Method, transmitter, and receiver in a communication system

Abstract

A method is disclosed in a communication system including a transmitter and a receiver with a high power amplifier communicating over a communication channel modeled as a continuum of a first linear filter, a nonlinear function, and a second linear filter, the method including obtaining a first received sequence corresponding to a transmission of a first pilot sequence by the transmitter to the receiver, estimating r in response to the first pilot sequence and the first received sequence, where r is equal to a convolution of the first linear filter and the second linear filter, and estimating r in response to the first pilot sequence and the first received sequence, where r is equal to a convolution of the first linear filter and the second linear filter. [0010] obtaining a plurality of candidates for the first linear filter in response to JPEG2024533863000055.jpg43; obtaining a second received sequence corresponding to transmission of a second pilot sequence by the transmitter to the receiver; and determining a nonlinear function and selecting one candidate for the first linear filter among the plurality of candidates in response to the second pilot sequence and the second received sequence.

Description

Technical Field

[0001] At least one of the present embodiments relates generally to a method in a communication system. The communication system includes a transmitter and a receiver that communicate through a communication channel. The communication channel is modeled as a continuum of a first linear filter, a non-linear function, and a second linear filter. The method is for determining the first linear filter, the non-linear function, and the second linear filter. At least one of the present embodiments also relates to a device configured to implement the method, such as a transmitter or a receiver.

Background Art

[0002] In a communication system, a transmitter is coupled to a receiver via a communication channel. The transmitter typically includes an encoder configured to encode input data into symbols. These symbols are then transmitted to the receiver via the communication channel. The receiver includes a decoder configured to decode the received symbols into output data. In the case of satellite communication, the communication channel includes a satellite transponder. FIG. 1 shows a satellite communication system 1 including a transmitter 10, such as a base station like a TV station, a satellite transponder 12, and a receiver 14, such as a TV receiver. The transmitter 10 is configured to transmit a signal (uplink communication) to the satellite transponder 12, and the satellite transponder 12 then transmits the amplified signal (downlink communication) to the receiver 14. To amplify the signal, the satellite transponder includes a high power amplifier also known as an HPA. Such an amplifier often exhibits non-linearity, especially when driven near saturation, which is a common occurrence due to efficiency reasons. Spectral spreading or regrowth and in-band distortion are examples of the effects of non-linear distortion that degrade the overall performance of the satellite communication system.

[0003] To compensate for these effects due to the non-linearity introduced by the HPA, well-known techniques include pre-distortion of the signal transmitted by the transmitter 10. That is, as shown in Figure 2, instead of directly transmitting the signal x, the transmitter 10 applies a pre-distortion function f'() to the signal x and transmits the signal x' such that the signal w = f(x') is received by the receiver 14. Here, w is very similar to the signal x. The purpose of applying the pre-distortion function f'() is to modify the input x so that the output x' approximates the desired response without non-linear distortion when processed by the non-linear system. To define the pre-distortion function f'(), it is necessary to model the non-linear channel 18. For example, as disclosed in Section 3.2.3 of the paper "Iterative predistortion algorithms adapted to the increasing throughput of satellite Communications" by N. Alibert, a direct learning architecture (DLA) adapts the pre-distortion based on the prior identification of the channel. In DLA, prior identification of the channel is used to adapt the pre-distortion function f'().

