Method and Device for Phase Modulation of a Carrier Wave and Application to the Detection of Multi-Level Phase-Encoded Digital Signals

Abstract

The method of phase modulating a carrier wave involves creating a set of signals s.sub.h(t) constituted by a carrier wave of frequency f.sub.C and of phase (t)=h.sub.0(t) that is modulated in time t in such a manner that s.sub.h(t)=cos(2f.sub.Ct+h.sub.0(t)), where h is an integer and where .sub.0(t)=2 arctan((tt.sub.0)/w.sub.0). The modulation corresponds to a single phase pulse centered on a time t.sub.0 of characteristic duration w.sub.0 that is positive, and incrementing the phase of the signal s.sub.h(t) by the quantity h2, in such a manner as to generate a single sideband frequency spectrum directly. The carrier wave may be of electromagnetic type or of acoustic type. The method applies in particular to transporting binary information by single sideband phase coding, to generating single sideband orthogonal signals, to detecting single sideband phase coded multiple-level digital signals, to transmitting single sideband phase coded binary signals in-phase and out-of-phase, and to single sideband combined amplitude-and-phase modulation.

Claims

1. A method of phase modulating a carrier wave, comprising the steps of creating a set of signals s.sub.h (t) constituted by a wave of carrier frequency f.sub.C and of phase (t)=h.sub.0(t) that is modulated in time t in such a manner that s.sub.h(t)=cos(2f.sub.Ct+h.sub.0(t)), where h is an integer and where .sub.0(t)=2 arctan((tt.sub.0)/w.sub.0), the modulation corresponding to a single phase pulse centered on a time t.sub.0, and of positive characteristic duration w.sub.0, and incrementing the phase of the signal s.sub.h(t) by the quantity h2, in such a manner as to generate a single sideband frequency spectrum directly.

2. The method according to claim 1, wherein the carrier wave is of electromagnetic type.

3. The method according to claim 1, wherein the carrier wave is of acoustic type.

4. A method of transporting binary information by single sideband phase coding by applying the method of modulation according to claim 1, the method including, for binary coding the phase, either establishing that the k.sup.th bit of duration T.sub.b contributes the quantity 2b.sub.k arctan((tkT.sub.b)/w) to the total phase (t) of the carrier, where b.sub.k=1 or 0, and where the width w is comparable to or smaller than the symbol duration T.sub.b, or else considering that the derivative of the phase is a sum of Lorentzian functions 2w/((tkT.sub.b).sup.2+w.sup.2) centered on kT.sub.b and weighted by the bit b.sub.k, and then integrating the phase, which is then added to the carrier using a phase modulation method, the quantities cos (t) and sin (t), which are the in-phase and quadrature components of the modulation signal, being calculated and combined with the in-phase amplitude cos 2f.sub.Ct and the quadrature amplitude sin 2f.sub.Ct of the carrier in order to obtain the signal for transmission in the form:
s(t)=cos(2f.sub.Ct+(t))=cos(2f.sub.Ct)cos t)sin(2f.sub.Ct)sin (t).

5. A method of generating single sideband orthogonal signals by applying the method of modulation according to claim 1, the method including, for generating a set of orthogonal functions u.sub.h (t), h=1, 2, 3, . . . , N over a finite duration T.sub.b for use in transmitting data at a rate of 1/T.sub.b per data channel, initially either considering the situation in which T.sub.b is infinite, thereby defining a single pulse, and establishing a base of orthogonal functions of the form: $u_h\left(t\right)=\frac{1}{\sqrt{2.Math.}}.Math.e^{\mathrm{ih}.Math..Math.{}_0\left(t\right)}.Math.\sqrt{\frac{d.Math..Math.{}_0\left(t\right)}{\mathrm{dt}}}=\frac{1}{\sqrt{2.Math.}}.Math.\frac{{\left(t+\mathrm{iw}\right)}^{h-1}}{{\left(t-\mathrm{iw}\right)}^h}$ where the phase is .sub.0 (t)=2 arctan(t/w), or else considering the signals $s_h\left(t\right)=e^{\mathrm{ih}.Math..Math.{}_0\left(t\right)}=\frac{{\left(t+\mathrm{iw}\right)}^h}{{\left(t-\mathrm{iw}\right)}^h}$ and then ensuring that two signals s.sub.h (t) and s.sub.h(t) are orthogonally separated by performing the following integration: $\frac{1}{2.Math.}.Math.{}_{-}^{+}.Math.s_{h^{}}^{*}\left(t\right).Math.s_h.Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}.Math.\mathrm{dt}={}_{h,h^{}}.Math..Math.\mathrm{where}.Math..Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}=2.Math.\mathrm{wl}\left(t^2+w^2\right)$ appears as a weight for the integration, the signals s.sub.h (t) being at constant amplitude, and the spectrum of the signal s.sub.h (t) being a single sideband spectrum.

6. The method according to claim 5, including generalizing to orthogonal functions over a time interval T.sub.b that is no longer infinite, but rather finite, by considering a periodic series of phase pulses spaced apart by the duration T.sub.b in order to obtain periodic signals of the following form: $s_h\left(t\right)=e^{\mathrm{ih}.Math..Math.{}_0\left(t,T_b\right)}={\left(\frac{\sin \left(\left(t+\mathrm{iw}\right).Math.\text{/}.Math.T_b\right)}{\sin \left(\left(t-\mathrm{iw}\right).Math.\text{/}.Math.T_b\right)}\right)}^h;$ where the derivative of the phase .sub.0 is a periodic sum of Lorentzian functions, which sum may be rewritten as a periodic function having the form: $\frac{d.Math..Math.{}_0}{\mathrm{dt}}=\frac{}{T_b}.Math.\frac{\mathrm{sh}\left(2.Math..Math..Math.w.Math.\text{/}.Math.T_b\right)}{{\sin }^2\left(.Math..Math.t.Math.\text{/}.Math.T_b\right)+{\mathrm{sh}}^2\left(.Math..Math.w.Math.\text{/}.Math.T_b\right)}$ where two signals differing by the integers h and h satisfy an orthogonality relationship over the time interval T.sub.b: $\frac{1}{2.Math.}.Math.{}_{-T_b/2}^{+T_b/2}.Math.s_{h^{}}^{*}\left(t\right).Math.s_h.Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}.Math.\mathrm{dt}={}_{h,h^{}};\frac{d.Math..Math.{}_0}{\mathrm{dt}}$ acting as a weight for the integration, such that by calculating $\frac{d.Math..Math.{}_0}{\mathrm{dt}}$ and then proceeding with integration in order to obtain the phase .sub.0 (t,T.sub.b) before synthesizing the signal s.sub.h (t)=e.sup.ih .sup.0.sup.(t,T.sup.b.sup.), a constant amplitude signal set is obtained presenting the characteristic of orthogonality, and a discrete spectrum is obtained that conserves the single sideband property, merely by performing a simple multiplication of the phase by an integer.

