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

0001The 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, each of said packets comprising a preliminary sequence by which a reference signal is available known on N bits which makes it possible to detect the synchro-frame and the synchro-bit by correlation with the differential phase and to initiate the estimation of the initial phase parameters ϑ_O and residual frequency difference Δf_O.

0002This 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.

0003In 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.

0004The 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.

0005The various known demodulation methods use differential or coherent methods. Particularly known from European patent application EP-A 0 091 167 is a method for correcting the frequency of the local carrier in the receiver of a data transmission system, in which a synchronization word is transmitted before the data. This synchronization word makes it possible, by correlation with the differential phase, to detect the synchro-frame and the synchro-bit.

0006The first method has the advantage of being relatively simple but the performance in terms of error rate is very degraded.

0007The coherent demodulation has better performance but it requires an additional device for recovering the carrier phase.

0008One 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.

0009Indeed 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. American patent N^o 4,570,125 from RBGibson and B. Hill).

0010The 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. .

0011The 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.

0012The last treatment is extremely important: it is the estimation of the initial phase and the residual frequency difference.

0013The 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.

0014The 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.

0015After having estimated the initial phase and the residual frequency deviation, compensation is carried out and finally the bits transmitted are decided.

0016A 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^o3, pages 521-554.

0017In 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.

0018After simulation, it turns out that the proposed algorithms do not adapt well to GMSK.

0019Indeed, 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.

0020In addition to estimate this frequency difference, it is no longer possible to eliminate the modulation by performing a squared elevation.

0021A 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^o5, May 1978, pages 552 to 564.

0022Inspired 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.

0023To this end, this method is remarkable in that the progressive refinement of the approximate values ​​is obtained by means of two nested digital loops: a second loop initialized beyond a threshold value for the detection of synchro-bit, and a first loop faster than the second loop, which processes the bits in successive blocks, each pass of which is initialized below said threshold value and which makes an intermediate decision of a block of bits to be added to the preceding bits of the information packet for the estimation of ϑ_O and Δf_O, until the packet bits are exhausted.

0024Said synchro-frame and synchro-bit detections are obtained by a first correlation on the differential phase making it possible to know the time of transmission of said packets ± 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.

0025Said 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.

0026Said estimates of the initial phase ϑ₀ and of the residual frequency difference Δf₀ following said filtering comprise the following steps:<ul id="ul0001" list-style="dash"><li>Elimination of the modulation term by carrying out the product of the signal received by the conjugate of the reference signal.</li><li>Phase sequence by eliminating the phase jumps of 2π to obtain a linear variation with the equation<maths id="math0001" num=""><math display="inline"><mrow><mtext>y = Δω₀x + ϑ₀ with Δω₀ = 2πΔf₀</mtext></mrow></math><img file="EP0349064B1_D0001.tif" /></maths> . </li><li>Calculation of the estimated parameters Δω̂₀ and ϑ̂₀ by a linear regression and deviation method<maths id="math0002" num=""><img file="EP0349064B1_D0002.tif" /></maths> between the points corresponding to said unwound phase and said regression line.</li></ul>

0027If 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.

0028If 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.

0029After 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.

0030The final decision is then made, then a differential decoding finally provides the sequence of binary information transmitted.

0031The 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:<dl id="dl0001"><dt>Figure 1 :</dt><dd>the block diagram of a modulator-demodulator device.</dd><dt>Figure 2:</dt><dd>phase pulse variations for GMSK, MSK and 2SRC type modulations.</dd><dt>Figure 3:</dt><dd>spectral occupancy diagrams for GMSK, MSK and 2SRC type modulations.</dd><dt>Figure 4:</dt><dd>the eye diagram for GMSK modulation.</dd><dt>Figure 5:</dt><dd>the temporal variations of the phase of the signal received after filtering during the evaluation sequence of ϑ₀ and Δf₀.</dd><dt>Figure 6:</dt><dd>the flow diagram of the entire demodulation treatment according to the method of the invention.</dd><dt>Figure 7:</dt><dd>the error rate curves noted in the literature for GMSK modulation.</dd><dt>Figures 8 and 9:</dt><dd>the error rate curves according to the demodulation method of the invention applied to a signal modulated in GMSK.</dd></dl>