[0004] So far, the non-linear channel 18 has been modeled using a Wiener-Hammerstein model as shown in Figure 3. The Wiener-Hammerstein model is directly applicable to satellite transponders as it matches the description of satellite transponders. The Wiener-Hammerstein model comprises three elements, namely, a first linear filter h of length L1, a non-linear function c(), and a second linear filter g of length L2. Commonly used as the representation of the Wiener-Hammerstein model is the Volterra model. The Volterra model does not represent each of the three elements h, c, and g individually, but directly represents the result of applying the three elements. This model can be described as the matrix operation w(n) = Q·Φ n (x). Here, Q is the kernel vector and Φn () is an operator that calculates a non - linear combination of the input vector x. On the one hand, using the Volterra model allows the vector Q to be estimated as Q=(Φ n (x) H Φ n (x)) (-1) Φ n (x) H w using the least - squares algorithm, which is convenient. On the other hand, this solution has the drawback of generating a very large number of coefficients. For example, if c(·) is modeled by a polynomial of degree K, the length of the vector Q will be longer than L2*(L1) K . Having such a large number of coefficients is a problem in both the estimation stage and the inference stage. That is, considering the estimation of the coefficients, inverting the matrix Φ n (x) H Φ n (x) induces a high complexity and requires a long training sequence x. In the case of the inference stage (e.g., predistortion), the transmitter will use this model for each sequence to be predistorted, and thus this involves high computational complexity.

SUMMARY OF THE INVENTION

PROBLEM TO BE SOLVED BY THE INVENTION

[0005] Therefore, it is desirable to find a method for estimating a non - linear channel that is less complex than methods based on the Volterra model.

MEANS FOR SOLVING THE PROBLEM

[0006] At least one of the embodiments of the present invention relates generally to a method in a communication system comprising a transmitter and a receiver that communicate through a communication channel, for example, a high - power amplifier in a satellite transponder. The communication channel is modeled as a cascade of a first linear filter of size L1, a non - linear function, and a second linear filter of size L2, where L1 and L2 are positive integers. The method a) Obtaining a first received sequence w1 corresponding to the transmission of a first pilot sequence x1 from a transmitter to a receiver, wherein the first pilot sequence is such that the output of a first linear filter obtained using the first pilot sequence as an input has a low peak amplitude with respect to the saturation level of a high-power amplifier, or a high peak amplitude with a low frequency; b) Estimating r according to the first pilot sequence and the first received sequence, where r is equal to the convolution of a first linear filter and a second linear filter; c) Determining a plurality of candidates for the first linear filter according to the estimated

Number

[0007] Advantageously, by this method, it is possible to estimate and simulate a non-linear channel using a smaller number of coefficients than the Volterra model. As a result, the complexity of predistortion on the transmitter side can be reduced.

[0008] In a specific embodiment, the first pilot sequence belongs to a codebook of the first pilot sequence, and after step b), the method

Number

[0009] In a specific embodiment, estimating r according to the first pilot sequence x1 and the first received sequence w1 uses the least squares method [Number] by [Number] to obtain, where x1 H represents the conjugate transpose of x1.

[0010] In a specific embodiment, the estimated [Number] Finding a plurality of candidates for the first linear filter according to [Number] Converting to the z domain to obtain R(z), Finding the roots of R(z), Factorizing R(z) into a plurality of products H(z)G(z), where each H(z) is a combination of L1 of the roots of R(z), and each H(z) corresponds to one candidate of the first linear filter, including.

[0011] In a specific embodiment, the non - linear function c() is defined as follows. [Number] Here, K is a positive integer. Finding the non-linear function and selecting one candidate of the first linear filter according to the second pilot sequence and the second received sequence among a plurality of candidates involves For each candidate h of the first linear filter i , calculate the corresponding second linear filter g i from H(z) and h i , and assuming * is the convolution operator, calculate u i = x2 * h i , and φ(u i ) = [u i , u i 2 ,..., u i K , and use the least squares method to obtain g’ i as follows, and

Number

Number

Number

[0012] At least one of the embodiments of the present invention comprehensively includes According to the method described in the above embodiment, obtaining a first linear filter, a non-linear function, and a second linear filter, adapting a predistortion function from the first linear filter, the non-linear function, and the second linear filter, relates to a predistortion method including the above.

[0013] At least one of the present embodiments comprehensively includes predistorting a signal by applying a predistortion function before transmission, and relates to a transmission method in which the predistortion function is adapted according to the predistortion method.

