Method and a system for digitally linearizing an amplifier

[0002] The invention relates to the field of amplifiers.

[0003] More precisely, the invention relates to a method and a system for linearizing an amplifier.

[0004] The invention is particularly, but not exclusively, applicable to linearizing amplifiers used in cellular radiocommunications systems (such as, in particular, the Global System for Mobile communications 900 MHz (GSM 900) (a public radiccommunications system operating in the 900 MHz band), the Digital Cellular System 1800 MHz (DCS 1800), the Personal Communication System (PCS), IS95, the Universal Mobile Telephone System (UMTS), the International Mobile Telecommunications 2000 system (IMT-2000), CDMA-One, etc.

[0005] A mobile terminal generally includes an amplifier in its transmission system.

[0006] Frequently, it is desirable to obtain linear amplification. Since an amplifier is never exactly linear, it is necessary to linearize the amplifier.

[0007] In radiocommunications systems, the non-linearity of the amplifier in the radio transmission system is detrimental because it distorts the transmitted signal, thereby degrading the spectrum of the signal, and thus causing interference signals to be generated in channels adjacent to the working channel. In other words, the non-linearity of the amplifier causes an increase in the level of transmission interference in the adjacent channels. This gives rise to failure to comply with specifications aiming to limit interference. Such failure to comply with the specifications is further accentuated when modulation having a non-constant envelope (which is more sensitive to non-linearity defects) is used upstream from the amplifier. It can thus be understood that it is necessary to linearize the amplifier.

[0008] Several solutions for performing such linearization are known from the state of the art.

[0009] A first known linearization solution consists in using the amplifier well within its capacities, so as to operate in a linear zone. The lower the power level, the smaller the non-linearity manifested by the amplifier. That first known solution suffers from the major drawback of degrading amplifier efficiency. Thus, if such an amplifier is included in a mobile terminal, it reduces battery life between charges.

[0010] A second known linearization solution consists in using an analog feedback correction loop (cf FIG. 1). Unfortunately, the correction loop requires additional analog components which firstly suffer from defects affecting the performance of the resulting overall system, and secondly are subjected to drift over time, causing the effectiveness of the correction to vary.

[0011] A third known linearization solution consists in using an analog feedforward correction loop (cf FIG. 2). The drawback with such a correction loop is that it requires a second amplifier which must be very linear.

[0012] A fourth known linearization solution is described in Patent Document EP 522 706 A2. That document describes a linearization method comprising a prior testing stage and an amplification stage. During the prior testing stage, the following are performed successively: a test signal is amplified; the corresponding output signal is sampled at a low rate over a long period; the Fourier transform is applied to the samples of the amplified test signal; the amplitudes of the harmonics are computed so as to reconstruct the output signal; the input signal is compared with the reconstructed output signal of the amplifier, so as to determine the distortion parameters specific to the amplifier, and a table is constructed of predistortion parameters which are such that they cancel the distortion parameters specific to the amplifier. During the amplification stage, the predistortion parameters are applied to the signal to be amplified, so that the amplified signal at the output of the amplifier is not distorted.

[0013] In other words, in the fourth solution, the principle of predistortion is used which consists in compensating for the non-linearity of the amplifier by adding a unit of inverse non-linearity at the input of the amplifier, so that the juxtaposition of the two non-linear units has a linear transfer function. The test stage which, in the fourth solution, consists in comparing the input test signal with the corresponding output signal, makes it possible to determine the distortion (or non-linearity) due to the amplifier.

[0014] The fourth known solution is not advantageous because it requires many samples to be taken at the output of the amplifier, and therefore requires a large amount of computation, considerable memory space, and a long time for acquiring (i.e. recovering) and processing the samples at the output of the amplifier. In addition, the results obtained are not accurate. It should be noted that a long acquisition time can be incompatible with certain operational constraints allowing only a small amount of time for acquisition: signals that are not steady, or time limits for transmission of test signals).

[0015] To sum up, it appears that all of the above-mentioned known solutions which aim to enable an amplifier to be linearized suffer from drawbacks, namely, in particular, their high cost considering the performance obtained.

[0016] A particular object of the present invention is to mitigate the various drawbacks of the state of the art.

