Method for the simulation of a nonlinear amplifier with envelope memory effect

A method for the simulation of responses of a nonlinear amplifier provides for measuring characteristics of nonlinearity of amplitude and of amplitude/phase-shift conversion of the amplifier, each measurement being made at an amplitude that is constant in input. The method further includes measuring the characteristics at different frequencies, developing the characteristics in sequences of direct transfer functions, computing frequency correctors for the direct transfer functions, measuring characteristics of distortion of amplitude modulation, each measurement being performed by modulating the input amplitude, computing modulation transfer functions reproducing the distortion amplitudes at output according to the input modulation amplitudes and correcting the direct transfer functions when the input amplitude is modulated in order to simulate the envelope memory effect. There is a direct application of the invention to the field of the simulation of high efficiency microwave amplification.

BACKGROUND OF THE INVENTION
 1. Field of the Invention
 The present invention relates to the field of the simulation of signal
 responses of nonlinear amplifiers. An object of the invention is a system
 for the simulation of the response signal of a nonlinear amplifier having
 a memory effect.
 A system of this kind can be applied to the simulation of high efficiency
 microwave amplification, especially in the AB, B or C class of
 amplification and more particularly to the simulation of the response of
 solid state power amplifiers (SSPA) and travelling wave tube amplifiers
 (TWT) used in land or satellite transmission radio links.
 At high frequencies or at high efficiency, amplifier devices of this kind
 have a nonlinear characteristic response curve.
 2. Description of the Prior Art
 FIG. 1 illustrates an exemplary input/output response curve of a nonlinear
 amplifier ANL. The curve giving the output signal level g of the amplifier
 ANL as a function of the input signal level x is typically inflected at
 the high amplitudes A of the input signal x because of saturation
 phenomena. When the amplifier is used in conditions such that the gain is
 not constant as a function of the input signal level x, it is said to be
 an amplifier working in nonlinear mode or more simply the amplification is
 called nonlinear amplification.
 Nonlinear devices can be divided into memoryless devices, quasi-memoryless
 devices and devices with memory.
 Memoryless amplifiers have high nonlinearity in amplitude and a lower phase
 distortion. The input/output response characteristic of a memoryless
 nonlinear amplifier ANL can then be reduced, as shown in FIG. 1, to a
 single curve g(x).
 It is possible to model and simulate the response, as a function of the
 time t, of an memoryless amplifier ANL to a sinusoidal input signal x,
 with a frequency f.sub.0, that is amplitude modulated A and phase
 modulated .phi., the signal having the following form:
EQU x(t)=A(t).cos(2.f.sub.0.t+.phi.(t)) (1)
 A(t) represents the envelope of the input signal, which is defined by the
 amplitude limits in which the sine signal x evolves, the envelope varying
 as a function of time. FIG. 3 illustrates a timing diagram of a signal
 x(t) having a constant envelope A.
 The output signal g of the memoryless amplifier then has the following
 form:
EQU g(t)=C(A(t)).cos(2.pi..f.sub.0.t+.phi.(t)) (2)
 It is useful to give a complex envelope to these signals x and g in
 abandoning any reference to the time t since the memoryless amplifiers
 have an instantaneous response.
 The input signal x of the equation (1) has a complex envelope X with the
 following form:
EQU X=A.exp(j..phi.) (3)
 All the useful information pertaining to the amplitude modulation A(t) and
 phase modulation .phi.(t) is recorded in this complex envelope X.
 The output signal g of equation (2) similarly has a complex envelope G with
 the following form:
EQU G=C(A).exp(j..phi.) (4)
 For a memoryless nonlinear amplifier ANL, it is shown that C(A) is the
 Chebyshev transform of the input/output response curve g(x).
 The response of a :memoryless nonlinear amplifier ANL to a modulated signal
 x can therefore be modeled and simulated simply by a single curve C(A),
 whose example is shown in an unbroken line in FIG. 6A. A curve of this
 kind giving the output amplitude C as a function of the input amplitude A
 is called a curve of nonlinearity in amplitude and is referenced in
 abbreviated form as an AM/AM curve.
 FIG. 2 illustrates a response characteristic of a nonlinear amplifier with
 memory ANLAM on which there appears a phenomenon of hysteresis prompted by
 a memorizing effect. It can be seen that the rising hysteresis curves m
 and m' are not superimposed on each other when the respective amplitudes A
 and A' of the input signal x are different. The variation in memorizing
 time related to the variation in amplitude prevents the curves m and m'
 from getting superimposed on each other.
 When the memorizing time of the amplifier ANLAM is negligible in comparison
 to the period of the amplitude variation A(t), it can furthermore be
 considered that the amplitude A is stable and the amplifier is called a
 quasi-memoryless amplifier.
 FIG. 4 shows a timing diagram of an output signal y of a quasi-memoryless
 amplifier to which there is applied the input signal x illustrated by the
 timing diagram of FIG. 3 whose formula is recalled here below:
EQU x(t)=A(t).cos(2.pi..f.sub.0.t+.phi.(t)) (1)
 For a quasi-memoryless nonlinear amplifier to which the signal x of
 equation (1) is applied, the output signal y takes the following form:
EQU y(t)=C(A).cos(2.pi..f.sub.0.t-.PHI.(A)+.phi.(t)) (5)
 where C(A) is the amplitude of the output signal y,
 and .PHI.(A) is the phase shift of the output signal y,
 which depends on the amplitude A(t) of the input signal x.
 Thus, at a given instant t, the amplitude C(A) and the phase shift .PHI.(A)
 of the output signal y depend solely on the amplitude A of the input
 signal x at this instant t. It is thus possible to overlook the amplitude
 variations A(t) as a function of time and consider that the amplitude A is
 almost constant as can be seen in FIG. 3.
 It is also useful to write in a complex form the envelopes of the signals x
 and y expressed here above, namely as the envelopes X and Y which take the
 following respective forms:
EQU X=A.exp(j..phi.) (3)
 (the envelope A considered as being constant in time)
EQU Y=C(A).exp(j..phi.-.PHI.(A)) (6)
 The response of a quasi-memoryless nonlinear amplifier can therefore be
 modeled and simulated simply on the basis of knowledge of the following
 two characteristic curves:
 1. A curve C(A) wherein the amplitude C of the output signal y is a
 function of the amplitude A of the input signal, illustrated for example
 by the curve AM/AM, in an unbroken line, of FIG. 6A (amplitude/amplitude
 conversion curve);
 2. A curve .PHI.(A) wherein the phase shift .PHI. of the output signal y
 with respect to the input signal x is a function of the amplitude A of the
 signal x, called an amplitude/phase conversion curve, abbreviated as
 AM/PM, an example of which is shown in an unbroken line in FIG. 6B.