[0014] At least one of the present embodiments comprehensively relates to a transmitter in a communication system including a receiver, wherein the transmitter and the receiver communicate through a communication channel including, for example, a high-power amplifier in a satellite transponder. The communication channel is modeled as a continuum of a first linear filter of size L1, a non-linear function, and a second linear filter of size L2, where L1 and L2 are positive integers. The transmitter a) obtaining a first received sequence w1 corresponding to the transmission of a first pilot sequence x1 from the transmitter to the receiver, wherein the first pilot sequence has an output of a first linear filter obtained using the first pilot sequence as an input having a low peak amplitude with respect to the saturation level of the high-power amplifier, or a high peak amplitude with a low frequency; b) estimating r according to the first pilot sequence x1 and the first received sequence w1, where r is equal to the convolution of the first linear filter and the second linear filter; c) estimating

Number

[0015] At least one of the embodiments of the present invention comprehensively relates to a receiver in a communication system including a transmitter, wherein the transmitter and the receiver communicate through a communication channel including a high-power amplifier in a satellite transponder, for example. The communication channel is modeled as a continuum of a first linear filter of size L1, a non-linear function, and a second linear filter of size L2, where L1 and L2 are positive integers. The receiver a) Obtaining a first received sequence w1 corresponding to the transmission of a first pilot sequence x1 from the transmitter to the receiver, wherein the first pilot sequence is such that the output of a first linear filter obtained using the first pilot sequence as an input has a low peak amplitude relative to the saturation level of the high-power amplifier, or a high peak amplitude with a low frequency; b) Estimating r according to the first pilot sequence x1 and the first received sequence w1, where r is equal to the convolution of the first linear filter and the second linear filter; c) The estimated

Number

[0016] At least one of the embodiments comprehensively includes a computer program product including program code instructions that can be loaded into a programmable device, and when the program code instructions are executed by the programmable device, cause the method described in any one of the previous embodiments to be implemented. It relates to a computer program product.

[0017] At least one of the embodiments comprehensively includes a storage medium storing a computer program including program code instructions, and when the program code instructions are read from the storage medium and executed by a programmable device, cause the method described in any one of the previous embodiments to be implemented. It relates to a storage medium.

[0018] The features of the present invention will become clearer by reading the following description of at least one example of the embodiment. This description is created with reference to the accompanying drawings.

Brief Description of the Drawings

[0019]

Figure 1

Figure 2

Figure 3

Figure 4A

Figure 4B

Figure 5

Figure 6

Figure 7

Figure 8

Embodiments for Carrying out the Invention

[0020] This embodiment can be implemented in a communication system such as the satellite communication system 1 shown in FIG. 1. The following embodiments disclosed for the satellite communication system can also be applied to other types of communication systems where a transmitter and a receiver communicate through a communication channel with a high - power amplifier. The communication channel is modeled as a continuum of a first linear filter, a non - linear function, and a second linear filter. Alternatively, the continuum of the first linear filter, the non - linear function, and the second linear filter models only the high - power amplifier with memory without modeling the entire communication channel.

[0021] According to the notation of FIG. 3, N is a positive integer, the input sequence of length N is x = [x(1)... x(n)... x(N)], the signal at the output of the first linear filter h of length L1 is u = [u(1)... u(n)... u(N)], the signal at the output of the non-linear function c(·) is y = [y(1)... y(n)... y(N)], and the signal at the output of the second linear filter g of length L2, which is also the output of the complete channel, is w = [w(1)... w(n)... w(N)]. The output of the first linear filter h is u(n) represented by the following equation.

Number

[0022] The non-linear function can be modeled as a polynomial of degree K as in the following equation.

Number

[0023] Also, when the amplitude |u(n)| is sufficiently small, i.e., when |u(n)| < μ, assuming this function is linear, c(u(n)) can be rewritten as in the following equation.

Number

[0024] When the amplitude of the input signal is not too large, i.e., smaller than the threshold μ, most high-power amplifiers (HPAs) exhibit linear amplification characteristics, so the above assumption is valid. The value μ depends on the HPA used. As an example, "input back-off" (IBO) is often regarded as the difference between the saturation level of the amplifier and μ. In some cases, the characteristics with small amplitudes may be only quasi-linear, in which case c(u(n)) = γ(1)u(n) + ε, which means that the influence of the remaining non-linearity is treated as the noise term ε.