7. A method of transmitting single sideband phase-coded binary signals in-phase and out-of-phase by applying the method of modulation according to claim 1, the method including independently modulating the in-phase component and the quadrature component of the carrier in order to double the bit rate, the signal under consideration having the following form and being constituted by the sum of two amplitudes and not being of constant amplitude:
s(t)=cos(2f.sub.Ct+.sub.1(t))+sin(2f.sub.Ct+.sub.2(t)) with the phases ${}_1\left(t\right)=\underset{k}{.Math.}.Math.b_{k,1}.Math.{}_0\left(t-{\mathrm{kT}}_b\right).Math..Math.\mathrm{and}.Math..Math.{}_2\left(t\right)=\underset{k}{.Math.}.Math.b_{k,2}.Math.{}_0\left(t-{\mathrm{kT}}_b\right),$ where two independent sets of bits b.sub.k,1(2) are used to double the bit rate, the spectrum of each of the out-of-phase and in-phase amplitudes being a single sideband spectrum, the total signal itself presenting the single sideband property.

8. A method of combined modulation of the signal of a carrier combining both amplitude modulation and phase modulation by applying the method of modulation according to claim 1, the method including, for pulses where the phase is expressed in the form (t)=h.sub.0(t) (h=1, 2, 3, . . . ), producing a signal of the form:
s(t)=cos(2f.sub.Ct)(1).sup.h cos(2f.sub.Ct+h.sub.a(t)) where the resulting spectrum is a single sideband spectrum.

9. A device for generating single sideband phase pulses for performing the method according to claim 1, the device comprising a dedicated fast DSP processor or a reconfigurable fast FPGA processor, a digital-to-analog converter, first and second modules respectively for determining the quantities sin (t) and cos (t) first and second mixers for multiplying the in-phase part and the phase quadrature part of the wave of carrier frequency f.sub.C respectively by said quantities sin (t) and cos (t), and an adder circuit for combining the signals delivered by said first and second mixers.

10. A device for generating single sideband phase pulses for performing the method according to claim 1, the device comprising an analog device for generating 2N periodic sequences of pulses d.sub.0,s (t)/dt of period 2NT.sub.b, each sequence being offset in time from the preceding sequence by T.sub.b, the analog device using an elementary phase .sub.0,s (t) such that the overlap between phase pulses separated by 2NT.sub.b is negligible, in order to synthesize d(t)/dt, and a device for generating frequency harmonics that are multiples of NT.sub.b in order to synthesize a periodic sequence of signals $d.Math..Math.{}_{0,s}^q\left(t\right).Math.\text{/}.Math.\mathrm{dt}=\underset{k=-}{\overset{+}{.Math.}}.Math.d.Math..Math.{}_{0,s}\left(t-\left(k+q\right).Math.T_b\right).Math.\text{/}.Math.\mathrm{dt},-Nq<N;$ and a demultiplexer configured to act in the time interval
(kN+)/2T.sub.bt<(k+N)/T.sub.b to demultiplex the bits in order to index them as b.sub.k+q, and by using the gate function (t) of width 2NT.sub.b to construct the total phase derivative: $d.Math..Math.\left(t\right).Math.\text{/}.Math.\mathrm{dt}=\underset{q=-N}{\overset{+N}{.Math.}}.Math.b_{k+q}.Math.\left(t-\left(k+q\right).Math.T_b\right).Math.d.Math..Math.{}_{0,s}\left(t-\left(k+q\right).Math.T_b\right).Math.\text{/}.Math.\mathrm{dt}$

11. A device for demodulating a single sideband phase coded signal, the device comprising a local oscillator of frequency f.sub.C, first and second mixers, and a 0-90 phase shifter for obtaining respectively the in-phase and quadrature components cos((t)) and sin((t)) of the modulation signal, a module for differentiating each of the in-phase and quadrature components cos((t)) and sin((t)) of the modulation signal and for multiplying each of the derivatives obtained by the other one of the in-phase and quadrature components cos((t)) and sin((t)) of the modulation signal in order to obtain the phase derivative:
d/dt=cos (t)d(sin (t))/dtsin (t)d(cos (t))/dt; and a module for reconstituting an initially generated series of Lorentzian function pulses, the module comprising a threshold detector with a value of half the amplitude of a single Lorentzian function pulse so as to discriminate the value of a bit b.sub.k=1 or 0 at a time t.sub.k=kT.sub.b.

12. A device for demodulating signals by a base of orthogonal periodic signals comprising four amplitude levels including zero amplitude, the device comprising a local oscillator of frequency f.sub.C, first and second mixers, and a 0-90 phase shifter serving to obtain respectively the in-phase and quadrature components cos((t)) and sin((t)) and of the modulation signal, a device for separately detecting the four levels h=0, 1, 2, and 3 of quaternary bits by using a demodulation module associated with a Lorentzian function generator of period T.sub.b, to form the following two quantities for each of the four amplitude levels: $R_h\left(t\right)=\left(\cos \left(\left(t\right)-h.Math..Math.{}_0\left(t,T_b\right)\right)\right).Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}.Math..Math.\mathrm{and}$ $I_h\left(t\right)=\left(\sin \left(\left(t\right)-h.Math..Math.{}_0\left(t,T_b\right)\right)\right).Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}$ a device for determining the convolution with a gate function of time width T.sub.b giving: $\stackrel{\_}{R_h}\left(t\right)={}_{t-T_b/2}^{t+T_b/2}.Math.R_h\left(t-\right).Math.d.Math..Math..Math..Math.\mathrm{and}$ $\stackrel{\_}{I_h}\left(t\right)={}_{t-T_b/2}^{t+T_b/2}.Math.I_h\left(t-\right).Math.d.Math..Math.t$ from the quantities R.sub.h(t) and I.sub.h(t), a device for calculating the quantity R.sub.h(t).sup.2+I.sub.h(t).sup.2, and a threshold detection device configured to determine that a peak observed in the quantity R.sub.h(t).sup.2+I.sub.h(t).sup.2 at an instant t=kT.sub.b for the level h=0, 1, 2, or else 3 indicates that the bit b.sub.k is equal to h.

13. A device for generating single sideband phase pulses in the optical domain, the device comprising a module for supplying data b.sub.k=1 or 0, a Lorentzian function generator, a phase generator module, a phase integrator module, a laser generator for generating a carrier frequency, and an electro-optical phase modulator configured to modulate the phase of the wave directly in such a manner that, wherein under the effect of a voltage proportional to the desired phase variation, an SSB phase modulation optical signal is generated in the modulator for transmission in an optical communications network.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Other characteristics and advantages of the invention appear from the following description of particular implementations of the invention, given as examples, and with reference to the accompanying drawings, in which:

[0049] FIGS. 1A to 1C show curves plotting the spectrum density of single sideband spectra for signals corresponding to a single phase pulse and in which the carrier is phase modulated for different values of the integer number h defining the modulation index, in accordance with an aspect of the invention;

[0050] FIGS. 2A and 2B show curves plotting the spectrum density of signal spectra for carriers phase modulated with different values of a non-integer number defining the modulation;

[0051] FIGS. 3A and 3B show curves plotting the spectrum density of single sideband spectra of signals for which the carrier is phase modulated with different Gaussian values defining modulation presenting an SSB nature that is close to 100%, in accordance with an aspect of the invention;

[0052] FIG. 4 shows curves plotting the relative amplitude of the signal as a function of frequency offset from a carrier defining a double sideband spectrum in accordance with the prior art;

[0053] FIGS. 5A and 5B show, in the context of a digital coding method using single sideband phase modulation in accordance with the invention, curves plotting firstly the phase derivative signal as generated and secondly the phase signal as integrated;