 It can be shown that, similarly, the curve C(A) is the norm of the complex
 Chebyshev transform of the response characteristic y(x), the curve
 .PHI.(A) being the argument of the complex transform.
 The complex envelope Y of the output signal can also be written in the form
 of two parts, namely a real part and an imaginary part, corresponding to
 an in-phase component P and a quadrature component Q, these components P
 and Q having the following forms:
EQU P(A)=C(A).cos(.PHI.(A)) (7')
EQU Q(A)=C(A).sin(.PHI.(A)) (7")
 FIG. 7 shows an example of curves P(A) and Q(A) equivalent to the curves
 C(A) and .PHI.(A) of FIGS. 6A and 6B.
 The known models of simulation of the response of quasi-memoryless
 amplifiers generally prefer to use characteristics in the form of pairs of
 curves P(A) and Q(A) rather than in the form of pairs of curves C(A) and
 .PHI.(A), although these pairs of curves are strictly equivalent.
 According to a known principle of the simulation of nonlinear amplifiers,
 the amplifier that is made is precharacterized on the test bench. The
 precharacterizing is done with a signal having a specified frequency and
 amplitude in order to then simulate the response to a signal of any
 frequency and amplitude.
 As shown schematically in FIG. 5, a signal with a single-frequency f.sub.0
 taking different amplitudes A', A", A'" is applied to the amplifier tested
 to obtain its characteristics, illustrated for example in FIG. 6 or 7.
 However, for amplifiers having a certain quantity of memory, it is observed
 that the characteristics vary to a major degree depending on the frequency
 f.sub.0.sup.-, f.sub.0 or f.sub.0.sup.+ of the signal to be amplified.
 A known system for the simulation of such amplifiers has been explained by
 H. B. Poza in an article entitled "A Wideband Data Link Computer
 Simulation Model", in the "NAECON'75 Record", page 71. The article
 proposes plotting of several pairs of curves AM/AM and AM/PM for several
 frequencies f.sub.0.sup.-, f.sub.0 or f.sub.0.sup.+ of operation of the
 amplifier. FIGS. 6A and 6B thus show an example of three pairs of curves
 AM/AM and AM/PM obtained respectively at three frequencies f.sub.0.sup.-,
 f.sub.0 and f.sub.0.sup.+ located in the useful band BU of a directional
 radio link amplifier.
 H. B. Poza's simulator stores only one pair of curves AM/AM and AM/PM, for
 example the pair of curves obtained at the frequency f.sub.0, and
 reconstitutes the other pairs of curves (not stored) corresponding to the
 other frequencies f.sub.0.sup.-, f.sub.0.sup.+ or to intermediate
 frequencies. A non-stored curve is deduced simply by translating the
 stored curve by an appropriate vector. The simulator computes the
 components along the axis (A) and along the axis (C) or (.phi.) of the
 translation vector to bring the curve for the frequency f.sub.0 ; shown in
 an unbroken line in FIG. 6A or 6B, as close as possible to the frequencies
 f.sub.0.sup.- or f.sub.0.sup.+, shown in dashed lines.
 A simulation system of this kind gives too much of an approximation to
 simulate the distortions that appear on a quasi-memoryless amplifier at
 different frequencies.
 Another drawback of a system of this kind is that it cannot be used to
 simulate the response of an amplifier with memory.
 FIG. 9 illustrates another known system of simulation according to the
 model of A. A. M. Saleh, described in an article "Frequency-Independent
 and Frequency-Dependent Models of TWTA Amplifiers" November 1982, in "IEEE
 Transactions on Communication", Volume Com-29, No. 11, page 1715. The
 computation of the response to an input signal x of a quasi-memoryless
 nonlinear amplifier can be subdivided into two steps P(A,f) and Q(A,f) for
 the computation of two respective components yp and yq of the output
 signal y. The component yp is in phase with the input signal x while the
 component yq is in quadrature.
 Saleh's stimulation system uses results for the characterization of the
 amplifier at several frequencies f, several pairs of curves P(A) and Q(A)
 being plotted at several frequencies f to compute transfer functions
 P(A,F) and Q(A,f) corresponding to each arm for the computation of the
 components yp and yq of the output signal y. Each of the transfer
 functions P(A,f) or Q(A,f) corresponds to the computation of a nonlinear
 response without memory effect, since no phase shift is introduced into
 each of these computation branches.
 A. A. M. Saleh's article points out that the model can be applied to the
 amplification of single-frequency signals and, by conjecture, assumes that
 it will be applicable to the simulation of signals of any form.
 Another drawback of this model is that it cannot be applied to amplifiers
 with memory, as the system does not realize distortions appearing at
 different frequencies when the amplitude of the signal varies at high
 speed with respect to the memorizing time constants.
 For nonlinear amplifiers with memory ANLAM, the memorizing effects are
 greater, as the memorizing time is not negligible in comparison with the
 time of variation of the amplitude A(t). In such cases, a more complex
 method such as that of nonlinear differential equations or the development
 of the characteristic curves in sequences of functions is necessary to
 simulate the response of the amplifier with memory ANLAM.
 An improved known model is proposed by M. T. Abuelma'atti in an article
 entitled "Frequency-Dependent Nonlinear Quadrature Model for TWT
 Amplifiers", in August 1984, in the journal IEEE Transactions on
 Communication, Volume Com. 32, No. 8, page 982. The modeling uses a
 development of the characteristic curves of a nonlinear amplifier in
 sequences of Bessel functions making it possible to simulate the response
 of amplifiers with memory.
 FIG. 10 illustrates the simulation system described by M. T. Abuelma'atti
 which, as here above, comprises two branches of computations of the in
 phase component yp and the quadrature component yq of the output signal y,
 each arm computing the contribution of a sequence of N Bessel functions J1
 depending on the amplitude A of the input signal x, in weighing each
 function J1 by a coefficient .alpha. and a factor G(f) of correction in
 frequency. The coefficients .alpha..sub.n and the factors G.sub.n (f) are
 computed after having established several pairs of curves P(A) and Q(A)
 that are characteristic of the nonlinear amplifier, the curves being
 plotted at several testing frequencies f of the amplifier.
 With FIG. 8, it can be seen that two weighted sums of N Bessel functions of
 the first kind of first order, referenced J1(n.pi.A/D), enable the very
 precise interpolation of the curves P(A) and Q(A), like those of FIG. 7,
 by adjusting the weighting coefficients .alpha..sub.np and .alpha..sub.nq
 of these functions J1.
