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Showing papers on "Harmonic published in 1986"


Journal ArticleDOI
TL;DR: In this article, the authors proposed a limit cycle control of both flux and torque using optimum PWM output voltage; a switching table is employed for selecting the optimum inverter output voltage vectors so as to attain as fast a torque response, as low an inverter switching frequency, and as low harmonic losses as possible.
Abstract: New quick-response and high-efficiency control of an induction motor, which is quite different from that of the field-oriented control is proposed. The most obvious differences between the two are as follows. 1) The proposed scheme is based on limit cycle control of both flux and torque using optimum PWM output voltage; a switching table is employed for selecting the optimum inverter output voltage vectors so as to attain as fast a torque response, as low an inverter switching frequency, and as low harmonic losses as possible. 2) The efficiency optimization in the steady-state operation is also considered; it can be achieved by controlling the amplitude of the flux in accordance with the torque command. To verify the feasibility of this scheme, experimentation, simulation, and comparison with field-oriented control are carried out. The results prove the excellent characteristics for torque response and efficiency, which confirm the validity of this control scheme.

3,316 citations


Journal ArticleDOI
TL;DR: Results of simulations indicate that the variances of the estimates are of the same order of magnitude as the CRB for sufficiently large data sets, and illustrate the performance in enhancing noisy artificial periodic signals.
Abstract: A new algorithm is presented for adaptive comb filtering and parametric spectral estimation of harmonic signals with additive white noise. The algorithm is composed of two cascaded parts. The first estimates the fundamental frequency and enhances the harmonic component in the input, and the second estimates the harmonic amplitudes and phases. Performance analysis provides new results for the asymptotic Cramer-Rao bound (CRB) on the parameters of harmonic signals with additive white noise. Results of simulations indicate that the variances of the estimates are of the same order of magnitude as the CRB for sufficiently large data sets, and illustrate the performance in enhancing noisy artificial periodic signals.

279 citations


Journal ArticleDOI
TL;DR: This experiment estimated the degree of mistuning required for this phenomenon to occur, for complex tones with 10 or 12 equal-amplitude components (60 dB SPL per component), in terms of a hypothetical harmonic sieve and mechanisms for the formation of perceptual streams.
Abstract: When a low harmonic in a harmonic complex tone is mistuned from its harmonic value by a sufficient amount it is heard as a separate tone, standing out from the complex as a whole. This experiment estimated the degree of mistuning required for this phenomenon to occur, for complex tones with 10 or 12 equal‐amplitude components (60 dB SPL per component). On each trial the subject was presented with a complex tone which either had all its partials at harmonic frequencies or had one partial mistuned from its harmonic frequency. The subject had to indicate whether he heard a single complex tone with one pitch or a complex tone plus a pure tone which did not ‘‘belong’’ to the complex. An adaptive procedure was used to track the degree of mistuning required to achieve a d’ value of 1. Threshold was determined for each ot the first six harmonics of each complex tone. In one set of conditions stimulus duration was held constant at 410 ms, and the fundamental frequency was either 100, 200, or 400 Hz. For most conditions the thresholds fell between 1% and 3% of the harmonic frequency, depending on the subject. However, thresholds tended to be greater for the first two harmonics of the 100‐Hz fundamental and, for some subjects, thresholds increased for the fifth and sixth harmonics. In a second set of conditions fundamental frequency was held constant at 200 Hz, and the duration was either 50, 110, 410, or 1610 ms. Thresholds increased by a factor of 3–5 as duration was decreased from 1610 ms to 50 ms. The results are discussed in terms of a hypothetical harmonic sieve and mechanisms for the formation of perceptual streams.

253 citations


Journal ArticleDOI
TL;DR: In this article, a control scheme based on the current deviation vector and satisfies both requirements was proposed, and the experimental results showed good agreement with the anticipated performance, however, in general, these requirements contradict each other.
Abstract: A high-performance current-controlled inverter must have a quick current response in transient state and low harmonic current content in steady state. However, in general, these requirements contradict each other. A novel control scheme is proposed which is based on the current deviation vector and satisfies both requirements. Experimental results showed good agreement with the anticipated performance.