[0054] FIG. 6 shows the block diagram of an example of a digital coder device using single sideband phase modulation of the invention;

[0055] FIG. 7 shows a curve plotting signal spectrum density as a function of frequency for signals that have been single sideband coded in accordance with the invention, with a phase increment that is exactly equal to 2;

[0056] FIGS. 8A and 8B show curves plotting signal spectrum density as a function of frequency for signals that have been single sideband coded in accordance with the invention, with respective phase increments that are equal to 0.9652 and to 0.91232;

[0057] FIG. 9 shows a curve plotting signal spectrum density as a function of frequency for coded signals that are not single sideband signals, since compared with the formula that gave rise to the curve of FIG. 7, Lorentzian functions have been replaced by Gaussian functions;

[0058] FIG. 10 shows the block diagram of an example device for demodulating a single sideband phase coded signal of the invention;

[0059] FIG. 11 shows a curve plotting the spectrum density of a multi-level phase coded signal that is averaged in accordance with an implementation of the invention;

[0060] FIG. 12 shows a curve plotting the spectrum density of a multi-level phase coded signal as obtained using prior art phase shift modulation, the spectrum being of the double sideband type;

[0061] FIG. 13 shows the block diagram of an example device for demodulating signals by a base of orthogonal periodic signals, of the invention;

[0062] FIGS. 14A to 14E show curves representing a starting signal, followed by four graphs representing the selective data detection signal at levels 3, 2, 1, and 0 respectively, in the context of a method of demodulating signals by a base of orthogonal periodic signals of the invention;

[0063] FIGS. 15A to 15C show curves representing firstly a phase derivative signal used for generating a signal to be detected, and secondly a signal for detecting bits of value 0 and of value 1 in the context of a method of demodulating a phase-coded binary signal of the invention;

[0064] FIGS. 16A and 16B show, for a method of transmitting single sideband phase-coded binary signals in-phase and out-of-phase in accordance with the invention, respectively the real and imaginary parts of a detected (or demodulated) signal for a series of bits in a first implementation, the phase derivative modulation of the coded signal being included in each graph;

[0065] FIG. 17 shows a curve plotting the spectrum density of single sideband phase coded binary signals of the invention, as shown in the examples of FIGS. 16A and 16B;

[0066] FIGS. 18A and 18B show, for a method of transmitting single sideband phase-coded binary signals in-phase and out-of-phase in accordance with the invention, respectively the real and imaginary parts of a detected (or demodulated) signal for a series of bits in a second implementation, the phase derivative modulation of the coded signal being included in each graph;

[0067] FIG. 19 shows a curve plotting the spectrum density of single sideband phase coded binary signals in accordance with the invention, as shown in the examples of FIGS. 18A and 18B;

[0068] FIGS. 20A to 20C show curves plotting a single sideband combined amplitude-and-phase modulated signal in an implementation of the invention, for values of the modulation index h that are respectively equal to 1, 2, and 3; and

[0069] FIG. 21 shows the block diagram of an example of a digital coder device using single sideband phase modulation of the invention in the context of an application to the optical field.

DETAILED DESCRIPTION OF IMPLEMENTATIONS

[0070] The invention relates to a method of modulating waves. Firstly, by means of an original time increment to the phase of a carrier wave, the method makes it possible to generate directly a signal having a single sideband (SSB) frequency spectrum, i.e. a signal having its frequency content lying either above or else below the frequency of the carrier wave, but not on both sides simultaneously.

[0071] Secondly, by conserving the same time form for the phase increment, but by multiplying it by an integer, the method of the invention makes it possible to generate an original base of mutually orthogonal time signals that conserve the SSB property.

[0072] Furthermore, the resulting frequency spectrum is very compact with an exponential decrease in spectral power in the single sideband.

[0073] The method may be applied to waves of any type, e.g. electromagnetic waves (from the lowest frequencies to the optical range), or indeed to sound waves.

[0074] An immediate application lies with physically coding information by phase modulation for transmitting digital data (e.g. GSM, Bluetooth, Wi-Fi, digital TV, satellite communications, RFID, etc., . . . for the microwave range, or for example for high data rate transmission in the optical range).

[0075] The invention proposes a particular form of modulation for modulating the phase of a carrier wave that, on its own, is capable of generating a single sideband frequency spectrum.

[0076] Consideration is given to a set of signals s.sub.h(t) constituted by a carrier wave of frequency f.sub.C, and of phase (t)=h.sub.0(t) that is modulated in time t:

s.sub.h(t)=cos(2f.sub.Ct+h.sub.0(t))

and where h is a positive integer or zero and where .sub.0(t)=2 arctan((tt.sub.0)/w.sub.0).

[0077] The modulation corresponds to a single phase pulse centered on time to of characteristic duration w.sub.0 (>0) and incrementing the phase of the signal s.sub.h(t) by the quantity h2.

[0078] With reference to the terminology used in the context of digital transmission based on phase modulation, h is known as the modulation index.

[0079] The spectrum density P.sub.h(f)=|{tilde over (s)}.sub.h(f)|.sup.2 of the signal, where {tilde over (s)}.sub.h(f) is the Fourier transform of s.sub.h(t), is shown in FIGS. 1A to 1C for values of h respectively equal to 1, 2, and 3.

[0080] It can be seen that the spectrum is a single sideband spectrum: the spectrum does not have any component in the frequency band lower than f.sub.C.

[0081] It should be observed that if the modulation index were selected so that h<0, then the spectrum would be a mirror image about the carrier frequency, and there would be no component in the upper band.

[0082] The choices for t.sub.0 and w.sub.0 can be arbitrary, but that does not change the single sideband property.

[0083] Explicitly, the spectrum density is given by a decreasing exponential multiplied by Laguerre polynomials L.sub.h(x) of degree h1.

P.sub.h(f)=[L.sub.h((ff.sub.C)w)].sup.2e.sup.4(ff.sup.C.sup.)w if ff.sub.C

P.sub.h(f)=0 if f<f.sub.C

[0084] In remarkable manner, the single sideband spectrum is conserved when the phase variation is generalized to the increment sum

[00013]$\left(t\right)=\underset{i}{.Math.}.Math..Math.h_i.Math.{}_i\left(t\right),$

where the h.sub.i are positive integers and .sub.i(t)=2 arctan((tt.sub.i)/w.sub.i) with arbitrary w.sub.i>0 and t.sub.i.

[0085] The condition of having the same sign for all of the integers h.sub.i is mandatory for conserving the SSB property.

[0086] The form

[00014]$\left(t\right)=\underset{i}{.Math.}.Math..Math.h_i.Math.{}_i\left(t\right)$

is taken advantage of and used below in application to examples relating to transporting binary information by single sideband phase coding.

[0087] There follow a few important properties of SSB phase modulation. [0088] For a single phase pulse, only the generic form .sub.0(t)=2 arctan((tt.sub.0)/w.sub.0) is capable of generating a single sideband, with arbitrary t.sub.0 and w.sub.0. Any other form of time variation will lead to a double sideband spectrum. The SSB property is conserved if the phase is a sum of phase pulses of form similar to .sub.0 and of arbitrary width, generated at arbitrary instants, and multiplied by an arbitrary positive integer. [0089] A single sideband spectrum that is a mirror image about the carrier frequency is obtained if the phase is of opposite sign. [0090] The multiplicative factor h (and more generally all h.sub.i, which must necessarily be of the same sign) must necessarily be an integer. [0091] FIGS. 2A and 2B show that the spectra obtained for signals s.sub.(t)=cos(2f.sub.Ct+.sub.0(t)) where (replacing h) is not an integer are spectra that are not single sideband. [0092] Multiplying .sub.0 by a positive integer makes it possible to generate a base of orthogonal functions, as explained below.

