Chirp travelling wave solutions and spectra

Abstract

Spectral components of waves having one or more properties other than phase and amplitude that vary monotonically with time at a receiver, and provide retardations or lags in the variation in proportion to the times or distances traveled from the sources of the waves to the receiver. The lags concern the property values at departure from a source and are absent in its proximity. Orthogonality of the lags to modulated information makes them useful for ranging and for separation or isolation of signals by their source distances. Lags in frequencies and wavelengths permit multiplication of capacities of physical channels. Constancy of the lagging wavelengths along the entire path from a source to the receiver enables reception through channels or media unusable at the source wavelengths, as well as imaging at wavelengths different from the illumination.

Claims

1. A receiver for obtaining a signal from electromagnetic, acoustic or other waves, the electromagnetic, acoustic or other waves arriving at the receiver from one or more sources of said waves, and the one or more sources being located at various distances from the receiver, wherein the receiver performs spectral decomposition or selection involving one or more reference quantities to construct spectral components having monotonic variation with time at the receiver in one or more time varying properties other than phase or amplitude, and the receiver varies the one or more reference quantities over time so as to induce the monotonic variation with time at a first set of rates of variation in the one or more time varying properties other than phase or amplitude, and the receiver obtains the signal from the constructed spectral components.

2. The receiver of claim 1, wherein instantaneous values of the one or more time varying properties at the receiver lag behind their values at the one or more sources in proportion to the first set of rates of variation in the one or more time varying properties and the distance of each source from the receiver.

3. The receiver of claim 2, wherein the receiver reconstructs time domain waveforms from the signal obtained from the constructed spectral components.

4. The receiver of claim 1, wherein the one or more time varying properties vary linearly.

5. The receiver of claim 1, wherein the one or more time varying properties vary exponentially.

6. The receiver of claim 1, wherein the waves are transverse, the one or more time varying properties include a plane of polarization, and a time variation in the plane of polarization is slow enough to be in effect monotonic over the various distances of the one or more sources from the receiver.

7. The receiver of claim 1, wherein the one or more time varying properties comprise frequency, wavelength or time scale, or a function thereof.

8. The receiver of claim 1, wherein the spectral components correlate at the receiver with other such spectral components having a second set of rates of variation of the one or more time varying properties.

9. The receiver of claim 7, wherein the spectral components are observable only at a first set of wavelengths in proximity to the one or more sources, are transmitted through or scattered by a physical medium, channel, object or scene en route from the one or more sources to the receiver, the physical medium, the channel, the object or the scene en route transmitting or scattering only a second set of wavelengths differing from the first set of wavelengths, wherein the spectral components travel to the receiver at the second set of wavelengths.

10. The receiver of claim 9, wherein the physical medium, the channel, the object or the scene en route bear transmission or scattering characteristics of interest at the second set of wavelengths and the spectral components arrive at the receiver bearing information of the transmission or scattering characteristics of interest at the second set of wavelengths.

11. The receiver of claim 1, wherein the spectral components are obtained using diffraction, refraction or a combination of diffraction and refraction.

12. The receiver of claim 1, wherein the spectral components are obtained using a digital transform.

13. The receiver of claim 12, wherein the spectral components are obtained using autocorrelation prior to the digital transform.

14. The receiver of claim 1, wherein the waves bear information modulated or encoded on one or more carrier frequencies.

15. A method for obtaining spectral components of electromagnetic, acoustic or other waves travelling at finite speeds from one or more sources of said waves to a receiver, the spectral components bearing one or more time varying properties, other than phase or amplitude, that vary monotonically with time at the receiver at a first set of rates of variation, the method comprising the step of: performing at the receiver spectral decomposition or selection involving one or more reference quantities.

16. The method of claim 15, wherein the step of spectral decomposition or selection comprises varying said one or more reference quantities with time.

17. The method of claim 16, wherein the waves bear information modulated or encoded on one or more carrier frequencies, the receiver includes one or more reference signals, the one or more reference quantities comprise frequencies of the one or more reference signals, and the method further comprises the step of: demodulating said modulated or encoded information using the one or more reference signals.

18. The method of claim 17, wherein the step of demodulating said modulated or encoded information using the one or more reference signals comprises “mixing” or down-translation.

19. The method of claim 17, wherein the method further comprises the steps of: locking a frequency or phase lock loop at the receiver to a locked one of said carrier frequencies to provide a provided one of said reference signals, and varying the frequency of the provided one of said reference signals from the locked one of said carrier frequencies.

20. The method of claim 19, wherein the varying the frequency step is performed using a control signal.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1a and 1b illustrate sinusoidal and chirp travelling waves, respectively, and their phase and frequency lags implied by the d'Alembertian principle of shape invariance of waves in travel.

(2) FIG. 2a is a distance-time-frequency diagram explaining the frequency lags and time dilations of the inventive chirp components as past states of the source chirp spectrum, and their occurrence as the result of the finite speed and shape invariance of travelling waves.

(3) FIG. 2b illustrates the source time-frequency plane of FIG. 2a in more detail.

(4) FIG. 3 is a schematic diagram illustrating a direct conversion radio receiver exploiting the chirp travelling wave components of the present invention.

(5) FIG. 4 is a schematic diagram illustrating a superheterodyne radio receiver exploiting the chirp travelling wave components of the present invention.

(6) FIG. 5 is a schematic diagram illustrating a heterodyne method for realizing the variation of one or more local oscillator frequencies needed in the radio receivers of FIGS. 3 and 4.

(7) FIG. 6 is a schematic diagram showing how the heterodyning method of FIG. 5 may be integrated into the radio receiver of FIG. 4.

(8) FIG. 7 is a schematic diagram showing an alternative arrangement for the radio receiver of FIG. 6.

(9) FIG. 8 is a schematic diagram illustrating a radio receiver using a phase locked loop for carrier recovery adapted to exploit the chirp travelling wave components of the present invention.

(10) FIG. 9a is a distance-frequency diagram reproduced from the Multiplexing Patent illustrating parallax in frequency domain provided by the inventive chirp solutions, and its use for separating signals by source distance, without depending on the carrier frequency, time slot, or code.

(11) FIG. 9b is a distance-frequency diagram related to FIG. 9a, illustrating phase correlation of the inventive chirp spectra with Fourier spectra of the same signals.

(12) FIG. 10a is a schematic diagram illustrating transmission of multiple signals over a common channel of fixed length and limited transmission bandwidth, using the fractional rate parameter of the inventive chirp spectral components to differentiate between the signals.

(13) FIG. 10b is a schematic diagram illustrating the use of the fractional rate parameter of the inventive chirp spectral components for characterizing a transmission channel or imaging objects at arbitrary wavelengths using waves of a fixed wavelengths from a wave source.

(14) FIG. 11 is a graph of the SSN range residuals during the 1998 earth flyby of NEAR, reproduced from Antreasian and Guinn's paper, to illustrate the delay of DSN Doppler data.

(15) FIG. 12a is a graph that compares the true approach and the delayed Doppler velocity profiles due to the DSN receiver design, according to the theory of the present invention, to explain both the positive and negative Δυ anomalies hitherto observed in various earth flybys.

(16) FIG. 12b is a related graph that explains the Δυ-adjusted post-encounter estimated trajectory and the apparent reducibility of the flyby anomalies by Δυ estimation.

DETAILED DESCRIPTION

(17) The core notions of travelling waves represented by d'Alembertian solutions, and the distinction of the travel invariance of frequency and wavelength from variation at a wave source or at a receiver, along with the origin of lags due to finite wave speeds, are explained first in Section A, with FIGS. 1a and 1b. The inventive travelling wave spectral components are described in terms of the spectral decompositions at the wave source and the receiver with FIGS. 2a and 2b in Section B.

(18) Applications of the distance information represented by the lags would include reliable ranging of distant sources without requiring a “ladder” of distant standard candles to construct the scale of distances, as in astrophysics, and multiplexing by source distances, including over air for WiFi and cellular communication, but without the power control constraints of FDMA (frequency division multiple access) or CDMA (code division), and independently of modulation or encoding.

(19) The distance information is revisited in Section D, with FIG. 9a from the Multiplexing Patent, to explain an indeterminacy of source distances with the inventive travelling wave spectral components bearing a single rate of change, and with FIG. 9b, illustrating correlation of the inventive travelling wave spectral components with simultaneously obtained Fourier components that resolves this issue. The treatment also shows that the correlation improves with the number of distinct rates of change used, and that physical access to the source for determining the lags is indeed obviated.

(20) Applications of the travel invariance obtained as a d'Alembertian characteristic include channel capacity multiplication, reception from sources transmitting at frequencies or wavelengths that may be obstructed or jammed en route, described with FIG. 10a, and imaging at arbitrary wavelengths independently of the illumination, described with FIG. 10b, in Section E.

(21) Mathematical treatment is given in Section C for exponential chirp travelling wave solutions as the simplest embodiments with continuous monotonic variation in frequency and wavelength, corresponding to constant fractional rates of change β in eq. (4). The treatment includes the generalized orthogonality and Parseval-Plancherel theorems, and establishes, with reasoning similar to Fermat's principle of optics, that the waveforms obtained would indeed always correspond to the fractional rates of change applied to reference quantities in the spectral decomposition or selection. An implication of direct interaction with sources is discussed in Section C-5 for electromagnetic and acoustic waves, relating the frequency lags to inherent independence of source and receiver clocks and the principle of scale treated in the Background. Equivalence of the optical and DFT embodiments of the Phase Gradient Patents to nonconstant binning is proved in Section C-7.

(22) Application to time varying resonant circuits or filters, and to communication in general, calls for different considerations of temporal signals as modulation would overlap the frequency variation in the chirp components. Treatment of down-translation, demodulation and phase lock loops used for carrier tracking or recovery is given in Section C-8, and shown consistent with the given explanation of the flyby anomaly, due to unintended phase lock of the DSN receiver to chirp spectral components of the downlink signal, in Section F. Properties and uses of the inventive travelling wave spectral components with continuous monotonic variation in polarization are discussed in Section G.

A. Generality of Lags in Travelling Waves

(23) FIGS. 1a and 1b are time-distance diagrams illustrating sinusoidal and chirp travelling wave solutions, respectively, that arrive at a receiver from a source S at distance r from the receiver R, and the lags that result from the invariance of their shapes and dimensions over travel. The time axis (12) marks time t increasing to the right in both figures. The instantaneous phase ϕ(t) has a leftward motion, as indicated by the horizontal arrows, as it evolves over time, in both waveforms (21) seen at the source, and (22) seen at the receiver. As the invention concerns travelling waves in general, including sound, relativistic space-time curvature is not of concern in the treatment. The invention concerns information a receiver may obtain or infer about wave sources through waves alone, hence all of its observations and inferences would be adequately represented by the receiver's clock. In particular, the time Δt=r/c required by the waves to travel from a source S′ to a receiver R, on the distance axis (11), must be accounted for by the receiver's clock in all received waves.

(24) The dotted lines (31) relate points on the source (21) and receiver waveforms (22) that represent the same positions on the respective waveforms. This correspondence, which also signifies causality, would be simple if the waveforms have definite start times, as depicted, but that is never the case for spectral components, which, as analytical constructs, extend to t=±∞. The impossibility of strictly monochromatic waves with finite start or end times is a known result of Fourier transforms. The proof, included as a lemma in the SPIE paper, lies in the equivalence of a waveform bounded in time to a product of the same waveform extending to ±∞ and a step function representing the time bounds. The Fourier spectrum of their product then becomes the convolution product of their spectra, and extends to ±∞, because the spectrum of the step function alone extends to ±∞.

(25) Transmission frequency or wavelength bands are chosen in wave guides and optical fibres, as well as in transmission lines on printed circuit boards (PCB) and in integrated circuit (IC) chips, to minimize dispersion effects. It is therefore generally safe to ignore the dispersion effects in such media in considering the present invention. The parallelism of the dotted lines (31) illustrates the resulting constancy of the wave speed for the period in consideration, as well as across wavelengths in FIG. 1b, where the wavelength A varies with time t along the chirp, so that λ.sub.3<λ.sub.2<λ.sub.1.

(26) The figures also illustrate a basic distinction between the invariance of d'Alembertian properties during travel from their evolution over time locally at the source and at the receiver, and the origin of the lags in these properties due to the constancy in travel. In the chirp waveform of FIG. 1b, its frequency and wavelength lags would vanish, notwithstanding the travel delay Δt=t.sub.2−t.sub.0, if, and only if the wavelength of its first cycle (23) decreased from λ.sub.1 at t.sub.0 to λ.sub.4 at t.sub.2 by the time it arrived as the first cycle (24) at the receiver. However, a change of wavelength in travel necessarily means that the successive wavefronts, i.e., surfaces denoting constant phase, had different speeds. For instance, the first cycle (23) would be shorter at t.sub.2, so as to match the gap between the leading (44) and trailing (43) edges of the source cycle at t.sub.2, if and only if its leading edge (42) travelled slower than its trailing edge (41), reducing their gap during travel.

(27) Such variations in speed would relate to dispersion, because the leading and trailing wavefronts belong to somewhat differing wavelengths in the chirp waveform of FIG. 1b. The common intuitions that find the distance proportional shifts of the Phase Gradient Patents impossible and violative of causality, instead themselves require assuming an inherent dispersion in nature, even for acoustic waves, to predicate the absence of such shifts in the inventive chirp travelling wave solutions. The hypothetical dispersion would have to be additionally proportional to β, because a chirp waveform with fractional rate of change β′>β needs to reduce from λ.sub.1 to λ′.sub.4<λ.sub.4 over the same travel, and the dispersion should vanish in the limit β.fwdarw.0, i.e., the prejudicial intuitions would further require the inherent dispersion to depend on the receiver's selection of β, and cannot be correct.

(28) Conversely, in absence of such hypotheses, the leading (46) and trailing (45) edges must exhibit the same separation upon arriving at the receiver as they did at the source ((42), (41) respectively), and thus yield a lag in the gap relative to the leading (44) and the trailing (43) edges at the source at the arrival time t.sub.2. Such lags are in any case known from CW-FM radars and echo-location in bats—the difference here is that the lags are obtained in chirp spectra.

