Method of coherently demodulating a continuous phase, digitally modulated signal with a constant envelope.

The invention relates to a method of coherent demodulation by digital processing of a digitally modulated signal in continuous phase and with constant envelope, the modulated term of said phase being equal to the convolution product of the phase pulse spread over several bit times by binary information transmitted in packets, the received signal transposed into baseband on two quadrature channels being converted to digital and transmitted to a signal processor which performs the processing of the demodulation process.

This process can be applied to any phase modulation of the aforementioned genre: (GMSK, MSK, 2SRC, TFM, GTFM ...) whose law of evolution following a progressive variation of the phase has the advantage of reducing spectrum. Another advantage results from the fact that the energy transmitted is constant.

In particular, modulation of the GMSK type, the phase variation of which is spread over the largest time interval (5 bit times), has the best spectral efficiency. Unfortunately, this has the effect of significantly increasing inter-symbol interference.

The use of this narrowband modulation can be envisaged in many fields such as protected communications systems in VHF and UHF, satellite transmissions or radio mobile networks. The aforementioned advantages made it retained by the Special Mobile Group (GSM) of CEPT to be used in the future Pan-European digital mobile network from 1992.

The various known demodulation methods use differential or coherent methods.

The first method has the advantage of being relatively simple but the performance in terms of error rate is very degraded.

The coherent demodulation has better performance but it requires an additional device for recovering the carrier phase.

One of the weaknesses encountered with this type of demodulation lies in the use of conventional synchronization methods which use phase locked loops to recover the carrier and the clock.

Indeed in the case of a system operating in Multiple Access with Time Distribution (TDMA) or in frequency evasion (EVF) and when the signal undergoes fading due to the channel, the resynchronization times of the analog loops become too long and reduce the useful life of the signal (cf. US Pat. No. 4,570,125 to RBGibson and B. Hill).

The main advantage of implementing a coherent demodulation method by digital signal processing offers the possibility of storing and processing the signal in packets for each of which a sequential processing must be carried out, ending with a decision on the binary information transmitted. .

The first operation in the sequence is to find the beginning of the packet; it is the synchro-frame. Then the synchro-bit makes it possible to determine the instants of decision and to ensure the correct temporal setting of the adapted filter. The role of this filter is to reduce noise without degrading useful information.

The last treatment is extremely important: it is the estimation of the initial phase and the residual frequency difference.

The initial phase is a parameter that is not controlled in a transmission system. A bad estimate of this parameter is disastrous on the error rate.

The residual frequency difference is the result of the frequency difference between the transmitter and the receiver and the frequency difference due to the Doppler effect. A faulty evaluation of this frequency difference results in decision errors on the last bits of the packet, when the phase has turned enough to cause such errors.

After having estimated the initial phase and the residual frequency deviation, compensation is carried out and finally the bits transmitted are decided.

A digital demodulation method used in 2SRC was proposed in the article by LOUBATON and VALLET entitled: "Pseudo-coherent demodulation of MSK type signals adapted to EVF transmissions" and published in the Thomson-CSF Technical Review, vol. 17, September 1985, N ° 3, pages 521-554.

In this method we find the following processing sequence: synchro-frame by partial correlation; synchro-bit by detection of zero crossing of the differential phase; suitable filtering; estimation of the residual frequency deviation by fast Fourier transformation on the squared samples; estimation of the initial phase by average; phase compensation.

After simulation, it turns out that the proposed algorithms do not adapt well to GMSK.

Indeed, because of the inter-symbol interference which can no longer be neglected, the determination of the synchro-bit is very degraded in the presence of a residual frequency deviation greater than 200 Hz.

In addition to estimate this frequency difference, it is no longer possible to eliminate the modulation by performing a squared elevation.

A demodulation technique for the transmission of packets by radio is given in the article by C. HEEGARD, JA HELLER and AJ VITERBI entitled: "A microprocessor-based PSK Modem for Packet Transmission over Satellite Channels" and published in IEEE, vol. COM-26, N ° 5, May 1978, pages 552 to 564.

Inspired by this technique, which applies only to PSK type modulations without intersymbol interference, the method of the invention aims to obtain a synchronization allowing the coherent demodulation to be carried out of any type of modulation having l intersymbol interference even with a high noise level and residual frequency deviation.