239 citations


Journal ArticleDOI
TL;DR: In this article, the nonlinear efficiency for a gyrotron oscillator operating at harmonics of the cyclotron frequency has been calculated and is presented as a function of generalized parameters for the second through fifth harmonics.
Abstract: The nonlinear efficiency for a gyrotron oscillator operating at harmonics of the cyclotron frequency has been calculated and is presented as a function of generalized parameters for the second through fifth harmonics. The numerical results are valid for a wide range of operating conditions, including voltage, current, beam radius, cavity dimensions, and operating mode. Relatively high efficiencies are found even at high harmonics; the maximum transverse efficiencies for harmonics 2, 3, 4, and 5 are 0.72, 0.57, 0.45, and 0.36, respectively. The calculation of the efficiency in terms of generalized parameters allows the straightforward design and optimization of harmonic gyrotrons. The influence of the axial profile of the rf field in the gyrotron cavity on the efficiency is also investigated. Improved efficiency can be achieved with asymmetric field profiles. The implications of these results for the generation of millimeter and submillimeter wave radiation by harmonic emission are discussed.

236 citations


Journal ArticleDOI
TL;DR: In this article, a response function is derived for an infinite, uniform, one-dimensional structure which is subjected to an array of harmonic forces or moments, spaced equidistantly, and which have a constant phase or ratio between any adjacent pair.

188 citations


Proceedings Article
01 Jan 1986
TL;DR: The results prove the excellent characteristics for torque response and efficiency, which confirm the validity of this quick-response and high-efficiency control of an induction motor.
Abstract: New quick-response and high-efficiency control of an induction motor, which is quite different from that of the field-oriented control is proposed. The most obvious differences between the two are as follows. 1) The proposed scheme is based on limit cycle control of both flux and torque using optimum PWM output voltage; a switching table is employed for selecting the optimum inverter output voltage vectors so as to attain as fast a torque response, as low an inverter switching frequency, and as low harmonic losses as possible. 2) The efficiency optimization in the steady-state operation is also considered; it can be achieved by controlling the amplitude of the flux in accordance with the torque command. To verify the feasibility of this scheme, experimentation, simulation, and comparison with field-oriented control are carried out. The results prove the excellent characteristics for torque response and efficiency, which confirm the validity of this control scheme.

117 citations


Journal ArticleDOI
TL;DR: In this paper, a method was developed which uses superposition of steady-state phasor solutions at the fundamental frequency and at the most important harmonic frequencies, with nonlinear inductances represented as harmonic current sources.
Abstract: Saturation effects in transformers and shunt reactors can produce harmonics in power systems. Their magnitude can sometimes be found with an electromagnetic transients program, by going from an approximate linear ac steady-state solution directly into a transient simulation in which the nonlinear effects are included. In lightly damped systems, such simulations can take a long time, however, before the distorted steady state is reached. Therefore, another method was developed which uses superposition of steady-state phasor solutions at the fundamental frequency and at the most important harmonic frequencies, with nonlinear inductances represented as harmonic current sources. This method can either be used by itself, or as an improved initialization procedure for electromagnetic transients programs.

104 citations


Journal ArticleDOI
23 Jan 1986-Nature
TL;DR: In this paper, the pattern of solar magnetic fields in spherical harmonics for a data set of 25 years and power spectrum analysis of the harmonic coefficients reveals a strikingly resonant modal structure.
Abstract: Decomposition of the pattern of solar magnetic fields in spherical harmonics for a data set of 25 years and power spectrum analysis of the harmonic coefficients reveals a strikingly resonant modal structure. The resonance frequencies contain information on the structure of the magnetic fields in the Sun's interior.

91 citations


Journal ArticleDOI
TL;DR: Observations in excised bovine liver using pulsed focused ultrasonic beams illustrate that finite-amplitude distortion may be of considerable significance in the transmission through tissue of ultrasonic pulses during diagnostic scanning.
Abstract: The progressive development of finite-amplitude distortion of ultrasonic pulses has been investigated in excised bovine liver using pulsed focused ultrasonic beams at nominal frequencies of 2.5 and 3.5 MHz. Both the transducers and the powers used were those which may be encountered with clinical imaging equipment. Significant distortion of the waveform was observed to occur, particularly at higher powers. For example, at 2.5 MHz, with a mean input pressure (p0) of 0.58 MPa, the second harmonic in the pulse spectrum showed a maximum value 10.5 dB below the fundamental and the highest third harmonic component was 19 dB below the fundamental. These particular observations illustrate that finite-amplitude distortion may be of considerable significance in the transmission through tissue of ultrasonic pulses during diagnostic scanning.