B. Relation of Spectra Across Space and Time

(29) FIG. 2a is a distance, time and frequency diagram explaining how the component frequencies relate over time and between the source and the receiver locations. Lines S-S.sub.T (13) and S-S.sub.Ω (15) denote the time and frequency axes, respectively, at the source location S. Lines R-R.sub.T (14) and R-R.sub.Ω (16) denote the corresponding time and component frequency axes, respectively, at receiver location R. The horizontal plane S-S.sub.Ω-R-R.sub.Ω denotes the present time corresponding to t.sub.2 in FIG. 1b. Past frequencies of the components are shown above the plane S-S.sub.Ω-R-R.sub.Ω in FIG. 2a, consistent with the downward direction of time along the time axes S-S.sub.T (13) and R-R.sub.T (14). Each of lines J-D (63) and I-E (62) thus denotes past frequencies of the respective chirp components, whose instantaneous present frequencies, at t.sub.2, are defined by the coordinate intervals S-D and S-E, respectively. The slopes of lines J-D (63) and I-E (62) denote the rates of change of frequency, {dot over (ω)}≡dω/dt=βω.

(30) The variation of component frequencies at a fixed value of the fractional rate of change β then yields a family of exponential curves covering the (vertical) time-frequency plane S-S.sub.T-S.sub.Ω at the source, and identified by their points of intersection with the present time plane S-S.sub.Ω-R-R.sub.Ω, such as points D and E, which identify the chirp components denoted by lines J-D (63) and I-E (62). The lines J-D (63) and I-E (62) denote segments of exponential curves, and are hence not parallel. This family of curves is clearly a single valued cover for the source time-frequency plane S-S.sub.T-S.sub.Ω, i.e., exactly one member of the family passes through each point in the plane S-S.sub.T-S.sub.Ω.

(31) The dotted line N-E (61) normal to the source frequency axis S-S.sub.Ω (15) at E likewise represents a sinusoid of frequency given by the coordinate interval |SE|, which is also the frequency of the chirp line I-E (62) at time t.sub.2. The completeness of a Fourier transform concerns the coverage of the frequency axis S-S.sub.Ω (15) by the normals, i.e., by the availability of a normal through each point on the frequency axis S-S.sub.Ω (15) contained in the transform. The family of normals is again a complete single-valued cover for the source time-frequency plane S-S.sub.T-S.sub.Ω The plane S-S.sub.T-S.sub.Ω thus denotes the Hilbert space of source waveforms representable by sinusoids, or equivalently by chirps. Thus, the chirp and sinusoidal spectra can equally describe arbitrary energy or power distribution on the present source frequency axis S-S.sub.Ω (15) at time t.sub.2, and must also bear the same coefficients, since the coefficients denote the same magnitudes and phases at the present time t.sub.2.

(32) However, the inclination of chirp lines J-D (63) and I-E (62) signifies distorted reconstruction of waveforms in past or future, due to time dilations or compressions, respectively, given by eq. (1) for the positive values of β represented by the depicted inclinations. The distortion is of the time domain representation of the local past and future at the source, as travel is not explicit in eq. (1).

(33) Component waveforms for the chirp lines and for the sinusoid represented by the normal N-E (61) are shown separately in FIG. 2b, to clarify how their phases evolve over time, and to illustrate how the chirp wavelengths expand in the past. Similar expansions occur along the chirp lines G-C (65) and H-F (64) in the receiver time-frequency plane R-R.sub.T-R.sub.Ω. The reconstructed time domain waveforms would be then incompatible in their time scales except on the present time axes S-S.sub.Ω (15) and R-R.sub.Ω (16), hence each family of chirp lines, defined by a common fractional rate β (and its derivatives) is orthogonal to all other such families differing in the fractional rate (or its derivatives). Their Hilbert spaces thus intersect only at the present frequency axes, ruling out transformations between such chirp basis sets. These past spectral states are not purely conceptual, since they must relate to actual past measurements of frequency if any, and their difference of time scale could be exposed by observations at a distance or across a round trip. For example, correction were needed for the uplink oscillator drift over the round trips of tens of hours in the downlink carrier frequency measurements in the Pioneer missions, according to Anderson, Laing et al. Such corrections are not required in most terrestrial communication systems because of the shorter trip times.

(34) In FIGS. 1a and 1b, the slope of the dotted connecting lines (31) represents the finite wave speed. The lines would be vertical, and the lags would vanish, if the speed were infinite. If the wave speed were zero, the lines would be horizontal and would not connect the source and receiver waveforms, leaving them causally unrelated. Corresponding connecting lines denoting travel in FIG. 2a must be again inclined in the direction of increasing time, and drawn parallel to the distance axis (11) to represent the invariance of frequencies during travel. Travel lines A-C (51) and K-L (52) starting from points A and K, respectively, on the source chirp line A-E, are therefore shown inclined at angle ∠DAC=tan.sup.−1 (|DC|/|AD|)=tan.sup.−1(r/Δt)≡tan.sup.−1(c).

(35) Linearity of the frequency lags is illustrated by the second travel line K-L (52) from point K at time t.sub.1>t.sub.0 on the same source chirp line A-E, and reaching point L at a shorter distance r.sub.1<r, for the correspondingly shorter delay Δt.sub.1=t.sub.2−t.sub.1, by the time t.sub.2. The linearity is due to similarity of the pairs of triangles {A-C-E, K-L-E} and {C-F-E, L-M-E}. The lags must be proportional to the inclinations ∠AEN≡∠CBF≡tan.sup.−1 (|CF|/|CB|)=tan.sup.−1(βΔt/Δt)=tan.sup.−1(β).

(36) The line A-C (51) would equally represent the sinusoid given by the normals A-D at the source and B-C at the receiver, which must deliver the same numerical amplitude or coefficient F(ω.sub.1) at C, signifying the same power or information, to the receiver at time t.sub.2. The coefficients F(ω) on the receiver frequency axis (16) must represent physical observations by the receiver at t.sub.2. Whether a coefficient F(ω.sub.1) belongs to a sinusoid B-C or to a chirp line G-C depends entirely on which line the receiver chooses to construct through the point C to represent the angular frequency ω.sub.1. If it constructs a chirp line, the source frequency implied by the connecting line A-C (51) would be still ω.sub.1 at time t.sub.0 at the source, as represented by point A, but the chirp would have evolved to ω.sub.4 at the source at time t.sub.2, as represented by point E, in the receiver's analysis, to pose a frequency lag Δω=|DE|=|CF|=(ω.sub.4-ω.sub.1). Conversely, the receiver can associate this chirp line and present angular frequency ω.sub.4 at E only with the coefficient received at point C.

(37) Without shifted past states, like A for point E, distant observations would be impossible even at frequency drifts too small to detect with existing nonzero Allan deviations. As the shifts depend only on the instantaneous source frequency ω.sub.4, the fractional rate β and the travel time Δt=r/c, the received chirp spectrum and its lags would be as time invariant as the Fourier spectrum.

C. Mathematical Description of the Invention

(38) The instantaneous angular frequency of an exponential chirp is given by ω(t)=ω.sub.0e.sup.βt, where ω.sub.0=ω(0). Its instantaneous phase is ϕ(t)≡∫ω(t)dt≡β.sup.−1 ω(t)=β.sup.−1ω.sub.0e.sup.βt≈β.sup.−1ω.sub.0+ω.sub.0t+ . . . where the constant and higher order terms arise from the nonconstancy of frequency. The phase reduces to the sinusoidal form, ω.sub.0t plus a constant, in the limit β.fwdarw.0 via L'Hôpital's rule.

(39) C-1 Chirp D'Alembertian Solutions

(40) The phase of an exponential chirp travelling wave would be ψ(r, t)=−ωβ.sup.−1 exp[β(t±r/c)], with the travel delay r/c. Using Δt≡(t−r/c), the first and second order derivatives would be

(41) $\begin{array}{cc}\frac{\partial \psi \left(r,t\right)}{\partial r}\equiv \frac{\partial e^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}}{\partial r}=e^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}\frac{\partial \left\{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]\right\}}{\partial r}=\psi \left(r,t\right).Math.\left(-i{\mathrm{\omega \beta }}^{-1}\right).Math.\left(-\beta /c\right)\exp \left[\mathrm{\beta \Delta }t\right]=\frac{i\omega }{c}\exp \left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)\mathrm{whence}\frac{{\partial }^2\psi \left(r,t\right)}{\partial r^2}\equiv \frac{{\partial }^2{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}}{\partial r^2}=\frac{\partial }{\partial r}\left[\frac{i\omega }{c}\exp \left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)\right]=\frac{-i\mathrm{\omega \beta }}{c^2}\exp \left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)-\frac{{\omega }^2}{c^2}\exp \left[2\mathrm{\beta \Delta }t\right]\psi \left(r,t\right),& \left(5\right)\end{array}$
and likewise,

(42) $\begin{array}{cc}\frac{\partial \psi \left(r,t\right)}{\partial t}\equiv \frac{\partial }{\partial t}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}=e^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}\frac{\partial }{\partial t}\left\{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]\right\}=\psi \left(r,t\right).Math.\left(-i{\mathrm{\omega \beta }}^{-1}\right).Math.\left(\beta \right)\exp \left[\mathrm{\beta \Delta }t\right]=-i\mathrm{\omega exp}\left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)\mathrm{yielding}\frac{{\partial }^2\psi \left(r,t\right)}{\partial t^2}\equiv \frac{{\partial }^2}{\partial t^2}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left[\mathrm{\beta \Delta }t\right]}=\frac{\partial }{\partial t}\left[-i\mathrm{\omega exp}\left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)\right]=-i\mathrm{\omega \beta exp}\left[\mathrm{\beta \Delta }t\right]\psi \left(r,t\right)-{\omega }^2\exp \left[2\mathrm{\beta \Delta }t\right]\psi \left(r,t\right),& \left(6\right)\end{array}$
implying [∂.sup.2/∂r.sup.2−c.sup.−2∂.sup.2/∂t.sup.2]e.sup.−iωβ.sup.−1.sup.exp[βΔt]=0 identically, so that the one-dimensional wave equation [∂.sup.2/∂r.sup.2−c.sup.−2∂.sup.2/∂t.sup.2]f(r,t)=0, would be satisfied by setting f(r, t)=e.sup.−iωβ.sup.−1.sup.exp βΔt.

(43) Extension of this result to the complementary chirp wave function g(r,t)=e.sup.−iωβ.sup.−1.sup.exp β(t+r/c) and to three dimensional space would be straightforward. The result establishes that the inventive chirp wave functions e.sup.±iωβ.sup.−1.sup.exp β(t±r/c) are equally valid d'Alembertian solutions for representing waves as the sinusoidal wave functions of current physics. These exponential chirp waveforms can be easily combined with exponential decays of amplitude to obtain chirped eigenfunctions for the Laplace and z transforms. Their validity as wave solutions cannot be surprising in any case because, given any “retarded” (travel-delayed) functions of time ψ(t−r/c), the relations

(44) $\begin{array}{cc}{\psi }_{,t}\equiv \frac{\partial \psi }{\partial t}\equiv \frac{\partial \psi }{\partial \tau }\frac{\partial \tau }{\partial t}=\frac{\partial \psi }{\partial \tau },{\psi }_{,\mathrm{tt}}\equiv \frac{{\partial }^2\psi }{\partial t^2}\equiv \frac{\partial }{\partial \tau }\left(\frac{\partial \psi }{\partial \tau }\frac{\partial \tau }{\partial t}\right)\frac{\partial \tau }{\partial t}=\frac{{\partial }^2\psi }{\partial {\tau }^2},\mathrm{and}{\psi }_{,r}\equiv \frac{\partial \psi }{\partial r}\equiv \frac{\partial \psi }{\partial \tau }\frac{\partial \tau }{\partial r}=-\frac{1}{c}\frac{\partial \psi }{\partial \tau },{\psi }_{,\mathrm{rr}}\equiv \frac{{\partial }^2\psi }{\partial r^2}\equiv \frac{\partial }{\partial \tau }\left(\frac{\partial \psi }{\partial \tau }\frac{\partial \tau }{\partial r}\right)\frac{\partial \tau }{\partial r}=-\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial {\tau }^2}& \left(7\right)\end{array}$
follow upon substituting τ=t-r/c, proving that ψ satisfies the wave equation. The only conditions on the solutions are the existence of first and second order derivatives, and the travel delays r/c.

(45) Decomposition into any family of orthogonal functions, such as Bessel functions and Laguerre polynomials, should yield a similar continuum of alternative spectra with lags, but these alternative functions show relatively little variation of wavelength, and become indistinguishable from sinusoids, away from the origin, so their lags also level off and become unusable as a distance measure. Only linear or exponential chirps assure indefinitely linear shifts and are therefore preferred.

(46) C-2 Hilbert Spaces of Travelling Wave Chirp Spectra

(47) The integral of the product of a travelling wave chirp spectral component of unknown initial angular frequency ω′.sub.0 and fractional rate of change β′, arriving from a source at distance r, with a reference chirp signal of initial angular frequency ω.sub.0 and fractional rate β at the receiver, evaluates, over a time interval T longer than several cycles and with the same notation Δt≡(t−r/c), as

(48) $\begin{array}{cc}{\int }_T^{}{e}^{i{\omega }_0^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }\Delta t\right)}{e}^{-i{\omega }_0{\beta }^{-1}\exp \left(\beta t\right)}\mathrm{dt}\simeq \delta \left({\omega }_0^{\prime }{e}^{{\beta }^{\prime }\Delta t}-{\omega }_0{e}^{\beta t}\right)\equiv \delta \left({\omega }_0^{\prime }{e}^{-{\beta }^{\prime }r/c}-{\omega }_0\right)\delta \left({\beta }^{\prime }-\beta \right),& \left(8\right)\end{array}$
using Dirac's delta function, defined as ∫δ(x) dx=1 if and only if the integration includes x=0, since the phase of the integrand is constant only for a component that matches the reference in both ω.sub.0 and β, and oscillates otherwise. The matching component then contributes in proportion to the number of cycles in the interval T≈(ω.sub.0T)/2π=v.sub.0T, whereas every other component contributes at most a cycle. Eq. (8) reduces to the Fourier orthogonality theorem in the limit β=β′.fwdarw.0, at which the component of matching frequency again contributes to the integral in proportion to v.sub.0T, but any other component contributes over at most one cycle.