To this end, this method is remarkable in that each of said packets comprises a preliminary sequence by which a known reference signal is available on N bits making it possible to approximately detect the synchro-frame and the synchro-bit by correlation with the differential phase and to initiate the estimation of the initial phase parameters ϑ₀ and of the residual frequency difference Δf₀, the gradual refinement of the approximate values ​​being obtained by means of two nested digital loops: a slow loop initialized beyond a threshold value for the detection of the synchro-bit, a fast loop carrying out intermediate decisions on additional blocks of bits for the estimation of ϑ₀ and Δf₀.

Said synchro-frame and synchro-bit detections are obtained by a first correlation on the differential phase making it possible to know the transmission time of said packets at ± T / 4, T being the duration of a bit, then by a second correlation on the differential phase performed with a reference signal shifted by T / 4. The corresponding correlation functions each have a peak independent of the initial phase and very little dependent on the residual frequency difference, the upper level peak and the lower level peak respectively defining a main SYNP bit and a SYNCHRO bit. secondary SYNS. The precision of ± T / 8 thus obtained on the synchro-bit is sufficient to know the sampling instant.

Said synchro-bit detection is followed by suitable filtering performed with the SYNP value by a finite impulse response filter of the Gaussian type in order to limit the noise band.

Said estimates of the initial phase ϑ₀ and of the residual frequency difference Δf₀ following said filtering comprise the following steps: - Elimination of the modulation term by performing the product of the signal received by the conjugate of the reference signal. - Phase sequence by eliminating the phase jumps of 2π to obtain a linear variation with the equation y = Δω₀x + ϑ₀ with Δω₀ = 2πΔf₀. - Calculation of the estimated parameters Δω̂₀ and ϑ̂₀ by a linear regression and deviation method<maths id="math0001" num=""><img file="EP0349064A1_D0001.tif" /></maths> between the points corresponding to said unwound phase and said regression line.

If said deviation ε is less than said threshold value, said estimation of Δf₀ and ϑ₀ is refined in a fast loop in several passes using the intermediate decisions on the N bits of the preliminary sequence to which a certain number of bits are added to each pass decisions.

If said deviation ε exceeds said threshold value due to a defective synchro-bit evaluation, said calculation process is reset according to a slow loop to redo the adapted filtering and the estimation of Δf₀ and ϑ₀ from the other synchro-bit value equal to said secondary SYNS value.

After the last pass, a compensation leaves only the phase component of the signal which is no longer affected by the residual frequency deviation or by the phase at the origin.

The final decision is then made, then a differential decoding finally provides the sequence of binary information transmitted.

The invention will be better understood with the aid of the following description given by way of nonlimiting example, said description being accompanied by drawings which represent:<ul id="ul0001" list-style="none"><li>Figure 1: the block diagram of a modulator-demodulator device.</li><li>Figure 2: phase pulse variations for GMSK, MSK and 2SRC type modulations.</li><li>Figure 3: Spectral occupancy diagrams for GMSK, MSK and 2SRC type modulations.</li><li>Figure 4: the eye diagram for GMSK modulation.</li><li>Figure 5: the temporal variations of the phase of the signal received after filtering during the evaluation sequence of ϑ₀ and Δf₀.</li><li>Figure 6: the flowchart of all demodulation processing according to the method of the invention.</li><li>Figure 7: the error rate curves found in the literature for GMSK modulation.</li><li>Figures 8 and 9: the error rate curves according to the demodulation method of the invention applied to a signal modulated in GMSK.</li></ul>

The demodulation of a GMSK signal according to the method of the invention has been simulated on a modulator-demodulator device whose FIG. 1 gives the diagram in the form of functional blocks comprising successively: - A set of frame generation 1 containing a polynomial generating a pseudo-random binary train at the rate of 16 kbits / s. The format of each transmitted frame is 128 bits with a known preliminary sequence of N = 16 or 32 bits which is placed at the start of the frame by means of a system of registers and flip-flops. There therefore remains 128-N bits available for the information to be transmitted. - A modulator 2 which generates a phase pulse with progressive variation of the GMSK type. The modulated signal is available in baseband from two quadrature I and Q channels. - Transposition elements at the intermediate frequency of 70 MHz. This transposition is carried out on transmission by means of mixers 3 and 4 respectively mixing the signals from channels I and Q with the signal from a local oscillator 5 at frequency F_e and the same if generally 90 ° out of phase in the phase shifter 6. After summing the signals from the two channels in the adder 7, the resulting signal from this adder successively crosses an attenuator 8, a white Gaussian noise generator 9 of N₀ spectral density to simulate real operating conditions and a broadband filter 10 centered on 70 MHz.