[0017] More precisely, one of the objects of the present invention is to provide a method and a system of digitally linearizing an amplifier, the method and system requiring lower computation power than the above-mentioned known solutions.

[0018] An additional object if to provide such a method and such a system that require only a small amount of memory space.

[0019] Another object of the invention is to provide such a method and such a system that make it possible to reduce the time required to acquire the samples at the output of the amplifier, as compared with the solutions known from the state of the art.

[0020] Another object of the invention is to provide such a method and such a system that make it possible to reduce the time required to process the output samples, as compared with the solutions known from the state of the art.

[0021] A further object of the invention is to provide such a method and such a system that make it possible to obtain results that are very accurate.

[0022] Another object of the invention is to provide such a method and such a system that are low in cost.

[0023] These various objects and others that appear below are achieved by the invention by means of a method of digitally linearizing an amplifier having non-linearity, the method being of the type comprising:

[0024] a prior determination stage in which correction parameters for correcting said non-linearity are determined, as a function of an amplified test signal present at the output of the amplifier when a test signal is applied to the input thereof; and

[0025] a linear amplification stage comprising the following steps:

[0026] causing an input signal which is to be amplified by said amplifier to be predistorted as a function of said correction parameters so as to obtain a predistorted signal; and

[0027] causing said predistorted signal to be amplified by said amplifier so that the amplified predistorted signal present at the output of said amplifier corresponds to said input signal as amplified linearly;

[0028] wherein said determination stage in which the correction parameters are determined comprises the following steps:

[0029] acquiring said amplified test signal in the form of samples in baseband;

[0030] defining a modelled amplifier, by computing parameters of a predetermined amplifier model on the basis of said samples of the amplified test signal; and

[0031] computing said correction parameters on the basis of said modelled amplifier so that they make it possible to linearize said modelled amplifier.

[0032] The present invention thus provides a novel way of determining the predistortion relationship (i.e. the correction parameters) to be applied upstream from the amplifier in order, ultimately, to obtain amplification that is linear.

[0033] More precisely, the general principle of the invention consists in modelling the real amplifier, and then in computing the predistortion relationship on the basis of the modelled amplifier (and not on the basis of the real amplifier) in order to compensate for its non-linearity.

[0034] In other words, the work is done on a model of the amplifier and not, as in above-mentioned Patent Document EP 522 706 A2, merely on a comparison between the input test signal and the corresponding output signal (i.e. the amplified test signal). It is important to note that the fact that a modelled amplifier is used makes it possible to obtain improved accuracy on the correction parameters to be applied to the input signal, and thus improved linearization of the amplifier. In general, the closer the modelled amplifier is to the real amplifier, the better the resulting linearization.

[0035] It is recalled that, when applied to a mobile terminal, the linearization method of the invention makes it possible to limit interference between channels. Clearly, however, the method of the invention is not limited to being applied to the case when the amplifier is incorporated in a mobile terminal, but rather it can be considered regardless of the use of the amplifier.

[0036] Advantageously, said step in which the modelled amplifier is defined comprises the following steps:

[0037] performing spectrum analysis on said amplified test signals to estimate the complex amplitudes of a predetermined number of harmonics at known frequencies, said harmonics being generated by said non-linearity of the amplifier; and

[0038] computing the parameters of the predetermined amplifier model by identifying firstly the estimated complex amplitudes and secondly mathematical expressions corresponding to complex amplitudes of harmonics at said known frequencies for a signal present at the output of the modelled amplifier, said mathematical expressions being computed on the basis of said predetermined amplifier model, by assuming that the test signal is applied to the input thereof.

[0039] Estimating the complex amplitudes of certain cross-modulation harmonics makes it possible to characterize the amplitude-amplitude (AM-AM) and amplitude-phase (AM-PM) non-linearities of the amplifier.

[0040] In addition, since a limit is set at a predetermined number of harmonics whose frequencies are known, the time taken to acquire the samples of the amplified test signal can be short.

[0041] Preferably, said spectrum analysis performed on the samples of the amplified test signal belongs to the group comprising:

[0042] parametric spectrum analyses; and

[0043] spectrum analyses by sampling followed by a Fourier transform.

