Amplifier linearization by adaptive predistortion

Description Submit all comments and votes

FIELD OF THE INVENTION

This application pertains to a method of adaptively predistorting a power amplifier to linearize the amplifier.

BACKGROUND OF THE INVENTION

Mobile communications systems, such as those used for cellular telephone communication, divide the available frequency spectrum into a multiplicity of individual signalling channels or frequency bands. Particular channels are allocated to individual users as they access the system. Each user's communications are routed by the system through the channel allocated to that user. Signals broadcast by the system must be carefully regulated so that they remain within the channels allocated to the various users. "Out-of-band" signals can spill over from one channel to another, causing unacceptable interference with communications in the other channels.

To date, mobile communications systems have generally employed frequency modulation ("FM") techniques which do not require variation of the amplitude of the transmitted signal. Thus, the frequency of the transmitted signal carrier changes, but the signal power level remains constant. Such systems have sufficed for voice communications. However, there is an increasing desire to expand the capabilities of mobile communications systems to encompass data as well as voice communications. Commercially worthwhile data transfer rates require the use of modulation techniques which are more spectrally efficient than the FM techniques used for voice communication. This necessitates the use of amplitude modulation techniques which in turn require linearized modulation.

The electronic amplifiers employed in any communications system inherently distort signals as they amplify the signals. FM techniques do not suffer from such distortion because of their constant amplitude. Amplitude modulation, however, causes the distortion to become dependent on the input signal, so that the amplifier output signal is no longer simply an amplified replica of the input signal. Although an amplifier's input signal may be confined within a particular channel or frequency band, the distorted, amplified output signal typically includes out-of-band frequency components which would overlap one or more channels adjacent to the channel within which the input signal was confined, thereby interfering with communications in the overlapped channel(s).

Some degree of channel signal overlap due to unregulated amplifier distortion is acceptable in some cases. However, mobile communications systems place very stringent restrictions on out-of-band signal emissions in order to minimize channel-to-channel interference.

To reduce out-of-band signal emissions to an acceptable minimum the amplifier input signal is conventionally "predistorted" before it is fed into the amplifier. Before the signal is amplified, an estimate is made of the manner in which the amplifier will inherently distort the particular input signal by amplifying that signal. The signal to be amplified is then "predistorted" by applying to it a transformation which is estimated to be complementary to the distorting transformation which the amplifier itself will apply as it amplifies the signal. In theory, the effect of the predistorting transformation is precisely cancelled out by the amplifier's distorting transformation, to yield an undistorted, amplified replica of the input signal. Such amplifiers are said to be "linearized" in the sense that the output signal is proportional to the input signal, thereby eliminating the generation of out-of-band components.

Unfortunately, amplifier distortion varies in a complex, non-linear manner as a function of a wide range of variables, including the amplifier's age, temperature, power supply fluctuations and the input signal itself. Accordingly, it is not possible to define a single predistortion transformation which will cancel out any and all distorting transformations applied by the amplifier.

One prior art approach to the problem (exemplified by U.S. Pat. No. 4,462,001 issued July 24, 1984 for an invention of Henri Girard entitled "Baseband Linearizer for Wideband, High Power, Nonlinear Amplifiers") has been to construct a look up table containing a multiplicity of entries which define predistortion transformation parameters appropriate for use with a corresponding multiplicity of different input signals. That is, the effects of the amplifier's distortion on a range of input signals are pre-measured, the complementary predistorting transformations corresponding to each input signal are calculated, and parameters defining the calculated complementary transformations are stored in the table. In operation, the fluctuating power level of the signal to be amplified is continuously measured. The power measurement is then applied to the electronic embodiment of the table, from which the corresponding predistortion parameters are derived, so that the input signal sample may be predistorted before it is fed to the amplifier. However, Girard's approach accounts only for variation of the input signal, not for variation of the amplifier's other distorting characteristics. Because the amplifier's other distorting characteristics in fact vary it is necessary to continuously "adapt" the lookup table parameters by changing them in response to changes in the amplifier's other distorting characteristics.