84 citations


Proceedings ArticleDOI
07 Apr 1986
TL;DR: A new frequency domain method for determining the fundamental frequency of speech using the information contained in the short-term phase spectral whereas the previous methods were limited to the amplitude spectrum.
Abstract: A new frequency domain method for determining the fundamental frequency of speech is presented in this paper. This method uses the information contained in the short-term phase spectral whereas the previous methods were limited to the amplitude spectrum. The short-term spectrum is computed by DFT and it is interpreted as the output of a bank of band-pass overlapping filters. Harmonic components are detected by searching for sets of three contiguous filters having the same instantaneous frequency. The frequency of a detected harmonic is given by the instantaneous frequency itself. A conventional harmonic numbering algorithm is used to convert the set of detected harmonics to a value of the fundamental frequency. Preliminary results show the validity of the method.

Journal ArticleDOI
TL;DR: Whether a harmonic sieve could also determine whether a particular harmonic contributes to the phonetic quality of a vowel is examined, which suggests similar grouping criteria to those suggested for pitch may operate for the determination of first formant frequency in voiced speech.
Abstract: The harmonic sieve has been proposed as a mechanism for excluding extraneous frequency components from the estimate of the pitch of a complex sound. The experiments reported here examine whether a harmonic sieve could also determine whether a particular harmonic contributes to the phonetic quality of a vowel. Mistuning a harmonic in the first formant region of vowels from an /I/–/e/ continuum gave shifts in the phoneme boundary that could be explained by (i) phase effects for small amounts of mistuning and (ii) a harmonic sievelike grouping mechanism for larger amounts of mistuning. Similar grouping criteria to those suggested for pitch may operate for the determination of first formant frequency in voiced speech.

Journal ArticleDOI
TL;DR: The first observations of waves at harmonics higher than the second of the electron plasma frequency were reported by the ISEE 1 spacecraft upstream of the earth's bow shock as discussed by the authors, with effective temperatures decreasing from 5 x 10 to 17th K to 10 billion K over this range.
Abstract: The first observations of waves at harmonics higher than the second of the electron plasma frequency are reported. The observations were made by the ISEE 1 spacecraft upstream of the earth's bow shock. The waves are interpreted as electromagnetic radiation at the fundamental and up to the fifth harmonic of the plasma frequency, with effective temperatures decreasing from 5 x 10 to the 17th K to 10 billion K over this range. Two models are proposed for the emission of the waves.

Journal ArticleDOI
TL;DR: In this paper, two different pulsewidth modulation (PWM) schemes for current source inverters (CSI) are described, one is based on off-line optimization of individual switching angles and requires a microprocessor for implementation and the second one uses a special subharmonic modulation and could be implemented with analog and medium-scale integration (MSI) digital circuits.
Abstract: Two different pulsewidth modulation (PWM) schemes for current source inverters (CSI) are described. The first one is based on off-line optimization of individual switching angles and requires a microprocessor for implementation and the second one uses a special subharmonic modulation and could be implemented with analog and medium-scale integration (MSI) digital circuits. When CSI's are used in ac motor drives, the optimal PWM pattern depends on the performance criteria being used, which in turn depend on the drive application. In this paper four different performance criteria are considered: 1) current or torque harmonic elimination, 2) current harmonic minimization, 3) speed ripple minimization, and 4) position error minimization. As an example a self-controlled synchronous motor (SCSM) supplied by the PWM CSI is considered. The performance of the CSI-SCSM with the optimal PWM schemes proposed herein are compared with that using a conventional 120° quasi-square wave current.

Journal ArticleDOI
TL;DR: In this article, a general expression for the harmonic mixing signal generated by an overdamped periodic potential subject to two alternating fields of force in the presence of white gaussian noise is obtained.