On reception, the transmitted signal is retransposed into baseband (real and imaginary parts on channels I ′ and Q ′ respectively) by means of mixers 3 ′ and 4 ′, the local oscillator 5 ′ at frequency F_r and the 6 ′ phase shifter. - A digital conversion assembly 11 comprising respectively for the two channels I ′ and Q ′ to be processed, the low-pass filters 12 and 13 which ensure the sampling while respecting the Shannon condition and the analog-to-digital converters 14 and 15 preceded sampler-blockers that maintain the signal level for the duration of the conversion.

Channels I ′ and Q ′ came out to carry out checks (among others the visualization of channels I and R after filtering) after reverse transformations operated through digital-analog converters 16 and 17 and filters 18 and 19. The binary train is also released after decoding through flip-flop 20. - A processing unit 21 comprising a signal processor in which the demodulation of the GMSK signal is carried out according to the method of the invention, this processor operating in complex mode and being controlled by a microprocessor.

When the digital information to be transmitted is carried by the phase, the modulated signal can take the form: S (t, B) = <maths id="math0002" num=""><img file="EP0349064A1_D0002.tif" /></maths> exp {j [2πf₀t + ϑ₀ + φ (t, B)]} t: time B: (B_i) continuation of binary information transmitted. E: signal energy T: duration of a bit f₀: carrier frequency (pulsation ω₀ = 2πf₀) ϑ₀: phase at the origin φ (t, B): phase varying according to the sequence of binary information:<maths id="math0003" num=""><img file="EP0349064A1_D0003.tif" /></maths> where q (t) is the phase pulse of finite duration.

The term 1/2 in the expression of the phase corresponds to the modulation index, that is to say the ratio of frequency excursion to frequency rhythm.

The function q (t) called phase pulse translates how the phase will vary.

FIG. 2 represents the variation of this function q (t) for the modulations of the GMSK, MSK and 2SRC type.

For the GMSK the phase variation is spread over 5 bit times against 2 bit times for the 2SRC and 1 bit time for the MSK.

This variation being slower for the GMSK, the occupied spectrum is less as shown by the curves of figure 3 representing the variations of the spectral power density (DSP) in dB according to the product (fT) of the frequency f by the duration T of a bit, for the modulations MSK (in solid line), 2SRC (in dotted lines) and GMSK (in dashed lines).

However, spreading the information over 5 bit times for the GMSK results in the presence of inter-symbol interference highlighted by the eye diagram of FIG. 4 obtained by the observation of the GMSK signal on a oscilloscope synchronized by the time-bit clock.

We will now explain the successive stages of the coherent demodulation of a digitally modulated signal in continuous phase and with constant envelope according to the method of the invention.

In order to be able to use a method which exploits the presence of inter-symbol interference, it appeared essential to insert at the head of each packet a preliminary sequence of length N.

This known sequence will make it possible to detect the start of the frame by correlation and then to initiate the estimation of ϑ₀ and Δf₀.

The processing can be broken down into four main parts: synchro-frame and synchro-bit, adapted filtering, estimation of ϑ₀ and Δf₀, and decision.

Synchro-frame and synchro-bit

The detection of the synchro-frame and the synchro-bit is carried out by correlation on the differential phase.

The standardized complex signal retransposed into baseband at reception has the expression: S (t) = exp {j [2πΔf₀t + ϑ₀ + φ (t)} in which Δf₀ represents the difference between the transmission frequency f_e and the reception frequency f_r to which is added the Doppler effect frequency f_d when the receiver is moving relative to the transmitter: Δf₀ = f_e - f_r + f_d.