0032The 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:<ul id="ul0002" list-style="dash"><li>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.</li><li>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.</li><li>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.</li></ul>

0033On 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.<ul id="ul0003" list-style="dash"><li>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-digital converters 14 and 15 preceded by '' sample and hold units that maintain the signal level for the duration of the conversion.</li></ul>

0034Channels 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.<ul id="ul0004" list-style="dash"><li>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.</li></ul>

0035When the digital information to be transmitted is carried by the phase, the modulated signal can take the form:<maths id="math0003" num=""><img file="EP0349064B1_D0003.tif" /></maths> 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="math0004" num=""><img file="EP0349064B1_D0004.tif" /></maths> where q (t) is the phase pulse of finite duration.

0036The 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.

0037The function q (t) called phase pulse translates how the phase will vary.

0038FIG. 2 represents the variation of this function q (t) for the modulations of the GMSK, MSK and 2SRC type.

0039For 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.

0040This 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).

0041However, 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.

0042We 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.

0043In 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.

0044This 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.

0045The 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

0046The detection of the synchro-frame and the synchro-bit is carried out by correlation on the differential phase.

0047The standardized complex signal retransposed into baseband at reception has the expression:<maths id="math0005" num=""><math display="inline"><mrow><mtext>S (t) = exp {j [2πΔf₀t + ϑ₀ + φ (t)}</mtext></mrow></math><img file="EP0349064B1_D0005.tif" /></maths> 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:<maths id="math0006" num=""><math display="inline"><mrow><msub><mrow><mtext>Δf₀ = f</mtext></mrow><mrow><mtext>e</mtext></mrow></msub><msub><mrow><mtext> - f</mtext></mrow><mrow><mtext>r</mtext></mrow></msub><msub><mrow><mtext> + f</mtext></mrow><mrow><mtext>d</mtext></mrow></msub></mrow></math><img file="EP0349064B1_D0006.tif" /></maths> .

0048Thanks to the preliminary sequence, there is a known reference signal over a duration NT, that is:<maths id="math0007" num=""><math display="inline"><mrow><mtext>R (t) = exp {jφ (t)} with tε [O, NT]</mtext></mrow></math><img file="EP0349064B1_D0007.tif" /></maths> We then define a signal S ′ equal to the product of the signal S by its conjugate delayed by two bit times:<maths id="math0008" num=""><math display="inline"><mrow><mtext>S ′ (t) = S (t) .S * (t-2T) = exp {j [4πΔf₀T + φ (t) - φ (t-2T)]}</mtext></mrow></math><img file="EP0349064B1_D0008.tif" /></maths>    By asking <maths id="math0009" num=""><math display="inline"><mrow><mtext>Δφ (t) = φ (t) -φ (t-2T)</mtext></mrow></math><img file="EP0349064B1_D0009.tif" /></maths> the differential phase between two bit times is shown, from where<maths id="math0010" num=""><math display="inline"><mrow><mtext>s' (t) = exp {j [4πΔf₀T + Δφ (t)]}</mtext></mrow></math><img file="EP0349064B1_D0010.tif" /></maths>    In this expression of S '(t), the initial phase term has disappeared and the frequency difference results in a constant phase shift.