[0093] Also explained below are SSB criteria for kinds of modulation that, although not perfect, come close to perfect modulation and consequently likewise come within the ambit of the present invention.

[0094] Modulation for which the phase increment is not a multiple of 2n, causes the second sideband to appear, even if the form of the modulation is unchanged, i.e. a Lorentzian function for the phase derivative. This is clearly apparent in FIGS. 2A and 2B.

[0095] In order to quantify the SSB nature, it is possible to define the ratio of the sum of the spectrum power for frequencies higher than the carrier divided by the sum of the total spectrum, as follows:

[00015]$c_{\mathrm{SBB}}={}_{f_c}^{}.Math.P\left(f\right)/.Math.{}_{-}^{}.Math.P\left(f\right).$

[0096] In FIGS. 2A and 2B, for modulation with respectively =0.5 and 1.5 the following are found respectively c.sub.SSB=56.9% and 61.7%, i.e. spectra that are very far from those expected for single sideband (c.sub.SSB=100%).

[0097] In certain application examples, the phase variation .sub.0(t)=2 arctan((tt.sub.0)/w.sub.0) may be considered as being too slow in order to reach the value 2. Specifically: .sub.0(t)w/t, t.fwdarw..

[0098] It may be useful to define an approximate form of .sub.0(t) in which the slow portion is truncated. That is done in the example described for an application to single sideband combined amplitude-and-phase modulation.

[0099] The Lorentzian function d.sub.0(t)/dt is multiplied by a Gaussian function of width s. The approximate phase derivative, written .sub.0,s(t) is then

d.sub.0,s(t)/dt=exp(t.sup.2/2s.sup.2)2w/(t.sup.2+w.sup.2)

where the parameter is a multiplier coefficient that makes it possible to conserve a total phase increment equal to 2. The spectral power of s(t)=cos(2f.sub.Ct+.sub.0,s(t)) is shown in FIGS. 3A and 3B for a Lorentzian function of width w=0.37 and for two Gaussian widths s=2.7 and 1.85 (respectively =1.112 and 1.165).

[0100] Since the derivative of the phase is now different from a Lorentzian function, a lower sideband appears. Nevertheless, values such that s>>w make it possible to conserve an SSB nature that is close to 100% (c.sub.SSB=95.9% and 95% respectively).

[0101] There follows a description of an example of an application of the present invention to transporting binary information by single sideband phase coding.

[0102] Known phase coding methods are summarized above with reference to FIG. 4, and they all give rise to a double sideband spectrum.

[0103] In application of the present invention, there follows a description of the digital coding principle making use are single sideband phase modulation.

[0104] As an application of the present invention, consideration is given to the following phase coding: the k.sup.th bit of duration T.sub.b contributes to the total phase (t) of the carrier by the quantity 2 b.sub.k arctan((tkT.sub.b)/w) where b.sub.k=1 or 0 and the width w is comparable to or smaller than the symbol time T.sub.b.

[0105] In practice, it is simpler to consider the derivative of the phase. This is then a sum of Lorentzian functions 2w/((tkT.sub.b).sup.2+w.sup.2) centered on kT.sub.b and weighted by the bit b.sub.k.

[0106] FIG. 5A shows the phase derivative signal that is generated. The phase is then integrated, as shown in FIG. 5B, and then applied to the carrier using a conventional phase modulation method.

[0107] The quantities cos (t) and sin (t) are calculated and combined with the in-phase and quadrature amplitudes cos 2f.sub.Ct and sin 2f.sub.Ct of the carrier in order to obtain the signal for transmission:

s(t)=cos(2f.sub.Ct+(t))=cos(2f.sub.Ct)cos (t)sin(2f.sub.Ct)sin (t).

[0108] A block diagram of a device enabling such digital coding to be performed is given in FIG. 6.

[0109] In FIG. 6, there can be seen a module 101 for supplying data b.sub.k=1 or 0, a Lorentzian function generator 102, a phase generator module 103, a phase integrator module 104, modules 105 and 106 respectively for generating the quantities cos (t) and sin (t), a carrier frequency generator 107, a phase shifter module 108, mixer circuits 109 and 110, and an adder circuit 111 for combining the quantities cos (t) and sin (t) with the in-phase and quadrature amplitudes cos 2f.sub.Cd and sin 2f.sub.Ct of the carrier in order to obtain the signal for transmission:

s(t)=cos(2f.sub.Ct+(t))=cos(2f.sub.Ct)cos (t)sin(2f.sub.Ct)sin (t).

[0110] An output amplifier 112 is connected to a transmit antenna 113.

[0111] Consideration is given to the spectra of the single sideband phase coded signals.

[0112] The spectral power of the signal is shown in FIG. 7 for a pulse width such that w/T.sub.b=0.37. The frequencies are in units of 1/T.sub.b. The carrier has a frequency equal to 10/T.sub.b. Selecting some other carrier frequency would give a similar SSB spectrum relative thereto.

[0113] The spectrum shows clearly the single sideband property. To the left of the carrier frequency, the spectrum decreases extremely rapidly, its finite value being due only to finite size effects. To the right of the carrier frequency, the spectral power decreases abruptly by 20 dB at the frequency f.sub.C+1/T.sub.b, and then by a further 20 dB at the frequency f.sub.C+2/T.sub.b and so on.

[0114] The compacting of the decrease at higher frequency is a result of the computation, which involves a finite number of samples (average of 32 spectra corresponding to independent draws of a series of 259-bit random numbers of duration T.sub.b).

[0115] If a smaller width w/T.sub.b is selected, then the Fourier components extend to higher frequency. Specifically, the power decreases exponentially by e.sup.4w/T.sup.b for an increase in frequency of 1/T.sub.b, i.e. 1/100 (20 dB) for w/T.sub.b=0.37.

[0116] The spectrum also shows narrow peaks, referred to as spectrum lines, centered on the frequencies f.sub.C, f.sub.C+1/T.sub.b, f.sub.C+2/T.sub.b, etc. These are due to selecting a phase increment that is exactly equal to 2. This effect has already been noted for conventional phase modulation methods for which the increment is 2, as indicated in the article by H. E. Rowe and V. K. Prabhu, entitled Power spectrum of a digital, frequency-modulation signal, published in The Bell System Technical Journal, 54, No. 6, pages 1095-1125 (1975).

[0117] In the present method, it is important not to depart from this value since that would lead to the reappearance of a lower sideband in the spectrum.

[0118] Nevertheless, in practice, this lower sideband in the spectrum remains negligible providing the increment is only a few percent less than or greater than 2, while the narrow peaks in the spectrum are reduced or even eliminated. This is shown in FIG. 8A for a spectrum in which the phase increment is 0.9652. The reduction in the narrow peaks is accompanied by a noisy portion of the curve for frequencies in the range 9.5 to 10, revealing the appearance of a small but non-zero contribution in the lower sideband. For a phase increment of 0.91232, the phenomenon is a little more marked (see FIG. 8B). For a phase increment less than 0.9 (2) or greater than 1.1 (2), it is considered that the SSB nature is lost.