Journal ArticleDOI
01 Jul 1986
TL;DR: Variation of the DC link current harmonics with power factor of the load and depth of modulation under these conditions is presented and discussed.
Abstract: A general method is described for calculating the current drawn by an inverter from its DC source. It has special relevance to pulse-width modulated inverters and allows the ripple current rating of the filter capacitor to be specified with some precision. The method is for an inverter of any number of phases and uses switching functions derived from the harmonic coefficients of the inverter waveform. Experimental results from single- and 3-phase inverters demonstrate the validity of the method. Some generalised results are presented for a 3-phase inverter for the special case where the output current can be assumed to be the fundamental component only. Variation of the DC link current harmonics with power factor of the load and depth of modulation under these conditions is presented and discussed.

Proceedings ArticleDOI
23 Jun 1986
TL;DR: In this article, a new harmonic current compensator composed of a static power converter was developed, which is composed of two voltage-type inverters in a multiplex configuration, this new compensator has a number of features that make it superior to the conventional passive filter composed of an AC reactor and a capacitor.
Abstract: A new harmonic current compensator composed of a static power converter was developed. It is composed of two voltage-type inverters in a multiplex configuration, this new compensator has a number of features that make it superior to the conventional passive filter composed of an AC reactor and a capacitor. This paper describes the principle of harmonic current compensation, the capability which the converter is required to have as compensator, the compensator's features and a trial production equipment. The important factor which are essential to a compensator, such as main circuit configuration, components, control circuit, control technique, etc. were discussed. Also the test results of a compensation characteristic were presented. Excellent characteristics of the new harmonic current compensator were verified through an experiment.

Patent
03 Nov 1986
TL;DR: In this article, amplitude detection in a transmission voltage regulator (static VAR generator, SVG) is provided based on real-time conventional, or fast, Fourier analysis, data samples being used to establish the fundamental voltage amplitude and collected during a specific, frequency tracking sliding window that precedes the amplitude reading by one line voltage cycle of the transmission line.
Abstract: In order to maintain low telephone interference (IT factor) during its normal operation and to establish rapid-post-fault voltage support to preserve the integrity of electric power transmission lines in the presence of degraded system impedance and harmonic instability, amplitude detection in a transmission voltage regulator (static VAR generator, SVG) is provided based on real time conventional, or fast, Fourier analysis, data samples being used to establish the fundamental voltage amplitude and collected during a specific, frequency tracking sliding window that precedes the amplitude reading by one line voltage cycle of the transmission line. The number of samples in the window is an integer multiple of the line frequency and, in the case of conventional analysis, the pulse number of the switching power circuit is used in the SVG. The amplitude value is updated after each sampling by shifting (sliding) the one cycle sample data base in a first in - first out manner.

Journal ArticleDOI
TL;DR: In this article, the characteristics of quasi-periodic (QP) VLF emissions observed near the equatorial plane by the GEOS satellites are first compared with simultaneous ground data obtained at L ≃ 6.
Abstract: The characteristics of quasi-periodic (QP) VLF emissions observed near the equatorial plane by the GEOS satellites are first compared with simultaneous ground data obtained at L ≃ 6. The frequency structure of QP elements is shown to be the same in the magnetosphere and at ground, whereas their amplitude measurement gives evidence of a ≃20-dB attenuation between GEOS and the ground. Different events are described as a function of their more or less complex relationships with concomitant Pc 3–4 magnetic pulsations. The spectral analysis of the simultaneous QP envelopes and ULF waves permits us to deduce that QP emissions are due to the modulation of a VLF hiss by a compressional component of ULF waves. This component is a harmonic of an ULF standing wave, showing that the classification between type I and type II is not so obvious in space. When QP envelope and ULF periodicities are different, they are shown to correspond to different harmonics of a same fundamental oscillation of the field line. This set of new results is discussed in the light of the quasi-linear theory of wave-particle interactions. The modulation rate of nondispersive QP type can be roughly well explained, and some ideas are given about a relaxation mechanism which is able to interpret the frequency modulation of the QP type rising tone.

Book ChapterDOI
01 Jan 1986
TL;DR: In this article, it was shown that the maximum reduction that can be achieved in the potential energy of the sound field is critically dependent on the relative locations of primary and secondary sources.
Abstract: The total time averaged acoustic potential energy in a steady harmonic sound field can be expressed as a positive definite quadratic function of the complex strengths of a number of “secondary sources” of sound introduced into the enclosure. For a given number and location of secondary sources, there is a unique set of complex source strengths which minimises this potential energy. This analysis is applied to the case of a lightly damped enclosure excited by a point primary source at a frequency well above the Schroeder large room frequency. It is demonstrated that in this case, the maximum reduction that can be achieved in the potential energy of the sound field is critically dependent on the relative locations of primary and secondary sources.