 It can be noted that the Bessel functions are the Chebyshev transforms of
 sine functions and that the development, in sequences of Bessel functions,
 of the curves P(A) and Q(A) corresponds to a Fourier development, in
 sequences of sine functions, of the curves y(x) of nonlinearity of the
 amplifier, which is elegantly suited to the sinusoidal form of the
 hysteresis curves y(x) as shown in FIG. 2.
 Theoretically, a system of this kind should enable the simulation of the
 amplification of multicarrier signals, namely signals comprising several
 sinusoidal components of distinct frequencies.
 However, the nonlinear amplification of a multicarrier input signal is
 complicated by the appearance of distortions known as intermodulation
 phenomena. When a nonlinear amplifier receives several carrier frequencies
 at input, there are obtained at output, in addition to the amplified
 carriers, undesired harmonics known as intermodulation products, each
 harmonic having a frequency distinct from the frequencies of the carriers.
 FIGS. 11-14 illustrate the appearance of the intermodulation phenomenon
 during the nonlinear amplification of a two-carrier signal.
 FIG. 11 is a graph pertaining to the frequency of a two-carrier input
 signal x representing, for example, two carrier components with respective
 frequencies f.sub.-1 and f.sub.1 having an equal input amplitude A. The
 two carrier frequencies f.sub.-1 and f.sub.1, located in the useful
 frequency band BU of the amplifier are separated by a frequency difference
 df.
 FIG. 14 is a graph pertaining to the frequency of an output signal y that
 corresponds to the amplification ANLAM with memory effect of the
 two-carrier input signal x of FIG. 11. It can be seen that the output
 signal y has a series of harmonics of various frequencies f and different
 amplitudes C.
 The frequency of each of these intermodulation components is an integer
 combination of the carrier frequencies at input.
 The detail of the output harmonics included in the useful band BU,
 illustrated in FIG. 14, is as follows:
 1. The carriers, with respective frequencies f.sub.-1 and f.sub.1, and
 amplitudes C.sub.-1 and C.sub.1 at output; and
 2. The third-order intermodulation components, with respective frequencies
 f.sub.-3, f.sub.3, and amplitudes C.sub.-3, C.sub.3 at output.
 FIG. 15 reproduces results according to Abuelma'atti's model which has
 enabled an estimation of the amplitudes C.sub.-1, C.sub.1, and C.sub.-3,
 C.sub.3 of carrier components and third-order intermodulation components.
 However, the estimation of the amplitude of the intermodulation components
 by Abuelma'atti's model corresponds poorly to the reality of the
 measurements of intermodulation distortions on nonlinear amplifiers in
 multicarrier operation with high efficiency.
 Indeed, a general phenomenon known as an envelope memory effect arises when
 a multicarrier signal is amplified. For this type of signal, the envelope,
 which is defined by the positive and negative limits in amplitude of the
 signal, is not constant.
 In this case, it can no longer be assumed that the envelope is constant as
 in the quasi-memoryless models. In fact, the memorizing time constants are
 no longer negligible with respect to the time of variation of the
 envelope.
 FIG. 12, which illustrates the temporal progress of the two-carrier input
 signal x of FIG. 11 shows, for example, that the envelope X(t) and -X(t)
 of the two-carrier signal varies very swiftly and in major proportions
 whereas the amplitude A of each carrier f.sub.1 and f.sub.-1 is assumed to
 be constant.
 FIG. 13 which illustrates the temporal progress of the output signal y
 corresponding to the preceding two-carrier input signal x, shows that the
 envelope Y(t) and -Y(t) of the output signal y is deformed by the
 intermodulation distortions. The appellation `envelope memory effect` is
 used to designate such deformations of amplitude of the signal by a
 nonlinear amplifier with memory.
 The known models do not take into account the two effects illustrated in
 FIGS. 16 and 17. In particular, FIG. 16 illustrates that the ratio C1/C3
 (comparing the amplitude C1 of carriers with respect to the amplitude C3
 of the intermodulation components) varies considerably depending on the
 frequency difference df of the carriers and depending on the precise value
 of the amplitude A of the carriers of the input signal.
 Through the application of Abuelma'atti's model, the ratio C1/C3 does not
 depend on the frequency difference and varies continuously according to
 the input amplitude A.
 The second observed effect is that the presence of a second carrier
 influences the output amplitude of the first carrier.
 FIG. 17 shows, for example; output amplitude attenuation or resonance peaks
 C1 or C-1 of each of the carriers with frequencies f.sub.1 or f.sub.-1,
 depending on the frequency difference df between the two carriers (df
 being equal to f.sub.1 -f.sub.-1).
 Therefore, the above-identified prior art models do not account for these
 effects. Moreover, in general, these models simply do not simulate any
 envelope memory effect.
 SUMMARY OF THE INVENTION
 It is therefore an object of this invention to provide for a precise
 simulation of the envelope memory effect on nonlinear amplifiers. It is a
 further object of this invention to provide a method and device enabling
 the very precise simulation of the response of a nonlinear amplifier to
 multicarrier signals.
 It is yet a further object of this invention to provide a method and device
 for the simulation of nonlinear amplifiers with memory effect.
 The invention achieves the foregoing objects by providing for a dynamic
 measurement of the characteristics of the amplifier under the conditions
 in which the distortions appear. According to the invention, the
 characteristics of distortion of amplitude modulation are measured by
 providing for the application of an amplitude-modulated signal, at each
 measurement, which allows the envelope of the signal to vary. The
 distortion characteristics make it possible to correct the simulated
 responses on the basis of characteristics plotted at constant amplitude.
 The characteristics are measured by applying a multicarrier signal which
 is therefore amplitude modulated.
 In the preferred embodiment of the invention, a method for the simulation
 of signal responses of a nonlinear amplifier showing a memory effect,
 includes measuring characteristics of amplitude/amplitude conversion and
 amplitude/phase-shift conversion of the amplifier, wherein each
 measurement is made at an amplitude that is constant in input. The
 characteristics in sequences of direct transfer functions are developed
 and characteristics of distortion of amplitude modulation are measured,
 wherein each measurement is performed by modulating the input amplitude.
 Finally, the modulation transfer functions reproducing the distortion
 amplitudes are computed at output according to the input modulation
 amplitudes and the direct transfer functions are corrected when the input
 amplitude is modulated in order to simulate the envelope memory effect.
 A measurement of characteristics of distortion of amplitude modulation
 consists of the measurement of the intermodulation distortions by the
 application at input of a signal comprising at least two carriers at
 different frequencies. In an alternate embodiment of the invention,
 characteristics of rejection of intermodulation components are measured at
 output, wherein each measurement is performed with a group of carriers at
 input having specified amplitudes.