(49) In contrast, Fourier transform theory involves convergence issues in the integration time limit T.fwdarw.∞ because, unlike a nonmatching chirp, every nonmatching sinusoid contributes periodically, after every so many cycles, indefinitely. The convergence is thus better assured for chirps.

(50) The fractional rate derivatives β.sup.(1), β.sup.(2), . . . contribute additional factors δ(β.sup.(1)-β′.sup.(1)), δ(β.sup.(2)-β′.sup.(2)), etc., hence travelling wave spectra cover a two dimensional product space Ω× *≡{ω}×{β, β.sup.(1), . . . } of instantaneous frequencies and their rates of change. Prior art transforms, including Fourier, Laplace and z, as well as wavelets, comprise the null subspace Ω×{0,0, . . . }, in which the fractional shifts z=βr/c representing distance information vanish. The distance information is conversely limited to the nonnull subspace Ω×[*−{0, 0, . . . }] describing the present invention.

(51) Eq. (8) also results naturally by relaxing the time invariance premise in the Fourier condition,

(52) $\begin{array}{cc}{\int }_T^{}{e}^{i{\omega }^{\prime }\left(t-r/c\right)}{e}^{-i\omega t}\mathrm{dt}\equiv {\int }_T^{}{e}^{-i{\omega }^{\prime }r/c}{e}^{i\left({\omega }^{\prime }-\omega \right)t}\mathrm{dt}\simeq \delta \left({\omega }^{\prime }-\omega \right){\int }_T^{}{e}^{-i{\omega }^{\prime }r/c}\mathrm{dt},& \left(9\right)\end{array}$
as the properties of oscillation and convergence responsible for the result in eq. (8) do not depend on constancy or exponential variation of frequencies. Eq. (9) thus more generally implies that if the reference angular frequency ω is varied slowly, the product integral would be nonzero only if the component angular frequency ω′ varied the same way over time, since the phase factor e.sup.i(ω-ω′)t in the integrand would otherwise oscillate. The result holds for T.fwdarw.±∞ and also for finite integration times T that span a large number of cycles, for the same reason as explained for eq. (8), and is the general basis of design for realizing the inventive spectral components, both in general spectrometry and in the reception of modulated signals, as treated ahead.

(53) As in Fourier theory, the orthogonality itself enables spectral decomposition, via the relation

(54) $\begin{array}{cc}{\int }_T^{}\left[\frac{1}{2\pi }{\int }_{{\Omega }_{\mathrm{\mu 0}}^{\prime }\times {ℬ}^{\prime }}^{}{F}_{{\beta }^{\prime }}\left({\omega }_{\mathrm{\mu 0}}^{\prime }\right)\exp \left\{i{\omega }_{\mathrm{\mu 0}}^{\prime }\frac{e^{{\beta }^{\prime }\left[t-r/c\right]}}{{\beta }^{\prime }}\right\}d{\omega }_{\mathrm{\mu 0}}^{\prime }d{\beta }^{\prime }\right]\exp \left\{-i{\omega }_0\frac{e^{\beta t}}{\beta }\right\}\mathrm{dt}\approx \frac{1}{2\pi }{\int }_{{\Omega }_{\mathrm{\mu 0}}^{\prime }\times {ℬ}^{\prime }}^{}{F}_{{\beta }^{\prime }}\left({\omega }_{\mathrm{\mu 0}}^{\prime }\right)\left[{\int }_T^{}\exp \left\{i{\omega }_{\mathrm{\mu 0}}^{\prime }\frac{e^{{\beta }^{\prime }\left[t-r/c\right]}}{{\beta }^{\prime }}-i{\omega }_0\frac{e^{\beta t}}{\beta }\right\}\mathrm{dt}\right]d{\omega }_{\mathrm{\mu 0}}^{\prime }d{\beta }^{\prime }=\frac{1}{2\pi }{\int }_{{\Omega }_{\mathrm{\mu 0}}^{\prime }\times {ℬ}^{\prime }}^{}{F}_{{\beta }^{\prime }}\left({\omega }_{\mathrm{\mu 0}}^{\prime }\right)\delta \left({\omega }_{\mathrm{\mu 0}}^{\prime }{e}^{{\beta }^{\prime }\left(t-r/c\right)}-{\omega }_0{e}^{\beta t}\right)d{\omega }_{\mathrm{\mu 0}}^{\prime }d{\beta }^{\prime }=F_{\beta }\left({\omega }_0\right),& \left(10\right)\end{array}$
showing that the coefficient F.sub.β(ω.sub.0) of a single chirp wave component would be extracted from a linear combination of chirp wave functions

(55) $F_{{\beta }^{\prime }}\left({\omega }_{\mathrm{\mu 0}}^{\prime }\right)e^{i{\omega }_{\mathrm{\mu 0}}^{\prime }{\beta }^{\prime -1}{e}^{{\beta }^{\prime }\left[t-r/c\right]}}.$
The square brackets demarcate the inverse transform yielding a time domain function ƒ(t−r/c) reconstructed from F.sub.β′.
C-3 Parseval-Plancherel Theorems for Exponential Chirp Spectra

(56) The generalized Parseval-Plancherel theorem follows again from the L.sup.2 norm, for a delay Δt≡r/c, for a complex valued time-domain signal f and its spectral coefficients F, as

(57) 0$\begin{array}{cc}{\int }_T^{}{.Math.f\left(t-\Delta t\right).Math.}^2\mathrm{dt}\equiv {\int }_T^{}{f}^{*}\left(t-\Delta t\right)f\left(t-\Delta t\right)\mathrm{dt}={\int }_T^{}\left[{\int }_{{\Omega }_{\mu }\times \beta }^{}\frac{F_{\beta }\left({\omega }_{\mu }\right)}{2\pi }{e}^{+i{\omega }_{\mu }\exp \left(\beta \left[t-\Delta t\right]\right)/\beta }d{\omega }_{\mu }d\beta {\int }_{{\Omega }_v^{\prime }\times {\beta }^{\prime }}^{}\frac{F_{{\beta }^{\prime }}^{*}\left({\omega }_v^{\prime }\right)}{{\left(2\pi \right)}^2}{e}^{-i{\omega }_v^{\prime }\exp \left({\beta }^{\prime }\left[t-\Delta t\right]\right)/{\beta }^{\prime }}d{\omega }_v^{\prime }d{\beta }^{\prime }\right]\mathrm{dt}={\int }_T^{}\left[{\int }_{{\Omega }_{\mu }\times \beta ,{\Omega }_v^{\prime }\times {\beta }^{\prime }}^{}\frac{F_{\beta }\left({\omega }_{\mu }\right)F_{{\beta }^{\prime }}^{*}\left({\omega }_v^{\prime }\right)}{{\left(2\pi \right)}^2}{e}^{i\left\{{\omega }_{\mu }\exp \left(\beta \left[t-\Delta t\right]\right)/\beta -{\omega }_v^{\prime }\exp \left({\beta }^{\prime }\left[t-\Delta t\right]\right)/{\beta }^{\prime }\right\}}d{\omega }_{\mu }d\beta d{\omega }_v^{\prime }d{\beta }^{\prime }\right]\mathrm{dt}={\int }_{{\Omega }_{\mu }\times \beta ,{\Omega }_v^{\prime }\times {\beta }^{\prime }}^{}\frac{F_{\beta }\left({\omega }_{\mu }\right)F_{{\beta }^{\prime }}^{*}\left({\omega }_v^{\prime }\right)}{{\left(2\pi \right)}^2}\left[{\int }_T^{}{e}^{i\left\{{\omega }_{\mu }\exp \left(\beta t^{\prime }\right)/\beta -{\omega }_v^{\prime }\exp \left({\beta }^{\prime }{t}^{\prime }\right)/{\beta }^{\prime }\right\}}{\mathrm{dt}}^{\prime }\right]d{\omega }_{\mu }d\beta d{\omega }_v^{\prime }d{\beta }^{\prime }\left(t^{\prime }=t-\Delta t\right)={\int }_{{\Omega }_{\mu }\times \beta ,{\Omega }_v^{\prime }\times {\beta }^{\prime }}^{}\frac{F_{\beta }\left({\omega }_{\mu }\right)F_{{\beta }^{\prime }}^{*}\left({\omega }_v^{\prime }\right)}{{\left(2\pi \right)}^2}\delta \left({\omega }_{\mu }{e}^{\beta t^{\prime }}-{\omega }_v^{\prime }{e}^{{\beta }^{\prime }{t}^{\prime }}\right)\hspace{1em}d{\omega }_{\mu }d\beta d{\omega }_v^{\prime }d{\beta }^{\prime }={\int }_{{\Omega }_{\mu }\times \beta }^{}\frac{{.Math.F_{\beta }\left({\omega }_{\mu }\right).Math.}^2}{2\pi }d{\omega }_{\mu }d\beta ,& \left(11\right)\end{array}$
omitting the redundant suffix 0 for initial angular frequencies. The result reduces to the Parseval theorem for Fourier transforms at β=β′.fwdarw.0, implying equality of chirp and Fourier coefficients at the source, as remarked with FIG. 2a, as they describe the same oscillations at the source, i.e.,

(58) $\begin{array}{cc}F\left({\omega }_{\mu }\right)\equiv \underset{t,r.fwdarw.0}{\lim }F\left({\omega }_{\mu }\right)e^{i{\omega }_{\mu }\left(t-r/c\right)}=\underset{t,r.fwdarw.0}{\lim }{F}_{\beta }\left({\omega }_{\mathrm{\mu 0}}\right)e^{i{\omega }_{\mathrm{\mu 0}}{\beta }^{-1}\exp \left(\beta \left[t-r/c\right]\right)}\equiv F_{\beta }\left({\omega }_{\mathrm{\mu 0}}\right),& \left(12\right)\end{array}$
using the convergence of chirp phases to Fourier values by L'Hôpital's rule.

(59) Eq. (12) implies that all information modulated or encoded at the source is preserved, i.e., the time dilation represented by the frequency shifts alters the transmission rate, but not the content.

(60) Eq. (11) means that the entire source energy ∫.sub.T|f(t)|.sup.2 dt, along with the modulated information contained in the coefficients F.sub.β(ω.sub.μ), would be available identically at all delays Δt, and thus over the continuum of lags and time dilations, because Δt is absent in the coefficients and vanishes from the norm.

(61) C-4 Analytical Origin of the Inventive Frequency Lags

(62) Real atomic emissions, as well as communication signals any kind of modulation, invariably comprise a linear combination of tones

(63) $\underset{j}{\overset{}{.Math.}}{F}_j{e}^{i{\omega }_j^{\prime }t+i{\varphi }_j^{\prime }}.$
At a recover at a distance r from the source, at rest relative to the source and at the same gravitational potential, the instantaneous arriving combined signal would be

(64) $\underset{j}{\overset{}{.Math.}}{F}_j{e}^{i{\omega }_j^{\prime }\left(t-r/c\right)+i{\varphi }_j^{\prime }}.$
Its chirp transform should therefore yield

(65) ${\int }_T^{}\underset{j}{.Math.}{F}_j{e}^{i{\omega }_j^{\prime }\left(t-r/c\right)+i{\varphi }_j^{\prime }}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left(\beta t\right)}\mathrm{dt}\simeq \underset{j}{.Math.}{\int }_T^{}{F}_j{e}^{-i{\omega }_j^{\prime }r/c+i{\varphi }_j^{\prime }}\delta \left({\omega }_j^{\prime }-\omega e^{\beta t}\right)\mathrm{dt}=e^{-i{\omega }_j^{\prime }r/c+i{\varphi }_j^{\prime }}{ℱ}_{\beta }\left(\omega \right),$
wherein the component angular frequencies ω′.sub.j are unchanged by travel. {.sub.β} denotes the ordinary chirp spectrum, which would be flat over ω as each tone ω′.sub.j closely matches every chirp component .sub.β(ω)e.sup.iωβ.sup.−1.sup.exp(βt) in phase over some cycle. The inverse transform, using the kernel e.sup.+iωβ.sup.−1.sup.exp(βt), would reproduce the combination of tones, with distortion only due to the sampling noise, the finite integration time and computational precision. The phase lags e.sup.−ω′.sup.j.sup.r/c denote advancements in component phases at the source while in transit, and their rates of advance, given by the angular frequencies ω′.sub.j, is constant. A similar decomposition at the source should yield

(66) $\begin{array}{cc}{\int }_T^{}\underset{j}{.Math.}{F}_{{\beta }^{\prime }j}{e}^{i{\omega }_j^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }t\right)+i{\varphi }_j^{\prime }}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left(\beta t\right)}\mathrm{dt}\simeq \underset{j}{.Math.}{\int }_T^{}{F}_{{\beta }^{\prime }j}{e}^{i{\varphi }_j^{\prime }}\delta \left({\omega }_j^{\prime }{e}^{{\beta }^{\prime }t}-\omega e^{\beta t}\right)\mathrm{dt}.& \left(13\right)\end{array}$

(67) The frequencies continue to evolve during the integration time T, but the local evolutions during T can be ignored if T<<C r/βc, i.e., if the integration is short, or r>>βcT, so that the travel exceeds the integration time. Eq. (13) would represent the receiver's analysis in accordance with all current ideas if the travel delay r/c were to affect only the additive phase ϕ′.sub.j, so as to yield

(68) ${\int }_T^{}\underset{j}{.Math.}{F}_{{\beta }^{\prime }j}{e}^{i{\omega }_j^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }t\right)+i\left({\varphi }_j^{\prime }-{\omega }_j^{\prime }r/c\right)}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left(\beta t\right)}\mathrm{dt}\simeq \underset{j}{.Math.}{\int }_T^{}{F}_{{\beta }^{\prime }j}{e}^{i\left({\varphi }_j^{\prime }-{\omega }_j^{\prime }r/c\right)}\delta \left({\omega }_j^{\prime }{e}^{{\beta }^{\prime }t}-\omega e^{\beta t}\right)\mathrm{dt},$
describing chirps constructed at the receiver from signals with no time dilation. However, the phase lags e.sup.−ω′.sup.j.sup.r/c cannot be correct as their rates of advancement at source are defined by ω′.sub.j, which are not constant over travel times r/c>>T. The chirp phase lag expression must represent the source frequency variation over the entire travel, but there is no way to add a travel delay r/c to the time t in the delta function in eq. (13) without implying frequency lags.