[0044] It should be noted that parametric spectrum analysis offers the advantage of requiring fewer calculations, and less memory space, and of requiring a shorter sample input time. In addition, it is possible to simplify this type of analysis considerably insofar as firstly the waveform of the signal is known (superposition of a plurality of sinewaves), and secondly the frequencies of the spectrum lines (or harmonics) whose amplitudes are to be determined are known.

[0045] In a preferred implementation of the invention, said step in which the complex amplitudes of harmonics are estimated consists in computing:

[0046] H{tilde over ( )}=M S

[0047] where

[0048] H{tilde over ( )}=[H₁{tilde over ( )} H₂{tilde over ( )} . . . H_p{tilde over ( )}]^Tis the vector of the complex amplitudes of p harmonics in question;

[0049] S=[s₀s₁. . . s_N]^Tis the vector of N samples of the amplified test signal; and

[0050] M is a predetermined matrix (p,N).

[0051] In this way, the vector H{tilde over ( )} can be computed very rapidly. The matrix M, which depends only on the frequencies of the harmonics that are to be characterized, is entirely known, and can be computed once and for all and stored in a memory. N is also referred to as the “sampling window”.

[0052] Advantageously, said matrix M can be written:

[0053] M=([Z^HZ]⁻¹Z^H)

[0054] where

[0055] Z^His the Hermitian transpose of Z;

[0056] z_i=exp(2jπf_i), f_ibeing the frequency of the i^thharmonic in question; and $Z=\left[\begin{array}{cccc}1& 1& \cdots & 1\\ z_1& z_2& \cdots & z_p\\ ⋮& ⋮& \cdots & ⋮\\ z_1^{N-1}& z_2^{N-1}& \cdots & z_p^{N-1}\end{array}\right]$

[0057] In a preferred implementation of the invention, said predetermined amplifier model is a polynomial model.

[0058] Clearly, the higher the degree of the polynomial that models the amplifier, the greater the accuracy of the modelled amplifier. The transfer function modelling the amplifier (voltage/voltage transfer function) may be given in RF by a polynomial, it nevertheless being possible to determine the complex coefficients of such a polynomial in baseband.

[0059] Preferably, said polynomial model is of odd order and does not take account of the even-order terms.

[0060] It is possible to take no account of even-order terms since they do not generate harmonics in the band of the high-frequency signal.

[0061] Preferably, said polynomial may be written y{tilde over ( )}=x{tilde over ( )}*G{tilde over ( )}_ain baseband, where:

[0062] the input may be written x{tilde over ( )}=x_i+j*x_q

[0063] the output may be written y{tilde over ( )}=y_i+j*y_q

[0064] G_a{tilde over ( )}(P_x)=a₁{tilde over ( )}+(3/4)·a₃{tilde over ( )}·(P_x)+(5/8)·a₅{tilde over ( )}·(P_x)²+(35/64)·a₇{tilde over ( )}·(P_x)³+ . . .

[0065] where:

[0066] P_x=x_i²+x_q²

[0067] a₁{tilde over ( )}=a_i·e^−jωτi

[0068] x_iand x_qare respectively the in-phase component and the quadrature component of x;

[0069] y_iand y_qare respectively the in-phase component and the quadrature component of y; and

[0070] the coefficients a_iand the associated delays τ_iare parameters of said predetermined amplifier model.

[0071] Going over to baseband makes it possible to facilitate computation of the parameters characterizing the modelled amplifier.

[0072] Preferably, said test signal is a signal having n tones whose frequencies and amplitudes are known.

[0073] In this way, it is possible to determine, a priori, the frequencies at which the cross-modulation products can appear in the amplified test signal. By means of this information, it is possible to use amplifier model identity algorithms that are much simpler and of higher performance than those generally used.

[0074] Preferably, the test signal is made up of two sinewaves (n=2) having the same amplitude.

[0075] The invention also provides a system for digitally linearizing an amplifier having non-linearity, in which system the means for determining correction parameters comprise:

[0076] means for acquiring said amplified test signal in the form of samples in baseband;

[0077] means for defining a modelled amplifier, by computing parameters of a predetermined amplifier model on the basis of said samples of the amplified test signal; and

[0078] means for computing said correction parameters on the basis of said modelled amplifier so that they make it possible to linearize said modelled amplifier.