[0119] Finally, FIG. 9 shows the spectrum that is obtained if the Lorentzian functions in d/dt are replaced by a Gaussian functions (while conserving the same phase increment and for comparable pulse width). The spectrum difference is striking. The presence of a lower sideband is very marked.

[0120] With reference to FIG. 10, there follows a description of an example method and device for demodulating the SSB phase coded signal in accordance with the invention.

[0121] On reception, by an antenna 201 and an amplifier 202, the first demodulation step for extracting the signal from the carrier is conventional. A local oscillator 203 of frequency f.sub.C associated with mixers 204 and 205 via a 0-90 phase shifter 206 serves to obtain the in-phase and quadrature components cos((t) and sin((t)) of the modulation signal. By differentiating them and multiplying them by their partners in a calculation module 207, the following phase derivative is obtained:

d/dt=cos (t)d(sin (t))/dtsin (t)d(cos (t))/dt.

[0122] This makes it possible to reconstitute the series of Lorentzian pulses as initially generated, such as those of FIG. 5A. By placing a threshold detector 208 at a value of half the amplitude of a single Lorentzian function, it is easy to discriminate between the value of a bit b.sub.k=1 or 0 at time t.sub.k=kT.sub.b. A clock 209 supplies pulses to the threshold detector 208 at a rate 1/T.sub.b.

[0123] In practice, detection noise is also added to the detected signal. Differentiating a signal, in this example sin and cos , has the effect of increasing the effect of noise. It is possible to use other demodulation means that do not make use of differentiation, as described below, with reference to the orthogonality property of the single sideband phase pulses.

[0124] There follows a description of a method of generating single sideband orthogonal signals.

[0125] The object is to generate a set of orthogonal functions u.sub.h(t), h=1, 2, 3, . . . , N over the finite duration T.sub.b, in order to use them, e.g. for data transmission at the rate 1/T.sub.b per data channel.

[0126] In order to construct these orthogonal functions, it is useful to begin by considering the situation in which T.sub.b is infinite (single pulse).

[0127] The base of orthogonal functions is then:

[00016]$u_h\left(t\right)=\frac{1}{\sqrt{2.Math..Math.}}.Math.e^{i.Math..Math.h.Math..Math.{}_0\left(t\right)}.Math.\sqrt{\frac{d.Math..Math.{}_0\left(t\right)}{d.Math..Math.t}}=\frac{1}{\sqrt{2.Math..Math.}}.Math.\frac{{\left(t+i.Math..Math.w\right)}^{h-1}}{{\left(t-i.Math..Math.w\right)}^h}$

where use is made of the above-defined phase .sub.0 (t)=2 arctan(t/w).

[0128] For reasons of simplicity, the functions are centered on t=0. It can be verified that

.sub..sup.+u.sub.h*(t)u.sub.h(t)dt=.sub.h,h.

[0129] In practice, it may be more advantageous to consider the signals

[00017]$s_h\left(t\right)=e^{i.Math..Math.h.Math..Math.{}_0\left(t\right)}=\frac{{\left(t+i.Math..Math.w\right)}^h}{{\left(t-i.Math..Math.w\right)}^h}$

and then to ensure two signals s.sub.h(t) and s.sub.h(t) are orthogonally separated by performing the following integration:

[00018]$\frac{1}{2.Math..Math.}.Math.{}_{-}^{+}.Math.s_{h^{}}^{*}\left(t\right).Math.s_h\left(t\right).Math.\frac{d.Math..Math.{}_0}{d.Math..Math.t}.Math.\mathrm{dt}={}_{h,h^{}}.Math..Math.\mathrm{where}.Math..Math.\frac{d.Math..Math.{}_0}{d.Math..Math.t}=2.Math..Math.w/\left(t^2+w^2\right)$

appears as a weight (or metric) for the integration.

[0130] With this definition, the signals s.sub.h(t) are at constant amplitude (unity modulus), which can present a practical advantage when generating them (constant transmission power). The spectrum of the s.sub.h(t) occupies a single sideband.

[0131] Generalization to orthogonal functions over a time interval T.sub.b that is no longer infinite, but rather finite, is obtained by considering the periodic series of phase pulses spaced apart by the duration T.sub.b. This gives the following periodic signals:

[00019]$s_h\left(t\right)=e^{i.Math..Math.h.Math..Math.{}_0\left(t,T_b\right)}={\left(\frac{\sin \left(\left(t+i.Math..Math.w\right)/T_b\right)}{\sin \left(\left(t-i.Math..Math.w\right)/T_b\right)}\right)}^h.$

The derivative of the phase .sub.0 is a periodic sum of Lorentzian functions.

[0132] This sum may be re-written in the form of a periodic function:

[00020]$\frac{d.Math..Math.{}_0}{d.Math..Math.t}=\frac{}{T_b}.Math.\frac{\mathrm{sh}\left(2.Math..Math..Math..Math.w/T_b\right)}{{\sin }^2\left(.Math..Math.t/T_b\right)+{\mathrm{sh}}^2\left(.Math..Math.w/T_b\right)}.$

[0133] Two signals differing by the integers h and h satisfy an orthogonality relationship over the time interval T.sub.b:

[00021]$\frac{1}{2.Math..Math.}.Math.{}_{-T_b/2}^{+T_b/2}.Math.s_{h^{}}^{*}\left(t\right).Math.s_h\left(t\right).Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}.Math.\mathrm{dt}={}_{h,h^{}}$

once more

[00022]$\frac{d.Math..Math.{}_0}{\mathrm{dt}}$

acts as a weight for the integration.

[0134] In practice

[00023]$\frac{d.Math..Math.{}_0}{\mathrm{dt}}$

is calculated (or generated) and then integrated in order to give .sub.0(t,T.sub.b) and then s.sub.h(t)=e.sup.ih.sup.0.sup.(t,T.sup.b.sup.) is synthesized. It can be seen that merely multiplying the phase by an integer makes it possible to obtain a set of constant amplitude signals that are orthogonal. Furthermore, the spectrum, which is now discrete, conserves the single sideband of property.

[0135] There follows an application example for detecting phase-coded multi-level digital signals.

[0136] Consideration is given initially to selecting the multi-level phase coding.

[0137] The object is to code 2 bits on four levels, e.g. like the 2Binary-1Quaternary (2B1Q) method of amplitude modulation, but transposed to phase modulation.

[0138] Naturally, it is possible to generalize to N levels (N-ary bits) with h=0, 1, . . . , N1 and N=2. The bit rate per second is no longer 1/T.sub.b, as above, but becomes p/T.sub.b.

[0139] It is possible to select

[00024]$\left(t\right)=\underset{k}{.Math.}.Math.b_k.Math.{}_0\left(t-{\mathrm{kT}}_b,T_b\right)$

phase coding where the bit b.sub.k has values b.sub.k=0, 1, 2, 3 (respectively for 00, 01, 10 and 11) and is defined in the time interval (k)T.sub.bt<(k+)T.sub.b.