[0025] When the amplitude |u(n)| is sufficiently small, for example, when |u(n)| < μ, the signal is said to have a low peak amplitude.

[0026] The output of the second linear filter g is,

Number

[0027] Finally, the output of the system is,

Number

Number

[0028] FIG. 4A shows a flowchart of a method for estimating the filters of a Wiener - Hammerstein model according to a particular embodiment, namely h and g, and the non - linear function c().

[0029] In step S40, the transmitter 10 transmits a first pilot sequence x1, and the receiver 14 receives a first sequence w1. The first pilot sequence x1 of length N is selected such that u = x1 * h has a low peak amplitude, that is, either u has a peak amplitude lower than the value μ or at least u has μ with a low frequency. By using the first pilot sequence with a low peak amplitude, it becomes possible to at least partially avoid the non - linear part of the amplifier. In one embodiment, x1 is broadband to identify the relevant spectrum of h.

[0030] The first pilot sequence x1 is, for example, in the case of a real signal,

Number

Number

[0031] Phase θ k is selected so that the peak amplitude is minimized. In the case of a real signal, to obtain the phase θ k can be done, for example, as disclosed in the literature "Synthesis of low-peak-factor signals and binary sequences with low autocorrelation" by Schroeder published in IEEE Trans. Inf. Theo., Vol 16, No 1, Jan 1970. This literature examines the problem of a method for adjusting the phase angle of a periodic signal having a given power spectrum so as to minimize its peak-to-peak amplitude. In this way, an expression for the phase angle that generally gives a low peak factor is derived. In the case of a complex signal, to obtain the phase θ k can be done, as disclosed in the literature "Polyphase codes with good periodic correlation properties" by Chu published in vol. 18, no. 4, pp. 531 - 532, Jul. 1972. When the filter h has a linear phase, the peak amplitude of u is the same as the peak amplitude of x1.

[0032] In another embodiment, the first pilot sequence x1 can be selected, for example, as a pseudo-white random pilot sequence.

[0033] Hereinafter, r is defined as the convolution product of filters h and g, i.e., r = h * g. In order to obtain an estimated value of the linear filter r with a given accuracy, for example, the length N of the sequence x1 can be determined as disclosed in the literature "FIR System Modeling and Identification in the Presence of Noise and with Band-Limited Inputs" by Rabiner et al. published in Aug. 1978. In this literature, the quality of the estimated value is evaluated through the following Q measure.

Number

Number

Number

[0034] Using the least squares method, the Q measure can be approximated as follows.

Number

[0035] In step S42, the linear filter r = h * g is estimated according to x1 and w1 using the least squares method. More precisely, the impulse response of the filter r is estimated as follows using the least squares method.

Number

[0036] Estimated filter [Mathematics] Once obtained, at step S44, Nb candidates h of h i are calculated. Here, Nb is a positive integer. The maximum number of candidates Nb max is the binomial coefficient [Mathematics] is given by. In the z-domain, R(z) = H(z)G(z). Therefore, the roots of the polynomial R(z) are derived from either H(z) or G(z). As a result, H(z) is obtained from one of the combinations of L1 roots out of the L1 + L2 roots of R(z).

[0037] This step is detailed in Figure 5. At step S440, [Mathematics] is transformed into R(z) by applying the z-transform. That is, [Mathematics] becomes. The roots of R(z) are obtained at step S442. At step S444, R(z) is factorized into the product H(z)G(z). Multiple factorizations of H(z)G(z) are possible, and each H(z) is obtained from L1 roots of R(z) out of the L1 + L2 roots of R(z). Each H(z) thus obtained corresponds to one candidate of the first linear filter.