Moreover, Girard's approach is based on separate tables containing amplitude and phase correction factors. This "polar coordinate" representation follows naturally from the common practice of representing amplifier distortion in terms of AM/AM and AM/PM characteristics. So far as the inventor is aware, all predistortion techniques prior to Girard's also attempted separate amplitude and phase correction.

Another prior art approach is exemplified by U.S. Pat. No. 4,700,151 issued Oct. 13, 1987 for an invention of Yoshinori Nagata entitled "Modulation System Capable of Improving a Transmission System". Nagata uses the real and imaginary quadrature components of the input signal sample to index into a lookup table containing predistortion transformation parameters. The real and imaginary components are each typically defined by at least 10 bits of information. Thus, Nagata employs a 20 bit index, which requires a lookup table containing 2.sup.20 entries (i.e. over 1 million entries). The lookup table entries are adaptively changed in response to variations in the amplifier's distorting characteristics. However, if the signalling channel is changed (a common occurrence in mobile communications systems) then every entry in Nagata's lookup table must be iteratively recalculated. This process can take 10 seconds or longer, which is unacceptable.

The present invention overcomes the disadvantages of the prior art. By storing table entries in rectangular coordinate format, it enables the subsequent predistortion operation to be performed more simply than Girard's polar coordinate approach. Further, it adapts to amplifier and oscillator changes, whereas Girard's predistorter does not. In comparison with Nagata's method, only a comparatively small lookup table is required. Signal phase rotators (required to stabilize Nagata's circuitry) are not required. Moreover, the lookup table entries are adaptively changed, within about 4 milliseconds ("msec.") in response to changes in the amplifier's distorting characteristics.

SUMMARY OF THE INVENTION

In accordance with the preferred embodiment, the invention provides a method of linearizing an amplifier to produce an amplified output sample v in response to a predistorted input sample v.sub.d derived from an input modulation sample v.sub.m, such that v.sub.a .perspectiveto. Kv.sub.m, where K is the amplifier's desired constant amplitude gain. The squared magnitude x.sub.m of the input modulation sample v.sub.m is first derived. A table entry F.sub.i is then selected from a table containing N.sub.t values F.sub.i where i=0, 1, . . . , N.sub.t -1. Each table entry corresponds to a squared magnitude value x.sub.mi ; and, for each table entry, F.sub.i G(x.sub.mi .vertline.F.sub.i .vertline..sup.2).perspectiveto.K, where G(x) is the complex gain of the amplifier. The table entry selected is the one for which the absolute value .vertline.x.sub.m -x.sub.mi .vertline. is minimized with respect to the table index i. The predistorted sample v.sub.d is then derived, in rectangular coordinates, as v.sub.d =v.sub.m F.sub.i, viz:

Re(v.sub.d)=Re(F.sub.i)-Im(V.sub.m)Im(F.sub.i)

Im(v.sub.d)=Re(v.sub.m)Im(F.sub.i)+Im(V.sub.m)Re(F.sub.i)

where Re(x) is the real component of x and Im(x) is the imaginary component of x.

The aforesaid steps are sequentially repeated k times. After each derivation of the predistorted sample v.sub.d (k), a sample v.sub.a (k) of the amplifier's output is derived. An error sample e(k) =v.sub.a (k)-Km.sub.v (k) is also derived. This facilitates derivation of an adjusted value F.sub.i (k+1) of the table entry F.sub.i (k) selected during the k.sup.th selecting step, where: ##EQU1## where .alpha. is an appropriately chosen constant. Table entry F.sub.i (k) is then replaced with the adjusted value F.sub.i (k+1), and all of the steps are again sequentially repeated.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a typical prior art adaptively linearized amplifier.

FIG. 2 is a block diagram of the mapping predistorter section of a prior art adaptively linearized amplifier.

FIG. 3 is a graph on which the AM/AM and AM/PM characteristics of a typical Class AB amplifier are shown with output power (dBm) plotted as the left ordinate, phase shift (degrees) plotted as the right ordinate, and input power (dBm) plotted as the abscissa. The solid line plots output power and the dashed line plots phase shift.