Thanks to the preliminary sequence, there is a known reference signal over a duration NT, that is: R (t) = exp {jφ (t)} with t∈ [0, NT] We then define a signal S ′ equal to the product of the signal S by its conjugate delayed by two bit times: S ′ (t) = S (t) .S * (t-2T) = exp {j [4πΔf₀T + φ (t) - φ (t-2T)]}

By setting Δφ (t) = φ (t) -φ (t-2T), the differential phase between two bit times appears, hence S ′ (t) = exp {j [4πΔf₀T + Δφ (t)]}

In this expression of S ′ (t), the initial phase term has disappeared and the frequency difference results in a constant phase shift.

We also define from the reference signal R another signal R ′ such that: R ′ (t) = R (t). R * (t-2T) = exp {jΔφ (t)}

The correlation function of the two complex signals S ′ and R ′ is written:<maths id="math0004" num=""><img file="EP0349064A1_D0004.tif" /></maths>

By taking the squared module of C (τ), the term exp {j4πΔf₀T} disappears:<maths id="math0005" num=""><img file="EP0349064A1_D0005.tif" /></maths>

The search for the maximum of the function | C (τ) | ² then makes it possible to determine the start of the packet, because | C (τ) | ² is maximum for τ = 0.

The advantage of performing the correlation on the differential phase is that the correlation peak is independent of the phase at the origin and not very dependent on the frequency difference as long as Δf₀T "1, that is to say as long that it is located in the transmission band of the filter placed at the outlet of the transmitter.

However, the level of the maximum of the correlation peak is more sensitive to noise (which amounts to having a 3 dB degradation of the signal to noise ratio).

Two aspects are involved in the choice of the preliminary sequence: its length (N = number of bits) and the configuration of the bits.

The longer the sequence, the better the Probabilities of False Alarm (PFA) and Non-Detection (PND).

The binary configuration of the sequence has an influence on the precision of the timing. The choice is not very easy, but we can still choose it while respecting the following constraints: - non-periodic sequence (otherwise several correlation peaks are formed), - non-constant sequence (otherwise there is a significant temporal spread), - sequence which does not have too many alternating values ​​(otherwise this leads to too small phase variations).

The correlation method described above can be applied to the calculation of the synchro-frame by performing this correlation at the rate of 2 samples per bit time. By designating the value S (i T / 2) of S (t) for t = i T / 2 and i integer by S (i) we have: S (i) = exp {j [2πΔf₀i <maths id="math0006" num=""><math display="inline"><mrow><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></math><img file="EP0349064A1_D0006.tif" /></maths> + ϑ₀ + φ (i <maths id="math0007" num=""><math display="inline"><mrow><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></math><img file="EP0349064A1_D0007.tif" /></maths>)]} R₁ (i) = exp {jφ (i <maths id="math0008" num=""><math display="inline"><mrow><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></math><img file="EP0349064A1_D0008.tif" /></maths>)} We calculate: S ′ (i) = S (i) .S<img file="EP0349064A1_D0009.tif" />i-4) = exp {j [4πΔf₀T + Δφ (i <maths id="math0009" num=""><math display="inline"><mrow><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></math><img file="EP0349064A1_D0010.tif" /></maths>)]} R ′ (i) = R₁ (i) .R<img file="EP0349064A1_D0011.tif" />(i-4) = exp {jΔφ (i <maths id="math0010" num=""><math display="inline"><mrow><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></math><img file="EP0349064A1_D0012.tif" /></maths>)} By asking :<maths id="math0011" num=""><img file="EP0349064A1_D0013.tif" /></maths> the search for the maximum of C₁ (j) makes it possible to detect the start of the packet. When this maximum is detected, the synchro-frame is acquired and the time of transmission of the packet is known to ± T / 4.

This precision is not sufficient to determine the synchro-bit.

To refine the estimate, a second correlation must be made with a reference signal shifted by T / 4. We define as for the first correlation: R₂ (i) = exp [jφ (i T / 2 + T / 4)] R′₂ (i) = exp [jΔφ (i T / 2 + T / 4)]<maths id="math0012" num=""><img file="EP0349064A1_D0014.tif" /></maths>

The two correlations C₁ (j) and C₂ (j) will present peaks for the indices J₁ and J₂ respectively. If C₁ (J₁) ≧ C₂ (J₂) we will take synchro-bit = J₁ T / 2 If C₂ (J₂)&gt; C₁ (J₁) we will take synchro-bit = J₂ T / 2 + T / 4

This double correlation makes it possible to have an accuracy of ± T / 8 on the synchro-bit.