[0079] Other characteristics and advantages of the invention appear on reading the following description of a preferred embodiment of the invention given merely by way of non-limiting example, and with reference to the accompanying drawings, in which:

[0080] FIGS. 1 and 2 show respective known systems for linearizing amplifiers, the systems respectively including an analog feedback loop (FIG. 1) and an analog feedforward loop (FIG. 2);

[0081] FIG. 3 is a simplified flow chart of a particular implementation of the method of the invention for digitally linearizing an amplifier; and

[0082] FIG. 4 is a simplified diagram showing a particular embodiment of the system of the invention for digitally linearizing an amplifier implementing the method shown in FIG. 3.

[0083] The invention relates to a method and a system for digitally linearizing an amplifier.

[0084] As shown in the flow chart of FIG. 3, the linearization method of the invention is of the type comprising two successive stages 1 and 2.

[0085] The purpose of the first stage 1 is to determine correction parameters for correcting the non-linearity of the amplifier. It is recalled that the correction defined by said parameters, which correction is intended to be used during the second stage 2 described in detail below, is predistortion having non-linearity that is the inverse of the non-linearity of the amplifier. The correction parameters are determined by applying a predetermined test signal to the input of the amplifier and by analyzing the signal obtained at the output (this signal being referred to as the “amplified test signal”). At the end of this first stage 1, the correction parameters are stored, e.g. in a “predistortion” table.

[0086] The purpose of the second stage 2 is to amplify a given signal linearly. It comprises the following two steps: a predistortion step 21 in which the signal to be amplified is predistorted as a function of the previously determined correction parameters so as to obtain a predistorted signal; and an amplification step 22 in which the predistorted signal is amplified by the amplifier. If the predistortion is well matched, the signal present at the output of the amplifier corresponds to the input signal as amplified linearly.

[0087] The present invention relates more precisely to a novel way of performing the first stage 1, in which correction parameters are determined for correcting the non-linearity of the amplifier.

[0088] In the invention, the first stage 1 comprises the following steps:

[0089] acquiring (11) the amplified test signal in the form of samples in baseband;

[0090] defining (12) a modelled amplifier, by computing parameters of a predetermined amplifier model on the basis of the samples of the amplified test signal; and

[0091] computing (13) correction parameters on the basis of the modelled amplifier so that they make it possible to linearize the modelled amplifier.

[0092] In the particular implementation described below, the step 12 in which the modelled amplifier is defined comprises the following steps:

[0093] estimating (121) the complex amplitudes of a predetermined number of harmonics at known frequencies; these harmonics are generated by the test signal applied to the input passing through the non-linearity of the amplifier; this estimation is performed by analyzing the spectra of the samples of the amplified test signal; and

[0094] computing (122) the parameters of the predetermined amplifier model by identifying firstly the estimated complex amplitudes and secondly mathematical expressions corresponding to complex amplitudes of harmonics at the known frequencies for a signal present at the output of the modelled amplifier. The mathematical expressions are computed on the basis of the predetermined amplifier model, by assuming that the test signal is applied to the input thereof.

[0095] A description follows of a particular implementation of the step 11 in which the amplified test signal is acquired (in the form of samples in baseband).

[0096] It is assumed that the test signal applied to the input of the amplifier is a two-tone signal that may be written as follows:

[0097] e_n=A₁·e^j(2πfnT+θ)+A₂·e^j(2πfnT+θ)

[0098] For reasons of simplification, the following are chosen:

[0099] A₁=A₂=1, which means that the two sinewaves have the same amplitude; and

[0100] θ₁=θ₂=0, which means that a good origin is taken for the times.

[0101] At the output of the amplifier, because of the non-linearity thereof, the amplified test signal has different harmonics. The amplified test signal is shifted to baseband (or to an intermediate frequency that is low enough), and is then filtered to remove the high-order harmonics. After filtering, only cross-modulation products of odd orders (orders 3, 5, 7, 9, . . . ) remain. This gives a sum of sinewaves at the frequencies f₁, f₂, 2*f₁−f₂, 2*f₂−f₁, 3*f₁−2*f₂, 3*f₂−2*f₁, 4*f₂−3f₁, 4*f₁−3*f₂, etc. The filtered signal is then sampled before passing through analog-to- digital converters.