0049Another signal R 'is defined from the reference signal R such that:<maths id="math0011" num=""><math display="inline"><mrow><mtext>R '(t) = R (t). R * (t-2T) = exp {jΔφ (t)}</mtext></mrow></math><img file="EP0349064B1_D0011.tif" /></maths>    The correlation function of the two complex signals S 'and R' is written:<maths id="math0012" num=""><math display="inline"><mrow><msub><mrow><mtext>C (τ) = </mtext></mrow><mrow><mtext>T</mtext></mrow></msub><msup><mrow><mtext>∫</mtext></mrow><mrow><mtext>NT</mtext></mrow></msup><mtext>S '(t) .R' * (t-τ) dt =</mtext></mrow></math><img file="EP0349064B1_D0012.tif" /></maths><maths id="math0013" num=""><math display="inline"><mrow><msub><mrow><mtext>exp {j4πΔf₀T}</mtext></mrow><mrow><mtext>T</mtext></mrow></msub><msup><mrow><mtext>∫</mtext></mrow><mrow><mtext>NT</mtext></mrow></msup><mtext>exp {j [Δφ (t) -Δφ (t-τ)]} dt</mtext></mrow></math><img file="EP0349064B1_D0013.tif" /></maths>    By taking the squared module of C (τ), the term exp {j4πΔf₀T} disappears:<maths id="math0014" num=""><math display="inline"><mrow><msub><mrow><mtext>| C (τ) | ² = |</mtext></mrow><mrow><mtext>T</mtext></mrow></msub><msup><mrow><mtext>∫</mtext></mrow><mrow><mtext>NT</mtext></mrow></msup><mtext>exp {j [Δφ (t) -Δφ (t-τ)]} dt | ²</mtext></mrow></math><img file="EP0349064B1_D0014.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.

0050The 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.

0051However, 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).

0052Two aspects are involved in the choice of the preliminary sequence: its length (N = number of bits) and the configuration of the bits.

0053The longer the sequence, the better the Probabilities of False Alarm (PFA) and Non-Detection (PND).

0054The 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:<ul id="ul0005" list-style="dash"><li>non-periodic sequence (otherwise several correlation peaks are formed),</li><li>non-constant sequence (otherwise there is a significant temporal spread),</li><li>sequence which does not have too many alternating values ​​(otherwise this leads to too small phase variations).</li></ul>

0055The 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<maths id="math0015" num=""><math display="inline"><mrow><mtext>t = i T / 2</mtext></mrow></math><img file="EP0349064B1_D0015.tif" /></maths> and i integer by S (i) we have:<maths id="math0016" num=""><math display="block"><mrow><mtext>S (i) = exp {j [2πΔf₀i </mtext><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac><mtext> + ϑ₀ + φ (i </mtext><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac><mtext>)]}</mtext></mrow></math><img file="EP0349064B1_D0016.tif" /></maths><maths id="math0017" num=""><math display="block"><mrow><mtext>R₁ (i) = exp {jφ (i </mtext><mfrac><mrow><mtext>T</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac><mtext>)}</mtext></mrow></math><img file="EP0349064B1_D0017.tif" /></maths> We calculate:<maths id="math0018" num=""><img file="EP0349064B1_D0018.tif" /></maths> By asking :<maths id="math0019" num=""><img file="EP0349064B1_D0019.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.

0056This precision is not sufficient to determine the synchro-bit.

0057To refine the estimate, a second correlation must be made with a reference signal shifted by T / 4. We define as for the first correlation:<maths id="math0020" num=""><math display="inline"><mrow><mtext>R₂ (i) = exp [jφ (i T / 2 + T / 4)]</mtext></mrow></math><img file="EP0349064B1_D0020.tif" /></maths><maths id="math0021" num=""><math display="inline"><mrow><mtext>R'₂ (i) = exp [jΔφ (i T / 2 + T / 4)]</mtext></mrow></math><img file="EP0349064B1_D0021.tif" /></maths><maths id="math0022" num=""><img file="EP0349064B1_D0022.tif" /></maths>    The two correlations C₁ (j) and C₂ (j) will present peaks for the indices J₁ and J₂ respectively.<maths id="math0023" num=""><math display="inline"><mrow><mtext>If C₁ (J₁) ≧ C₂ (J₂) we will take synchro-bit = J₁ T / 2</mtext></mrow></math><img file="EP0349064B1_D0023.tif" /></maths><maths id="math0024" num=""><math display="inline"><mrow><mtext>If C₂ (J₂)> C₁ (J₁) we will take synchro-bit = J₂ T / 2 + T / 4</mtext></mrow></math><img file="EP0349064B1_D0024.tif" /></maths>    This double correlation makes it possible to have an accuracy of ± T / 8 on the synchro-bit.

0058Given the slowness of the phase variation for the GMSK modulation, this precision is good enough to know the sampling instant.

0059It 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).