[0038] As an example, considering the roots z1 and z2, H(z) = (z - z1)(z - z2) = z 2It becomes -(z1 + z2)*z + z1*z2. In this case, the coefficients of the candidate filter h i are [1; z1 + z2; z1*z2]. For each candidate h i there is a corresponding unique g i . When R(z) = (z - z1)(z - z2)(z - z3)(z - z4), if H(z) is set to be equal to (z - z1)(z - z2), then G(z) = (z - z3)(z - z4). Therefore, the coefficients of the filter g i are [1; z3 + z4; z3*z4].

[0039] In a specific embodiment, in order to reduce the number of candidates, that is, to make Nb < Nb max , a priori information regarding the filter h is considered. For example, the following information can be used. · Whether the estimated filter h is a low-pass filter or a high-pass filter; and · The symmetry of the complex roots of the filter.

[0040] If it is known that the estimated filter h is a low-pass filter or a high-pass filter, some values cannot be taken by the roots. As a result, some roots of R are directly excluded, and consequently, some candidate h i are excluded.

[0041] If there are some symmetries, some roots are grouped and thus belong to either H(z) or G(z).

[0042] Also, as done in the Volterra method, the filter sizes L1 and L2 can be chosen to be shorter than their true values (i.e., the values of the estimated filter) to reduce complexity if necessary.

[0043] In step S45, the transmitter 10 transmits a second pilot sequence x2, and the receiver 14 receives a second sequence w2. The second pilot sequence x2 is defined such that its peak amplitude is higher than the peak amplitude of x1. More precisely, x2 is defined such that u = x2 * h has a high peak amplitude, that is, u has a peak amplitude greater than μ, or at least u exceeds μ with a high frequency.

[0044] In step S46, the non-linearity c(·) is determined, and among the set of Nb candidates calculated in step S42, one candidate h is selected according to x2 and w2. i As a result, since a unique g i corresponds to each candidate h i , one g i is also determined. This step is detailed in FIG. 6.

[0045] Steps S42 to S46 can be performed in the transmitter 10 or the receiver 14.

[0046] In one embodiment, the receiver 14 that knows x1, x2, w1, and w2 performs steps S42 to S46 to estimate h, c, and g. Then, the receiver 14 can transmit the estimated h, c, and g to the transmitter 10 using a reliable return channel or coefficients representing the predistortion function if applicable. The above coefficients are obtained according to the estimated h, c, and g. The transmitter 10 is then responsible for adapting the predistortion function f'() to be considered or selecting the predistortion function according to the estimated h, c, and g. In the latter case, each predistortion function in the codebook is defined, for example, by a look-up table.

[0047] In one variant, when a codebook of the predistortion function is used, the receiver 14 can select an appropriate predistortion function in the codebook according to the estimated h, c, and g, and then transmit an index identifying the selected predistortion function to the transmitter 10.

[0048] In another variant, the receiver 14 estimates h, c, and g, and then applies an equalization function whose weights are adapted according to the estimated h, c, and g. This equalization function replaces the predistortion applied on the transmitter side.

[0049] In another embodiment, the receiver 14 uses a reliable feedback channel to send w1 and w2 back to the transmitter 10. In this case, the transmitter 10, which knows x1, x2, w1, and w2, performs steps S42 to S46 to estimate h, c, and g, and then adapts a predistortion function f'() from the estimated h, c, and g, or selects a predistortion function in the codebook of predistortion functions according to the estimated h, c, and g. The transmitter 10 can then apply the predistortion function f'() to each signal to be transmitted.

[0050] In all embodiments, h, c, and g can be finally estimated before any transmission of relevant signals. In one variant, h, c, and g can be re - estimated periodically. Each time h, c, and g are estimated, the predistortion function is adapted or a new predistortion function is selected. In one embodiment, the sequences x1 and x2 can be uniquely defined in the standard.

[0051] In one variant shown by FIG. 4B, while a plurality of sequences x1 can be defined in the standard as a codebook, for example, the sequence x2 is finally fixed. The selection of one sequence x1 in the codebook (for estimating h, c, g) can be performed using the feedback path from the receiver 14 to the transmitter 10. The pilot sequences in the codebook are arranged, for example, in a pre-defined transmission order. The steps in FIG. 4B that are the same as the steps in FIG. 4A are identified using the same reference numerals.