FIG. 4 is a graph on which the complex gain characteristics of a typical Class AB amplifier are shown with complex gain magnitude plotted as the left ordinate, complex gain phase (degrees) plotted as the right ordinate, and input power (mwatts) plotted as the abscissa. The solid line plots magnitude and the dashed line plots phase shift.

FIG. 5 is a block diagram of a gain based predistorter amplifier constructed in accordance with the invention.

FIG. 6 is a graph illustrating the manner in which look up table entries F(x) are optimized according to the relationship F(x.sub.m)G(x.sub.m .vertline.F(x.sub.m).vertline.)=K.

FIG. 7 is a graph illustrating the performance of the predistorter amplifier of FIG. 5 in a two tone test, with relative power (dB) plotted as the ordinate and the ratio f/f.sub.s plotted as the abscissa. The long dashed line plots the output of the power amplifier ("PA") only with PBO=0.22 dB; the short dashed line plots the output of the PA only with PBO=24.6 dB; and, the solid line plots the output of the combined PA and predistorter ("PD") with PBO=0.22 dB.

FIG. 8 is a graph illustrating the performance of the predistorter amplifier of FIG. 5 in a noise loading test, with power spectral density (dB) plotted as the ordinate and the ratio f/f.sub.s plotted as the abscissa. The line labelled "A" plots the output of the PA with no PD and OBO =13.2 dB; the line labelled "B" plots the output of the PA with no PD and OBO=33.7 dB; the line labelled "C" plots the output of the PA with a 32 point PD and OBO=15.0 dB; and, the line labelled "D" plots the output of the PA with a 64 point PD and OBO=15.0 dB.

FIG. 9 is a graph illustrating the response of the predistorter amplifier of FIG. 5 to a 16QAM data signal, with power spectral density (dB) plotted as the ordinate and the ratio f/f.sub.s plotted as the abscissa. The line labelled "E" plots the output of the PA with no PD and PBO =0.22 dB; the line labelled "F" plots the output of the PA with a 32 point PD and PBO=0.22 dB; the line labelled "G" plots the output of the PA with a 64 point PD and PBO=0.22 dB; and, the line labelled "H" plots the output of the PA with no PD and PBO =30.24 dB.

FIG. 10 illustrates partitioning of the input space (v.sub.m space) of the mapping predistorter of FIG. 2 into quantization cells.

FIG. 11 is a graph on which the distribution of intermodulation power with instantaneous output power of the combined amplifier of FIG. 5 and 32 point predistorter is shown, with quantization noise power (mwatts) plotted as the left ordinate, relative error variance plotted as the right ordinate, and output power (watts) plotted as the abscissa. The solid line plots quantization noise power; the short dashed line plots relative error variance .times.1000; and, the long dashed line plots relative error variance .times.100,000.

FIG. 12 is a block diagram of a prior art adaptively linearized amplifier of the type exemplified by U.S. Pat. No. 4,700,151 Nagata.

FIG. 13 is a graph on which the convergence behaviour of the linear and secant update algorithms is shown, with squared magnitude output error (watts) plotted as the ordinate, and iteration number plotted as the abscissa. The long dashed lines illustrate the linear convergence algorithm's performance at 30 watts. The short dashed lines illustrate the linear convergence algorithm's performance at 15 watts. The solid lines illustrate the secant algorithm's performance. The numbers applied to the dashed lines denote the feedback phase shift whereas those applied to the solid lines denote the output power in watts.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

I INTRODUCTION

In the past, both theory and practice of mobile communications have emphasized constant envelope modulation, such as FM or Gaussian minimum shift keying ("GMSK"). These techniques allow power amplifiers to be operated in the nonlinear region near saturation, for power efficiency, yet they do not generate intermodulation products in nearby channels. However, continuing pressure on the limited spectrum available is forcing the development of more spectrally efficient linear modulation methods, such as 16-ary quadrature amplitude modulation ("16QAM") and quadrature phase shift keying ("QPSK") with pulse shaping. Since their envelopes fluctuate, these methods generate intermodulation products in a nonlinear power amplifier. In the mobile environment, restrictions on out-of-band emissions are stringent, and the designer is faced with two alternatives: back off an inefficient Class A amplifier to an even more inefficient, but linear, operating region; or linearize the amplifier.