(69) As d'Alembertian solutions, the chirp components must also retain their frequencies and rates of change all the way from the source. The only possible inclusion of travel delay then yields

(70) $\begin{array}{cc}{\int }_T^{}\underset{j}{.Math.}{F}_{{\beta }^{\prime }j}{e}^{i{\omega }_j^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }\left[t-r/c\right]\right)+i{\varphi }_j^{\prime }}{e}^{-i{\mathrm{\omega \beta }}^{-1}\exp \left(\beta t\right)}\mathrm{dt}\simeq \underset{j}{.Math.}{\int }_T^{}{F}_{{\beta }^{\prime }j}{e}^{i{\varphi }_j^{\prime }}\delta \left({\omega }_j^{\prime }{e}^{{\beta }^{\prime }\left[t-r/c\right]}-\omega e^{\beta t}\right)\mathrm{dt}& \left(14\right)\end{array}$
same as eq. (8) but for involving multiple components, and showing that the same coefficients F.sub.β′j appear at frequencies offset by the lag factor e.sup.−βr/c≈1−βr/c. Polarization is usually represented by separate coefficients F.sub.β′j for the two transverse coordinate planes at the receiver. An effectively continuous monotonic variation of the plane of polarization should then manifest as a corresponding transfer of amplitudes from one transverse coordinate plane to the other, with corresponding lags.
C-5 Physical Basis of the Inventive Frequency Lags

(71) As shown in standard texts, the electromagnetic wave equations are obtained in terms of the scalar potential ϕ (not to be confused with phase) and the vector potential A, from Maxwell's equations describing forces on a test charge or a current (at the receiver) of unit magnitude exerted directly by source charges and currents. In vector form (due to Heaviside), Maxwell's equations are

(72) $\begin{array}{cc}\nabla .Math.E=\frac{\rho }{{ϵ}_0},\nabla .Math.B=0,\nabla \times E+\frac{\partial B}{\partial t}=0\mathrm{and}\nabla \times B-{ϵ}_0{\mu }_0\frac{\partial E}{\partial t}={\mu }_{0}J,& \left(15\right)\end{array}$
where E denotes the electric field intensity; B, the magnetic flux density; ϵ.sub.0, the susceptibility of free space; μ.sub.0, its permeability; and ρ and J, source charge and current, respectively. The potentials become B=∇×A and ∇ϕ=−E−∂A/∂t, and yield, in the Lorentz gauge ∇.Math.A=(ϵ.sub.0μ.sub.0).sup.−1∂ϕ/∂t,

(73) $\nabla .Math.E=\nabla .Math.\left(-\nabla \varphi -\frac{\partial A}{\partial t}\right)=-{\nabla }^2\varphi -\frac{\partial }{\partial t}\nabla .Math.A=-{\nabla }^2\varphi +{ϵ}_0{\mu }_0\frac{{\partial }^2\varphi }{\partial t^2}=\frac{\rho }{{ϵ}_0}$$\mathrm{and}$$\nabla \times B-{ϵ}_0{\mu }_0\frac{\partial E}{\partial t}=\nabla \times \nabla \times A-{ϵ}_0{\mu }_0\frac{\partial }{\partial t}\left(-\nabla \varphi -\frac{\partial A}{\partial t}\right)={\mu }_{0}J.$
The identity ∇×∇×A=∇(∇.Math.A)−∇.sup.2A then leads to the electromagnetic wave equations

(74) 0$\begin{array}{cc}{\nabla }^2\varphi -{ϵ}_0{\mu }_0\frac{{\partial }^2\varphi }{\partial t^2}=-\frac{\rho }{{ϵ}_0}\mathrm{and}{\nabla }^{2}A-{ϵ}_0{\mu }_0\frac{{\partial }^{2}A}{\partial t^2}=-{\mu }_{0}J,& \left(16\right)\end{array}$
of the same form as the one-dimensional scalar wave equation in Section C-1. The frequency lags in vacuum could be thus attributed to direct interaction with the source distributions ρ and J.

(75) In acoustic interactions, forces on the receiver are delivered by the medium. The inventive chirp travelling wave solutions satisfy the scalar equation for pressure waves, ∇.sup.2p−c.sup.−2∂.sup.2p/∂t.sup.2=0, where p is the pressure change [§ I-47-2, Feynman's Lectures in physics (Addison, 1969)], as well as the vector wave equation μ∇×∇×u−(λ+2μ)∇∇.Math.u+ρü=f for an isotropic elastic medium, where ρ and u denote its local density and displacement; λ and μ are the Lame parameters for bulk and shear moduli of elasticity, respectively; and f is a driving function analogous to the electromagnetic source distributions ρ and J. However, the same frequency lags are implied by the travelling wave solutions, so the lags do not inherently depend on direct interaction with wave sources.

(76) The frequency lags more particularly relate to the absence of a physical mechanism in eqs. (15) through (16), connecting the rates of change of source charges and currents, or of source forces and displacements in the acoustic wave equations, to the clock rate at the receiver, since all quantities are local intensities or densities, except the wave speed c, which merely relates the local scales of distance and time. The scale of time kept by the receiver's clock is local. For example, changes in the local gravitational potential should cause it to vary at the rate β=−{dot over (Φ)}/c.sup.2≈−d/dt(GM.sub.e/r.sub.ec.sup.2)={dot over (r)}GM.sub.e/r.sub.e.sup.2c.sup.2≡g.sub.e{dot over (r)}/c.sup.2, along with the contraction of comoving measuring rods, during a fall, where G is the gravitational constant, M.sub.e is the earth's mass, and g.sub.e≡GM.sub.e/r.sub.e.sup.2 is the gravitational force on ground. Setting this to H.sub.0 yields {dot over (r)}=H.sub.0c.sup.2/g.sub.e≈0.02 m s.sup.−1, as the rate of descent of a telescope to double the observed Hubble shifts, or of its ascent to cancel them out. Jet airliners ascend or descend at close to 1000 m min.sup.−1 or 15 m s.sup.−1, so the challenge for a flying telescope like SOFIA would be in maintaining a vertical rate of just 1.2 m min.sup.−1 for exposures lasting hours.

(77) C-6 Fermat's Principle in the Frequency Domain

(78) The enormous value of the speed of light makes large lags difficult to realize. The cumulative chirp phase of an exponential chirp wave solution over an integration time T would be
ω.sub.0β.sup.−1[e.sup.βT−e.sup.β0]=ω.sub.0β.sup.−1(1+βT+(βT).sup.2/2!+ . . . −1)=ω.sub.0T+ω.sub.0βT.sup.2/2+ . . . .  (17)
The difference from sinusoidal phase ω.sub.0T would be small, on the order of 10.sup.−25 rad over a 5 mm diffraction grating at β=10.sup.−18 s.sup.−1 and 5×10.sup.14 Hz, and of 10.sup.−7 rad at β=1 s.sup.−1, as mentioned. To compare, a full cycle of phase difference results if ω′.sub.0e.sup.−βr/c differed from ω.sub.0 by just (v.sub.0T).sup.−1≡2π/ω.sub.0T˜ 10.sup.−3, denoting a typical integration frames of 10.sup.2 to 10.sup.3 cycles. The larger difference across frequencies makes the first delta factor in eq. (8) a stronger selector. The second delta factor δ(β′−β) appears weak for selecting between components differing only in β, and also for preferring sinusoids over chirps. It suffices for strong selectivity via a variational argument, as follows.

(79) Existence of chirp components is assured for all fractional rates β since lines can be constructed at arbitrary inclinations tan.sup.−1(β)≡∠NEA=∠CBF to the time axis R-R.sub.T (14) at every point F on the receiver frequency axis R-R.sub.Ω (16) in FIG. 2a. For each fractional rate β′=β+δβ, where δβ>0, the starting angular frequency ω′.sub.1 at time t.sub.0 that leads to ω.sub.4 (point E) would be a point A′ left of A along the line A-N. If the phase lag going from A to E is Δϕ, the phase lag going from A′ to E would be Δϕ′<Δϕ, since ω′.sub.1<ω. For every such point A′ a component can be constructed starting at a point A″ to the right of A along A-N, whose phase lag at E would be Δϕ″=−Δϕ, if A′ and A″ are close enough to A to be within the band of frequencies admitted by the receiver.

(80) As their phase lag differences cancel out, each such pair simply adds to the amplitude at β. This constructive interference, which feeds into and strengthens the spectral selection, assumes that the condition β″.sup.−1 [e.sup.β″T−1]+β′.sup.−1[e.sup.β′T−1]=0 can be solved for β″ for all combinations of real values of β′ and β. For small δβ, the condition reduces to (β″−β)T.sup.2/2+(β′-β)T.sup.2/2=0 from eq. (17), yielding (β″−β)=−(β′−β). The condition would be clearly solvable with higher order terms at larger δβ. If A′ is so far from A that its phase complement A″ is outside of the admitted frequency band, its contribution would be limited to a few cycles by its increasing frequency difference over the integration. Fermat's principle depends on constructive interference across neighbouring paths at a fixed frequency, instead of neighbouring frequencies over a fixed path considered here. Similar reasoning should hold for a rotating plane of polarization, assuring the corresponding result.

(81) C-7 Application to Time Varying Spectrometry

(82) Diffractive embodiments in the Phase Gradient Patents involve a diffraction grating of width L and integration times T=L sin θ/c, where θ is the angle of diffraction, as T is the maximum travelled time difference between the interfering wavefronts in the resulting diffraction pattern. Regardless of whether the intervals of the diffraction grating or the refractive index of its surrounding medium are varied, the wave crests and troughs arrive, at each point in the diffraction pattern, at nonconstant intervals, and thereby constitute a chirp waveform. Both schemes are described by the relation

(83) $\begin{array}{cc}n\frac{d\lambda }{\mathrm{dt}}=\left[\eta \frac{dl}{\mathrm{dt}}+l\frac{d\eta }{\mathrm{dt}}\right]\sin \theta .& \left(18\right)\end{array}$
where n is the order of diffraction, λ is the instantaneous value of the diffracted wavelength, l is the grating interval, and η is the refractive index of the medium following the grating. Dividing by the grating equation, nλ=ηl sin θ, and setting l.sup.−1dl/dt≡β with dη/dt=0, or η.sup.−1dη/dt≡β with dl/dt=0, to describe either approach, or a combination l.sup.−1dl/dt=γβ with η.sup.−1dη/dt=(1−γ)β, where γ∈[0,1), would ensure the wavelength at a subsequent detector varies as λ.sup.−1dλ/dt≡β.

(84) At each angle of diffraction θ, the interference comprises N=L/l samples of the waveform from successive slits of the grating, separated in time by delays of τ=l sin θ/c, so the decomposition corresponds to a discrete Fourier transform (DFT) of the waveform function ƒ(t) as

(85) $\begin{array}{cc}F\left(m{\omega }_{\tau }\right)=\underset{n=0}{\overset{N-1}{.Math.}}{e}^{im{\omega }_{\tau }n\tau }f\left(n\tau \right),\mathrm{with}\mathrm{the}\mathrm{inverse}f\left(n\tau \right)=\frac{1}{N}\underset{m=0}{\overset{N-1}{.Math.}}{e}^{-im{\omega }_{\tau }n\tau }F\left(m{\omega }_{\tau }\right),& \left(19\right)\end{array}$
where ω.sub.τ=2π/N.sub.τ. The corresponding discrete orthogonality condition would be

(86) $\begin{array}{cc}\underset{n=0}{\overset{N-1}{.Math.}}{e}^{im{\omega }_{\tau }\left(n\tau -r/c\right)}{e}^{-\mathrm{il}{\omega }_{\tau }n\tau }\equiv \frac{1-e^{i\left(m{\omega }_{\tau }-l{\omega }_{\tau }\right)}}{1-e^{i\left(m{\omega }_{\tau }-l{\omega }_{\tau }\right)/N}}\underset{n=0}{\overset{N-1}{.Math.}}{e}^{-im{\omega }_{\tau }r/c}=N{\delta }_{\mathrm{ml}}\underset{n=0}{\overset{N-1}{.Math.}}{e}^{-im{\omega }_{\tau }nr/c},& \left(20\right)\end{array}$
where δ.sub.ml denotes the Kronecker delta, defined as 1 if m=l and as 0 otherwise.

(87) Varying the grating intervals l would cause the sample intervals τ to vary at the same fractional rate. The result must be identical to varying either the sampling intervals or the angular frequencies ω.sub.τ in the transform kernel, in a digital signal processing (DSP) time varying sampling embodiment, also treated in the Phase Gradient Patents. The orthogonality condition, relating to eq. (8), is

(88) $\begin{array}{cc}\underset{n=0}{\overset{N-1}{.Math.}}{e}^{im{\omega }_{\tau 0}^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }\left[n\tau -r/c\right]\right)}{e}^{-\mathrm{il}{\omega }_{\tau 0}{\beta }^{-1}\exp \left(\beta n\tau \right)}=N{\delta }_{\mathrm{ml}}\delta \left({\omega }_{\tau 0}^{\prime }{e}^{{\beta }^{\prime }\left[t-r/c\right]}-{\omega }_{\tau 0}{e}^{\beta t}\right),& \left(21\right)\end{array}$
which holds because the delta function effectively equates frequencies corresponding to the summed phases, and each frequency yields a sum of N terms of unity value when their phases cancel out, but provides at most a cycle otherwise, obviating an intermediate closed form as in eq. (20). The time varying selection replaces the Fourier kernel e.sup.−imω.sup.τ.sup.nτ with the chirp e.sup.−ilω.sup.τ0.sup.β.sup.−1.sup.exp(βnτ). The result also shows that the components would be received at shifted frequencies, consistent with the analysis in FIG. 2a indicating angular frequency ω.sub.1 would be received for source frequency ω.sub.4.

