra is fleshed out in BLACK in the figure above. The effective g(x) is shown in the figure below, where a slower modulation is superimposed over a faster modulation.
So essentially what this is suggesting is that a few tidal factors modulated by two sinusoids produces enough spectral detail to easily account for the ~40 peaks in the ENSO power spectra. It can do this because a modulating sinusoid is an efficient harmonics and cross-harmonics generator, as the Taylor’s series of a sinusoid contains an effectively infinite number of power terms.
To see this process in action, consider the following three figures, which features a slider that allows one to get an intuitive feel for how the LTE modulation adds richness via harmonics in the power spectra.
Start with a mild LTE modulation and start to increase it as in the figure below. A few harmonics begin to emerge as satellites surrounding the forcing harmonics in RED.
drag slider right for less modulation and to the left for more modulation
2. Next, increase the LTE modulation so that it models the slower sinusoid — more harmonics emerge
3. Then add the faster sinusoid, to fully populate the empirically observed ENSO spectral peaks (and matching the time series).
It appears as if by magic, but this is the power of non-linear harmonic generation. Note that the peak labeled AB amongst others is derived from the original A and B as complicated satellite-cross terms, which can be accounted for by expanding all of the terms in the Taylor’s series of the sinusoids. This can be done with some difficulty, or left as is when doing the fit via solver software.
To complete the circle, it’s likely that being exposed to mind-blowing Fourier series early on makes Fourier analysis of climate data less intimidating, as one can apply all the tricks-of-the-trade, which, alas, are considered routine in other disciplines.
GeoEnergy Math 