PatentDOI
TL;DR: In this article, a transmission from a hydroacoustic source, especially for underwater geophysical exploration, is generated in accordance with two binary codes (provided by two pulse trains or sequences having two levels and which provide a spectrum of harmonically related components).
Abstract: A transmission from a hydroacoustic source, especially for underwater geophysical exploration, is generated in accordance with two binary codes (provided by two pulse trains or sequences having two levels and which provide a spectrum of harmonically related components). One of these codes has one more or one less harmonic component than the other. The transmission is provided with a frequency spectrum which extends between an upper and lower frequency limit by sweeping the pulse trains over an octave (2:1) frequency range, far shorter than the range of the spectrum, and utilizing one of the codes and then the other of the codes during the sweep. The spectrum of the transmission extends between two arbitrary frequencies which need not be harmonically related, notwithstanding that the sweep is limited to an octave in frequency range. By adjusting the rate at which the pulse train which defines the first code sweeps with respect to the rate at which the pulse train which defines the second code sweeps, the energy of the spectrum will be smooth and continuous, thereby avoiding introducing undesirable undulations in the autocorrelation function of the transmission.

Journal ArticleDOI
TL;DR: In this paper, optical second harmonic generation offers a unique method of probing the structure of the buried metal-semiconductor interface and unambiguously identifies the interface as the source of the second harmonic signal, which persists through at least 30 A of gold.

Journal ArticleDOI
TL;DR: In this article, the second-harmonic signals of the Lorentzian absorption line at the line center were calculated for various modulation waveforms and the suppression efficiency of the fine-pitch etalon fringe signals by those waveforms was discussed.
Abstract: Second-harmonic signals of the Lorentzian absorption line at the line center are calculated for various modulation waveforms. The suppression efficiency of fine-pitch etalon fringe signals by those modulation waveforms is discussed. It is found that there are several modulation waveforms that give larger absorption signals and suppress fine-pitch fringe signals more efficiently than does the sinusoidal waveform. The dependence of the efficiency of the fringe-signal suppression on different jitter waveforms is also calculated.

Patent
17 Jun 1986
TL;DR: In this paper, the authors proposed a higher harmonic wave generator, which consists of a first optical waveguide made of a nonlinear optical material, having a structure which can guide the incident fundamental wave; and a second optical waveguaranteed to be connected to the first waveguide optically.
Abstract: A higher harmonic wave generator, comprising: (1) a first optical waveguide made of a nonlinear optical material, having a structure which can guide the incident fundamental wave; and (2) a second optical waveguide made of nonlinear optical material, connected to the first optical wave guide optically, having a structure which can guide a higher harmonic wave of the fundamental wave; wherein the effective refractive index of the first optical waveguide is made to be equal to the effective refractive index of the second optical waveguide.

Proceedings ArticleDOI
03 Nov 1986
TL;DR: In this article, the use of periodic modulation of lithium niobate waveguide properties to achieve phase-matching for second harmonic generation has been investigated both theoretically and experimentally, and a compact expression for the conversion efficiency has been derived for the case when simultaneous modulation of both the linear and nonlinear properties are taken into account.
Abstract: Use of periodic modulation of lithium niobate waveguide properties to achieve phase-matching for second harmonic generation has been investigated both theoretically and experimentally. The mathematical description has been reduced to the same form of differential equations as for the unperturbed case, but with a modified coupling coefficient. A compact expression for the conversion efficiency has been derived for the case when simultaneous modulation of both the linear and nonlinear properties are taken into account. For the case of modulation of only the linear properties, second harmonic light has been obtained experimentally.