(89) Autocorrelation followed by a DFT is preferred in radio astronomy, as used in the Arecibo radio telescope and the Herschel mission, to obtain the power spectrum directly, as

(90) $\begin{array}{cc}R\left(\tau \right)={\int }_{T}f\left(t\right)f^{*}\left(t-\tau \right)\mathrm{dt}\mathrm{and}{.Math.F\left(m\omega \right).Math.}^2=\underset{n=0}{\overset{N-1}{.Math.}}{e}^{\mathrm{im}\omega t_n}R\left(t_n\right),& \left(22\right)\end{array}$
where R(τ) denotes the autocorrelation. Each Fourier component Ae.sup.−iωt within the arriving signal contributes Ae.sup.−ωt.A*e.sup.iω(t−τ)≡A.sup.2e.sup.iωt to the autocorrelation, where A* is the complex conjugate of the amplitude A, even if f and its component amplitude A are random variates. The second of eqs. (22), known as the Wiener-Khintchine theorem in the theory of random processes, thus always yields the power spectrum. When autocorrelation is applied to sinusoidal waves, the travel delay r/c enters both factors, as R(t)=∫.sub.T f(t−r/c)f*(t−τ−r/c) dt. The contribution of a sinusoidal wave component is then Ae.sup.iω(t−r/c), A*e.sup.iω(t−τ−r/c)≡A.sup.2e.sup.−i ωτ, so the travel delay r/c drops out of the power spectrum. Chirp frequency lags might thus seem unavailable via autocorrelation.

(91) However, the corresponding convolution product for a travelling chirp wave would be
A exp(iωβ.sup.−1e.sup.β[t−r/c]).Math.A*exp(−iωβ.sup.−1e.sup.β[t−τ−r/c])=A.sup.2 exp(iωβ.sup.−1e.sup.β[t−r/c][1−e.sup.−βτ]),  (23)
so the frequency lags would indeed survive convolution and shift the power spectrum.
C-8 Application to Modulated Signals

(92) Reception of radio frequency (RF) modulated signals is constrained by receiver integrations to less than the shortest modulation cycle, unlike astronomical observations, particularly at high redshifts, that can be integrated long enough to average out much longer fluctuations. Secondly, most radio receivers are designed to select only the carrier frequencies, and admit side bands bearing modulated information only as a result of the carrier selection, and the carrier itself is reconstructed from side bands in suppressed-carrier systems, as in deep space telemetry, so assurance is needed that the admitted side bands would belong to the selected carrier spectrum even when it comprises chirps.

(93) Most receiver designs involve multiplying and integrating the radio frequency (RF) signal with the output of a local oscillator (LO) for either down-translating to an intermediate frequency (IF), to simplify side band filtering, or demodulation by removing the carrier altogether, as in a Costas loop for carrier recovery in frequency modulation (FM), wherein the error signal itself serves as the demodulated signal. The orthogonality condition for FM reception is then the expectation value

(94) $\begin{array}{cc}{\int }_T.Math.\exp \left[i\left({\omega }_c+{\Omega }_m\right){\beta }^{\prime -1}{e}^{{\beta }^{\prime }\left[t-r/c\right]}\right]\exp \left[-i{\omega }_o{\beta }^{-1}{e}^{\beta t}\right].Math.\mathrm{dt}\simeq \delta \left(\left[{\omega }_c+.Math.{\Omega }_m.Math.\right]e^{-{\beta }^{\prime }r/c}-{\omega }_o\right)\delta \left({\beta }^{\prime }-\beta \right),& \left(24\right)\end{array}$
where ω.sub.c is the nominal carrier angular frequency; Ω.sub.m, a random variable denoting the instantaneous modulation; and ω.sub.o, the reference angular frequency provided by the LO. The condition Ω.sub.m=0 is generally required, even in the limit β.fwdarw.0 representing ordinary (Fourier) selection, to guarantee absence of a d.c. (direct current) modulation component, which would convey no information and complicate carrier recovery. Eq. (24) can also represent amplitude modulation (AM) signals, which generally involve multiple side band frequencies, if Ω.sub.m represents their overall effect. Variation of the reference angular frequency ω.sub.o should suffice via eq. (24) and the reasoning in Section C-6, for selecting chirp components of the carrier with the applied fractional rate.

(95) Then, if the carrier remains well within the admitted frequency band, and its frequency changes slowly relative to the integration, the only side band components also admitted would be also around the lagging carrier frequency. The condition Ω.sub.m=0 cannot hold for side band components that do not belong to the same source emission, so the obtained side band components always belong to the selected carrier. This is the only condition constraining the sinusoidal side bands in current receivers, and should therefore equally suffice to constrain the chirp side bands.

(96) Demodulation in direct conversion can be described by a similar integral relation
∫.sub.T[(ω.sub.c[τ−r/c]±Ω.sub.m[τ−r/c])−ω.sub.o[τ]]dτ≈∫.sub.T±Ω.sub.m[τ−r/c]dτ,  (25)
where the square bracket subscripts denote the time parameter, to avoid confusion with time factors. The integral represents an LPF (low pass filter) typically applied after a multiplier to multiply the RF signal, corresponding to the sum of ω.sub.c and Ω.sub.m, with the LO signal bearing the reference angular frequency ω.sub.0. The LPF suppresses components at the sum of the carrier and reference frequencies, whose combination with those of their difference frequencies would otherwise mathematically keep the signal modulated on the carrier and is thus critical. The integration time T corresponds to the time constant of the LPF, and would be short enough to not average out the modulation.

(97) Eq. (25) describes direct conversion (homodyne) receivers, and this use of multiplication and filtering is known as “mixing”. If the LO frequency is varied exponentially, eq. (25) becomes
∫.sub.T[(ω.sub.c[τ−r/c]±Ω.sub.m[τ−r/c])−ω.sub.o[τ]e.sup.βτ]dτ≈∫.sub.T±Ω.sub.m[τ−r/c]dτ.
Since the modulation frequency Ω.sub.m is in general unrelated to the carrier, eq. (9) requires, as in the argument following eq. (9), that ω.sub.c vary the same way as ω.sub.0. The result is therefore
∫.sub.T[ω.sub.c[τ−r/c]]dτ−ω.sub.o[τ]β.sup.−1e.sup.βτ≈0 or ω.sub.c[τ−r/c]≈ω.sub.o[τ]e.sup.βτ.  (26)
Substituting t=τ−r/c then leads to the frequency condition
ω.sub.c[t]e.sup.−βr/c≈ω.sub.c[t](1−βr/c)≈ω.sub.o[t]e.sup.βτ,  (27)
proving the inevitability of frequency lags in direct conversion with a time varying LO.

(98) FIG. 3 illustrates an inventive direct conversion receiver, in which the arriving signal (111) from the antenna (71), bearing instantaneous angular frequency ω.sub.d=ω.sub.c±Ω.sub.m, as the sum of the carrier and modulation angular frequencies, is multiplied at a multiplier (83) by the output signal of an LO (88), providing the reference angular frequency ω.sub.o. The resulting components of the sum frequencies ω.sub.c±Ω.sub.m+ω.sub.o are suppressed by an LPF (75), leaving only the components of difference frequencies ω.sub.c±Ω.sub.m−ω.sub.o in the result (128). The principle of direct conversion requires setting ω.sub.o=ω.sub.c, so that the result signal (128) comprises only the modulation frequency Ω.sub.m, and in effect reconstructs the modulating signal at the source, with time dilation per eq. (1).

(99) The direct conversion principle includes limiting the pass band of the LPF (75) to a modulation bandwidth W/2. By setting ω.sub.o=ω.sub.c, components of difference frequencies outside of ω.sub.c-ω.sub.o±W/2 at the output of the multiplier (83), due to components outside of ω.sub.c±W/2 in the Fourier spectrum of the arriving signal (111), would be effectively eliminated in the output (128) of the LPF (75). The antenna (71) is usually designed to receive over a wide range of wavelengths, so an RF tuner (98) would be included in the antenna path as shown. The variation of the angular frequency ω.sub.o of the LO (88) needed for a nonzero β in eq. (24) can be realized using a voltage controlled oscillator (VCO) as the LO (88), and applying a ramp or sawtooth signal v(t) (121) to the frequency control input of the VCO, as indicated in the figure.

(100) Heterodyne receivers involve one or more IF stages. The carrier down-translation and filtering at each stage, other than the last, provides the next stage IF carrier and modulation frequencies. Denoting the carrier RF and the IFs by the sequence {ω.sub.0, ω.sub.1, . . . , ω.sub.n}, where ω.sub.0≡ω.sub.c is the original (RF) carrier, and ω.sub.n=0 signifies the last stage, the LO reference angular frequencies at each stage would be ω.sub.o.sup.(j)=ω.sub.j-1−ω.sub.j, for j=1 . . . n. Exponential variation of the LO frequencies yields

(101) $\begin{array}{cc}{\int }_T\left({\omega }_{c\left[\tau -r/c\right]}\pm {\Omega }_{m\left[\tau -r/c\right]}\right)d\tau -{\omega }_{o_{.Math.\tau .Math.}}^{\left(1\right)}\frac{e^{{\beta }_1\tau }}{{\beta }_1}\approx {\int }_{T_1}\left({\omega }_{1\left[\tau \right]}\pm {\Omega }_{m\left[\tau -r/c\right]}\right)d\tau ,\mathrm{and}{\int }_{T_j}\left({\omega }_{j-1_{\left[\tau \right]}}\pm {\Omega }_{m\left[\tau -r/c\right]}\right)d\tau -{\omega }_{o_{.Math.\tau .Math.}}^{\left(j\right)}\frac{e^{{\beta }_j\tau }}{{\beta }_j}\approx {\int }_{T_j}\left({\omega }_{j\left[\tau \right]}\pm {\Omega }_{m\left[\tau -r/c\right]}\right)d\tau ,& \left(28\right)\end{array}$
for j=2 . . . n, in which the integration time T.sub.j is the jth stage filter time constant, and the LO frequencies refer to current time τ at the receiver, and not a retarded time (τ−r/c). By L'Hôpital's rule, the cumulative phase factors β.sub.j.sup.−1e.sup.β.sup.j.sup.τ reduce to unity in the limit β.sub.j.fwdarw.0 describing current systems. The independence of Ω.sub.m allows dropping it altogether in eqs. (28), yielding
ω.sub.c[τ−r/c]−ω.sub.o.sub.[τ].sup.(1)e.sup.β.sup.1.sup.τ≈ω.sub.1[τ] and ω.sub.j-1.sub.[τ]−ω.sub.o.sub.[τ].sup.(j)e.sup.β.sup.j.sup.τ≈ω.sub.j.sub.[τ]for j=2 . . . n.  (29)
The main difference from the homodyne constraint, eq. (26), is that the form of the selected IF or RF carrier component supplying ω.sub.j-1 is undetermined until the last stage. Eqs. (29) lead to
ω.sub.c[t]e.sup.β(t−r/c)≈ω.sub.c[t]e.sup.βt(1−βr/c)≈ω.sub.o.sub.[t].sup.(1)e.sup.β.sup.1.sup.t+ . . . +ω.sub.o.sub.[t].sup.(n)e.sup.β.sup.n.sup.t,  (30)
corresponding to eq. (27). The carrier frequency lag then corresponds to an effective β given by

(102) $\begin{array}{cc}{e}^{\beta \left(t-r/c\right)}\approx \frac{{\omega }_o^{\left(1\right)}}{{\omega }_c}{e}^{{\beta }_{1}t}+\mathrm{.Math.}+\frac{{\omega }_o^{\left(n\right)}}{{\omega }_c}{e}^{{\beta }_{n}t}\mathrm{or}\beta \approx \frac{{\omega }_o^{\left(1\right)}}{{\omega }_c}{\beta }_1+\mathrm{.Math.}+\frac{{\omega }_o^{\left(n\right)}}{{\omega }_c}{\beta }_n& \left(31\right)\end{array}$
to a first order, where the IF ratios ω.sub.o.sup.(j)/ω.sub.c may need adjustment to support practical fractional rates β.sub.j. All of the LOs can participate in determining the carrier β this way because the processing is phase-coherent all the way to the demodulation. As the desired value of β can be obtained by varying a single LO, the result allows a choice of which LOs to vary, and to make the best use of the range, speed and precision of variations provided by the component technologies.

(103) FIG. 4 illustrates an inventive heterodyne receiver, in which the signal (111) from antenna (71), of instantaneous angular frequency ω.sub.d=ω.sub.c±Ω.sub.m, is multiplied at a first stage multiplier (84) by the output of a first stage LO (89) providing a reference angular frequency ω.sub.o.sup.(1) and then filtered at a first stage band-pass filter (BPF) (93), to suppress components of the sum frequencies ω.sub.c±Ω.sub.m+ω.sub.o.sup.(1), as well as those of difference frequencies ω′.sub.c±Ω.sub.m−ω.sub.o.sup.(1) from other carrier frequencies ω′.sub.c, that would otherwise cause interference. The BPF output (129) is then the IF signal of instantaneous frequency ω.sub.c±Ω.sub.m−ω.sub.o.sup.(1)=ω.sub.1±Ω.sub.m, where ω.sub.1 is typically below the original carrier ω.sub.c.