0060To make the final decision, we will define SYNP the synchro-bit value called "main" and SYNS the synchro-bit value called "secondary".<dl id="dl0002"><dt>If C₁ (J₁) ≧ C₂ (J₂) then SYNP:</dt><dd>J₁ T / 2</dd><dt>SYNS:</dt><dd>J₂ T / 2 + T / 4</dd><dt>If C₂ (J₂)> C₁ (J₁) then SYNP:</dt><dd>J₂ T / 2 + T / 4</dd><dt>SYNS:</dt><dd>J₁ T / 2</dd></dl>    The appropriate filtering will be carried out with the SYNP value.

0061The 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.

0062The 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.

0063This 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

0064We can show (cf. PALAURENT: "Interpretation of half-integer index modulations. Extension to neighboring indices and applications", 9^th GRETSI conference, Nice, May 1983, pages 503 to 509) that all numerical modulations of the form <maths id="math0025" num=""><math display="inline"><mrow><mtext>S (t, B) = exp.j.φ (t, B)</mtext></mrow></math><img file="EP0349064B1_D0025.tif" /></maths> can be represented in the form of amplitude modulation according to the following expression:<maths id="math0026" num=""><img file="EP0349064B1_D0026.tif" /></maths> in which F_p(t) is the main function.

0065The decomposition of the GMSK modulation into an amplitude modulation is particularly interesting, because it makes it possible to easily determine the suitable filter.

0066The latter has an impulse response equal to F_p(t - synchro-bit).

0067The term synchro-bit takes into account the position of the received signal relative to the sampling clock.

0068The 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

0069The method envisaged is based on the exploitation of the preliminary sequence.

0070At 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:<maths id="math0027" num=""><math display="inline"><mrow><mtext>Z (t) = exp {j [2πΔf₀t + ϑ₀ + φ (t)}</mtext></mrow></math><img file="EP0349064B1_D0027.tif" /></maths>    The preliminary sequence being known, one can easily calculate the evolution of the signal over an interval [O, NT], N being the number of bits of the preliminary sequence.

0071We then know the reference signal whose figure 5b shows the variation as a function of time and whose normalized expression can be written:<maths id="math0028" num=""><math display="inline"><mrow><mtext>Z₀ (t) = exp {jφ (t)} for tε [O, NT]</mtext></mrow></math><img file="EP0349064B1_D0028.tif" /></maths>    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).<maths id="math0029" num=""><img file="EP0349064B1_D0029.tif" /></maths>

0072The next step is to transform the complex signal obtained into a linear variation reflecting the evolution of the phase.<maths id="math0030" num=""><img file="EP0349064B1_D0030.tif" /></maths>    For that, it is necessary to unroll the phase by eliminating the phase jumps of 2π.

0073Figure 5d represents this variation, the equation of which is:<maths id="math0031" num=""><math display="inline"><mrow><mtext>y = Δω₀.x + ϑ₀ with Δω₀ = 2πΔf₀</mtext></mrow></math><img file="EP0349064B1_D0031.tif" /></maths>    By a linear regression method, we can calculate the estimated parameters Δω̂₀ and ϑ̂₀. This calculation is systematic and therefore simple to implement.