The present invention provides a method for linearizing a power amplifier by predistorting its input. It is particularly well suited to baseband implementation using digital signal processing techniques, though hybrid variants are easily concocted. It has a number of advantages compared to previously published techniques; it is powerful and economical, and it adapts rapidly to amplifier or oscillator changes.

FIG. 1 shows a generic model for many prior art adaptive amplifier linearization methods. All signal designations employed herein refer either to complex baseband signals or the to the complex envelope of bandpass signals. The linearizer or predistorter 10 creates a predistorted version v.sub.d (t) of the desired modulation v.sub.m (t), making use of its measurements v.sub.f (t) of the actual amplifier output v.sub.a (t). Quadrature modulator 12 creates a real bandpass signal from the components of v.sub.d (t) for input to power amplifier ("PA") 14. The feedback path (incorporating bandpass filter 16) directs a portion of the real bandpass PA output to quadrature demodulator 18 for recovery of the complex envelope. Its output v.sub.f (t) is a scaled, rotated, and possibly delayed version of v.sub.a (t). Note that the same oscillator 20 is used in up and down conversion for coherence, and that some methods require a phase shifter 22 for stability.

Linearization by Cartesian feedback [see generally Y. Akaiwa and Y. Nagata, "Highly Efficient Digital Mobile Communications with a Linear Modulation Method", IEEE J. Sel. Areas in Comms., vol SAC-5, no 5, pp 890-895, June 1987; A. Bateman, D.M. Haines, and R.J. Wilkinson, "Linear Transceiver Architectures", Proc. IEEE Vehic. Tech. Conf., pp 478-484, Philadelphia, 1988; and, S. Ono, N. Kondoh and Y. Shimazaki, "Digital Cellular System ith Linear Modulation", Proc. IEEE Vehic. Tech. Conf., pp 44-49, San Francisco, 1989] has the great virtue of simplicity; the amplifier input complex envelope v.sub.d (t) is proportional to the difference between the desired v.sub.m (t) and the measured amplifier output v.sub.f (t), as in a classical control system. However, its linearity and its bandwidth (i.e. the gain-bandwidth product) are critically dependent on loop delay. Linearization by Cartesian feedback is therefore ineffective for travelling wave tube amplifiers, or if there is additional analog or digital filtering at any point in the loop. Moreover, the stability of a system dependent upon linearization by Cartesian feedback depends on precise adjustment of the phase shifter, an adjustment that depends on the current channel in use.

A more robust alternative to Cartesian feedback is a predistorter ("PD"), in which the linearizer produces v.sub.d (t) by applying v.sub.m (t) to a memoryless nonlinearity complementary to that of the power amplifier. Feedback is used only for adaptation of the predistorter nonlinearity, rather than for real time calculation of v.sub.d (t). Because it is insensitive to loop delay, predistortion has attracted considerable attention.

Several prior art predistorters [e.g. M. Nannicini, P. Magni, and F. Oggioni, "Temperature Controlled Predistortion Circuits for 64 QAM Microwave power Amplifiers", IEEE MTT-S Digest, pp 99-102, 1985; and, J. Namiki, "An Automatically Controlled Predistorter for Multilevel Quadrature Amplitude Modulation", IEEE Trans. Commun., vol COM-31 no. 5, pp 707-712, May 1983] adaptively cancel the dominant 3rd order component of the PA nonlinearity. The complexity of such structures increases rapidly if extended to 5.sup.th or 7.sup.th order terms. Moreover, Class AB amplifiers have significant kinks in the transfer characteristic at low levels which are not well modelled by a cube law. The predistorter of Nannicini et al is noteworthy, because the quantity fed back is not required to be an accurate replica of the PA output v.sub.a (t), but rather a simple estimate of the power in the 3rd order spectral skirts, as measured by narrowband filters and envelope detection. This, however, restricts its use to modulation of a specific bandwidth.