Given the slowness of the phase variation for the GMSK modulation, this precision is good enough to know the sampling instant.

It has already been specified that in certain cases where the signal to noise ratio is low (Eb / N₀ &lt;6 dB), the choice of synchro-bit could be wrong. This results from the uncertain appreciation of the level of correlation peaks (for example if we choose J₂ T / 2 + T / 4 instead of J₁ T / 2).

To make the final decision, we will define SYNP the synchro-bit value called "main" and SYNS the synchro-bit value called "secondary". <tables id="tabl0001" num="0001"><table frame="all"><tgroup cols="4" colsep="1" rowsep="0"><colspec colnum="1" colname="col1" colwidth="39.37mm" /><colspec colnum="2" colname="col2" colwidth="39.37mm" /><colspec colnum="3" colname="col3" colwidth="39.37mm" /><colspec colnum="4" colname="col4" colwidth="39.37mm" /><tbody valign="top"><row><entry namest="col1" nameend="col1" align="left">If C₁ (J₁) ≧ C₂ (J₂)</entry><entry namest="col2" nameend="col2" align="left">so</entry><entry namest="col3" nameend="col3" align="left">SYNP:</entry><entry namest="col4" nameend="col4" align="left">J₁ T / 2</entry></row><row><entry namest="col1" nameend="col1" /><entry namest="col2" nameend="col2" /><entry namest="col3" nameend="col3" align="left">SYNS:</entry><entry namest="col4" nameend="col4" align="left">J₂ T / 2 + T / 4</entry></row><row><entry namest="col1" nameend="col1" align="left">If C₂ (J₂)&gt; C₁ (J₁)</entry><entry namest="col2" nameend="col2" align="left">so</entry><entry namest="col3" nameend="col3" align="left">SYNP:</entry><entry namest="col4" nameend="col4" align="left">J₂ T / 2 + T / 4</entry></row><row rowsep="1"><entry namest="col1" nameend="col1" /><entry namest="col2" nameend="col2" /><entry namest="col3" nameend="col3" align="left">SYNS:</entry><entry namest="col4" nameend="col4" align="left">J₁ T / 2</entry></row></tbody></tgroup></table></tables>

The appropriate filtering will be carried out with the SYNP value.

The possible modification will be made in the algorithm for estimating the frequency deviation and the initial phase on the basis of an error criterion which will be defined later.

The error on the synchro-bit has little effect on the adapted filtering; on the other hand, this greatly degrades the estimate of Δf₀ and ϑ₀.

This method of double correlation on the differential phase is judicious because it makes it possible to determine the start of the frame and to carry out a first estimate of the synchro-bit. This last parameter will be confirmed or adjusted during the estimation of the carrier phase.

Adapted filtering

We can show (cf. PALAURENT: "Interpretation of half-integer index modulations. Extension to neighboring indices and applications", 9th GRETSI Symposium, Nice, May 1983, pages 503 to 509) that all numerical modulations of the form S (t, B) = exp.j.φ (t, B) can be represented in the form of amplitude modulation according to the following expression:<maths id="math0013" num=""><img file="EP0349064A1_D0015.tif" /></maths> in which F_p(t) is the main function.

The decomposition of the GMSK modulation into an amplitude modulation is particularly interesting, because it makes it possible to easily determine the suitable filter.

The latter has an impulse response equal to F_p(t - synchro-bit).

The term synchro-bit takes into account the position of the received signal relative to the sampling clock.

The adapted filter is produced in the form of a Finite Impulse Response filter with 11 coefficients.

Estimation of the initial phase and the frequency deviation

The method envisaged is based on the exploitation of the preliminary sequence.

At the output of the matched filter there is a received signal whose variation as a function of time is shown in FIG. 5a. After normalization, this signal has the expression: Z (t) = exp {j [2πΔf₀t + ϑ₀ + φ (t)}

The preliminary sequence being known, it is easy to calculate the evolution of the signal over an interval [0, NT], N being the number of bits of the preliminary sequence.