(104) This IF signal then passes through zero or more further IF stages, each comprising a multiplier (85), an LO (90) providing reference angular frequency ω.sub.o.sup.(j), and a BPF (94) to produce a modulated IF signal of a still lower angular frequency ω.sub.j±Ω.sub.m as its output (130). The last stage includes a final multiplier (86) that multiplies the IF signal from the preceding stage, of angular frequency ω.sub.n-1±Ω.sub.m, with the output of the final LO (91) of angular frequency ω.sub.o.sup.(n), and is followed by a final LPF (75), whose output (128) has only the modulation term Ω.sub.m. The heterodyne design allows setting the LOs in one or more stages to above the IF carrier at each stage. The BPFs allow better rejection of adjacent carriers than direct conversion, as well as “crystal radios”, which comprise only RF tuning and envelope detection. An RF tuner (98) is also ordinarily needed to suppress the mirror carriers ω.sub.c ∓ω.sub.o.sup.(1) that would yield the same IF and thus pass through the BPFs.

(105) The required varying of one or more of the LO frequencies ω.sub.o.sup.(j)(j=1 . . . n) may be achieved again using ramp or sawtooth signals υ.sub.j(t), as frequency control input (122) to the first LO (89), the frequency control input (123) to an intermediate stage LO (90), or the frequency control input (124) to the last LO (91), respectively, as shown. The last stage may terminate in an envelope detector for AM reception, or be replaced by a phase locked loop (PLL), Foster-Seely, quadrature or ratio detector in the case of FM, and so on, as would be apparent to those skilled in the related arts. When using a PLL with a VCO for FM demodulation, the VCO would be locked to the IF output of the preceding stage, and the loop frequency error signal is the demodulated output, as in Costas receivers. This VCO could be also varied to contribute to the overall β, per eq. (31).

(106) The LOs would generally need to be electrically varied in general to achieve the fractional rates β needed in most applications, as calculated in the Phase Gradient Patents and the SPIE paper. The electrical variation range of variable capacitors, known as varicaps and varactors, though typically less than 100 pF, would be adequate at higher frequencies. However, the tuning speed gets limited by residual inductances, and simultaneous variation of multiple LOs poses problems of control.

(107) FIG. 5 illustrates one way how multiple LO frequencies and their variation can be achieved with “mixing” to keep the control simple. The figure shows the output of a single control LO (99), varied by a single control signal ϵ(t) (125) at instantaneous rate {dot over (ω)}.sub.0≡β.sub.oω.sub.o, that supplies the time-varying LO frequencies for ω.sub.o.sup.(j)(t), j=1 . . . n, needed in FIG. 4, with the help of frequency multiplier or divider means (100), (101) and (102), to construct varying frequency differences δω.sup.(1)(t), δω.sup.(j) (t) and δω.sup.(n)(t) that are multiplied by fixed LO frequencies ω.sub.o0.sup.(1), ω.sub.o0.sup.(j) and ω.sub.o0.sup.(n), generated by LOs (89), (90) and (91), respectively, at multipliers (103), (104) and (105), at the first, jth, and last stage, respectively, as shown. The band-pass filters (95), (96) and (97) reject the difference frequencies ω.sub.o0.sup.(1)−δω.sub.o0.sup.(1)(t), ω.sub.o0.sup.(j)−δω.sup.(j)(t) and ω.sub.o0.sup.(n)−δω.sup.(n)(t), producing the sum frequencies ω.sub.o0.sup.(1)+δω.sup.(1)(t), ω.sub.o0.sup.(j)+δω.sup.(j)(t) and ω.sub.o0.sup.(n)+δω.sup.(n)(t), with the effective instantaneous fractional rates

(108) $\begin{array}{cc}{\beta }_j\equiv {\omega }_{o0}^{\left(j\right)-1}\frac{d}{\mathrm{dt}}\left[\delta {\omega }^{\left(j\right)}\left(t\right)\right]={\rho }_j{\stackrel{.}{\omega }}_o/{\omega }_{o0}^{\left(j\right)}={\rho }_j{\beta }_o{\omega }_o/{\omega }_{o0}^{\left(j\right)},\mathrm{for}j=1\mathrm{.Math.}n,& \left(32\right)\end{array}$
where ρj=δω.sup.(t)/ω.sub.o is the net multiplication factor from the LO (99) to the jth multiplier (104).

(109) The scheme enables uniform variation of multiple LO frequencies using a single frequency control signal (125). The frequency multiplier or divider means (100), (101) and (102) would likely contribute phase noise, but similar phase noise is also likely with the independently varied LOs of FIG. 4. The band-pass filters (95), (96) and (97) are not critical, since with a judicious choice of the IFs and the sum frequencies ω.sub.o0.sup.(1)+δω.sup.(1)(t), ω.sub.o0.sup.(j)+δω.sup.(j)(t) and ω.sub.o0.sup.(n)+δω.sup.(n)(t), BPFs (93) and (94), and the LPF (75) present in the receiver of FIG. 4 could be also used to eliminate the difference frequencies ω.sub.o0.sup.(1)−δω.sup.(1)(t), ω.sub.o0.sup.(j)−δω.sup.(j)(t) and ω.sub.o0.sup.(n)−δω.sup.(n) (t). FIG. 6 illustrates the simplification. The multipliers (103), (104) and (105) are retained because analogue multipliers typically allow only two inputs. The receiver could alternatively designed to utilize the difference frequencies ω.sub.o0.sup.(j)−δω.sup.(j)(t), by instead rejecting the sum frequencies ω.sub.o0.sup.(j)+δω.sup.(j)(t) at the filters.

(110) Two further variations could offer substantial advantages, in terms of component availabilities, operational stability and other practical considerations. First, the control LO (99) could be designed to operate at close to the lowest LO frequency ω.sub.o0.sup.(n), i.e., from the right-most LO (91) in FIG. 6, with the frequency multiplier or divider means (100), (101) and (102) connected in reverse to up-convert the output of the control LO (99) to the multipliers (103), (104) and (105), going from right to left, as shown in FIG. 7. This allows operating at very high RF at which direct control of varicaps become impractical due to leakage reactances. The second variation concerns choosing ρj=ω.sub.o0.sup.(j)/ω.sub.o in the receiver of FIG. 6, so that all of the achieved fractional rates as well as the effective β (eq. 31) become equal to β.sub.o, and in feeding the outputs of the frequency multiplier or divider means (100), (101) and (102) directly to the multipliers (84), (85) and (85), respectively. The LOs (89), (90) and (91), and multipliers (100), (101) and (102) are eliminated, reducing the total number of parts.

(111) Eqs. (25-32) govern all such analogue radio receivers using one or more LOs, as well as digital receivers in which the down-translation and filtering are performed digitally. Only for crystal radio receivers, which use envelope detection and a single RF tuning stage for carrier selection, varying the RF tuning, as described in U.S. Pat. No. 7,180,580, is the only means for chirp carrier selection.

(112) The phase lock condition applicable to phase locked loops (PLL) used for carrier recovery, and also for demodulation of FM signals, as in Costas loop designs, is fundamentally given by
ϵ(t)=∫.sub.T′[(ω.sub.c−ω.sub.o)τ+Φ.sub.c+Φ.sub.m−Φ.sub.o]dτ≈0,  (33)
where ϵ(t) is the phase error signal generated by a phase comparator in the loop; ω.sub.0 is the angular frequency of the VCO used in the loop; T′ is the time constant of an LPF that invariably follows the phase comparator; Φ.sub.m is the instantaneous phase deviation due to the modulation; and Φ.sub.c and Φ.sub.o denote carrier and VCO phase noises, respectively. The phase noise and the modulation phase terms qualify as random variables uniformly distributed over [0, 2π), since T′ is generally set much longer than the longest modulation component cycle in order to ensure carrier recovery. The error signal ϵ(t) then locks the VCO to the carrier phase to within a cycle, hence barring occasional cycle slips, the VCO frequency ω.sub.0 should track the arriving carrier through all frequency variations.

(113) However, all known treatments of PLLs interpret the carrier frequency as ordinarily constant, and its variations as merely shifts of the Fourier spectrum. The possibility of identical phase lock to a nonsinusoidal spectrum of the carrier has never been considered, so the usual assumption that the result would be at most a changing sinusoidal behaviour cannot be correct. Since the object of carrier recovery is to suppress modulation as well as phase noise, its success signifies the stronger condition ∫.sub.T′[Φ.sub.c−Φ.sub.o+Φ.sub.m] dτ ≈0, i.e., ∫.sub.T′ (ω.sub.c-ω.sub.o)τdτ≈0, with T′>>T, where T denotes the integration time as in eqs. (25-31). This does not require constancy of ω.sub.c or ω.sub.o. Further, ω.sub.c and ω.sub.o denote peak or close to peak frequencies in the carrier and VCO spectra, respectively, and the LPF acts only on the phase error signal, so the input and VCO spectra are also not constrained.

(114) The loop constraint with phase differences computed individually at each frequency would be

(115) 0$\begin{array}{cc}{\int }_{T^{\prime }}\underset{j}{.Math.}{F}_{{\beta }^{\prime }j}^{\prime }\left[{\omega }_j^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }\left[t-r/c\right]\right)+{\varphi }_j^{\prime }-{\omega }_o\right]\tau d\tau \approx 0,& \left(34\right)\end{array}$
where F′.sub.β′j are weights relating to the coefficients F.sub.β′j of the carrier chirp spectrum for an arbitrary fractional rate β′, from Section C-4. A similar spectral expression for ω.sub.o is not appropriate because ω.sub.o represents the angular frequency set by the receiver in order to select from the carrier spectrum.

(116) Eq. (34) covers the steady-state scenario of stationary carrier and phase-locked VCO frequencies at β′=0, and is thus more general than an expression involving only the carrier's Fourier spectrum. It is not the only possible representation, as the precise function of the phase differences minimized depends on the PLL design—as the time factor τ is linear, any polynomial expression of the phase differences could be minimized to achieve the lock. The lock of itself means that ω.sub.0 tracks variations of the carrier spectrum and is not inherently constant. Eq. (34) thus resembles eq. (9) in requiring that the VCO and the carrier spectrum vary similarly. However, since the VCO is needed to track the carrier frequency, it is not obvious that it can be independently varied like the LO of FIG. 3 to force an analogous phase lock over the chirp spectrum of the carrier.

(117) FIG. 8 illustrates a PLL with two possible modifications to modulate the VCO while maintaining phase lock. The PLL comprises a phase comparator (72), a first low-pass filter (LPF) (76), the VCO (73), and a feedback circuit (74) feeding the output signal (119) of the VCO (73) back into the phase comparator (72) for comparing with a modulated input signal (115). The first LPF (76) suppresses high frequency variations and phase noise in the output of the phase comparator (72), to produce the frequency error signal ϵ(t) (114) at the frequency control input of the VCO (73).

(118) The first mechanism included for modulating the VCO comprises a summing device (87) to add a first control signal υ(t) (126) to the error signal ϵ(t) (114) to change the input (116) to VCO (73) to ϵ′(t)≡ϵ(t)+υ(t). If this first control signal υ(t) (126) is a ramp or sawtooth waveform, the VCO output (119) should drift as ω′.sub.o(t)=ω.sub.o+Kϵ′(t)=ω.sub.o+K[ϵ(t)+υ(t)], in which the coefficient K is determined by the PLL design. The second mechanism comprises a multiplier (82) multiplying the VCO output (119) by a second control signal v′(t) (127), followed by a second LPF (79), needed to suppress either the sum or the difference frequency component in the output of the multiplier (82), and to thereby modify the feedback signal (118) input to the comparator (72). Again using a ramp or a sawtooth frequency modulated signal as the second control signal v′(t), the result would be a drift ω′.sub.o(t)=ω.sub.o+ω′(t), where ω′(t) is the (angular) frequency of the second control signal v′(t). With either or both mechanisms, i.e., υ(t)≠0 or v′(t) varying, the phase lock condition becomes

(119) $\begin{array}{cc}{\int }_{T^{\prime }}\underset{j}{.Math.}{F}_{{\beta }^{\prime }j}^{\prime }\left[{\omega }_j^{\prime }{\beta }^{\prime -1}\exp \left({\beta }^{\prime }\left[t-r/c\right]\right)+{\varphi }_j^{\prime }-{\omega }_o{\beta }^{-1}\exp \left(\beta t\right)\right]\tau d\tau \approx 0,& \left(35\right)\end{array}$
where the reference frequency term ω.sub.oβ.sup.−1 exp (βt) denotes the drifting VCO angular frequency, and implies that the phase lock can indeed only occur at the applied rate β, consistent with eq. (27).

(120) The difference is that eq. (35) concerns signal differences rather than products as in eq. (8), so the pairs of neighbouring fractional rates β±δβ in the Fermat's principle reasoning of Section C-6, should cancel each other, instead of adding constructively at the fractional rate β, and thus imply an overall amplitude of zero measure, occurring only at β, as the net result. This unphysical result simply means that the ideal phase comparator assumed in eq. (34) does not exist.

(121) Real PLLs typically use a multiplier as the phase comparator element (72), and the subsequent LPF (76) suppresses the sum frequency components; the distinction from “mixing” in the homodyne and heterodyne receivers of FIGS. 3-7 is that as the frequency difference in lock would be extremely small, the LPF would be set to reject as little as 1 Hz even at a 20 MHz IF, as notably done in the DSN according to the DSN Handbook. The underlying physics therefore still involves constructive interference over a differential neighbourhood of β, and conforms to the reasoning in Section C-6.

(122) C-9 Other Practical Considerations

(123) It would be apparent to those skilled in the related arts that variation of variable tuning elements, of the coefficients of convolution filters, and of the kernel in the digital spectral transform of Section C-7 could be combined with the variation of LOs for realizing the inventive chirp components.

(124) In particular, convolution filters with time varied coefficients would in effect directly extract the dilated (or compressed) waveforms corresponding to the inventive chirp components with lags. It would be further apparent that these techniques can be easily combined with, or incorporated in, orthogonal frequency division multiplexing (OFDM), ultra-wideband (UWB), frequency hopping, and other such advanced schemes, or combined with digital processing, as in software defined radio (SDR). Varying of LOs would be also applicable to optical communication using lasers as LOs and optical “mixing” for phase-coherent down-translation or demodulation.

