)escription
The invention will be better understood and other characteristics and idvantages will be better apparent on reading the description which follows, iven without limiting effect, and by virtue of the appended figures among /hich:
• Figure 1a describes the main elements of a transmission apparatus according to the prior art not implementing any predistortion;
• Figure 1b describes the main elements of a transmission apparatus according to the prior art implementing a digital predistortion of the signal;
• Figure 2 describes the main elements of a signal transmission apparatus comprising a device for predistorting the power amplifier according to one embodiment of the invention;
• Figures 3a and 3b are representations, given by way of illustration, of signals which are resynchronized respectively without implementing and by implementing the invention;
• Figure 4 is a representation in the form of a state chart of the steps of an embodiment of the method of time synchronization according to the invention.
Figure 2 describes the main elements of a signal transmission apparatus comprising a device for predistorting the power amplifier according o one embodiment of the invention.
This embodiment comprises the whole set of elements of the ransmission chain according to the prior art described in Figure 1b. In )articular, it implements a so-called coarse time-synchronization mechanism 114.
This mechanism is carried out according to the known procedures of he prior art, such as for example:

- by characterization of the group time of the transmission chain and of the predistortion calculation loop, using the mean group times of each of the digital and analog elements of which it is composed;
- by measurement of the group time of the transmission chain and of the predistortion calculation loop, using for example a particular signal such as a Dirac and by measuring the offset in position of this Dirac on the predistorted signal and the downsampled emitted signal;
- by correlations between the predistorted signal and the emitted signal, undertaken in the downsampled domain, on sequences of reasonable size so as not to be too expensive implementationally.
The aim of the first time-synchronization mechanism is to compensate the biggest of the time offset between the two signals to be synchronized. The total group propagation time (from emission to reception) generally being very large in relation to the predistortion estimation capability, it is expected of the first synchronization mechanism that it offers sufficient precision to be able to calculate the coefficients of the predistortion, and not that it makes it possible to achieve the optimal performance of the predistortion algorithm. This is why one speaks of coarse time-synchronization substantially correcting the offset between the signals. In practice, the expected precision of this first synchronization is that it allows registration of the signals to within a few samples, and preferably to within plus or minus one sample, at the sample tempo of the signals which reach it.
The predistortion chain 201 of this embodiment furthermore comprises a so-called fine time-synchronization mechanism 202.
The aim of the fine time-synchronization mechanism is to correct the residual time gap between the two signals after compensation of the coarse time offset. The precision expected by the fine time-synchronization is less than the duration of a sample in the downsampled domain. In contradistinction to the coarse synchronization, the fine synchronization is not based on an analysis of the signals but on the coefficients of the predistortion. Thus, the predistortion calculation loop 201 comprises a

module for analyzing the coefficients 203, which takes as input the coefficients calculated by the module 115 to determine the offset that has to be applied by the fine synchronization to the digitized emitted signal. This offset being dependent on that applied by the coarse synchronization, it may involve a delay or an advance.
The choice of the location where the fine synchronization in the predistortion calculation loop is performed has no effect on the predistortion performance. Indeed, it can be situated either before or after the downsampler 113, or even be situated after the coarse time-synchronization. However, the fine synchronization is aimed at offsetting the signals with a granularity of less than the sample time in the downsampled domain. If it is carried out after the downsampling 113, it is then necessary to interpolate the signals so as to advance them or delay them, and this may be expensive in terms of calculations to be carried out. If it is carried out before the downsampler 113, it then requires only the implementation of an advance line or a delay line. Its cost of implementation is therefore lower when it is carried outbefore.. the .downsampling of the emitted signal, as is the case in the embodiment represented in Figure 2.
In order to calculate the offset that has to be applied by the fine time-synchronization 202, the invention relies on the specific features of the linearization. Indeed, apart from for very narrow-band applications, or else for very relaxed specifications in respect of linearities and out-of-band emissions, the linearization method must take account of the power amplifier's memory effects. The coefficients of the predistortion therefore intrinsically have a time dimension when the memory effects are considered. Consequently, the temporal information in respect of synchronization is available in these coefficients.
Taking the amplifier's memory effects into account in the predistortion coefficients allows the algorithm to partly compensate the lack of precision of the coarse time-synchronization. The synchronization information present in the coefficients is then more precise than the initial synchronization.