0074From the estimated parameters, we can now compensate the signal by performing a complex multiplication:<dl id="dl0003"><dt>Signal received:</dt><dd><maths id="math0032" num=""><math display="inline"><mrow><mtext>Z (t) = exp {j [Δω₀t + ϑ₀ + φ (t)]}</mtext></mrow></math><img file="EP0349064B1_D0032.tif" /></maths></dd><dt>Compensated signal:</dt><dd><maths id="math0033" num=""><math display="inline"><mrow><mtext>S (t) = Z (t) exp [-j (A</mtext><mover accent="true"><mrow><mtext>ω</mtext></mrow><mo>ˆ</mo></mover><mtext>₀t +</mtext><mover accent="true"><mrow><mtext>ϑ</mtext></mrow><mo>ˆ</mo></mover><mtext>₀)]</mtext></mrow></math><img file="EP0349064B1_D0033.tif" /></maths><maths id="math0034" num=""><math display="inline"><mrow><mtext>S (t) = exp {j [(Δω₀-Δ</mtext><mover accent="true"><mrow><mtext>ω</mtext></mrow><mo>ˆ</mo></mover><mtext>₀) t + ϑ₀-</mtext><mover accent="true"><mrow><mtext>ϑ</mtext></mrow><mo>ˆ</mo></mover><mtext>₀ + φ (t)]}</mtext></mrow></math><img file="EP0349064B1_D0034.tif" /></maths></dd></dl>    If the estimate is correct,<maths id="math0035" num=""><math display="inline"><mrow><mtext>Δ</mtext><mover accent="true"><mrow><mtext>ω</mtext></mrow><mo>ˆ</mo></mover><mtext>₀ = Δω₀, </mtext><mover accent="true"><mrow><mtext>ϑ</mtext></mrow><mo>ˆ</mo></mover><mtext>₀ = ϑ₀ and S (t) = exp {jφ (t)}</mtext></mrow></math><img file="EP0349064B1_D0035.tif" /></maths>    The signal obtained is no longer affected by a frequency deviation or by the phase at the origin.

0075The estimation of Δω₀ and ϑ₀ is sensitive to three parameters: noise, timing and the length of the preliminary sequence.

0076As 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.

0077The 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.

0078To 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.

0079On 128 bits, this results in a maximum transmission efficiency of 50%.

0080Such efficiency is completely incompatible with packet transmission.

0081If 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.

0082However, the precision of the estimate is not sufficient to perfectly correct the phase when the noise level is high.

0083A 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.

0084It is therefore the bits near the end of the packet that are most affected.

0085The 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.

0086In four new passes, we get an accuracy of a few Hertz for <maths id="math0036" num=""><math display="inline"><mrow><msub><mrow><mtext>E</mtext></mrow><mrow><mtext>b</mtext></mrow></msub><mtext>/ N₀ = 6 dB</mtext></mrow></math><img file="EP0349064B1_D0036.tif" /></maths> . This makes an estimate on N + 64 bits at the end.

0087The 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.

0088However, as already mentioned, the estimation of Δω₀ and ϑ₀ is sensitive to synchro-bit.

0089A 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="math0037" num=""><img file="EP0349064B1_D0037.tif" /></maths>    In this case, ε will increase more and more.

0090Very quickly (on the first or second pass) ε will exceed a threshold value and will order a change of synchro-bit.

0091The secondary value SYNS will then be taken as the synchro-bit value.

0092The 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

0093After 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).

0094To calculate the transmitted bits, it suffices to finally do a differential decoding.

0095Figure 6 provides a flowchart of the entire calculation process.

0096The 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.

0097We 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).

0098The rest of the demodulation process can then be described as a system of two nested digital loops:<ul id="ul0006" list-style="dash"><li>a first loop for the estimation of Δω₀ and ϑ₀ closing by the link 31,</li><li>a second loop for the synchro-bit estimation closing by the link 32.</li></ul>

0099At 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.

0100At 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).

0101There are two possible cases in the treatment:<ul id="ul0007" list-style="dash"><li>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.</li><li>calling into question 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.</li></ul>

0102The 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).

0103As an indication, FIG. 7 shows the theoretical error rate (BER) curves A and B noted in the literature for the GMSK and MSK modulations.

0104For 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.

0105This type of demodulation cannot therefore operate, either in EVF or in TDMA. In addition, these curves are obtained without frequency deviation.

0106Curve 3 corresponds to the result obtained with an analog differential demodulator.

0107At 10⁻² of error rate, the degradation compared to the theory is very important (about 7 dB).

0108For 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).

0109For Δf₀ = 800 Hz the results are quite good.

0110At 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.

0111The results are not very sensitive to the frequency difference as long as Δf₀ &lt;1000 Hz. Beyond, the results are slightly degraded.

0112The method of the invention has made it possible to implement a coherent demodulation algorithm for digital modulation of the GMSK type.

0113The simulation results show that this method resists noise well and even a significant frequency difference between the transmitter and the receiver.

0114This method is therefore entirely compatible with TDMA or EVF operation and it can be applied to any modulation exhibiting inter-symbol interference.