 Another alternative embodiment of the invention includes measuring
 characteristics of interaction of carriers at output, wherein each
 measurement is performed with a group of carriers at input having
 specified amplitudes.
 Yet another alternative embodiment of the invention includes measuring
 output modulation noise characteristics, wherein each measurement is
 performed with specified input modulation amplitudes.
 Another object of the invention provides a device for the simulation of
 signal responses of a nonlinear amplifier having a memory effect, wherein
 the device includes a module for the computation of the response of a
 nonlinear amplifier to a single-carrier signal. The module includes at
 least one filter having a transfer function prepared from real
 characteristics of the amplifier, including characteristics of
 nonlinearity of amplitude and amplitude/phase shift conversion, wherein
 the transfer function is corrected in frequency by a corrector computed on
 the basis of characteristics measured at different single-carrier signal
 frequencies. The device also includes means for the generation of
 harmonics in response to a multicarrier signal.
 The harmonic generation means of the device preferably includes a module
 for the Fourier analysis of a signal into frequency components and/or a
 combinational module computing the combinations of frequency of
 intermodulation components and/or a normative module computing an r.m.s.
 value of signal amplitude.
 In the preferred embodiment of the invention, a filter for the modulation
 of the response of the computation module includes a transfer function
 prepared from characteristics of rejection of intermodulation components
 when the amplification of a multicarrier signal is simulated by the
 device.
 In an alternative embodiment of the invention, a filter for the correction
 of the response of the computation module includes a transfer function
 prepared from carrier interaction characteristics, so that the amplitude
 of a carrier given by the computation module is corrected by the
 correction filter when the amplification of a multicarrier signal is
 simulated by the device.
 Other objects, features and advantages of the present invention will become
 apparent to those skilled in the art from the following detailed
 description and the accompanying drawings. It should be understood,
 however, that the detailed description and specific examples, while
 indicating preferred embodiments of the present invention, are given by
 way of illustration and not of limitation. Changes and modifications may
 be made within the scope of the present invention without departing from
 the spirit thereof, and the invention includes all such modifications.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
 Referring to the drawings, and more particularly to FIGS. 21-26, the first
 embodiments of the invention will be described. A simulation device 10
 (FIG. 24) simulates the amplification of signals of any form including
 signals comprising several frequency components. In this regard, device 10
 includes computation means 12 configured to simulate the response of a
 nonlinear amplifier (FIG. 25). Computation means 12 implements the process
 steps iterated in FIG. 26. Device 10 further includes means for the
 generation of harmonics in response to a multicarrier signal.
 Referring to FIGS. 21-23, device 10 according to the invention includes an
 output signal computation block 14, 16, 18 that computes an output signal
 y and a frequency computation block 20, 22, 24, 26 for the generation of
 harmonics that computes the frequencies of the intermodulation components
 before the computation of their precise amplitude to correct the output
 signal y. Output signal computation blocks 14, 16 and 18 implement a known
 simulation model such as the memoryless nonlinear model MNLSM by H. B.
 Poza (FIG. 21), the memoryless quadratic nonlinear model MNLSM by A. A. M.
 Saleh (FIG. 22) and the model with memory MM by M. T. Abuelma'atti (FIG.
 23). Other models can be implemented, and the description of the
 implementation of the invention will be developed essentially on the basis
 of the improved model by M. T. Abuelma'atti.
 The parallel structure of output computation blocks 14 and 16 (MNLSM) or 18
 (MM) and of harmonics generation blocks 20, 26 (Harm) or 22 (Squa) and 24
 (FFT) is given by way of example. The structure of the blocks could be
 serial, and the effective making of the computation blocks could be done,
 by preference, by computer programs, wherein the blocks provide a
 schematic representation of the functions of computation subprograms.
 FIG. 23 provides a detailed view of an alternative embodiment of harmonics
 generation for the computation of the improved intermodulation components
 on the basis of the Abuelma'atti model shown in FIG. 10. Blocks 26 (Harm)
 and 18 (MM) enable the computation of the amplitudes of intermodulation
 components . . . , Y.sub.010, Y.sub.1-11, Y.sub.-120, Y.sub.-111,
 Y.sub.0-12, Y.sub.-102, Y.sub.20-1, Y.sub.2-10, Y.sub.11-1, Y.sub.02-1, .
 . . when the input signal x comprises a multitude of frequency components
 A, B, . . . , M. This complex structure of blocks 26 (Harm) and 18 (MM) is
 designed to simulate the amplification of signals of any form, and
 especially multicarrier signals.
 As illustrated in FIG. 20, the harmonics generation block 26 (Harm)
 includes an FFT Fourier series analysis filter 28 to analyze the
 multiple-frequency input signal x into frequency components with
 amplitudes A, B, . . . , M. These components are applied to the input of
 elementary filters grouped in tables corresponding to the computation of
 an intermodulation component, for example Y.sub.010. Each elementary
 filter has a transfer function derived from a Bessel function J,
 generalized on the example of the filters of FIG. 10. The detail of the
 transfer functions of the filter tables and of the generalized Bessel
 functions shall be described in detail after the description of models.
 The architecture of device 10 according to the invention comprises means
 for the generation of harmonics, preferably with a Fourier series analysis
 module and a known simulation model.
 A development of the formulae proposed by the most improved of the known
 models, the Abuelma'atti model, leads to simulation errors. These
 simulation errors are corrected using computation means 12. In particular,
 following the above equations 7' and 7", Abuelma'atti (FIGS. 10 and 23)
 analyzes the nonlinearity characteristics in phase P and in quadrature Q
 separately into a sequence of N Bessel functions according to the
 following formulae:
 ##EQU1##
 where
 2D is the dynamic range at input of the amplifier, and
 .alpha..sub.np, .alpha..sub.nq are coefficients computed by a minimum
 mean-square-error method presented in the article quoted.
 This analysis and the corresponding Abuelma'atti simulation device are
 illustrated in FIG. 10.
 The input signal x has two carriers with frequencies f.sub.-1 and f.sub.1
 and equal amplitudes referenced A, i.e. that the signal x has the
 following form:
EQU x(t)=A.cos(2.pi..f.sub.-1.t)+A.cos(2.pi..f.sub.1.t) (9)
 The input signal x then has an envelope X that is highly modulated in
 amplitude as is shown by FIG. 12 and the following formula:
EQU X(t)=2A cos(.pi..df.t) (10)
 where
 df=f.sub.1 -f.sub.-1 represents the frequency difference between the two
 carriers f.sub.-1 et f.sub.1.