[0052] Steps S40 to S42 are applied using the first sequence x1 of the codebook.

[0053] After S42, the receiver 14, in step S43, compares the error

Number

Number

[0054] Steps S40 to S42 are thus repeated until a sequence x1 with an error

Number

[0055] In another variant, the receiver 14 calculates the error

Number

Number

[0056] FIG. 6 shows a flowchart detailing step S46 of FIG. 4A or FIG. 4B according to a particular embodiment.

[0057] In step S460, the index i is first initialized to a first value i0, for example i0 = 0.

[0058] In step S462, the corresponding unique g i is calculated from h i . G(z) = R(z) / Z(h i ). As an example, when R(z) = (z - z1)(z - z2)(z - z3)(z - z4), if H(z) is set to be equal to (z - z1)(z - z2), then G(z) = (z - z3)(z - z4). Thus, the coefficients of the filter g i are [1; z3 + z4; z3 * z4]. Also, u i = x2 * h i is also calculated.

[0059] In the Hammerstein model, w2 is obtained from u i as follows.

Number

[0060] Therefore, in step S464, the coefficient g' of the Hammerstein model i , that is, c*g, is obtained from (u i , w2) as follows through the least squares method.

Equation

[0061] In step S466, the coefficient γ i (k) is then calculated from g’ i and g i by averaging the estimated values as follows.

Equation

[0062] In step S468, the corresponding w 2,i is calculated as w i = c i (x1*h 2,i )*g i using the estimated model (h i , γ i ). Here, c i is defined from γ i by Equation (1).

[0063] In step S470, (i - i0) is compared with Nb - 1. If (i - i0) < Nb - 1, then i is incremented by 1 in S472 and the method continues in S462. Otherwise, the method continues in S474.

[0064] In step S474, the minimum

Equation

[0065] FIG. 7 schematically shows an example of the hardware architecture of the transmitter 10 according to a particular embodiment.

[0066] The transmitter 10 includes at least a set of a processor or CPU (acronym for Central Processing Unit) 101, a random access memory RAM 102, a read only memory ROM 103, a hard disk or a storage medium reader such as an SD (acronym for Secure Digital) card reader or the like storage unit 104, and a communication interface COM 105 that enables the transmitter 10 to transmit and receive data, connected by a communication bus 110.

[0067] The processor 101 is capable of executing instructions loaded into the RAM 102 from the ROM 103, from an external memory (such as an SD card), from a storage medium (such as an HDD), or from a communication network. When the power of the transmitter 10 is turned on, the processor 101 can read instructions from the RAM 102 and execute them. These instructions form a computer program that causes the processor 101 to implement the methods described with respect to FIGS. 4A - 6.

[0068] The methods described with respect to FIGS. 4A-6 can also be implemented in the form of software by execution of a set of instructions by a programmable machine, such as a DSP (acronym for Digital Signal Processor), a microcontroller, or a GPU (acronym for Graphics Processing Unit), or in the form of hardware by a machine or dedicated components (chip or chipset), such as an FPGA (acronym for Field-Programmable Gate Array) or an ASIC (acronym for Application-Specific Integrated Circuit). Generally, the transmitter 10 includes electronic circuits adapted and configured to implement the methods described with respect to FIGS. 4A-6.

[0069] FIG. 8 schematically shows an example of the hardware architecture of the receiver 14 according to a particular embodiment.

[0070] The receiver 14 includes at least a set of a processor or CPU (acronym for Central Processing Unit) 201 connected by a communication bus 210, a random access memory RAM 202, a read only memory ROM 203, a hard disk or a storage medium reader, such as an SD (acronym for Secure Digital) card reader or the like storage unit 204, and a communication interface COM 205 that enables the receiver 14 to transmit and receive data.