Journal ArticleDOI
TL;DR: Mise en evidence de l'accrochage de mode dans le cas of commande impulsionnelle, lorsque the relaxation of l'ODC entre les impulsions est suffisante.
Abstract: We study how a deformable charge-density wave in the presence of random impurities responds to driving fields with both ac and dc components. Features in the differential resistivity are found when an internal frequency proportional to the velocity is either a harmonic or subharmonic of the driving frequency. Even infinitely large systems can exhibit true mode locking to the driving frequency. The behavior cannot be described by an equation of motion for a single effective degree of freedom.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss the construction and operation of a superconducting microwave cavity parametric converter suitable for the measurement of small harmonic displacements, such as the displacement of one endwall of a piezoelectric ceramic and of order δx = 3 × 10−17 cm.
Abstract: We discuss the construction and operation of a superconducting microwave cavity parametric converter suitable for the measurement of small harmonic displacements. Displacements of one endwall of the cavity produced by a piezoelectric ceramic and of order δx = 3 × 10−17 cm have been detected. Such a device is presently used to search for anomalous long range interactions of a beam of stored high energy particles.

Journal ArticleDOI
TL;DR: In this article, the authors derived an equation relating the change in resonant frequency resulting from the nonlinearity to the current through the crystal, independent of the external circuitry, which is employed in the evaluation of the coefficient of nonlinear resonance for SC-cut quartz from measurements of the shift in resonance frequency with current level in contoured resonators.
Abstract: Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of nonlinear resonance in doubly rotated contoured‐quartz resonators and all terms present in the general anisotropic case are included. Since the modes of motion in contoured‐quartz resonators are essentially thickness modes varying slowly along the plate, only the thickness dependence of the elastic nonlinearities are retained in the equations, as in earlier work. The linear portions of the equations are the same as those that have recently been derived and employed in the analysis of doubly rotated contoured‐quartz resonators. The steady‐state solutions are obtained by means of an asymptotic iterative procedure and an expansion in the linear eigensolutions while retaining the nonlinear correction to the eigensolution that has a resonant frequency in the vicinity of the driving frequency. The slow variations in the mode along the plate are included in the nonlinear correction by averaging over the plate. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance and relate the amplitude of the mode nonlinearly to the voltage across the electrodes, are obtained. In each instance the expression for the current through the crystal is determined, the external circuitry is incorporated in the description and an equation relating the mode amplitude nonlinearly to the driving voltage and other circuit parameters is obtained. The analysis holds for the fundamental and odd harmonic overtones. Nonlinear resonance curves are calculated for AT‐cut quartz using the known nonlinear coefficients. In particular, it is shown that the order of the harmonic has a more significant influence on the shift in resonant frequency from the linear value than the current through the crystal. An equation relating the change in resonant frequency resulting from the nonlinearity to the current through the crystal, independent of the external circuitry, is derived. This latter equation is employed in the evaluation of the coefficient of nonlinear resonance for SC‐cut quartz from measurements of the shift in resonant frequency with current level in contoured resonators.

Journal ArticleDOI
TL;DR: In this paper, the beam-steering grid is a programmable reflector, where the diode bias controls the phase shift of the reflection, and the variation of the phase across the grating sets the direction of the reflected beam.
Abstract: Loading a grid with diodes offers the possibility of two-dimensional control of millimeter waves that is analogous to holography and nonlinear optics. These grids are attractive because they are suitable for monolithic integration with gallium-arsenide Schottky diodes and for high-power operation. Here we present grid designs for electronic beam-steering and harmonic generation. The beam-steering grid is a programmable reflector, where the diode bias controls the phase shift of the reflection. The variation of the phase across the grating sets the direction of the reflected beam. The reflection loss in computer simulations is 3dB at 90GHz. The harmonic-generating grid acts as a nonlinear reactive surface, where the nonlinear capacitance of the diodes produces the harmonic frequencies. Quasioptical filters select the desired harmonic. Computer simulations predict that a 65GHz-to-130GHz doubler would have an output power of 0.56W/cm2 and a conversion efficiency of 35%.

Journal ArticleDOI
TL;DR: The right-hand side of the differential equations is modified in such a way that the exact solution of the nonlinear initial-value problem is known, so that the truncation error of the numerical scheme can be studied in detail.
Abstract: The implementation of a technique for locating programming errors in shallow-water codes, establishing the correctness of the code, and assessing the performance of the numerical model under various flow conditions is described. The right-hand side of the differential equations is modified in such a way that the exact solution of the nonlinear initial-value problem is known, so that the truncation errors of the numerical scheme can be studied in detail. The exact solution is prescribed to be any linear combination of Hough harmonics which propagate in time according to their natural frequencies.