We then know the reference signal whose figure 5b shows the variation as a function of time and whose normalized expression can be written: Z₀ (t) = exp {jφ (t)} for t∈ [0, NT]

By performing the product of the received signal Z (t) by the conjugate of the reference signal Z₀ (t) we eliminate the term φ (t) due to the modulation (FIG. 5c). Z (t) .Z₀<img file="EP0349064A1_D0016.tif" />t) = exp {j [2πΔf₀t + ϑ₀]}.

The next step is to transform the complex signal obtained into a linear variation reflecting the evolution of the phase. Ø (t) = Arctg [Z (t) .Z<img file="EP0349064A1_D0017.tif" />(t)] = [2πΔf₀t + ϑ₀] modulo 2π

For that, it is necessary to unroll the phase by eliminating the phase jumps of 2π.

Figure 5d represents this variation, the equation of which is: y = Δω₀.x + ϑ₀ with Δω₀ = 2πΔf₀

By a linear regression method, we can calculate the estimated parameters Δω̂₀ and ϑ̂₀. This calculation is systematic and therefore simple to implement.

From the estimated parameters, we can now compensate the signal by performing a complex multiplication: <tables id="tabl0002" num="0002"><table frame="all"><tgroup cols="2" colsep="1" rowsep="0"><colspec colnum="1" colname="col1" colwidth="78.75mm" /><colspec colnum="2" colname="col2" colwidth="78.75mm" /><tbody valign="top"><row><entry namest="col1" nameend="col1" align="left">Signal received:</entry><entry namest="col2" nameend="col2" align="left">Z (t) = exp {j [Δω₀t + ϑ₀ + φ (t)]}</entry></row><row><entry namest="col1" nameend="col1" align="left">Compensated signal:</entry><entry namest="col2" nameend="col2" align="left">S (t) = Z (t) exp [-j (Δω̂₀t + ϑ̂₀)]</entry></row><row rowsep="1"><entry namest="col1" nameend="col1" /><entry namest="col2" nameend="col2" align="left">S (t) = exp {j [(Δω₀-Δω̂₀) t + ϑ₀-ϑ̂₀ + φ (t)]}</entry></row></tbody></tgroup></table></tables>

If the estimate is correct, Δω̂₀ = Δω₀, ϑ̂₀ = ϑ₀ and S (t) = exp {jφ (t)}

The signal obtained is no longer affected by a frequency deviation or by the phase at the origin.

The estimation of Δω₀ and ϑ₀ is sensitive to three parameters: noise, timing and the length of the preliminary sequence.

As the noise increases (E_b/ N₀ &lt;6 dB), this can cause sudden phase variations which result in jumps of 2 π on the unwound phase. This problem was eliminated by using a 2 π phase jump detection and correction technique.

The sensitivity to timing is linked to the evaluation of the synchro-bit. If this parameter is incorrectly estimated, the modulation is not perfectly eliminated; this results in an unwound phase affected by a modulation residue. The estimate of Δω₀ and ϑ₀ is therefore degraded.

To have a sufficiently good frequency estimate (error less than 10 Hz) which does not lead to a decision error, it is necessary to use a preliminary sequence of length greater than or equal to 64 bits.

On 128 bits, this results in a maximum transmission efficiency of 50%.

Such efficiency is completely incompatible with packet transmission.

If a preliminary sequence of shorter length N = 16 or 32 bits is adopted at the start, the method described above makes it possible to obtain compensated samples.

However, the precision of the estimate is not sufficient to perfectly correct the phase when the noise level is high.

A 20 Hz error between the start of the message and the end results in a phase rotation of 58 °, which leads to decision errors on the end of the packet.

It is therefore the bits near the end of the packet that are most affected.

The idea of ​​the invention consists in deciding a certain number of bits, for example the 16 bits following the preliminary sequence and to redo the estimation process by considering a new reference sequence corresponding to the N bits of preliminary sequence plus 16 new ones. bits decided.

In four new passes, we can get an accuracy of a few Hertz for E_b/ N₀ = 6 dB. This makes an estimate on N + 64 bits at the end.

The length of the preliminary sequence could thus be notably reduced by this method of estimation in several passes which exploits the intermediate decisions on blocks of bits, which withstands noise very well and whose convergence is rapid.

However, as already mentioned, the estimation of Δω₀ and ϑ₀ is sensitive to synchro-bit.