[0071] The processor 201 is capable of executing instructions loaded into the RAM 202 from the ROM 203, from an external memory (such as an SD card), from a storage medium (such as an HDD), or from a communication network. When the receiver 14 is powered on, the processor 201 is capable of reading and executing instructions from the RAM 202. These instructions form a computer program that causes the processor 201 to implement the methods described with respect to FIGS. 4A-6.

[0072] The methods described with respect to FIGS. 4A-6 can also be implemented in the form of software by execution of a set of instructions by a programmable machine, such as a DSP (acronym for Digital Signal Processor), a microcontroller, or a GPU (acronym for Graphics Processing Unit), or in the form of hardware by a machine or dedicated components (chip or chipset), such as an FPGA (acronym for Field-Programmable Gate Array) or an ASIC (acronym for Application-Specific Integrated Circuit). Generally, the receiver 14 includes electronic circuitry adapted and configured to implement the methods described with respect to FIGS. 4A-6.

Claims

1. A method in a communication system comprising a transmitter and a receiver that communicate through a communication channel having a high-power amplifier, wherein the communication channel is modeled as a continuum of a first linear filter of size L １ , a non-linear function, and a second linear filter of size L ２ , where L １ and L ２ are positive integers, and the method comprises a) obtaining a first received sequence w corresponding to transmission of a first pilot sequence x from the transmitter to the receiver, wherein the first pilot sequence has a peak amplitude lower than a threshold value dependent on the high power amplifier, or a peak amplitude higher than the threshold value with a low frequency, where the output of the first linear filter obtained using the first pilot sequence as an input has the peak amplitude; １ １ ​​ b) The first pilot sequence x １ and the first received sequence w １ to estimate r in response thereto, where the non-linear function c() is defined as follows 【Number 1】 K is a positive integer, and r is equal to the convolution of the first linear filter and the second linear filter, c) obtaining a plurality of candidates for the first linear filter according to the estimated 【Number 2】 by converting to the z-domain to obtain R(z), 【Mathematics 3】 finding the roots of R(z), factorizing R(z) into a plurality of products H(z)G(z), where each H(z) is a combination of L1 of the roots of R(z), and each H(z) corresponds to one candidate of the first linear filter, including obtaining, for each of the candidates hi of the first linear filter, d) A second received sequence w ２ corresponding to the transmission of a second pilot sequence x ２ from the transmitter to the receiver is obtained, where the second pilot sequence is such that the output of the first linear filter obtained using the second pilot sequence as an input has a high peak amplitude relative to the threshold of the high-power amplifier.​​​​ e) obtaining the non-linear function and selecting, among the plurality of candidates, one candidate of the first linear filter according to the second pilot sequence x ２ and the second received sequence w ２ wherein the second linear filter is uniquely determined from the one candidate of the selected first linear filter, and the selecting comprises calculating the corresponding second linear filter gi from H(z) and hi, and calculating ui = x2 * hi, where * is the convolution operator, assuming φ(ui) = [ui, ui2,..., uiK], using the least squares method to obtain g’i as follows, obtaining γi from g’i and gi, for each integer j in the range [1; K], [Number 4] such that 【Number 5】 calculating w2,i from hi and γi by w2,i = ci(x1 * hi) * gi, selecting the pair (hi, γi) that results in the minimum including selecting, 【Number 6】 including a method.

2. The communication system is a satellite communication system, and the high-power amplifier is within a satellite transponder. The method according to claim 1.

3. The first pilot sequence belongs to the codebook of the first pilot sequence. After step b), the method further includes comparing the error with the threshold value, 【Number 7】 if it exceeds the threshold value, returning to step a) with the next first pilot sequence of the codebook, and if not, proceeding to step c). The method according to claim 1. 【Number 8】

4. by the first pilot sequence x １ and the first received sequence w １ estimating r according to involves using the least squares method 【Number 9】

5. 【Number 10】 including obtaining, where x １ Ｈ is the conjugate transpose of x １ The method according to any one of claims 1 to 3, where １ and Ｈ denote the conjugate transpose of １ . According to the method according to any one of claims 1 to 3, obtaining the first linear filter, the non-linear function, and the second linear filter, adapting a predistortion function from the first linear filter, the non-linear function, and the second linear filter, including a predistortion method.