There exist numerous algorithms for digital predistortion of a signal (Digital Pre-Distortion, or DPD in English), which take into account the memory effects of the power amplifier. Most, such as for example the MP (Memory Polynomial in English), GMP, DDR (Dynamic Deviation Reduction i in English) and VS (Volterra Series) algorithms, model the non-linearities of the amplifier by a polynomial model. This polynomial model is expressed by coefficients that may be classed into two categories:
- linear coefficients, which are expressed in a linear manner with respect to the input, and
i - nonlinear coefficients.
The nonlinear coefficients act in particular on the spectral regrowths away from the carrier frequency, generally at very low power levels. Conversely, the effects of the time-synchronization offsets are predominantly borne by the linear coefficients of the predistortion.
■> The invention therefore makes the simplifying assumption that the effect
of the time-offset is ..borne by the linear coefficients alone. Hereinafter, the GMP algorithm is used by way of example to illustrate the calculation of the offset on the basis of predistortion coefficients, but the invention applies in an equivalent manner whatever predistortion algorithm is used, provided it is
) expressed according to a model that takes memory effects into account.
In GMP, the predistorted signal y(n), resulting from the predistortion of the signal to be transmitted x(n), can be expressed in the following manner as a function of the coefficients akh bklrn, cklm.

The coefficients ak], bKlm, and cklm are calculated by the module 115. They can be obtained, for example, using the least squares procedure.
In the proposed approach, only the linear terms are preserved, that is to say, for GMP, the La coefficients akl for k = 0, which corresponds to the coefficients of type "A" (first line of equation (1)) of the GMP algorithm. Indeed, the La coefficients akl for k = 0 are the only ones which vary linearly as a function of the signal x(n) to be transmitted, the other coefficients corresponding to nonlinear terms. Thus, equation (1) can be simplified into:
La-1
y(n) = 2^ «o,i • x(n - I). (2)
1 = 0
To deduce therefrom the offset At, the Z-transform of equation (2) is calculated, i.e.:
La-1
' H(Z)= ^Z-'-a0,,. (3)
1 = 0
The normalized angular frequency response con = oj/fDPD (with fDPD the sampling frequency at which the predistortion operates) is obtained by replacing the Z operator with eiU)n:
La-1
ff{ei**) = V e~joJnl ■ a0il. (4)
1=0
The sought-after delay A£ corresponds to the opposite of the group propagation time GPT:
At - -GPT | Wn=0 = — , (5)
n (on=0
with Arg[X] the argument of X.

The relation connecting the Napierian logarithm of a complex number and its argument is:
ln{x) = ln\x\ +j ■ Arg[x\. (6)
The delay is then expressed in the following manner:
d(Arg[H(ei0J»)])
At — " -
d (/ma#[/n(//(e-^))])
d "n .n=o (7)
d (fa(//(y6,'>)))
= Imag -
a o)n
a>n=0-
with Imag[X] the imaginary part of the complex number^. The derivative of In (tf (e;^n) j equals:
da)n H(eJ"n) (8)
j^a-ie-jconi . aQ]
The signal being in baseband, i.e. at the zero frequency, one seeks the group time at the centre of the useful signal, that is to say at the angular frequency con = 0:
d(fa("(e^))) tg-j-l-aoi
Of course, when the signal is not in baseband, the con considered can be adapted accordingly.

The delay At is expressed by virtue of the previous formulation in the following manner:
At = Imag
a>n=0- (10)
. T,{=0 I " ao,l
lma9 -J ■ yLa-l n
2->i=o ao,l Finally, the simplified expression for the delay At is:
* r, i £f=0 ' ' a0,l
A. = - Real ° , (11)
with Rear[JP] the real part of the complex number X.
The offset At that has to be applied by the fine time-synchronization can therefore be calculated very simply on the basis of the La predistortion coefficients a0i\.
Equation (11) can be generalized by taking into account negative index linear coefficients, that is to say for indices / = [-L'a; La - 1] instead of l = [Q;La-l],
When the'predistortion model used does not take memory effects into account, such as for example the algorithms based on a simple polynomial model such as that described in equation (12):
y(n)= 2^ akx(n)\x(n)\k, (12)
k=o
it is impossible to calculate the group time on the basis of the linear coefficient a0 alone. A few linear coefficients can then be added to take into account the delay between the two signals, for example:

k=Q 1=0
This model, on which the invention can be implemented simply, pertains to that presented in equation (2).
Equation (11), making it possible to express the fine propagation delay on the basis of the predistortion coefficients, is applicable to all the predistortion models derived from the various Volterra series (SV, MP, GMP, DDR, etc...)- However, the invention applies for all the other predistortion families. For example, the relation between the input and the output of the predistortion of DVR type (English acronym for Decomposed Vector Rotation) described in A. Zhu, "Decomposed Vector Rotation-Based Behavioral Modeling for Digital Predistortion of RF Power Amplifiers," in IEEE Transactions on Microwave Theory and Techniques, vol. 63, no. 2, pp. 737-744, Feb. 2015, is expressed in the following manner:
M
In the same manner as for the GMP model, if one considers only the linear terms at of the DVR model, the expression for the delay At may be written:

The device for transmitting a signal according to the invention therefore makes it possible to exploit the low complexity of implementation of a coarse synchronization, and then utilizes the time dimension of the calculated predistortion coefficients to refine an optimal sampling instant. It is very simple to implement and does not require any complex calculations.
Since this involves a digital predistortion algorithm, all of the processing, that is to say the analysis of the coefficients 203 and the application of the fine time-synchronization 202, is carried out by a digital calculation circuit. For example, it may be carried out by a reprogrammable calculation machine such as a processor, a DSP (English acronym for Digital Signal Processor), or a micro controller. It can also be carried out by a dedicated calculation machine, such as an FPGA (English acronym for Field-Programmable Gate Array) or an ASIC (English acronym for Application Specific Integrated Circuit), or by any other hardware module allowing the execution of calculations.
The digital calculation circuit may be embedded in an RF or wired transmission apparatus, or in any other apparatus intended to emit a modulated signal, under real conditions or in the laboratory, such as for example a signal generator or a computer.
Figures.3a .and 3b are representations, given by way of illustration, of signals which are re synchronized respectively without implementing and by implementing the invention
In Figure 3a, the signal 301 to be transmitted and the signal 302 emitted and downsampled are resynchronized temporally according to the low-complexity procedures known from the prior art. The temporal compensation is comparable to the temporal compensation carried out by the coarse time-synchronization carried out by a transmission apparatus according to one embodiment of the invention. A delay, corresponding to the fractional delay

related to the differences in sampling instants between the oversampler 101 and the downsampler 113, implies that the two signals are not exactly in phase, this having consequences on the predistortion coefficients calculated and therefore on the spectral regrowths of the emitted signal.
Conversely, in Figure 3b, the signal to be transmitted 301 and the downsampled emitted signal 303 are resynchronized such as described in the invention. This ^synchronization comprises coarse time-resynchronization, making it possible to resynchronize the signals to within a few samples, and fine resynchronization, estimated on the basis of the predistortion coefficients. The two signals are then perfectly synchronized, thus making it possible to optimize the calculation of the predistortion coefficients, and therefore the shape of the spectrum emitted after amplification by an amplifier whose response is not necessarily linear.
The invention also pertains to a method for temporally synchronizing two signals in a signal transmission device, the two signals being used by a predistortion calculation loop to calculate predistortion coefficients applied to the signal to be transmitted before its amplification and its emission.
According to the embodiment, the two signals can be the predistorted signal and the emitted signal, taken either in the downsampled domain or in the oversampled domain. It may also entail the signal to be transmitted (that is to say before predistortion) and the emitted signal.
Figure 3 is a representation in the form of a state chart of the steps of an embodiment of the method of time synchronization according to the invention.
This method comprises a first step 301 of applying a first, coarse, time offset 114. This first time offset corresponds substantially to the gap between the two signals to be synchronized, although, in order for its implementation cost to be as low as possible, it does not seek to estimate the offset with significant precision. This offset may then be an offset calculated theoretically, measured on the basis of a calibration signal, or obtained by correlation between the two signals to be temporally synchronized, preferably