[0140] Thus, full use could be made of the orthogonality of the signals s.sub.b.sub.k(tkT.sub.b)=e.sup.ib.sup.0.sup.(tkT.sup.b.sup.,T.sup.b.sup.) over each time interval [k,k+]T.sub.b in order to recover the value of the quaternary bit b.sub.k on demodulation. That is one possible application of the present invention.

[0141] However, for two consecutive bits b.sub.k and b.sub.k+1 of different values, the derivative of the phase has a discontinuity equal to (b.sub.k+1b.sub.k)d.sub.0(T.sub.b/2,T.sub.b)/dt. Such discontinuities generate spectrum tails that decrease slowly. In the presently selected application example, preference is given to spectrum compactness at the cost of making poorer use of the orthogonality property. For this purpose, phase is coded as stated above with

[00025]$\left(t\right)=\underset{k}{\overset{}{.Math.}}.Math.b_k.Math.{}_0\left(t-{\mathrm{kT}}_b\right).Math..Math.\mathrm{where}$${}_0\left(t\right)={}_0\left(t,T_b=\right)=2.Math..Math.\arctan \left(t.Math.\text{/}.Math.w\right).Math..Math.\mathrm{and}$$b_k=0,1,2,3.$

[0142] The derivative of the phase is thus a sum of Lorentzian functions of amplitude that takes on four value levels randomly. This coding ensures there is no phase discontinuity. Nevertheless, the signals e.sup.ih.sup.0.sup.(t) do not satisfy orthogonality relationships with the functions e.sup.ih.sup.0.sup.(t,T.sup.b.sup.) that are used on demodulation for retrieving the b.sub.k, but rather they satisfy orthogonality relationships that are only approximate. Nevertheless, demodulation remains effective in practice.

[0143] The principle for data transmission is similar to that shown in FIGS. 5A and 5B, but in which the binary bits 0, 1 are replaced by the quaternary bits b.sub.k=0, 1, 2, 3.

[0144] There follows a description of the spectrum of the multi-level phase-coded signal.

[0145] As a proposed application example, consideration is given to the spectrum of a signal made up of a run of 33 quaternary bits of duration T.sub.b. The signal that is generated is as follows:

[00026]$s\left(t\right)=\cos \left(2.Math..Math..Math.f_C.Math.t+\left(t\right)\right),\mathrm{where}.Math..Math.\left(t\right)=\underset{k=-16}{\overset{+16}{.Math.}}.Math.b_k.Math.{}_0\left(t-{\mathrm{kT}}_b\right).$

The quaternary bits b.sub.k=0, 1, 2, or 3 (corresponding to the binary bits 00, 01, 10, 11) are selected by using a pseudorandom number generator to represent a data sequence. The rate is 2/T.sub.b bits per second. The frequency of the carrier is selected as f.sub.C=10/T.sub.b and the width is selected as w=0.371.

[0146] FIG. 11 shows the average frequency spectrum corresponding to 32 different sequences of quaternary bits.

[0147] The single sideband nature is clearly apparent. The spectrum does not have any significant component for frequencies lower than the average carrier frequency f=f.sub.C+b.sub.k/T.sub.b=11.5/T.sub.b.

[0148] For frequencies higher than f+2/T.sub.h, the spectrum decreases rapidly and exponentially at about 10 dB for each 1/T.sub.b (20 dB for a frequency increase equal to the bit rate 2/T.sub.b). A greater width w would give an even faster exponential decrease.

[0149] By way of comparison, the following graph shown in FIG. 12 is the result of modulation of the frequency shift keying (FSK) type in which frequency is modulated on four levels

[00027]$\left(i.e..Math.d.Math..Math.\left(t\right)/\mathrm{dt}=2.Math..Math.\underset{k}{.Math.}.Math.b_k\left(1/T_b\right)\right).$

[0150] In such a configuration that does not form part of the present invention, it can be seen that the spectrum is of the double sideband type on either side of the average carrier frequency f=f.sub.C+b.sub.k/T.sub.b=11.5/T.sub.b. Its main width is 2/T.sub.b, but the spectrum is flanked by spectrum tails that decrease slowly and not exponentially.

[0151] There follows a description of a method and a device for demodulating signals on a base of orthogonal periodic signals.

[0152] On reception, the first step of demodulation for extracting the signal from the carrier is conventional and similar to the example given above with reference to FIG. 10. The signal received by an antenna 301 is amplified in an amplifier 302. A local oscillator 303 of frequency f.sub.C associated with mixers 304 and 305 via a 0-90 phase shifter 306 serves to obtain the in-phase and quadrature components cos((t) and sin((t)) of the modulation signal. In a first embodiment, it would be possible to use the same scheme as in FIG. 10 to obtain the derivative of the phase of the modulated signal. Nevertheless, detecting for amplitude levels (including zero amplitude) would appear to be difficult because of the overlap between the Lorentzian functions of different attitudes transmitted at neighbouring instants.

[0153] Thus, a preferred solution is to use the following base of periodic orthogonal signals

[00028]$s_h\left(t\right)=e^{\mathrm{ih}.Math..Math.{}_0\left(t,T_b\right)}={\left(\frac{\sin \left(\left(t+\mathrm{iw}\right)/T_b\right)}{\sin \left(\left(t-\mathrm{iw}\right)/T_b\right)}\right)}^h.$

[0154] In practice, the four levels h=0, 1, 2, and 3 of quaternary bits are detected separately. This is achieved by using appropriate demodulation means given reference 307 in FIG. 13 to form the four quantities

[00029]$R_h\left(t\right)=\left(\cos \left(\left(t\right)-h.Math..Math.{}_0\left(t,T_b\right)\right)\right).Math..Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}.Math..Math.\mathrm{and}$$I_h\left(t\right)=\left(\sin \left(\left(t\right)-h.Math..Math.{}_0\left(t,T_b\right)\right)\right).Math.\frac{d.Math..Math.{}_0}{\mathrm{dt}}$

and then by using a module 308 to perform convolution with a gate function of time width T.sub.b giving:

[00030]$\stackrel{\_}{R_h}\left(t\right)={}_{t-T_b/2}^{t+T_b/2}.Math.R_h\left(1-\right).Math.d.Math..Math..Math..Math.\mathrm{and}.Math..Math.\stackrel{\_}{I_h}\left(t\right)={}_{t-T_b/2}^{t+T_b/2}.Math.I_h\left(t-\right).Math.\mathrm{dt}$

[0155] Thereafter, in a module 309, the quantity R.sub.h(t).sup.2+I.sub.h(t).sup.2 is calculated. A peak observed in this quantity (which constitutes a bit level h detection signal) at an instant t=kT.sub.b for the level h=0, 1, 2, or indeed 3 indicates that the bit b.sub.k is equal to h.

[0156] Four threshold detectors 310 to 313 are thus used for the levels h=0, 1, 2, and 3 respectively.

[0157] A clock 314 serves to deliver pulses at a rate of 1/T.sub.b.

[0158] FIG. 13 also shows a generator of periodic Lorentzian functions of period T.sub.b and a calculation module 316 for supplying the module 307 with the values sin(h.sub.0(t,T.sub.b)) and cos(h.sub.0(t,T.sub.b)).