Another prior art predistorter [H. Girard and K. Feher, "A New Baseband Linearizer for More Efficient Utilization of Earth Station Amplifiers Used for QPSK Transmission", IEEE J. Selected Areas in Commun., vol. SAC-1, no. 1, pp 46-56, January 1983; see also U.S. Pat. No. 4,462,001 supra] is based on complex gain, as is the present invention. However, it requires a dynamic phase shifter, and is not adaptive.

Yet another prior art predistorter [exemplified by U.S. Pat. No. 4,412,337 issued Oct. 25, 1983 for an invention of Robert H. Bickley et al entitled "Power Amplifier and Envelope Correction Circuitry"] differs from the present invention in that it makes no attempt to correct distortion due to AM/PM conversion, and is restricted to digital signalling consisting of a sequence of pulses.

A number of prior art predistorters [e.g. J. Graboski and R.C. Davis, "An Experimental MQAM Modem Using Amplifier Linearization and Baseband Equalization Techniques", Proc. Natl. Commun. Conf., pp E3.2.1-E3.2.6, 1982; U.S. Pat. No. 4,291,277 issued Sept. 22, 1981 for an invention of Robert C. Davis et al entitled "Adaptive Predistortion Technique For Linearizing a Power Amplifier For Digital Data Systems"; and, A.A.M. Saleh and J. Salz, "Adaptive Linearization of Power Amplifiers in Digital Radio Systems", Bell Syst. Tech. J., vol. 62, no. 4, pp 1019-1033, April 1983] are restricted to particular modulation formats. Implemented in digital baseband as RAM lookup tables ("LUTs"), with an entry for each predistorted point in the signal constellation, they are fast and require very little memory. However, they are limited to rectangular pulses, or to pulse shaping implemented by filtering following the PA, an unattractive arrangement. Saleh et al provide an adaptive version, which requires conversion between polar and rectangular representations. Davis et al (supra) also include an adaptation algorithm of the linear type discussed in Section IV A below. Its convergence is therefore slower than the present invention, and it requires a phase shifter in the feedback path for stability (though this is not noted in the Davis et al '277 patent).

The most general and powerful predistorter to date was reported by Y. Nagata, "Linear Amplification Technique for Digital Mobile Communications", Proc. IEEE Vehic. Tech. Conf., pp 159-164, San Francisco, 1989 (see also U.S. Pat. No. 4,700,151 supra). Nagata generalizes the table lookup approach of Grabowski et al and Saleh et al to provide the predistorted equivalent v.sub.d of any input value v.sub.m (FIG. 2), thereby mapping the complex plane into itself. Nagata's approach is therefore unrestricted by order and type of PA nonlinearity (provided it is memoryless). Since this permits pulse shaping prior to predistortion, it is not restricted by modulation format, either. Nagata also provided an update algorithm for adaptation of the table, and a delay compensation algorithm so v.sub.m (t) and v.sub.a (t) can be compared. Nagata's technique is hereinafter referred to as the "mapping predistorter" ("mapping PD").

Powerful though it is, the mapping PD has several drawbacks. The lookup table ("LUT") is 2 Mword long for 10 bit representation of the real and imaginary parts of v.sub.m, and increases to 8 Mword for 11 bit representation. It requires a phase shifter in the feedback path for stability in the adaptation update, and convergence is very slow (10 sec, at 16 ksym/sec). Moreover, switching to a new channel requires readjustment of the phase shifter, and reconvergence of the table over another 10 sec.

The present invention, like the mapping PD, is unrestricted by modulation format, or by order of PA nonlinearity. In addition, it has some major advantages compared with the mapping PD. It requires over 4 orders of magnitude less table memory (typically under 100 complex word pairs). It reduces convergence time by a similar factor, to a few msec at 16 ksym/sec. It eliminates the reconvergence time following a channel switch, since the table for each channel is so small that it can be downloaded or simply retained in memory. Finally, it eliminates the need for a phase shifter in the feedback loop.