(125) In narrow band applications like radar, linear chirps may be used instead of the exponential form to exploit known methods to keep the absolute rate {dot over (ω)} steady. Though the fractional shifts then vary with time, the actual shifts, i.e., the lags, would be constant over fixed round trip distances r, as δω={dot over (ω)}r/c, as in CW-FM radar theory. The advantages remain that the inventive shifts increase with r indefinitely beyond the transmitter's frequency range, and would not be limited to echoes.

(126) The Phase Gradient and Multiplexing Patents allow for a possibility that the inventive spectral selection or decomposition could be repeatedly applied to the same signal or waves so as to multiply, or negate, the inventive spectral shifts. The inventive selection or decomposition was denoted by an operator H(β), so that successive applications could be described by a product law H(β.sub.1)H(β.sub.2)=H(β.sub.1+β.sub.2), with the inverse H.sup.−1(β)≡H(−β). Source separation was then succinctly described by the product H.sup.−1GH having the form of a projection operator, where G denoted a filter admitting the shifted frequency band. However, eq. (8) implies H(β) H(β′)=H(β)δ(β-β′), so the inverse only exists at β=0. The function of H.sup.−1, translating the output of filter G back to the original frequency band, is digitally trivial, however, and can be conceivably achieved in analogue receivers by other means including “mixing”, as mentioned in the Multiplexing Patent.

(127) Multiplication of the shifts across receivers is permitted with retransmission. A signal at angular frequency ω.sub.1=ω.sub.0(1−β.sub.1r.sub.1/c) from a source at distance r.sub.1 emitting instantaneously at an angular frequency ω.sub.0 with fractional rate β.sub.1 received by a first receiver, upon retransmission, at the (shifted) received frequency, can be received at an additional distance r.sub.2 at a different fractional rate β.sub.2 at angular frequency ω.sub.2=ω.sub.1(1−β.sub.2r.sub.2/c)=ω.sub.0(1−β.sub.2r.sub.2/c/)(1−β.sub.1r.sub.1/c), since there is nothing to stop a receiver from retransmitting. This notion is better expressed by incorporating the distances into the operator notation, as ω.sub.2=H(β.sub.2,r.sub.2)H(β.sub.1, r.sub.1)ω.sub.0. A receiver having an extended internal path length r.sub.2 could incorporate both operators, and set β.sub.2=−β.sub.1r.sub.1/r.sub.2, in order to realize H.sup.−1(β.sub.1,r.sub.1) as H(−β.sub.1r.sub.1/r.sub.2, r.sub.2) for the source separation. The retransmission itself could also occur at a different frequency than received, including passively as in fluorescence.

(128) The orthogonality between fractional rates, by eq. (8), incidentally also means that the existence of travelling chirp wave solutions cannot be verified by simulation, as the only spectral components detected would be those specifically constructed in the simulation.

(129) Fast Fourier transforms (FFT) with close to O(N log N) performance on nonequispaced data are now available [D Potts, G Steidl and M Tasche, “Fast Fourier transforms for nonequispaced data: A tutorial”, Modern Sampling Theory: Math, and Appl., Birkhauser (2001); J Keiner, S Kunis and D Potts, “Using NFFT3—a software library for various nonequispaced fast Fourier transforms”, ACM Trans Math, Software, 36, pages 1-30 (2009)]. These permit time-varied phase factors in FFT over uniform sample data streams, equivalent to refractive index variation in eq. (18), as described in the second of the Phase Gradient Patents, as another practical route to the invention in digital receivers, especially those using integrated RF tuners that disallow continuous variation.

D. Applications of the Distance Information

(130) Orthogonality of the lags in the inventive travelling wave solutions to source information, including modulated or encoded information, is best illustrated by application to source distance-based separation of signals arriving simultaneously or bearing the same encoding, so they occupy the same channels in time domain multiplexing (TDM) or code division multiplexing (CDM), respectively.

(131) The scheme exploits the similarity of the inclination of chirp lines J-D (63) and I-E (62) to an angle of view, tan.sup.−1(β)=∠DAE≡∠NEA, of source frequencies at D and E, respectively, by combining it with the inclination ∠DAC=tan.sup.−1(|DC|/|AD|)=tan.sup.−1(r/Δt)≡tan.sup.−1(c) of the travel lines like A-C (51) to obtain an angle of view ∠ECD=tan.sup.−1(|DE|/|DC|)=tan.sup.−1(Δω/r)≡tan.sup.−1(β/c) across space, denoting a frequency domain analogue of parallax, as illustrated in FIG. 9a reproduced from the Multiplexing Patent. In the figure, receiver R (181) must distinguish signals from two transmitters S.sub.1 (182) and S.sub.2 (183) at distances r′ and r″, respectively, transmitting over the same (angular) frequency band of width W, includes suitable guard bands, and the same carrier frequency ω.sub.c, so they would not be separable by frequency domain techniques.

(132) The scheme was formalized in the Multiplexing Patent in terms of a linear operator H(β), which, when applied to the received combined signal spectrum F′(ω)+F″(ω), yields H(β)F(ω)=H(β)F′(ω)+H(β)F″ (ω), where H(β)F′(ω) and H(β)F″(ω) would be found shifted in proportion to their respective source distances r′ and r″, as shown. The figure shows that the bandwidths of the shifted spectra would be proportionally scaled as well, so the shifts more truly represent scaling of frequencies, consistent with a reverse scaling of the receiver clock rate (see Section C-5). The Multiplexing Patent includes a subband strategy to address signals of large bandwidth or sources close in range, in which cases, the scaled spectra would otherwise still overlap.

(133) The net result is a separation of the signal spectra along the frequency axis (16) at the receiver, so that either signal can be then selected using a band-pass filter whose pass-band covers the scaled desired frequency band HF′ or HF″, respectively. The filtered products G′HF′≡G′(ω)H(β)F′(ω) and G″HF″≡G″(ω)H(β)F″(ω) would be separate, but shifted and magnified, so a reverse scaling operation H.sup.−1(β)≡H(−β) is necessary to complete the recovery of the desired source signal. The overall process is thus an operator product H.sup.−1GH, where G=G′ or G″, in the form of a projection operation in the frequency-distance domain, although the H.sup.−1 operation must be realized by other means, as remarked in Section C-9. An asymptotic rule derived in the SPIE paper,

(134) $\begin{array}{cc}\frac{\delta r}{r}\simeq \frac{2W}{f_c-W}\approx \frac{2W}{f_c},& \left(36\right)\end{array}$
where f.sub.c≡ω.sub.c/2π denotes the centre frequency, also holds for the present invention.

(135) The Multiplexing Patent proposed a pre-filter S around ω.sub.c to avoid interference from any Fourier components arriving in the pass-bands of filters G′ or G″, as the Fourier components would be also admitted by these filters, but all Fourier components would be eliminated by the orthogonality of the inventive chirp components to sinusoids in the spectral decomposition or selection process. The only additional possibility of interference is from chirp components from unwanted sources bearing the same fractional rates which would not eliminated by the spectral decomposition or selection.

(136) FIG. 9b illustrates this additional problem of “chirp mode interference”, depicted by the shaded inclined region (142), showing that signals at carrier angular frequency ω.sub.c from the first transmitter S.sub.1 (182) and at ω′.sub.c<ω.sub.c from the second transmitter S.sub.2 (183) would both arrive in the pass-band of filter G′ at receiver R (181) at the receiver's applied fractional rate β. These signals ordinarily do not interfere, as their Fourier interference regions (143) and (144) do not overlap. This chirp mode interference would not be an issue in deep space telemetry, but would affect terrestrial applications due to the multiplicity of sources on earth. The pre-filter in the Multiplexing Patent would block the Fourier interference region (141), which is already rejected by the selection of chirp components represented by the inclined region (142), however. The Fourier interference regions (143) and (144), corresponding to signals that interfere only in the chirp mode represented by inclined region (142), are non-interfering, and thus present a way around chirp mode interference, just as the chirp mode presents a way around the interference of Fourier spectra, according to FIG. 9a.

(137) If the receiver R (181) correlates chirp mode information arriving through filter G′ with Fourier information arriving through filter G, the correlated signal would be the signal of the first transmitter S.sub.1 (182) emitted at carrier angular frequency ω.sub.c, since the interference in the chirp mode from the second transmitter (183) would be a signal transmitted at carrier angular frequency ω′.sub.c, which would be generally unrelated to, and therefore not correlate with, the Fourier mode interference from the second transmitter S.sub.2 (183) or any other transmitter, emitted at carrier angular frequency ω.sub.c. Any chirp mode components starting at angular frequency ω.sub.c at the second transmitter S.sub.2 (183) would arrive at frequencies beyond the pass-band of filter G′ and thus not enter the correlation. Likewise, any Fourier mode interference at carrier angular frequency ω′.sub.c from the first transmitter S.sub.1 (182) would arrive in the pass-band of filter G″, and not of filter G, and would be thus excluded.

(138) Conversely, receiver R (181) could correlate the chirp mode information arriving through filter G′ with the Fourier information arriving via filter G″, to obtain the signal of the second transmitter S.sub.2 (183) emitted at carrier angular frequency ω′.sub.c, free of both chirp and Fourier mode interference from the first transmitter S.sub.1 (182), as the latter's chirp mode interference would have been emitted at carrier angular frequency ω.sub.c, and its Fourier interference, at carrier angular frequency ω′.sub.c. To jam a signal of carrier angular frequency ω′.sub.c by the second transmitter S.sub.2 (183) from receiver R (181), the first transmitter S.sub.1 (182) must transmit its jamming signal not only at the carrier angular frequency ω′.sub.c of the second transmitter S.sub.2 (183) but also at ω.sub.c=ω′.sub.c(1−βr″)/(1−βr′), in order to also jam chirp mode reception of the second transmitter S.sub.2's signal in the pass-band of filter G′.

(139) Since β is chosen, and can be varied arbitrarily, by receiver R (181), there is no way the first transmitter S.sub.1 (182) could anticipate and jam all chirp mode reception. The correlation raises the challenge for jamming, as the first transmitter S.sub.1 (182) must emit the same jamming signal at both ω.sub.c and ω′.sub.c in order to defeat correlation, and the value of ω.sub.c depends on the receiver's choice of β.

(140) Denoting the time domain signal f(t) reconstructed from the Fourier components admitted by filter G″ as f(t)=f.sup.(2)(t)+f.sup.(1)(t), where f.sup.(2) is the signal of the second transmitter S.sub.2 (183) and f.sup.(1) is the interference from the first transmitter S.sub.1 (182); and the time domain signal reconstructed from the chirp spectrum admitted by filter G′, after correcting for frequency shift and time dilation, as f.sub.β(t)=f.sub.β.sup.(2)(t)+f.sub.β.sup.(1)(t), where f.sub.β.sup.(1) again denotes the interference from transmitter S.sub.1 (182); the expectation value of their product would be
f(t)f.sub.β(t)≡[f.sup.(2)+f.sup.(1)][f.sub.β.sup.(2)+f.sub.β.sup.(1)]≃f.sup.(2)(t)f.sub.β.sup.(2)(t)=|f.sup.(2)(t)|.sup.2,  (37)
because equality of Fourier and chirp coefficients is assured if and only if they belong to the same transmission, hence only for coefficients F.sup.(2)(ω) and F.sub.β.sup.(2)(ω) from transmitter S.sub.2 (183). The result corresponds to the square of the amplitude of the original signal as the reconstructions f.sup.(2)(t) and f.sub.β.sup.(2)(t) would be each independently identical to this original signal from transmitter S.sub.2 (183). The similar term f.sup.(1)(t)f.sub.β.sup.(1)(t) for the first transmitter S.sub.1 (182) must vanish as the factors belong to generally different signals originating at ω′.sub.c and ω.sub.c from the first transmitter S.sub.1's location. The cross terms f.sup.(1)(t)f.sub.β.sup.(2)(t) and f.sup.(2)(t)f.sub.β.sup.(1)(t) vanish as the factors come from different transmitters. The time domain multiplications correspond to phase correlations in the frequency domain. The result can be clearly improved by extending to m signals and different fractional rates, as
f(t)f.sub.β.sub.1(t) . . . f.sub.β.sub.m(t)≃(f.sup.(2)(t)f.sub.β.sub.1.sup.(2)(t) . . . f.sub.β.sub.m.sup.(2)≡|f.sup.(2)(t)|.sup.m.  (38)

E. Applications of the Wavelength Transformation

(141) A fundamental significance of the signal separation scheme above is it allows reuse of the bandwidth W around the angular frequency ω.sub.c by indefinitely many transmitters, and in effect thus multiplies the channel capacity indefinitely, though the channel capacity realized between the receiver R (181) and each of transmitters S.sub.1 (182) and S.sub.2 (183) remains equal to the signal bandwidth W. Capacity multiplication over point-to-point links like optical fibre, coaxial cable, and microwave, would have much wider applicability, from transmission lines in integrated circuits to large cable networks.

(142) FIG. 10a illustrates how capacity multiplication can be obtained between a receiver R (181) and a source S (184) with a link (191) of optical length r and having a narrow transmission band around a specific wavelength λ.sub.1. Multiple signals are shown entering one end A of the link (191) close to the source S (184) and leaving the other end B near the receiver R (181).

(143) The wavelength axis (17) is drawn at the source S (184), with arrows to explain that signals emitted by the source (184) at diverse wavelengths λ.sub.0, λ.sub.1, λ.sub.2, . . . can be coupled into and transmitted via the link (191) by applying different corresponding fractional rates β.sub.0, β.sub.1, β.sub.2, . . . , respectively, at the receiver R (181), as marked on the fractional rate axis (18). Over the given path length r, the signals arrive at the receiver R (181) at lagging wavelengths λ.sub.j[1+β.sub.jr/c+o(β.sub.j)], for j=0, 1, 2 . . . , hence by setting β.sub.j≈(λ.sub.1/λ.sub.j−1) c/r, these arriving wavelengths can each be made λ.sub.1, which is transmitted by the link (191). The result depends only on the total effective path length r and not on the specifics of the physical channels or media along the path, and exploits the d'Alembertian character of the inventive travelling wave solutions not anticipated in the prior patents.