 According to the quasi-memoryless quadratic models, the envelope Y of the
 output signal corresponding to the input signal x of the equations 9 and
 10 takes the following quadratic form:
EQU Y.sub.p (t)=C[2A.cos(.pi..df.t)].cos(.PHI.[2A.cos(.pi..df.t)]) (11')
EQU Y.sub.q (t)=C[2A.cos(.pi..df.t)].sin(.PHI.[2A.cos(.pi..df.t)]) (11")
 where
 Y.sub.p is the in phase component of the envelope Y of the output signal y
 corresponding to the preceding input x, and
 Y.sub.q is the quadrature component of the envelope Y.
 Formulae 11 and 11' cannot be applied directly because the simulation of
 nonlinear amplifiers ANLAM with memory for the input envelope X is
 sinusoidal with a frequency half the difference df and varies far too
 rapidly in time as compared with the time constants of memorization.
 According to Abuelma'atti's improved model, the envelope Y of the output
 signal corresponding to the two-carrier input signal x of the equations 9
 and 10 comprises a series of components with a frequency referenced
 f.sub.i and an amplitude referenced C.sub.i, i being a positive or
 negative integer.
 As illustrated in FIG. 14, the frequency components of the output signal
 therein include the following frequencies and amplitudes:
 1. A carrier component, with a frequency f.sub.1 having an output amplitude
 C.sub.1 ;
 2. A carrier frequency, with a frequency f.sub.-1 having an output
 amplitude C.sub.-1 ;
 3. Second-order intermodulation components with frequencies respectively
 equal to 2.f.sub.-1, f-.sub.1 +f.sub.1, 2.f.sub.1 as well as f.sub.-1
 -f.sub.1, having non-referenced amplitudes;
 4. A third-order intermodulation component with a frequency referenced
 f.sub.3 equal to 2.f.sub.1 -f.sub.-1 and an amplitude of C.sub.3 ;
 5. A third-order intermodulation component with a frequency referenced
 f.sub.-3 equal to 2.f.sub.-1 -f.sub.-1 and an amplitude C.sub.-3 ;
 6. A fifth-order intermodulation component with a frequency referenced
 f.sub.5 equal to 3.f.sub.1 -2.f.sub.-1 and an amplitude C.sub.5 ;
 7. A fifth-order intermodulation component with a frequency referenced
 f.sub.-5 equal to 3.f.sub.-1 -2.f.sub.1 and an amplitude C.sub.-5 ; and
 8. Other odd-order components of smaller value and other even-order
 components generally rejected out of the useful band BU.
 Only the third-order and possibly fifth-order intermodulation components
 are located in the useful band BU and have amplitudes C.sub.-3, C.sub.3,
 C.sub.-5, C.sub.5 sufficient for examination in the response of the
 amplifier. The order of the intermodulation components is the sum of
 absolute values of the integer coefficients of combination of the carrier
 frequencies.
 The corresponding output signal y is the sum of the amplitudes of the
 frequency components according to the following formula:
EQU y(t)=.SIGMA.2.C.sub.i (A.sub.i).cos(2.pi..f.sub.i.t) (12)
 where
 i takes all the negative and positive odd-order integer values, and
 C.sub.i is a complex amplitude that is analyzed into a real part C.sub.ip
 (in phase) and an imaginary part C.sub.iq (in quadrature) according to the
 following formula:
EQU C.sub.i (A)=C.sub.ip (A)+C.sub.iq (A) (13)
 Based on Abuelma'atti, the computation of the amplitudes C.sub.i of the
 intermodulation components coming from the amplification of a two-carrier
 signal are performed on the basis of a summation of Bessel functions Jn of
 the first kind but of various nth orders according to the following
 formulae:
 ##EQU2##
 where
 .alpha..sub.np, .alpha..sub.nq are the coefficients obtained from the
 equations 8' and 8", by developing the characteristics P and Q of
 nonlinearity in first-order Bessel functions J1 only,
 and
 J0, J1, J2 are the first kind of zero order, first order and second order
 Bessel functions respectively.
 The zero order, first order, second order or Nth order of a Bessel function
 J0, J1, J2 or JN is chosen so as to correspond to the absolute value of
 the integer coefficient of combination of a carrier frequency f.sub.-1 or
 f.sub.1 in the frequency f.sub.i of the intermodulation component with an
 amplitude C.sub.i. The .vertline.i.vertline. order of each intermodulation
 component C.sub.i therefore corresponds to the sum of the zero, first,
 second, or Nth order of the Bessel functions J0, J1, J2 or JN involved in
 each formula.
 In the general case where the input signal x has m frequency components
 (carriers) of any amplitude, referenced A, B, . . . , M, there is obtained
 a multitude of intermodulation products each having a frequency given by
 the following formula:
EQU f.sub.i =.alpha..f.sub.A +.beta..f.sub.B + . . . +.mu..f.sub.M
 where
 .alpha., .beta., . . . , .mu. are relative integers, and
 f.sub.A, f.sub.B, . . . , f.sub.M are the frequencies of the carriers A, B,
 . . . , M respectively present in the input signal x.
 The amplitude C.sub.i of each intermodulation component with a frequency
 f.sub.i follows the following general formulae:
 ##EQU3##
 where
 J.vertline.n.vertline. is a Bessel function of the first kind and of an
 order corresponding to the absolute value of n (generalization).
 FIG. 20 illustrates a sophisticated embodiment of means MM for the
 computation of intermodulation components that has been developed from
 these formulae of a model with memory. In particular, FIG. 20 is a
 schematic view in the form of tables of elementary filters connected in
 series horizontally to compute the product of the generalized Bessel
 functions J.vertline..alpha..vertline. modulated by the amplitude of the
 components A, B, . . . , M of the multicarrier signal. Each horizontal
 result is then added up with the other horizontal results of the table and
 with the results of a similar table for the components in quadrature
 before giving, at output, the amplitude of a harmonic Y.alpha.,.beta., . .
 . , .mu..
 In this sophisticated exemplary embodiment, on the basis of a central
 carrier B and two side carriers A and M, apart from the amplified carrier
 Y.sub.010 with a frequency f.sub.B, there are at least nine
 intermodulation products Y.sub.20-1, Y.sub.2-10, Y.sub.11-1, Y.sub.02-1,
 Y.sub.1-11, Y.sub.-120, Y.sub.-111, Y.sub.0-12 et Y.sub.-102 in the
 immediate vicinity of the central frequency f.sub.B.
 Therefore, the number of intermodulation products increases exponentially
 with the number m of carriers, so that the performance of a simulation for
 a signal with more than two carriers cannot be detailed in the present
 application. Those skilled in the art can generalize the computations
 described for two carriers.
 A decisive factor of quality in the designing and choice of a microwave
 amplification system is the ratio of the amplitude C.sub.3 of the greatest
 intermodulation component at output to the amplitude C.sub.1 at output of
 a carrier component.