[0102] Ultimately, the amplified test signal is represented in the form of digital samples in baseband.

[0103] A description follows of a particular implementation of the estimation step 121 in which the complex amplitudes of certain harmonics are estimated. In this particular implementation, parametric spectrum analysis is performed on the samples of the amplified test signal, e.g. using an Auto-Regressive (AR) technique or an Auto-Regressive Moving Average (ARMA) technique.

[0104] In addition, a predetermined signal model is used that is derived from the Prony model and is such that the n^thsample of the amplified test signal can be estimated by: ${\hat{s}}_n=\sum _{i=1}^{i=p}{H}_i^{~}{e}^{j\left(2\pi f_i\mathrm{nT}\right)}+e_n$

[0105] where:

[0106] p is the number of harmonics in question;

[0107] f_iis the frequency of i^thharmonic in question;

[0108] H_i{tilde over ( )} is the complex amplitude of the i^thharmonic in question;

[0109] T is the sampling period; and

[0110] e_nis the noise, e.g. the quantification noise of the analog-to-digital converters, affecting the sample s_n.

[0111] For example, if the harmonics generated up to the fifth order are to be estimated, it is necessary to take p=6. The following complex amplitudes are then sought: H₁{tilde over ( )}, H₂{tilde over ( )}, H₃{tilde over ( )}, H₄{tilde over ( )}, H₅{tilde over ( )}, and H₆{tilde over ( )}. The associated frequencies f_iare as follows: f₁, f₂, 2*f₁−f₂, 2*f₂−f₁, 3*f₁−2*f₂, 3*f₂−2*f₁.

[0112] As explained below, the complex amplitudes H_i{tilde over ( )} of the harmonics make it possible to construct the modelled amplifier (i.e. to determine the parameters thereof) and thus the predistortion table. These amplitudes characterize the AM-AM and AM-PM distortion of the amplifier.

[0113] If S=[s₀s₁. . . s_N]^Tis the vector of the samples acquired during the acquisition step 11 in which the amplified test signal is acquired, and Ŝ=[ŝ₀ŝ₁. . . ŝ_N]^Tis the vector of the samples estimated using the above-mentioned signal model, and by minimizing (e.g. using the least squares method) the distance ∥S−Ŝ∥, it can be shown that:

[0114] H{tilde over ( )}=M S

[0115] where

[0116] H{tilde over ( )}=[H₁{tilde over ( )} H₂{tilde over ( )} . . . H_p{tilde over ( )}]^Tis the vector of the complex amplitudes of the p harmonics in question;

[0117] S=[s₀s₁. . . s_N]^Tis the vector of the N samples of the amplified test signal taken into account for computing the complex amplitudes of harmonics. The number N depends on the duration of the sampling window and on the over-sampling factor. For example, with a window of 1 ms, and a factor of 2, N=36 (for a symbol rate of 18 kHz); and

[0118] M is a predetermined matrix (p,N) that can be written: M=([Z^HZ]⁻¹Z^H), where z_i=exp(2jπf_i), Z^His the Hermitian transpose of Z, and $Z=\left[\begin{array}{cccc}1& 1& \cdots & 1\\ z_1& z_2& \cdots & z_p\\ ⋮& ⋮& \cdots & ⋮\\ z_1^{N-1}& z_2^{N-1}& \cdots & z_p^{N-1}\end{array}\right]$

[0119] A description follows of a particular implementation of the step 122 in which the parameters of the predetermined amplifier model are computed.

[0120] For example, the predetermined amplifier model may be a polynomial model of odd order (5, 7, or even 9, depending on the desired degree of accuracy) which can be written in the following form (e.g. by going up to the 7^thorder):

[0121] y(t)=a₁*x(t−τ₁)+a₃*x(t−τ₃)³+a₅*x(t−τ₅)⁵+a⁷*x(t−τ₇)⁷  (1)

[0122] where x(t) and y(t) are respectively the input signal and the output signal. The coefficients a_iand the delays τ_iare the parameters of the predetermined amplifier model. This model therefore does not take into account the terms of even order because such terms generate no harmonic in the band of the Radio Frequency (RF) signal.