[0159] FIGS. 14A to 14E show from bottom to top: the starting signal

[00031]$d.Math..Math./\mathrm{dt}=\underset{k=-16}{\overset{+16}{.Math.}}.Math.b_k.Math.d.Math..Math.{}_0\left(t-{\mathrm{kT}}_0\right)/\mathrm{dt}$

(FIG. 14A); followed by four graphs showing the selective detection signals for detecting bits of levels h=3, 2, 1, and 0 (FIGS. 14B to 14E).

[0160] The information to be taken into consideration for a multi-level bit b.sub.k is given by the values of the detection signals taken at exactly t=kT.sub.b. For example, for k=8, the level of the signal detected for h=3 (FIG. 14B) takes on a large value and shows a peak, while for the graphs corresponding to h=0, 1, and 2 (FIGS. 14E, 14D, and 14C), the signal levels are low: the bit b.sub.k=8 thus has the value 3 (or 11). For the levels h=1 and 2 (FIGS. 14D and 14C), there can sometimes also be found significant values for integer t/T.sub.b that do not correspond to peaks, but rather to troughs, and that are not taken into consideration for identifying the value of b.sub.k. For example for k=2, b.sub.k=0 is identified by a peak for the detection signal with h=0 (FIG. 14E), while the value of the detection signal with h=1 (FIG. 14D) gives a value that is not zero but that is associated with a trough.

[0161] In spite of the large amount of overlap of the Lorentzian functions, it can be seen that the method consisting in projecting the signal onto the base of periodic signals makes it possible to sort the bits selectively depending on their levels in a manner that is very effective.

[0162] Consideration is given once more to demodulating the phase coded binary signal as given above by way of example.

[0163] As mentioned above, reconstituting the in-phase derivative cannot be effective when the detected signals are noisy. The demodulation method making use of the orthogonality property, as explained above for quaternary bits, is preferable and applies even more effectively for a binary signal.

[0164] As before, detection consists in calculating:

(.sub.tT.sub.b.sub./2.sup.t+T.sup.b.sup./2 cos((t)h.sub.0(t,T.sub.b))d.sub.0/dt).sup.2+(.sub.tT.sub.b.sub./2.sup.t+T.sup.b.sup./2 sin((t)h.sub.0(t,T.sub.b))d.sub.0/dt).sup.2

where now h=1 or 0 and

[00032]$\left(t\right)=\underset{k=-16}{\overset{+16}{.Math.}}.Math.b_k.Math.{}_0\left(t-{\mathrm{kT}}_b\right).$

[0165] Once again, .sub.0(t) does not satisfy an orthogonality relationship with .sub.0(t,T.sub.b) but the overlap is sufficient for effective demodulation. FIGS. 15A to 15E show, for the last 38 bits of the sequence of 259 bits shown in FIG. 4, the detection signal for bits of value 0 (FIG. 15A) and of value 1 (FIG. 15B). FIG. 15C reproduces the in-phase derivative signal that was used for generating the signal to be detected.

[0166] The present invention lends itself to various other applications, and in particular to in-phase and out-of-phase transmission of phase-coded binary signals in single sideband.

[0167] In this application, it is proposed to take advantage of the possibility of modulating the in-phase and the quadrature components of the carrier independently in order to double the data rate (i.e. in order to have a bit rate that is equal to twice the symbol rate).

[0168] In the above examples, the signals are of constant power (or amplitude) with:

[00033]$s\left(t\right)=\cos \left(2.Math..Math..Math.f_C.Math.t+\left(t\right)\right).Math..Math.\mathrm{and}.Math..Math.\left(t\right)=\underset{k}{.Math.}.Math.b_k.Math.{}_0\left(t-{\mathrm{kT}}_b\right)$

[0169] In the present example, since the signal is the sum of two amplitudes, it is no longer at constant amplitude:

s(t)=cos(2f.sub.Ct+.sub.1(t))+sin(2f.sub.Ct+.sub.2(t))

[0170] In this example

[00034]${}_1\left(t\right)=\underset{k}{.Math.}.Math.b_{k,1}.Math.{}_0\left(t-{\mathrm{kT}}_b\right).Math..Math.\mathrm{and}.Math..Math.{}_2\left(t\right)=\underset{k}{.Math.}.Math.b_{k,2}.Math.{}_0\left(t-{\mathrm{kT}}_b\right),$

where two independent sets of bits b.sub.k,1(2) are used in order to double the rate.

[0171] Since the spectra for the out-of-phase and in-phase amplitudes are both single sideband, the total signal continues to present the single sideband property (see FIGS. 17 and 19).

[0172] There follows an explanation of the demodulation procedure for retrieving the information on the transmitted bits.

[0173] For simplification purposes, this explanation is restricted to binary bits. For good demodulation, it is shown that the relative phase variations of .sub.1(t) and of .sub.2(t) need to remain small. These variations come from the interference (overlap) between adjacent phase pulses (also known as intersymbol interference (ISI)), a constraint that does not apply to the example of FIGS. 3A and 3B.

[0174] On demodulating the carrier, the in-phase and out-of-phase parts are obtained, i.e. respectively:

Re(t)=cos(.sub.1(t))sin(.sub.2(t))

Im(t)=sin(.sub.1(t))+cos(.sub.2(t))

[0175] When w<<T.sub.b (no ISI), for t=kT.sub.b, the real part Re(kT.sub.b)=cos(b.sub.k,1)sin(b.sub.k,2) gives 1 or 1 for b.sub.k,1=0 or 1 respectively and independently of the value of b.sub.k,2.

[0176] Likewise, the imaginary part: Im(kT.sub.b)=sin(b.sub.k,1)+cos(b.sub.k,2 ) gives 1 or 1 for b.sub.k,2=0 or 1 respectively, independently of the value of b.sub.k,1.

[0177] R.sub.e gives information about the first set of bits and I.sub.m gives information about the second set of bits. When w/T.sub.b is greater, an additional phase

[00035]${}_1=\underset{k^{}k}{.Math.}.Math.b_{{}^{}.Math.k,1}.Math.{}_0\left(\left(k^{}-k\right).Math.T_b\right)$

is added to the expected phase in T.sub.b .sub.1(kT.sub.b)=b.sub.k,1+0. In similar manner, a phase .sub.2 affects .sub.2. This gives:

Re(kT.sub.b)=cos(b.sub.k,1+.sub.1)sin(.sub.2))

Im=(kT.sub.b)=sin(.sub.1)+cos(b.sub.k,2+.sub.2)

[0178] In order to recover each of the bits transmitted at time kT.sub.b without error, it is essential for |.sub.1|<</4 and |.sub.2<</4 (i.e. to ensure that Re and Im always have a value that is significantly positive (bit 0) or negative (bit 1) but never close to 0).

[0179] If time filtering of d/dt is used to limit the ISI to bits transmitted at times lying in the range (kN)T.sub.b, then:

|.sub.1,2|.sup.MAX(+ln(N))w/T.sub.b<</4;

where =0.577 . . . is Euler's constant. In practice this gives N<<4.7 for w/T.sub.b=0.37, N<<6.5 for w/T.sub.b=0.32 and N<<39 for w/T.sub.b=0.185. Under all circumstances, time filtering is necessary to limit interference between adjacent phase pulses.

[0180] Some examples are given below.