Section II describes the predistorter structure, and demonstrates its ability to suppress intermod products using only a small table. Section III analyzes the effect of PD nonidealities (especially limited table size) on the PA output; to the inventor's knowledge, such an analysis is missing from all previously published descriptions of linearizers. Section IV introduces a fast adaptation algorithm. Finally, Section V summarizes the results and their implications.

II THE GAIN BASED PREDISTORTER

A. Basic Structure

The amplifier can be modelled as a memoryless nonlinearity in several ways. The most productive for present purposes is as a level dependent complex gain. That is, the complex envelope of the amplifier's input v.sub.d and output v.sub.a are related by:

v.sub.a =v.sub.d G(.vertline.v.sub.d .sup.2)=v.sub.d G(x.sub.d)(1)

where x.sub.d denotes the squared magnitude of v.sub.d ; and G(x.sub.d), the complex gain of the amplifier, summarizes its AM/AM and AM/PM characteristics. Note that G(x.sub.d) depends only on the instantaneous power of the input, not on its phase. FIGS. 3 and 4 show the relation between bench measurements and the complex gain for a typical Class AB amplifier. The effect of compression is clearly evident at high input levels, as is the loss of gain at low levels, because of the crossover point between the push and pull halves of the amplifier.

The gain based PD of the present invention (FIG. 5) is not just modelled, but actually constructed, according to the complex gain formulation. Its input and output complex envelopes are related by:

v.sub.d =v.sub.m F(.vertline.v.sub.m .vertline..sup.2)=v.sub.m F(x.sub.m)(2)

where x.sub.m denotes the squared magnitude of v.sub.m.

For any input power, the optimum value of the PD complex gain F is determined by equating the composite PD/PA nonlinearity to a nominal constant amplitude gain K. Normally, K is selected to be a little less than the amplifier's midrange gain. However, as shown in Section III, the choice of K has no effect on the signal to quantization noise ratio at the amplifier output. Combining (1) and (2), we define F implicitly by:

v.sub.m F(.vertline.v.sub.m .vertline..sup.2)G(.vertline.v.sub.m .vertline..sup.2 .vertline.F(.vertline.vm.vertline..sup.2).vertline..sup.2)=Kv.sub.m(3)

or:

F(x.sub.m)G(x.sub.m .vertline.F(x.sub.m).vertline.)=K (4)

A fast technique for adaptive calculation of F(x.sub.m) is described in Section IV. Note that the PD complex gain F(x.sub.m) has a real domain, so that it can be represented by a one-dimensional LUT, rather than the two-dimensional LUT required by the mapping PD.

In practice, there is little point in trying to linearize an amplifier up to its saturated output power P.sub.sat, because the distortion increases drastically in this region, and we are faced with rapidly diminishing returns on linearization effort. Accordingly, define the span S as the fraction of saturated power over which linearization is attempted. Thus the maximum output power is given by S P.sub.sat. Realistic values for the span are in the range 0.95 to 0.98. An alternative description is the peak backoff ("PBO") of the PA in dB:

PBO=-10log(S) (5)

The span in turn limits the domain of the linearizer:

0.ltoreq.x.sub.m .ltoreq.p.sub.mm (6)

where the maximum power of v.sub.m is given by:

p.sub.m =S P.sub.sat /K.sup.2 (7)

The implementation of F(x.sub.m) as a LUT with entries equispaced in input power x.sub.m will now be examined. Although a good case can be made for nonuniform spacing, as shall be seen in Section III, it would be at the cost of more computation when implemented digitally. With N table entries, the step size is given by:

D.sub.g =P.sub.mm /N.sub.t (8)

The range and midpoint of each step, and the corresponding table entries, are given for i =0, 1, . . . , N -1 as:

X.sub.mi ={x.sub.m :iD.sub.g .ltoreq.x.sub.m <(i+1)D.sub.g }(9)

X.sub.mi =D.sub.g (i+1/2) (10)

F.sub.i =F(X.sub.mi) (11)

That is, the table is optimized according to (4) for the midpoint of each step, as shown in FIG. 6.