A bad synchro-bit estimate will result in a significant difference between the points corresponding to the unfolded phase and the regression line: Is<maths id="math0014" num=""><img file="EP0349064A1_D0018.tif" /></maths>

In this case, ε will increase more and more.

Very quickly (on the first or second pass) ε will exceed a threshold value and will order a change of synchro-bit.

The secondary value SYNS will then be taken as the synchro-bit value.

The entire calculation process is then reset to redo the adapted filtering and the estimation of Δω₀ and ϑ₀ with the new synchro-bit value.

Decision

After compensation, the decision is made on the expression of the signal put in the form of an amplitude modulation involving the main function F_p(t).

To calculate the transmitted bits, it suffices to finally do a differential decoding.

Figure 6 provides a flowchart of the entire calculation process.

The procedure for estimating the parameters Δω₀ and ϑ₀ from the BEGINNING of the program (box 22) may seem cumbersome to implement, but it is systematic and relatively simple.

We first perform a rough estimate of the synchro-frame (box 23) and the synchro-bit (box 24) by correlation, calculation of the SYNP and SYNS values ​​for the synchro-bit, and the initial choice SYN = SYNP for the value of the SYN synchro-bit, then the adapted filtering (box 25), then the approximate determination of the carrier phase by linear regression on 16 or 32 bit times: suppression of the modulation (box 26), progress of the phase (box 27), estimation of Δω₀ êt ϑ₀ and calculation of the difference ε (box 28). Finally, we compare the deviation ε calculated with a threshold value S (box 29), and we decide whether it is the last pass (box 30).

The rest of the demodulation process can then be described as a system of two nested digital loops: a first loop for the estimation of Δω₀ and ϑ₀ closing by the link 31, - a second loop for the synchro-bit estimation closing by the link 32.

At each passage in the first digital loop, the N bits which follow the preliminary sequence are redecided in box 31 in order to refine the estimation of Δω₀ and ϑ₀ progressively. These decisions are said to be intermediate.

At each passage in the second digital loop, the deviation ε is calculated in box 29 which is compared to the threshold value. This criterion will make it possible to validate or readjust the synchro-bit. In the latter case, the SYNS secondary value will be taken for the value of the synchro-bit SYN (box 32).

There are two possible cases in the treatment: - no questioning of the synchro-bit. The determination of Δω₀ and ϑ₀ is then carried out in a few passes by rapid convergence of the first loop. - questioning of the synchro-bit. In this case, the whole filtering and demodulation process is started again. The processing time via the second loop then becomes longer.

The rest of the organization chart from the last pass (box 30) includes the implementation of the final decision (box 33) and the END of the program (box 34).

As an indication, FIG. 7 shows the theoretical error rate (BER) curves A and B noted in the literature for the GMSK and MSK modulations.

For GMSK modulation, curves 1 and 2 correspond to a coherent demodulator with carrier recovery by an analog servo loop, of bandwidths B_L = 460 Hz and 920 Hz respectively.

This type of demodulation cannot therefore operate, either in EVF or in TDMA. In addition, these curves are obtained without frequency deviation.

Curve 3 corresponds to the result obtained with an analog differential demodulator.

At 10⁻² of error rate, the degradation compared to the theory is very important (about 7 dB).

For GMSK modulation, the error rate curves obtained with the demodulation method of the invention are shown in FIGS. 8 and 9 for preliminary sequences of 32 and 16 bits respectively and with frequency differences Δf₀ = 800 Hz (curves 1) and Δf₀ = 1600 Hz (curves 2).

For Δf₀ = 800 Hz the results are quite good.

At an error rate of 10⁻², there is a degradation of 1.2 dB for N = 32 bits and 1.4 dB for N = 16 bits compared to the theoretical error rates including curves A and B (already shown in Figure 7) are also shown in said figures.

The results are not very sensitive to the frequency difference as long as Δf₀ &lt;1000 Hz. Beyond, the results are slightly degraded.

The method of the invention has made it possible to implement a coherent demodulation algorithm for digital modulation of the GMSK type.

The simulation results show that this method resists noise well and even a significant frequency difference between the transmitter and the receiver.

This method is therefore entirely compatible with TDMA or EVF operation and it can be applied to any modulation exhibiting inter-symbol interference.