[0123] The modulated signal x(t) may be written as follows, as a function of its in-phase component x_iand of its quadrature component x_q, and with ω=2πft, where f is the frequency of the carrier:

[0124] x(t)=x_i* cos(ωt)−x_q* sin(ωt)  (2)

[0125] Similarly, the signal y(t) at the output of the amplifier may be written as follows, as a function of its in-phase component y_iand of its quadrature component y_q:

[0126] y(t)=y_i* cos(ωt)−y_q* sin(ωt)  (3)

[0127] By replacing x(t) in (1) by its expression obtained in (2), the following is obtained after computation:

[0128] y(t)=x_i*Re [e^jωt*G_a{tilde over ( )}]−x_q*Im[e^jωt*G_a{tilde over ( )}]  (4)

[0129] where:

[0130] G_a{tilde over ( )}=a₁·e^−jωτ1+(3/4)·a₃·e^−jωτ3·(x_i²+x_q²)+

[0131] (5/8)·a₅·e^−jωτ5·(x_i²+x_q²)²+(35/64)·a₇·e^−jωτ7·(x_i²+x_q²)³  (5)

[0132] i.e.:

[0133] G_a{tilde over ( )}=a₁{tilde over ( )}+(3/4)·a₃{tilde over ( )}*(P_x)+(5/8)·a₅{tilde over ( )}*(P_x)²+(35/64)·a₇{tilde over ( )}*(P_x)³  (5)

[0134] by writing:

[0135] a₁{tilde over ( )}=a_i·e^−jωτ1

[0136] and:

[0137] P_x=x²_I+x²_q

[0138] By combining (3) and (4), and by writing:

[0139] G{tilde over ( )}_a=G_ai+j*G_aq

[0140] the following are obtained:

[0141] y_i=x_i*G_ai−x_q*G_aq

[0142] and

[0143] y_q=x_i*G_aq−x_q*G_ai

[0144] which can also be written:

[0145] y{tilde over ( )}=x{tilde over ( )}*G{tilde over ( )}_a  (6)

[0146] where: x{tilde over ( )}=x_i+j*x_qand y{tilde over ( )}=y_i+j*y_q

[0147] This is the baseband representation of the RF polynomial modelling (1) of the amplifier. It can be noted that the phase and amplitude distortions of the amplifier depend only on the instantaneous power P_xof the input signal. This fact is used to construct the predistortion table, as explained below.

[0148] For greater clarity, and if necessary, we replace

[0149] G{tilde over ( )}_a=G_ai+j*G_aqwith G{tilde over ( )}_a(P_x)=G_ai(P_x) +j*G_aq(P_x).

[0150] The amplitude and phase distortions of the amplifier can be written as follows in baseband:

[0151] amplitude distortion (AM-AM) : p_y=p_x*|G{tilde over ( )}_a(P_x)|²; and

[0152] phase distortion (AM-PM):

[0153] arg(y{tilde over ( )})=arg(x{tilde over ( )})+arg(G{tilde over ( )}_a(P_x))

[0154] To “construct” the modelled amplifier, it is necessary to determine the values of the parameters a_iand τ_i. For this purpose, as explained above, a two-tone test signal is caused to pass through the non-linear amplifier. Thus:

[0155] x_test(t)=A₁cos(ω₁t+θ₁)+A₂cos(ω₂t+θ₂)  (7)

[0156] To keep the model simple, let: A₁=A₂=1, and θ₁=θ₂=0, as indicated above.