B. Performance of Basic Predistorter

This section demonstrates that even very small PD gain tables can produce major reductions in intermodulation products, using in all cases the Class AB amplifier of FIGS. 3 and 4 with a PBO of about 0.22 dB, giving a 95% span. First, define the output backoff ("OBO") as the dB difference between P.sub.sat and the average signal output power. Clearly OBO depends on the peak to average power ratio of the signal, hereinafter called the "signal backoff"("SBO"):

OBO=PBO+SBO (12)

Large OBO values imply inefficient operation, due either to amplifier peak backoff or to the signal format. Note that compression lowers the SBO below the value characteristic of the signal in a linear regime.

The first example is a two-tone test. Although it does not fully exercise the nonlinearities, it is widely used. Its SBO is 3 dB in the absence of distortion. FIG. 7 shows the results of two-tone tests of the amplifier of FIG. 5 with PBO values of 0.22 dB and 24.63 dB (i.e. SBO values of 2.53 and 27.67 dB, respectively). All spectra are normalized to 0 dB for the desired components at f/f.sub.s =.+-.0.0156, and the frequency is normalized by the sampling rate. Even with a PBO of 25 dB, the dominant IM products of the uncorrected amplifier are only 40 dB down. In contrast, the third curve on FIG. 7 shows that a 64 point predistortion table achieves the desired PBO of 0.22 dB (OBO of 3.22 dB) with IM products reduced below 60 dB.

The second example (FIG. 8) is a more demanding noise loading test. Thirty sine waves with equal amplitudes and randomly selected phases are located fifteen on each side of a center channel spectral null. The degree to which the null fills in at the output of the PA indicates the total power in the nonlinear products. Averaging of a few such output spectra, each with a new set of phases, is sufficient. The SBO for this signal is 14.8 dB. With a SBO of 13.2 dB (PBO of 0.22 dB), wide IM skirts about 14 dB down are evident in the uncorrected amplifier, and even an OBO of 33.7 dB lowers the IM by only about 8 dB. In contrast, both a 32 point and a 64 point PD table introduce a major reduction in intermod power at an OBO of 15 dB (PBO of 0.22 dB).

The final example is more closely related to data transmission. The input signal is 16QAM, with a square root spectral raised cosine pulse with 25% rolloff, Hamming windowed to 7 symbols. The SBO is about 6 dB. As shown on FIG. 9, the 3.sup.rd, 5.sup.th and 7.sup.th order skirts are clearly visible when the uncorrected amplifier is operated at 0.22 dB PBO (3.0 dB OBO). Even with an OBO of 33.7 dB (PBO of 30.24 dB), out-of-band emission is still about 56 dB down, and the power efficiency of the amplifier is hopelessly small. Again a striking improvement is seen with the gain based PD operated at a PBO of 0.22 dB: even a 32 point gain table brings the skirts below 60 dB, and a 64 point table lowers the IM by another 6 dB (see Section III).

The foregoing examples demonstrate that major improvements in linearity can be achieved with very small tables: N.sub.t =64 entries, compared with over 1 million entries in the mapping PD. This dramatic reduction is due to exploitation of the rotational invariance of the amplifier nonlinearity. Of the two dimensions required to specify a point in the plane, only one (radius) need be quantized; the other (phase) remains continuous.

The cost, compared with the mapping PD, is computation. Both the squared magnitude of v.sub.m (for the table index) and the complex multiplication of v.sub.m by the table entry have to be performed at a sampling rate adequate to represent the highest order nonlinearity of interest. Such a PD has been implemented on a TMS320C25 at 240 kHz, which is sufficient for 7.sup.th order nonlinearities when the nominal channel bandwidth is 30 kHz.