(144) The receiver R (181) would be then able to receive data from the source (184) at as many times the capacity of the link (191) as the number of fractional rates β.sub.j simultaneously implemented at the receiver R (181), provided the source (184) can transmit at each of the wavelengths λ.sub.j. The only limit on the channel capacity is the totality of chirp mode signals that could be accommodated with adequate noise margin in the linear range of the physical media comprising the link, in which the electric field or acoustic displacements would be a superposition of the concurrent signals.

(145) These ideas clearly also enable receiving signals emitted at the wavelengths λ.sub.1, λ.sub.2 . . . through a restrictive channel, such as a narrow aperture or a long tunnel or waveguide, that ordinarily blocks these wavelengths, but would admit wavelength λ.sub.0. Such a receiver could be further coupled with a transmitter to re-transmit the received signals, to serve as a repeater, and combined with additional similarly multiplexed hops, following the retransmission considerations in Section C-9.

(146) FIG. 10b illustrates a converse application of the d'Alembertian behaviour that all transmission, absorption, scattering and diffraction behaviour over the entire path from the source must conform to the lagging wavelengths seen by the receiver. The link (191) of FIG. 10a could be then replaced by an object or test material (192) and the source S (184) needs to radiate only a narrow spectrum around wavelength λ.sub.1, in order to enable measurement or imaging of the transmission, absorption, scattering or diffraction characteristics of the object or test material (192) to the receiver R (181), at arbitrary wavelengths λ.sub.j using corresponding fractional rates β.sub.j, as in FIG. 10a. This supports and refines the capabilities envisaged in the Frequency Generation Patent.

(147) In retrospect, all physical channels have been assumed to have finite capacities in communication theory, as in C E Shannon's “A Mathematical Theory of Communication” [Bell Sys Tech J, 27:379-423, 623-656, 1948], due to the assumption that only sinusoidal travelling wave solutions exist, which dates back not just to Fourier's treatment of the heat equation ca. 1807 closely following Fresnel's treatment of diffraction ca. 1805, but further back to the “vibrating string controversy” involving Euler and others regarding d'Alembert's solutions, whose generality was thus, in hindsight, occluded by concerns of continuity, eventually addressed by the Dirichlet conditions without allowing for time dilations, as remarked, and of standing wave modes of vibration, rather than travelling waves.

(148) FIG. 10a establishes that current notions of limits on communication, including signal isolation and transmission capacity in space or over a medium, cannot be fundamental. Shannon's theorem C=W log.sub.2(1+S/N) relating physical channel capacity C to the bandwidth W and signal-to-noise ratio S/N, and related considerations of time-bandwidth products, would still govern each selection or decomposition process providing a chirp or Fourier mode input in eqs. (38)-(38), and each index j in FIGS. 10a and 10b. The multiplexing and capacity multiplication capabilities result from the multiplicity of chirp modes, which complement spatial modes in waveguides and optical fibres. The decoupling of the wavelengths of observation or imaging from those of illumination, in FIG. 10b, concerns a similar complementarity to the spatial diversity of rays, and is equally fundamental.

F. Illustrative Application to the Flyby Anomaly

(149) FIG. 11 reproduces the graph of the SSN range residuals against DSN data in the 1998 NEAR flyby from Antreasian and Guinn's paper, of values as large as 1 km, inconsistent with their independently established resolutions. Current focus on the 760 mHz mismatch between pre- and post-encounter DSN data as velocity gain admits exotic explanations like dark matter [S L Adler, arXiv:1112.5426, 2011], even though the anomalies are within the range of satellite orbits that show no such issues.

(150) The difference in the residuals of the two SSN stations cannot have come from the SSN, whose data represents true two-way round trip times. Antreasian and Guinn state that the residuals were computed by “passing the (SSN) data through a trajectory estimated with pre-encounter DSN data”, so each station's data was checked separately. The range errors precluded further check of the differenced SSN range data against the DSN Doppler data. The output of the DSN carrier loop would be a chirp during the pre-encounter acceleration, implying a chirp form of eq. (9),
∫.sub.T exp(−ω.sub.0β.sup.−1e.sup.β[t−r/c])e.sup.iω.sup.r.sup.tdt≃δ(ω.sub.0e.sup.−βr/c−ω.sub.r),  (39)
where the integral would again vanish unless the reference angular frequency ω.sub.r varies the same way during integration as the arriving chirp, as in eq. (35). The DSN reference is the uplink transmitter signal, synchronized with atomic reference clocks [§ III-A, Anderson, Laing et al.], and translated to the downlink frequency, so the integral is only satisfied by chirp spectral components of this uplink reference signal. The lag is still contained in the downlink signal, and yields both the Doppler error Δυ=aΔt and a range error Δr=υΔt accounting for the SSN residuals, as explained.

(151) FIG. 12a illustrates this error between the true velocity profile in approach (171), which would have otherwise matched the differenced SSN range data, and the Doppler signal (172) expected for the DSN cycle counts over the duration of the flyby. The error is equivalent to an unexpected delay of Δt, and the Doppler information of closest approach (174) thus lags the true periapsis (173). The curves are monotonic, but cannot coincide over the SSN tracking period, indicated on the time axis (14), under vertical displacements, because of curvature due to acceleration. This explains the irreducibility of these residuals through Δυ estimation noted by Antreasian and Guinn.

(152) The anomaly, as currently defined, refers to inconsistency of the post-encounter segment of the DSN Doppler profile (172) captured at Canberra to the right of AOS (acquisition of signal) in the figure, with trajectory estimated from the pre-encounter segment of the same profile (172) obtained at Goldstone prior to LOS (loss of signal). As more clearly depicted in FIG. 12b, the pre-encounter based estimated trajectory (175) could pass through the true periapsis (173) and closely follow the pre-encounter segment of the DSN Doppler signal profile (172) after LOS, but it would then exhibit a larger velocity deviation Δυ from the DSN Doppler signal profile (172) near periapsis, as shown by its dotted extension right of the true periapsis (173), because the estimation adjusts the velocity down to fit the DSN Doppler signal profile (172) at LOS. The estimation error Δυ depends on when the DSN Doppler signal is measured. Antreasian and Guinn reported the anomaly as reducible by estimation since a reestimated post-encounter trajectory (176), adjusted for the velocity difference Δυ at AOS, fits the post-AOS segment of the profile (172) to within expected margins of error.

(153) The apparent velocity gain Δυ is therefore positive if the DSN Doppler profile (172) is observed before the peak of the true velocity profile (171) left of true periapsis (173) in FIG. 12a, or after the DSN Doppler's post-encounter negative peak right of its apparent periapsis (174). The negative Δυ in the Galileo's second flyby corresponds to observations through or very close to the periapsis since atmospheric drag was considered a significant contributor. The delayed DSN Doppler signal profile (172) would have implied the spacecraft was slow reaching periapsis, where the signal changes sign. In the Cassini flyby, for which a negative anomalous Δυ is also reported despite masking by the firing of attitude control jets, the tracking was continuous through the true periapsis (173).

(154) The quantitative explanation of the SSN residuals in the Background was arrived at as follows. As the net speed change was merely (6.87-6.83)/6.87≈0.6%, and most acceleration was close to earth in increasingly tangential motion, the hyperbolic excess speed at periapsis V.sub.∞≡6.851 km s.sup.−1, which is also the mean of its asymptotic values, should be close enough for the present estimates.

(155) This notion is supported by the uniformity of the 10 min ticks in the equatorial view, given as FIG. 1 by Anderson, Campbell et al. The large difference in the latter before and after the flyby in similar 10 min ticks in the north polar view (FIG. 9 in Antreasian and Guinn) is partly due to the projection. The polar and equatorial views more particularly show the motion was mostly radial. The 219 min gap in DSN tracking thus signifies 6.851 km s.sup.−1×219 min=90, 000 km.

(156) According to Antreasian and Guinn, the SSN tracking ended at 06:51:08 at Altair 32 min before periapsis, i.e., at about 6.851 km s.sup.−1×32 min≈ 13,150 km, signifying a one-way delay of 43.9 ms and range error of 300 m, neglecting errors due to the earth's radius and the inclination of spacecraft motion to the radial, which would both require more detailed trajectory information. The tracking started at 06:14:28 at Altair and 06:12:22 at Millstone, representing ranges of 28, 230 km˜ 94.2 ms and 29, 090 km˜ 97 ms, hence range errors 645 m and 665 m, respectively. The actual residuals are closer to 950 m, about 30% larger. Corrections for the earth's radius and the tangential component of the velocity vector would both reduce the range error estimates, so the larger residuals point to an underestimation of range in these calculations using V.sub.∞. The peak velocity V.sub.f=12.739 km s.sup.−1, at periapsis indeed leads to 26% and 30% overestimations for Altair and Millstone, respectively.

(157) The rate of decrease due to diminishing range would have been 6.851 km s.sup.−1×6.851 km s.sup.−1/c≈0.157 m s.sup.−1, across 1187 s from 06:25:25 to 06:45:12, implying 0.157×1187≈186 m. This matches the 200 m decrease in the Millstone residual in this period quite closely, as shown. The differences between the residuals in slope and magnitude in FIG. 11 follow the radial ranges and accelerations discernible in the equatorial and polar views and the ground track diagram (FIG. 7 in Antreasian and Guinn's paper). The trajectory is almost equidistant from the two SSN stations in the ground track diagram, initially moving away from Altair towards Millstone, so the delay Δt for Millstone initially decreased faster. The subsequent southward turn indicates a sustained greater acceleration towards Altair. The Millstone residuals indeed start out larger and cluster around Altair's, and are subsequently smaller, decreasing slower than Altair's, consistent with the turn.

(158) The explanations given in the Background for the anomalies in the DSN data are based on the Canberra AOS range estimate of 62, 070 km from the stated AOS time of 2 h 31 min after periapsis, implying 207 ms excess one-way delay, which should cause 21.4 mm s.sup.−1 actual velocity error, that would be interpreted as 11.7 mm s.sup.−1 from the two-day data, as stated.

(159) For the post-encounter diurnal oscillations, Canberra's latitude of 35.2828° places it 6371 km×cos(35.2828°)=5201 km off the earth's axis. The declination, i.e., the angle from the equatorial plane, for the outgoing asymptotic velocity was −71.96°, according to Anderson, Campbell et al, so the diurnal range variation would have been 5201 km×cos(71.96°)≈ 1611 km along the asymptote, which would imply a 15.6 mHz oscillation in the post-encounter Doppler. Anderson, Campbell et al. attributed the entire oscillation to inadequacy of the pre-encounter trajectory for predicting the post-encounter direction. The excess one-way delay of 94 ms at Goldstone LOS would cause such an issue, but a part of the difference would be due to a slightly smaller declination at AOS.

(160) Every detail of the flyby anomaly is thus well explained by the Doppler lag, which belongs to the Doppler signal and is independent of the inventive chirp travelling wave components. The lag was observable only by decomposition of the uplink reference signal into a chirp spectrum (eq. 39), however, and the DSN range data showed the same anomaly only by demodulation from the chirp spectrum. The anomaly proves the reality of such lags and the realizability of chirp spectra. Further support lies in the broader consistency of astrophysical and geophysical data with the Hubble shifts themselves being lags due to a residual systematic, as explained in the Background.

G. Variations

(161) Numerous variations of the present invention, by itself and in combination with other technologies, are envisaged and intended within its scope. For example, since the frequency lags in the inventive chirp components are primarily due to travel time and not the distance, ultra-slow light techniques [“Light speed reduction to 17 metres per second . . . ”, L V Hau, S E Harris, Z Dutton and C H Behroozi, Nature, 1999] could be exploited to achieve large shifts in very short scale applications.

(162) Likewise, commercial tunable lasers are mechanically tuned external cavity devices intended for spectroscopy, and as shown in the SPIE paper, mechanical tuning is too slow for the fractional rates required in terrestrial applications. A vertical cavity surface emitting laser (VCSEL) described by C Gierl et al. [“Surface micromachined tunable 1.55 μm-VCSEL with 102 nm continuous single-mode tuning”, Optics Exp, 19(18), pp. 17336-17343 (2011)] achieves linear tuning at to 215 Hz, and could be used for coherent selection of the chirp spectral components in optical fibres.

(163) In addition, distance information at short range yielded by the frequency lags, as in a studio or a theatre, may be captured and encoded alongside conventional modulation, for subsequent analysis or reconstruction of the spatial source distribution, thereby generalizing over the “scatter plot” of sources described in the Multiplexing Patent. Reverse exploitation of the lags, by measuring shifts in frequency or wavelength of known sources at known distances to calibrate the realized fractional rate of a receiver, is also envisaged, generalizing over reverse use of the Hubble shifts for estimating residual drifts in existing instruments, as proposed in the first Phase Gradient Patent.

(164) Use of the inventive lags in the plane of polarization might seem redundant for some applications like receiving a signal through a polarization constraining channel, or for imaging at a different angle of polarization, in analogy to FIG. 10b, or for capacity multiplication, discussed in FIG. 10a, due to the nature of polarization. However, the lags in polarization could be exploited by themselves, with the mentioned advantage that the implementation of spectral decomposition or selection, including demodulation, could be kept unchanged, for source ranging and for separation or discrimination of signals by source range, just as described for the chirp frequency lags in Section D. The polarization lags may be also combined with the chirp frequency lags in all applications.

(165) Lastly, the lags and distance information in polarization, frequency or wavelength would be also independent of the angular momentum of the received photons, so all of the properties, advantages and applications described for the inventive travelling wave spectral components remain applicable in combination with angular momentum multiplexing.

(166) Many other modifications and variations may be devised given the above description of various embodiments for implementing the principles in the present disclosure. It is intended that all such modifications and variations be considered within the spirit and scope of this disclosure as defined in the following claims.