[0157] In the expression (1), by replacing x(t) with x_test(t) as defined in the expression (7), and:

[0158] by taking account of the fact that:

[0159] ω_i*τ_i≈ω₂*τ_i≈ω*τ_i;

[0160] by assuming that:

[0161] ω₁=ω−Δω and ω₂=ω+Δω, where Δω<<ω;

[0162] by noting that, for conventional amplifiers, the τ_kvalues are of the order of one nanosecond, and, as far as possible, Δω is chosen such that Δω·τ_k<<1; and

[0163] by removing the harmonics which are not in the working band, the following is obtained after computation: $\begin{array}{cc}\begin{array}{c}{y\left(t\right)}_{\mathrm{filtered}}=\mathrm{Re}\left[H_1^{~}*\left(e^{{\mathrm{j\omega }}_{1}t}+e^{{\mathrm{j\omega }}_{2}t}\right)\right]+\\ \mathrm{Re}\left[H_3^{~}*\left(e^{j\left(2{\omega }_2-{\omega }_1\right)t}+e^{j\left(2{\omega }_1-{\omega }_2\right)t}\right)\right]+\\ \mathrm{Re}\left[H_5^{~}*\left(e^{j\left(3{\omega }_2-2{\omega }_1\right)t}+e^{j\left(3{\omega }_1-2{\omega }_2\right)t}\right)\right]+\\ \mathrm{Re}\left[H_7^{~}*\left(e^{j\left(4{\omega }_2-3{\omega }_1\right)t}+e^{j\left(4{\omega }_1-3{\omega }_2\right)t}\right)\right]\end{array}& \left(8\right)\end{array}$

[0164] where:

[0165] H₁{tilde over ( )}=a₁{tilde over ( )}+(9/4)a₃{tilde over ( )}+(25/4)a₅{tilde over ( )}+(1225/64)a₇{tilde over ( )}  (9)

[0166] H₃{tilde over ( )}=(3/4)a₃{tilde over ( )}+(25/8)a₅{tilde over ( )}+(735/64)a₇{tilde over ( )}  (10)

[0167] H₅{tilde over ( )}=(5/8)a₅{tilde over ( )}+(245/64)a₇{tilde over ( )}  (11)

[0168] H₇{tilde over ( )}32 (35/64)a₇{tilde over ( )}  (12)

[0169] The complex amplitudes H_i{tilde over ( )} are estimated previously, during the step 121. It is thus possible, by inverting the system formed by the equations (9) to (12), to find the coefficients a_iand τ_i, i.e. the parameters of the modelled amplifier. Hence:

[0170] a₁{tilde over ( )}=(H₁{tilde over ( )}−3*H₃{tilde over ( )}+5*H₅{tilde over ( )}−7*H₇{tilde over ( )})  (13)

[0171] a₃{tilde over ( )}=(4/3)(H₃{tilde over ( )}−5*H₅{tilde over ( )}+14*H₇{tilde over ( )})  (14)

[0172] a₅{tilde over ( )}=(8/5)(H₅{tilde over ( )}−7*H₇{tilde over ( )})  (15)

[0173] a₇{tilde over ( )}=(64/35)(H₇{tilde over ( )})  (16)

[0174] Thus, the modelled amplifier is well defined in baseband. It is then used to construct the predistortion table.

[0175] A description follows of a particular implementation of the step 13 in which the correction parameters are computed.

[0176] The signal y{tilde over ( )}_linearthat is desired at the output, and that would be obtained if the amplifier were linear may be written as follows:

[0177] y{tilde over ( )}_linear=K*x{tilde over ( )}  (17)

[0178] where K is the ideal gain of the amplifier, and x{tilde over ( )} is the baseband modulated signal at the input of the amplifier.

[0179] It should be noted that the gain K of the amplifier is constant: it does not therefore depend on the instantaneous power of the signal x{tilde over ( )} at the input of the amplifier. For reasons of simplicity, the gain K is assumed to be real (no phase shift between the input and the output of the amplifier that is to be linearized).

[0180] A signal z{tilde over ( )}=z_i+j*z_qis sought that is to be applied in place of x{tilde over ( )} to the input of the non-linear amplifier, and that enables y{tilde over ( )}_linearto be obtained at the output. In other words, the signal z{tilde over ( )} is sought that corresponds to the predistorted signal x{tilde over ( )}, such that:

[0181] z{tilde over ( )}*G{tilde over ( )}_a(P_z)=y{tilde over ( )}_linear=K*x{tilde over ( )}  (18)

[0182] where P_z=z²_i+z²_q

[0183] It is not possible to invert the relationship (18) directly, i.e; to compute z{tilde over ( )} on the basis of x{tilde over ( )}. However, it is easy to deduce from (18) that:

[0184] if P_zis known, then z{tilde over ( )} is fully determined:

[0185] z{tilde over ( )}=(K/G{tilde over ( )}_a(P_z))*x{tilde over ( )}  (19)

[0186] the choice of P_zdepends on P_xonly; on the basis of (18):

[0187] P_z*|G{tilde over ( )}_a(P_z)|²=K²*P_x  (20)

[0188] This relationship is the only condition on P_z.