 In the preferred embodiment of the present application, this factor is an
 intermodulation rejection factor that is expressed by the following
 formula:
EQU P/I=C1/C3 (18)
 The greater the ratio P/I, the higher is the quality of this system, with
 the carrier C1 being distinguished more clearly from the intermodulation
 noise represented by C3.
 In the microwave domain, the rejection factor is generally expressed in
 terms of power. This corresponds to the following ratio:
EQU C1.sup.2 /C3.sup.2 referenced herein P/I.sup.2.
 In quadratic form, the intermodulation rejection factor can be written as
 follows:
 ##EQU4##
 By using the above results of the equations 14', 14" and 16', 16"
 established by Abuelma'atti, the intermodulation rejection factor
 simulated for a microwave amplifier varies very little according to the
 frequency.
 Indeed, the frequency correctors G.sub.n (f.sub.i) have a low correction
 effect, the curve of the correctors G.sub.n being computed from
 measurements with a single carrier, hence without modulation of the input
 envelope. In particular, the intermodulation rejection factors P/I thus
 computed are not a function of the frequency difference df of the carriers
 contrary to the effects of resonance observed and illustrated for example
 in FIG. 16.
 According to a first embodiment of the invention, the characteristics of
 intermodulation rejection are precisely measured and these measurements
 are used to correct the output signal y computed on the basis of a known
 model.
 In particular, the signal y(t) computed according to the Abuelma'atti model
 is compared with a signal z(t) comprising intermodulation components as
 really measured. This comparison results in modulation signal u(t) which
 will be simulated by a modulation filter 30 according to the invention.
 FIG. 21 illustrates a schematic view of a first embodiment of device 10
 according to the invention, including modulation filter 30. Filter 30 has
 a transfer function E(f,H) that computes the signal u as a function of the
 input signal. The signal y of the known model is then corrected by the
 modulator signal u in a Mix block 32 to produce a signal z which
 reproduces the modulation distortions actually measured.
 It is necessary to establish certain relationships that can be used to
 construct the transfer function E(f,H) of filter IMF 30 for the
 computation of the intermodulations. In this regard, the modulator signal
 u is useful only if the input signal has an amplitude-modulated envelope
 X. Indeed, when the input signal x is not modulated, the known model has
 an output signal y that appropriately simulates the response of the
 amplifier.
 In the preferred embodiment of the present invention, the envelope of the
 signal is computed by device 10 by taking the square of the norm of the
 signal x. Taking into account equation 10 of the envelope X of the
 two-carrier signal with an amplitude A and a frequency difference df, the
 square X.sup.2 of the following envelope is obtained:
EQU X.sup.2 (t)=4.A.sup.2.cos.sup.2 (.pi..df.t) (20)
 The square X.sup.2 given by equation 20 is split up into a constant
 amplitude A.sup.2 and a modulation factor given by the following equation:
EQU X.sup.2 (t)=2.A2.+2.A.sup.2.cos(2.pi..df.t) (20')
 FIG. 21 illustrates that filter 30 receives, at input, the r.m.s. value
 X.sup.2 (t) of the envelope of the input signal x.
 According to the invention, the modulator signal u generated by filter 30
 enables the writing of a correction formula as follows:
EQU Z(t)=Y(t).[1+U(t)] (21)
 where
 Z is the envelope of the output signal z with the actually measured
 intermodulations,
 Y is the envelope of the signal y computed by a known model; and
 U is the envelope of the signal u.
 As a result, when there is no modulation at input, the signal z simulated
 according to the invention has an envelope Z that corresponds to the
 envelope Y of the output signal y simulated by the known model MNLSM 14,
 16 or MM 18, and the modulator signal u is advantageously zero.
 By assumption, the envelope U of the modulator signal is expressed
 similarly to the equation 20' as the result of the conversion of the
 r.m.s. value X.sup.2 by the transfer function E(f,H) which depends on the
 frequency f and the amplitude H of the harmonics h at input, namely in the
 following form:
EQU U(t)=2A.sup.2.E(0,A)+2A.sup.2.E(df,A).cos(2.pi..df.t) (22)
 where
 E(df,A) is the response of the transfer function to a signal with a
 frequency df and an amplitude A,
 and
 E(0,A) is the response of the transfer function with a zero frequency
 difference.
 Conventionally, E(0,A) has a zero value since it is unnecessary to have a
 modulator signal u when there is no difference in carrier frequency.
 K is used as a reference for the following variable:
EQU K=A.sup.2.E(df,A)
 The envelope U of the modulator signal given by the equation 22 is
 expressed in the following complex and exponential form:
EQU 1+U(t)=K.exp(-j.2.pi.df.t)+1+K.exp(j.2.pi..df.t) (23)
 In a manner similar to the equation (12), the envelope Y of the output
 signal y of the known model is expressed in the following complex and
 exponential form:
 ##EQU5##
 where i is an odd integer taking all the negative and positive values.
 ##EQU6##
 Hence, from the equations (21), (23) and (24'), Z(t) is expressed as
 follows:
 ##EQU7##
 FIRST ALTERNATIVE EMBODIMENT OF THE INVENTION
 In the first alternative method of the invention, only the amplitude of the
 third-order intermodulation components is modulated by carrying out the
 minimum modification of the first-order carrier components computed by the
 known model. This is obtained by minimizing the value of K so that the
 terms K.C.sub.-3, K.C.sub.1, K.C.sub.-1 and K.C.sub.3 are negligible. In
 this case, the two central lines of the equation (25) indicate that the
 carrier components with a frequency f.sub.-1 and f.sub.1 truly have an
 amplitude substantially equal to C.sub.-1 and C.sub.1, respectively in the
 modulated output signal z.
 In a first stage, it may be assumed that the amplitudes C.sub.i of
 intermodulation components do not depend on the frequency f.sub.i, by
 eliminating the frequency corrections G.sub.np,q (f.sub.i) provided for in
 the computation formulae 12' to 15" . The computation is then simplified
 for the components that are symmetrical in frequency f.sub.-i and f.sub.i
 around the carriers f.sub.-1 and f.sub.1 have equal amplitudes according
 to the following identity:
EQU C.sub.-i =C.sub.i
 The last two lines of the equation (23) represent the frequency components
 f.sub.1 and f.sub.3 respectively, enabling the computation of the
 rejection factor P/I which finally has the following value:
 ##EQU8##
 In operation, after having determined the amplitudes C.sub.1 (A), C.sub.3
 (A) and C.sub.5 (A) of the components according to the known methods and
 after having measured the value of the rejection factor P/I for this value
 with an amplitude A of the carriers, the equation (26) is resolved to
 determine the value of the variable K.