6. ​ A transmission method including pre-distorting a signal by applying a pre-distortion function before transmission, wherein the pre-distortion function is adapted according to the pre-distortion method of claim 5. Claim 7. A communication system comprising a transmitter and a receiver, wherein the transmitter and the receiver communicate through a communication channel including a high-power amplifier, and the communication channel is modeled as a continuum of a first linear filter of size L １ , a non-linear function, and a second linear filter of size L ２ , where L １ and L ２ are positive integers, and the system a) The receiver receives a first sequence corresponding to a first pilot sequence transmitted from the transmitter, wherein the first pilot sequence has a peak amplitude lower than a threshold value dependent on the high-power amplifier, or a peak amplitude higher than the threshold value with a low frequency, where the output of the first linear filter obtained using the first pilot sequence as an input. b) Estimating r according to the first pilot sequence and the received first sequence, wherein the non-linear function c() is defined as follows: 【Number 11】 K is a positive integer, and r is equal to the convolution of the first linear filter and the second linear filter. c) Obtaining a plurality of candidates for the first linear filter according to the estimated 【Number 12】 including converting to the z-domain to obtain R(z), 【Number 13】 finding the roots of R(z), factoring R(z) into a plurality of products H(z)G(z), where each H(z) is a combination of L1 of the roots of R(z), and each H(z) corresponds to one candidate of the first linear filter. d) The receiver receives a second sequence corresponding to a second pilot sequence transmitted from the transmitter, wherein the second pilot sequence has a peak amplitude higher than the threshold value of the high-power amplifier, where the output of the first linear filter obtained using the second pilot sequence as an input. e) Obtaining the non-linear function and selecting one candidate of the first linear filter according to the second pilot sequence and the received second sequence among the plurality of candidates, wherein the second linear filter is uniquely obtained from the selected one candidate of the first linear filter, and the selecting includes: for each candidate hi of the first linear filter ​ Calculate the corresponding second linear filter \(g_i\) from \(H(z)\) and \(h_i\), and calculate \(u_i = x_2 * h_i\), where \(*\) is the convolution operator, assuming \(\varphi(u_i)=[u_i, u_i^2,..., u_i^K]\), obtain \(g'_i\) as follows using the least squares method, 【Number 14】 obtaining \(\gamma_i\) from \(g'_i\) and \(g_i\), for each integer \(j\) in the range \([1;K]\), 【Number 15】 such that, calculating \(w_{2,i}\) from \(h_i\) and \(\gamma_i\) by \(w_{2,i}=c_i(x_1 * h_i)*g_i\), the minimum 【Number 16】 selecting the pair \((h_i, \gamma_i)\) that yields, and including selecting, a communication system configured to perform. **Claim 8** A computer program product including program code instructions loadable onto a programmable device, wherein said program code instructions, when executed by said programmable device, cause the method according to any one of claims 1 to 3 to be implemented. **Claim 9** A computer program product including program code instructions loadable onto a programmable device, wherein said program code instructions, when executed by said programmable device, cause the predistortion method according to claim 5 to be implemented. **Claim 10** A computer program product including program code instructions loadable onto a programmable device, wherein said program code instructions, when executed by said programmable device, cause the transmission method according to claim 6 to be implemented. **Claim 11** A storage medium storing a computer program including program code instructions, wherein said program code instructions, when read from said storage medium and executed by a programmable device, cause the method according to any one of claims 1 to 3 to be implemented. **Claim 12** A storage medium storing a computer program including program code instructions, wherein the program code instructions cause the programmable distortion method according to claim 5 to be implemented when the program code instructions are read from the storage medium and executed by a programmable device.

13. A storage medium storing a computer program including program code instructions, wherein the program code instructions cause the transmission method according to claim 6 to be implemented when the program code instructions are read from the storage medium and executed by a programmable device.

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