[0189] To compute z{tilde over ( )} on the basis of x{tilde over ( )}, the following are then performed:

[0190] the instantaneous power P_xof the signal x{tilde over ( )} is computed: P_x=x_i²+x_q²;

[0191] P_zis determined, satisfying (20); and

[0192] z{tilde over ( )} is then computed by means of (19):

[0193] z{tilde over ( )}=G{tilde over ( )}_p(P_x)*x{tilde over ( )}  (21)

[0194] where:

[0195] G{tilde over ( )}_p(P_x)=K/G{tilde over ( )}_a(P_z)  (22)

[0196] The contents of a predistortion table including the correction parameters follow naturally from the above. The input of the table is P_xand it outputs the value G{tilde over ( )}_p(P_x), which appears as a “complex predistortion gain” on the basis of (21). The value of the predistorted signal is obtained by applying the complex gain G{tilde over ( )}_p(P_x) to x{tilde over ( )}.

[0197] The invention also relates to a system of digitally linearizing an amplifier. A particular embodiment of the system is described below with reference to FIG. 4.

[0198] The system comprises:

[0199] a digital-to-analog converter 41 receiving either a predistorted signal 47 or a test signal 49, depending on the operating mode of the system (as explained below);

[0200] means 42 for putting the signal on a carrier, on two paths I and Q in quadrature, the means being placed downstream from the digital-to-analog converter 41;

[0201] an amplifier 43 placed downstream from the means 42 for putting the signal on a carrier, which amplifier delivers an amplified signal 44; it is recalled that the amplifier 43 has non-linearity that is to be compensated by predistortion;

[0202] predistortion means 45 receiving a digital signal to be amplified 46 and delivering a predistorted signal 47 (to be applied to the digital-to-analog converter 41);

[0203] means 48 for generating a test signal 49 to be applied to the digital-to-analog converter 41;

[0204] means 50 for determining the correction parameters used by the predistortion means 45;

[0205] first switch means 51 making it possible to apply either the predistorted signal 47 or the test signal 49 to the input of the digital-to-analog converter 41;

[0206] optional second switch means 52 making it possible to direct the amplified signal 44 either to the output of the system (corresponding, for example, to an antenna 54) or to the means 50 for determining the correction parameters; and

[0207] third switch means 53 making it possible to make or break a connection between the predistortion means 45 and the means 50 for determining the correction parameters.

[0208] The system has two operating modes corresponding to respective ones of the two stages 1, 2 of the method described above with reference to FIG. 3, namely:

[0209] an “acquisition” mode, in which the digital-to-analog converter 41 receives the test 49, and the means 50 for determining the correction parameters receive the amplified signal 44 (which is then the amplified test signal 55), and are connected to the predistortion means 45 (which they inform of the new correction parameters; and

[0210] an “amplification” mode in which the digital-to-analog converter 41 receives the predistorted signal 47, and the amplified signal 44 is transmitted to the antenna 54.

[0211] In a particular embodiment of the invention, the means 50 for determining the correction parameters comprise:

[0212] demodulation means 56 for demodulating the amplified test signal 55;

[0213] an analog-to-digital converter 57 placed downstream from the demodulation means 56;

[0214] means 58 for defining a modelled amplifier; and

[0215] means 59 for computing correction parameters on the basis of the modelled amplifier; in other words these are means for constructing the predistortion table used by the predistortion means 45 in amplification mode.

[0216] The definition means 58 and the computation means 59 are, for example, contained in a digital signal processor (DSP) 60. They and the predistortion means 45 operate in a manner corresponding to implementation of the steps bearing the same respective names, as described above with reference to FIG. 3. The manner in which they operate is therefore not described again.