 For the resolution, the variable K is preferably in the following complex
 form:
EQU K=.vertline.K.vertline..exp(j..theta.)
 The argument .theta. is made to vary in order to compute the values of the
 norm .vertline.K.vertline. as a function of the argument .theta..
 The minimization sought here above of the modification of amplitude
 C.sub.-1 and C.sub.1 of the carriers f.sub.-1 and f.sub.1 is obtained by
 using the argument .theta. enabling the minimum value of the norm
 .vertline.K.vertline. to be obtained.
 FIG. 18 illustrates a schematic view of the system of measurements
 performed according to the invention. The characteristics are determined
 by taking a series of measurements with a multicarrier signal f.sub.-1 and
 f.sub.1, in obtaining a variation of both amplitude A, A', A" of the
 carriers as well as the frequency difference df'=f.sub.1 -f.sub.-1 of the
 carriers. Furthermore, it is possible to obtain a variation in the basic
 frequency f of the carrier f.sub.1 or f.sub.-1 used as a reference for the
 frequency difference df.
 For each frequency difference df, as shown in FIG. 16, a characteristic
 curve P/I is measured as a function of the amplitude A or the power of the
 input carrier. By applying the above computation for each frequency
 difference df, a curve of the values of the variable K is obtained as a
 function of the amplitude A of the carriers. By calculating this variation
 in the frequency differences df, df', df", df'", d"" of the carriers,
 there is thus obtained a table of curves of values of the variable K. This
 table of values of the variables K establishes the frame of the transfer
 function E(df,H) of filter 30.
 The continuity of the transfer function E(df,H) according to the amplitude
 H of the harmonics h at input or according to the power X.sup.2 of the
 envelope of an input signal x is obtained by interpolating the discrete
 values of the table of the values of the variable K.
 The operations of computation and interpolation are performed also for all
 the frequency difference values df, the values of the variable K possibly
 being also interpolated between the chosen values df', df", df'", df""
 with a frequency difference df. Computation means 12 preferably performs
 these computations and stores the values of the variable K in matrix form.
 At a second stage, it is assumed that the amplitudes C.sub.i of the
 intermodulation components depend on the frequency f.sub.i according to
 the complete formulae of the equations 14' to 17". There is then no longer
 any identity between C.sub.i and C.sub.-i.
 The dissymmetries in frequency of the response of the amplifier are then
 simulated by this variant of the first alternative method. In this case,
 the transfer function E'(df,H) of the filter 14 may be dissymmetrical.
 The hypotheses of the equation 22 and of the preceding computation are then
 modified according to the following complex formula:
 ##EQU9##
 where
 E'(O,A) has a zero value for the filter does not transfer any signal
 continuous component,
 and
 E'(-df,A) is a response of the transfer function E' to a frequency
 difference -df, which is the response distinct from the response E'(df,A)
 to a frequency difference df, owing to the dissymmetry of the filter E'.
 The terms of the equation 22' are simplified as follows:
EQU K.sup.+ =A.sup.2.E'(df,A)
 and
EQU K.sup.- =A.sup.2.E'(-df,A)
 The equation 23 is then modified according to the following formula:
EQU 1+U'(t)=K.sup.-.exp(-j.2.pi..df.t )+1+K.sup.+.exp(j.2.pi..df.t) (28)
 Let C.sub.i ' denote the modulated components of the output signal z'
 obtained according to this variant of the first alternative method. The
 output signal z' is then obtained by modulating in Mix block the signal y
 of a known model by a modulator signal u' computed by the transfer
 function E' of filter 36 of FIG. 23. This makes it possible to distinguish
 the modulated components C'.sub.i according to the invention from the
 components C.sub.i computed by the known memoryless model MNLSM.
 According to the above equations 21, 24' and 28, the modulated components
 have amplitudes C.sub.-3 ', C.sub.-1 ', C.sub.1 ', C.sub.3 ' determined by
 the following formulae:
EQU C.sub.1 '=K.sup.+.C.sub.-1 +C.sub.1 +K.sup.-.C.sub.3 (29)
EQU C.sub.3 '=K.sup.+.C.sub.1 +C.sub.3 +K.sup.-.C.sub.5 (30)
EQU C.sub.-1 '=K.sup.+.C.sub.-3 +C.sub.-1 +K.sup.-.C.sub.1 (31)
EQU C.sub.-3 '=K.sup.+.C.sub.-5 +C.sub.-3 +K.sup.-.C.sub.-1 (32)
 Two more complex results of intermodulation rejection factors, referenced
 P/I.sup.+ and P/I.sup.-, are obtained depending on whether the rejection
 factor is measured on the higher carrier and intermodulation frequencies
 f.sub.1 and f.sub.3 or on the lower carrier and internodulation
 frequencies f.sub.-1 and f.sub.-3. The two rejection factors have the
 following shapes:
 ##EQU10##
 In a known way, the two equation system 33 and 34 are resolved with two
 variables K.sup.- and K.sup.+.
 Thus, when the amplifier has a dissymmetrical response around carrier
 frequencies, the intermodulation rejection characteristics are measured by
 providing for the measurement of the two upper and lower rejection factors
 P/I.sup.+ and P/I.sup.- for each amplitude A and frequency difference df
 of carriers, in order to simulate the modulation distortions at output.
 SECOND ALTERNATIVE EMBODIMENT OF THE INVENTION
 A second alternative method according to the invention is used to simulate
 the second effect of intermodulation distortion.
 The second effect observed is the variation of the gain of the carriers as
 a function of the envelope frequency df/2 illustrated for example in FIG.
 17. By measuring the gain of the amplifier for a carrier f.sub.1 with an
 output amplitude C.sub.1 in the presence of another carrier f.sub.-1, it
 is realized that the gain may have major variations as a function of the
 frequency difference df of the two carriers f.sub.-1 and f.sub.1.
 Computation means 14 allows adjustment of the output amplitudes C.sub.1 and
 C.sub.-1 of the carriers computed by a known model in taking account of
 measurements of characteristics C1/C-1 of interaction of carriers.
 Preferably, the adjustment of the second alternative method is combined
 with the adjustment of the main intermodulation components C3 and C-3
 planned in the first alternative method of the invention explained here
 above. It is possible to furthermore provide for an adjustment of the
 secondary intermodulation components C5, C-5.
 In the second embodiment of a device according to the invention, the
 adjustment is done by correcting the output signal y or preferably the
 modulated signal z or z' by means of a correction signal v coming from a
 filter 38 having a correction transfer function F that is summed with z'
 in a block 40.