[0181] One way of limiting ISI is to use a Lorentzian-Gaussian function for the phase derivative, as mentioned above with reference to FIGS. 3A and 3B, with the elementary phase pulse obtained by integrating:

d.sub.0,s(t)/dt=exp(t.sup.2/2s.sup.2)w/(t.sup.2+w.sup.2)

where the parameter p is a coefficient that makes it possible to conserve a total phase increment equal to 2.

[0182] FIGS. 16A and 16B show the signals Re (FIG. 16A) and Im (FIG. 16B) in arbitrary units for a series of bits with w/T.sub.b=0.32 and s/T.sub.b=3.2 (=1.0811) together with:

[00036]$d.Math..Math.{}_1\left(t\right)/\mathrm{dt}=\underset{k}{.Math.}.Math.b_{k,1}.Math.d.Math..Math.{}_{0,s}/\mathrm{dt}.Math..Math.\mathrm{and}.Math..Math.d.Math..Math.{}_2\left(t\right)/\mathrm{dt}=\underset{k}{.Math.}.Math.b_{k,2}.Math.d.Math..Math.{}_{0,s}/\mathrm{dt}$

[0183] The corresponding frequency spectrum for a carrier frequency f.sub.C=13 (units of 1/T.sub.b) is shown in FIG. 17.

[0184] The SSB nature is well conserved, apart from a small spectrum component in the lower sideband, given that the elementary phase derivative is no longer strictly a Lorentzian function. It can also be seen that 90% of the spectrum is concentrated in a 1/T.sub.b frequency band, i.e. half the bit rate.

[0185] The following example shows that it is possible to reach 98% of the spectrum in a frequency band equal to half the bit rate using the following parameters: w/T.sub.b=0.37 and s/T.sub.b=2.7 (=1.112). FIGS. 18A and 18B show the signals Re (FIG. 18A) and Im (FIG. 18B) and FIG. 19 shows the spectrum.

[0186] These two examples show that a very high spectrum efficiency (ratio of the bit rate over the spectrum width) of about 2 bits per second per hertz (bit/s/Hz) can be obtained with a very compact SSB spectrum.

[0187] There follows a description of an application to single sideband combined amplitude-and-phase modulation.

[0188] A direct application of the present invention consists in modulating the carrier signal simultaneously in amplitude and in phase.

[0189] In the description above, consideration is given only to phase modulation (t). The principle is to manage one pulse of the signal, i.e. a signal that starts and then returns to zero. For a single pulse is centered on t.sub.0 and of width w.sub.0, and for the elementary phase pulse (t)=.sub.0 (t)=2 arctan((tt.sub.0)/w.sub.0):

s(t)=cos(2f.sub.Ct)+cos(2f.sub.Ct+.sub.a(t))

[0190] The signal may also be written in the form of amplitude modulation cos(.sub.0(t)/2) and of phase modulation .sub.0(t)/2:

s(t)=2 cos(.sub.0(t)/2)cos(2f.sub.Ct+.sub.0(t)/2).

[0191] It is thus possible to generalize for pulses where (t)=h.sub.0(t) (h=1, 2, 3, . . . ) with:

s(t)=cos(2f.sub.Ct)(1).sup.h cos(2f.sub.Ct+h.sub.a(t))

FIGS. 20A to 20C show the signal s(t) respectively for h=1, 2, and 3 and pulse width modulation w/T.sub.b=0.37 with a carrier f.sub.C=13. The time signals are mutually orthogonal.

[0192] The resulting spectrum is given by the sum of the SSB spectrum (cos(2f.sub.Ct+(t) term) plus the spectrum localized at the frequency f.sub.C (cos(2f.sub.Ct) term), so it is indeed a single sideband spectrum. It is identical to the spectrum given in FIGS. 1A to 1C, for h=1, 2, and 3 respectively, with the exception of a strengthening at the frequency f.sub.C.

[0193] There follow a few practical examples of single sideband phase pulse generators.

[0194] It is possible to synthesize the carrier and its modulation in all-digital manner: in the present state of the art, for phase pulses generated at a rate of up to several million pulses per second, and for carriers up to GHz order, digital methods are available making use of dedicated fast processors (known as digital signal processors (DSPs)), or of reconfigurable fast processors (known as field programmable gate arrays (FPGAs)).

[0195] At lower bit rates, at present less than 1 million pulses per second, but potentially increasing with technological progress, it is possible to use inexpensive solutions based on software radio cards. After digital-to-analog conversion, the quantities sin (t) and cos (t) are generated and then sent separately to the mixers, as in the embodiment of FIG. 6 in order to multiply the in-phase part and the quadrature part of the carrier.

[0196] By way of alternative, still using digital synthesis, the phase (t) is calculated followed by digital-to-analog conversion and then sent to a voltage-controlled phase shifter or an oscillator.

[0197] It is also possible to perform analog synthesis. Under such circumstances, by using an elementary phase .sub.0,s(t) such that the overlap between phase pulses separated by 2NT.sub.b is negligible, d(t)/dt is synthesized by generating 2N periodic sequences of pulses d.sub.0,s(t)/dt of period 2NT.sub.b, each sequence being offset in time from the preceding sequence by T.sub.b. The periodic sequence

[00037]$d.Math..Math.{}_{0,s}^q\left(t\right).Math.\text{/}.Math.\mathrm{dt}=\underset{k=-}{\overset{+}{.Math.}}.Math.d.Math..Math.{}_{0,s}\left(t-\left(k+q\right).Math.T_b\right).Math.\text{/}.Math.\mathrm{dt},-Nq<N;$

is easy to synthesize by generating frequency harmonics that are multiples of NT.sub.b with the appropriate phase and amplitude.

[0198] In the time interval

(kN+)/2T.sub.bt<(k+N)/T.sub.b

the bits are de-multiplexed in order to index them as b.sub.k+q and by using the gate function (t) of width 2NT.sub.b it is possible to construct the total phase derivative:

[00038]$d.Math..Math.\left(t\right).Math.\text{/}.Math.\mathrm{dt}=\underset{q=-N}{\overset{+N}{.Math.}}.Math.b_{k+q}.Math.\left(t-\left(k+q\right).Math.T_b\right).Math.d.Math..Math.{}_{0,s}\left(t-\left(k+q\right).Math.T_b\right).Math.\text{/}.Math.\mathrm{dt}$

[0199] This procedure for generating periodic pulses by synthesizing harmonics at frequencies that are multiples of NT.sub.b can easily be performed in the frequency domain up to tens of GHz by cascading frequency multipliers, or by using frequency comb generators for generating base harmonics.

[0200] In the optical domain, it is possible to modulate the phase of the wave directly with electro-optical modulators, the voltage applied to the modulator being proportional to the phase variation, as shown in the embodiment of FIG. 21.

[0201] In FIG. 21, there can be seen a module 401 for supplying data b.sub.k=1 or 0, a Lorentzian function generator 402, a phase generator module 403, a phase integrator module 404, a laser generator 406 for generating the carrier frequency, and an electro-optical phase modulator 405 that serves to modulate the phase of the wave directly in such a manner that, under the effect of a voltage representing the desired phase variation, an SSD phase modulated optical signal is generated in the modulator 405 for transmission in the optical communications network.

[0202] Various modifications and additions may be applied to the embodiments described without going beyond the ambit defined by the accompanying claims.

[0203] In particular, various embodiments may be combined with one another, providing there is no mention to the contrary in the description.