C. Nonuniform Spacing of Table Entries

The discussion and examples to this point have assumed a table (equations 9-11) optimized according to (4) for the midpoint of each step. In practice, however, the table entries may not be perfectly adjusted; indeed, Section III(C) makes an explicit calculation concerning this point. Although such departures from the condition (4) may degrade linearization, they nevertheless fall within the scope of the invention. Accordingly, the condition to be satisfied by the table entries is:

F.sub.i G(X.sub.mi .vertline.F.sub.i .vertline..sup.2).perspectiveto.K(12a)

The foregoing discussion and examples have also assumed table entries equispaced in power x.sub.mi. The generalization to nonuniform spacing is straightforward. The table contains N.sub.t distinct values of input power x.sub.mi, each with an associated F.sub.i value satisfying (12a). In the case of nonuniform spacing, the table index i is determined for an arbitrary input power x.sub.m as the one for which the x.sub.mi value is closest to the x.sub.m value; that is, the table index i minimizes the absolute value .vertline.x.sub.m -x.sub.mi .vertline.. The range of each step is therefore given by:

X.sub.mi ={x.sub.m :(x.sub.m,i-1 +x.sub.mi)/2.ltoreq.x.sub.m <(x.sub.mi +x.sub.m,i+1)/2) (12b)

The ranges and the table entries selected by the predistorter are identical to those given earlier in (9-11) if the table entries are uniformly spaced in x.sub.mi.

D. Alternative Selection Criteria

Equations (9-11), which define selection of the best table entry, are equivalent to:

i=round(x.sub.m /D.sub.g) (12c)

where round(x) is the closest integer to x. Alternative selection criteria may be desirable in the interests of implementation simplicity:

i=floor(x.sub.m /D.sub.g) (12d)

and:

i=ceil(x.sub.m /D.sub.g) (12e)

where floor (x) and ceil (x) represent, respectively, the greatest integer less than or equal to x, and the smallest integer greater than or equal to x. Although these latter two selection criteria will, in general, degrade linearization accuracy, doubling the table size restores the performance, as hereinafter shown in Section III(B). All three criteria are considered to fall within the scope of the present invention.

The general form of the selection criterion, including nonuniform spacing, follows from (12b-12c). The selected table index is either of the two values:

1. i, such that x.sub.mi is the largest table entry less than or equal to x.sub.m ; or,

2. i, such that x.sub.mi is the smallest table entry greater than or equal to x.sub.m. That is to say, we can select either of the table entries bracketing x.sub.m.

E. Interpolation of Table Entries

Better use of the table can be made, at the cost of additional computation, by interpolating a value from the table. By way of example, a linearly interpolated value results from the calculation: ##EQU2## where i is such that x.sub.mi is the smallest table entry greater than or equal to x.sub.m. This, and other standard interpolation formulae [see, for example, Sgermund Dahlquist and Ake Bjorck, Numerical Methods, Prentice-Hall, 1974], are also within the scope of the present invention.

III ERROR ANALYSIS OF THE PREDISTORTERS

Section II presented experimental evidence that the gain based PD of FIG. 5 can achieve significant reduction in IM products with a table four orders of magnitude smaller than that of the mapping PD. This section provides an analytical explanation for the results, by exploring the relation between PA characteristics and table size. Among other findings, it will be demonstrated that table size effects show up as relative error in the gain based PD, compared with absolute error in the mapping PD; that error power in the gain based PD decreases inversely as (N.sub.t).sup.2, rather than inversely as N.sub.t in the mapping PD; and that the jitter in the gain table entries caused by adaptation can be compensated by a relatively small increase in table size.

A. The Mapping Predistorter

In the mapping PD, the input v.sub.m, representing the desired output of the amplifier, is quantized to several bits in its real and imaginary parts separately, and acts as an index to the table. The table size is determined by the number of bits; for example, with 10 bit accuracy, 2.sup.20 or about 1 million table entries are needed. Note that the number of bits in each table entry is a different issue. For simplicity, assume that the table has infinite precision.

Analysis of the effects of input quantization through two nonlinearities is surprisingly straightforward. Imagine the input space (v.sub.m space) as partitioned into quantization cells, as shown in FIG. 10. If there are N.sub.t entries in the two-dimensional table, then each cell has width: ##EQU3## since we associate maximum power P.sub.mm with the extreme corner cells. All v.sub.m values in a cell are mapped onto a single v.sub.d value, which we assume to be the correct predistorter value for the center of the cell. Thus the amplifier output v.sub.a has its desired value Kv.sub.m o