 The transfer function F is computed similarly to the function E by assuming
 a correction of the modulated components C.sub.i ' formulated here above
 in the equations 29 to 32 according to the following formulae:
EQU C.sub.3 "=K.sup.+.C.sub.1 +C.sub.3 +K.sup.-.C.sub.5 (35)
EQU C.sub.-3 "=K.sup.+.C.sub.-5 +C.sub.-3 +K.sup.-.C.sub.-1 (36)
EQU C.sub.1 "=K.sup.+.C.sub.-1 +C.sub.1 +K.sup.-.C.sub.3 +L.sup.+ (37)
EQU C.sub.-1 "=K.sup.+.C.sub.-3 +C.sub.-1 +K.sup.-.C.sub.1 +L.sup.- (38)
 wherein
EQU L.sup.+ =A.F(df,A)
EQU L.sup.- =A.F(df,A)
 F being the correction function transfer expressed as a function of the
 modulation frequency of the envelope, namely the frequency difference df
 of the amplitude carriers A at input.
 The components C.sub.3 ", C.sub.-3 ", C.sub.1 " and C.sub.-1 " are called
 corrected components.
 The equations 35 to 38 form a system of four equations with four variables
 K.sup.+, K.sup.-, L.sup.+, L.sup.- that is resolved in a known manner. The
 resolution of the last two equations 37 and 38 give the values of the
 correction variables L.sup.+ and L.sup.-.
 The precise measurement of the real amplitudes of the third-order
 intermodulation components f.sub.3, f.sub.-3 and of the carriers f.sub.1,
 f.sub.-1 thus make it possible, after computation of the first order,
 third order and fifth order components C.sub.-5, C.sub.-3, C.sub.-1,
 C.sub.1, C.sub.3, C.sub.5 according to a known model, to compute the
 modulation transfer function E(df,A) and frequency correction function
 F(df,A).
 Full measurements of amplitudes of components at output of an amplifier of
 this kind comprising both measurements of third-order and fifth-order
 intermodulation component rejection at output and measurements of carrier
 interaction thus enable the total simulation of the signal response with
 the distortions due to the envelope memory effect of a nonlinear memory
 amplifier. It will be noted however that partial measurement such as the
 third-order intermodulation rejection measurements C1/C3 or the carrier
 interaction measurements C1/C-1 are enough to characterize certain effects
 of intermodulation distortion.
 A second alternative method makes it possible to validate the model by the
 measurement of the signal-to-noise ratio in the presence of a large number
 of carriers whose power spectral density DSP is modeled by a white noise.
 FIGS. 19A and 19B illustrate a noise frequency graph respectively applied
 at input and measured at output of a nonlinear memory amplifier in order
 to characterize it.
 According to the invention, as shown in FIG. 19A, there is applied a white
 noise having a power spectral density DSP with a level NP that is constant
 in a frequency band BW except in a frequency window BH where the noise is
 absent. This input signal although it has a constant spectral density has
 a variable amplitude. It therefore enables a measurement of the
 distortions of modulation appearing at output of the amplifier. The
 distortions appear at output of the amplifier in the form of a carryover
 of noise to the above-defined window BH.
 The characterization of the amplifier consists in measuring its capacity to
 limit the noise carryover into the window BH by measuring the difference
 in noise level NPR at output in the frequency band BW and in the window
 BH.
 These measurements of characteristics are performed at different noise
 levels NP and introduced into the simulator so that it reproduces its
 noise characteristics according to the conditions of operation.
 A third mode of control of the validity of the model consists in measuring
 the binary error rate during an amplification of modulated carriers.
 OPERATION OF PREFERRED EMBODIMENT OF INVENTION
 FIG. 24 illustrates device 10, according to the invention, to make
 measurements of characteristics of a nonlinear amplifier DUT with memory
 effect. The operation of test bench device 10 is within the scope of those
 skilled in the art. Device 10 includes a Calc. block 42, a signal unit 44,
 a noise unit 46, and an error rate unit 48. Device 10 further includes
 known structures such as a Band. Transp. block 50, gates 52, 54, 56, an
 Att. unit 58, an Amp. unit 60, a DUT unit 62, a SIM unit 64 and a NA unit
 66.
 In particular, unit 10 is configured for the generation of amplitude
 modulated signals with, for example, two generators SG1 and SG2 having
 frequencies that may be distinct to apply a multicarrier signal to
 amplifier device DUT unit 62 to be characterized. It furthermore comprises
 noise measurement unit 46 with white noise generator NG with a window and
 possibly unit 48 to measure the binary error rate with a generator MOD of
 signals modulated by a random binary message and a comparator Comp to
 compare the original signal and the return signal from the amplifier.
 FIG. 25 illustrates a drawing of the files of characteristics implemented
 by computational means 12 to simulate the nonlinear amplification
 according to the method of the invention.
 A computational program 68 receives input from a file 70 that includes the
 results of measurements of static characteristics AP, namely curves
 C,.PHI. or curves P,Q drawn up with constant amplitude (in a known way).
 Program 68 further receives input from a series of files 72 that include
 the results of measurements of dynamic characteristics P/I such as
 intermodulation rejection curves C1/C3 or carrier interaction curves
 C1/C-1. Each file 72 corresponds to, for example, a frequency difference
 df of carriers. Program 68 is launched after the input of data is read
 from a general data Dat file 74 into the amplifier to be simulated.
 After program 68 processes files 70, 72 and 74, program 68 calculates and
 produces a file 76 of the coefficients a of sequences of direct transfer
 functions, especially Bessel functions. Program 68 further calculates and
 produces a file 78 and a file 80 that contain the data of the
 above-mentioned modulation transfer function E and correction function F,
 respectively.
 FIG. 26 illustrates an exemplary drawing of a flow diagram of the process
 steps implemented by program 68 including a start block 82, a read dat
 block 84, a read AP block 86, a dvlpt block 88, an increment block 90, a
 read P/I block 92, an increment block 94, a calc block 96, a calc block
 98, an assignment block 100, an assignment block 102, and an end block
 104.
 After reading data and initializing p, f and i, dvlpt block 88 calculates a
 as an output. After incrementing f in block 90, reading P/I in block 92,
 incrementing i in block 94 and calculating y in block 96, calc block 98
 calculates values for E and F as outputs. If i then equals points P/I, the
 condition f=files P/I is tested in block 102. If the result of block 102
 is positive, then the loop is ended at block 104, otherwise the process
 proceeds to block 90 and f is incremented. If i does not equal points P/I,
 then the process proceeds to block 94 and I is incremented.
 Other exemplary embodiments, characteristics and advantages may be
 developed by those skilled in the art without going beyond the scope of
 the invention, defined especially by the following claims.