in the downsampled domain by using sequences of reasonable size. In practice, the signal to be transmitted, or the predistorted signal, will generally be stored in a memory buffer, the predistortion loop retrieving the signal from the memory buffer at a position which depends on the present instant and on the calculated time offset.
It also comprises a second step 302 of applying a second, fine, time offset 202. This second time offset seeks to compensate the residual gap between the signals after application of the coarse time offset. It is calculated on the basis of the predistortion coefficients, and seeks to achieve a precision of less than the duration of a sample in the downsampled domain. Advantageously, only the linear predistortion coefficients are used to calculate the value of this second offset. Advantageously, so as to reduce the complexity of implementation of the method, the second offset is applied before the downsampling 113 of the emitted signal.
The two steps are carried out consecutively, their order being unimportant, on all the data used by the predistortion calculation loop.
Various embodiments of the method are possible.
According to a first embodiment, the coarse synchronization is a fixed value, calculated theoretically or in the factory. In this embodiment, the offset carried out by this synchronization does not vary over time.
According to a second embodiment, the coarse synchronization is calculated by virtue of a calibration signal, or by correlation of the signals to be transmitted (predistorted or not) with the digitized emitted signal. It can then be calculated at precise moments, such as for example during initialization of the apparatus, or at regular intervals, or calculated on the fly continuously or by data blocks to be transmitted.
In all these embodiments, the role of the fine synchronization is to correct the defects of this synchronization, and to follow the variations of the group time. It too can be carried out at specific instants, at regular intervals, or in a continuous manner throughout transmission.

When the predistortion coefficients applied to the data to be transmitted do not modify them, which may be the case when the first few data are passing through the predistortion calculation loop, the offset At applied by the fine synchronization is zero.

1. Method for temporally synchronizing two signals in a signal
transmission device, the two signals being used by a predistortion calculation
loop (201) to calculate predistortion coefficients (115) applied to the signal to
be transmitted before its amplification (103) and its emission, the method
being characterized in that it comprises consecutively:
the application (601) of a first time offset (114) corresponding substantially to the time gap between the two signals, and
the application (602) of a second time offset (202) corresponding substantially to the time gap between the two signals after application of the first time offset, the second time offset being calculated on the basis of the said predistortion coefficients.
2. Method for temporally synchronizing two signals according to Claim 1, in which the first time offset is an offset calculated theoretically, measured on the basis of a calibration signal, or obtained by correlation between the two signals to be temporally synchronized.
3. Method for temporally synchronizing two signals according to one of the preceding claims, in which the two signals to be temporally synchronized are:
a first signal corresponding to the signal to be transmitted before or after application of the predistortion,
a second signal corresponding to the emitted signal.
4. Method for temporally synchronizing two signals according to one of
the preceding claims, in which the second time offset is calculated on the
basis of the predistortion coefficients which are expressed linearly as a
function of the signal to be transmitted.

5. Method for temporally synchronizing two signals according to Claim 4,
in which the second time offset At equals:
A* D «» ^i=o l-ao.i
At = —Real ———z ,
Z»{==0 ao,i
where aoi represents La predistortion coefficients expressed linearly as a function of the signal to be transmitted, and Real(X) the real part of the complex number X. "'""
6. Method for temporally synchronizing two signals according to Claim 4,
in which the second time offset At equals:
At = - Real —^ ,
where ax represents M predistortion coefficients expressed linearly as a function of the signal to be transmitted, and Real(X) the real part of the complex number JT.
7. Method for temporally synchronizing two signals according to one of the preceding claims, in which the processings carried out by the predistortion calculation loop comprise the downsampling of the signal emitted, and where the second time offset is applied before the said downsampling.
8. Device for transmitting a signal comprising a predistortion calculation loop (201).for,calculating predistortion coefficients applied to the signal to be transmitted, and an amplifier (103) for amplifying and emitting the said predistorted signal, characterized in that it comprises:
an application means (114) for applying a first time offset of two signals used by the predistortion calculation loop, configured to substantially compensate the time gap between the two signals, and
an application means (202) for applying a second time offset of the two signals used by the predistortion calculation loop, configured to

substantially compensate the time gap between the two signals after application of the first time offset, the second time offset being calculated on the basis of the said predistortion coefficients.