Abstract:
A system and method of linearizing the gain error of a power amplifier. The method involves the steps of generating a signal representing a directional derivative in the phase-magnitude space of an error signal of said power amplifier and modulating the gain of said power amplifier with said directional derivative signal.

Description:
This application claims the benefit of priority from U.S. provisional Application Ser. No. 60/074,593 file Feb 13, 1998. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to a system and method for linearizing the gain error of a power amplifier. More particularly, the invention relates to directional derivative and gradient methods for estimating and correcting complex gain error. 
     BACKGROUND OF THE INVENTION 
     Radio frequency power amplifiers have a nonlinear power transfer function. In other words, both the amplitude and phase components of the amplifier&#39;s gain depend on the power of the amplifier&#39;s input signal. This amplifier non-linearity is undesirable because it distorts the output waveform, broadens the output waveform&#39;s spectrum, and generates interference within, adjacent channels. 
     Generally, power amplifier linearization is sought using one of three techniques: vector feedback, adaptive predistortion, and adaptive feedforward compensation. In each of the three techniques, the amplifier&#39;s complex gain error must be estimated, preferably in real-time. It is toward this estimation step that the present invention is directed. 
     Conventionally, the gain error is estimated by comparing the amplifier&#39;s output signal with a reference signal, usually the amplifier&#39;s input signal. This comparison measures quantities such as the correlation or the error power of the two signals. 
     Unfortunately, conventional correlation circuits introduce DC offsets and common-mode feedthrough errors. Also problematic, conventional error power measurement circuits use a search process that disturbs the gain along the amplification path and yields a signal convergence that is too slow for vector feedback systems. 
     It would be desirable to have a gain error estimator that does not suffer from these problems. 
     SUMMARY OF THE INVENTION 
     The illustrated embodiments avoid conventional common-mode feedthrough problems by measuring the error signal across a single square-law detector in a single processing path. 
     Instead of conventionally varying the gain along the amplifier path to search for the minimum error, the embodiments vary the amplitude and/or phase along a reference path. In other words, differential measurements needed to estimate the gain error gradient are obtained by amplitude and/or phase modulating the reference signal. The modulated error power measurement is later demodulated, so that the amplifier gain is unaffected by the search process. 
     Although modulating the reference signal is only a one-dimensional search of the amplitude phase (δa-δø) space, by implementing two concurrent (and preferably orthogonal) searches, one can resolve the gain error gradient from the directional derivatives obtained. 
     Thus, according to one embodiment of the invention, there is provided a method of linearizing the vector gain of a power amplifier including generating a directional derivative signal representing a phase-amplitude space directional derivative of an error signal of the power amplifier and modulating the gain of the power amplifier in response to the directional derivative signal. Generating a directional derivative signal might include locating in the phase-amplitude space of the error signal a vector that is substantially equal to the directional derivative. By extension, locating a vector that is substantially equal to the directional derivative might include searching for the best approximation of the directional derivative along a bounded one-dimensional path in the phase-amplitude space of the error signal. 
     Alternatively, locating a vector that is substantially equal to the directional derivative might include: searching for the best approximation of the directional derivative along a bounded one-dimensional path in the phase-amplitude space of the error signal, the path being parallel to the phase axis of the phase-amplitude space; searching for the best approximation of the directional derivative along two bounded one-dimensional paths in the phase-amplitude space of the error signal; searching for the best approximation of the directional derivative along two substantially orthogonal bounded one-dimensional paths in the phase-amplitude space of the error signal; searching for the best approximation of the directional derivative alternately along each of two substantially orthogonal bounded one-dimensional paths in the phase-amplitude space of the error signal; searching for the best approximation of the directional derivative simultaneously along each of two substantially orthogonal bounded one-dimensional paths in the phase-amplitude space of the error signal and then combining into a weighted average the best approximation of the directional derivative along each of the two substantially orthogonal bounded one-dimensional paths; or searching at a first frequency for a first component of the best approximation of the directional derivative along a first bounded one-dimensional path in the phase amplitude space of the error signal and then searching at a second frequency for a second component of the best approximation of the directional derivative along a second bounded one-dimensional path in the phase amplitude space of the error signal. 
     Desirably, the method includes: receiving a feedback signal corresponding to an output signal output from the power amplifier, receiving a reference signal corresponding to an input signal input to the power amplifier, modulating the phase of the reference signal through a bounded range of phase shift angles, subtracting the feedback signal from the modulated reference signal to produce a difference signal, and rectifying the difference signal to produce a rectified difference signal that corresponds to the magnitude of a directional derivative oriented parallel to the phase-amplitude space phase axis. 
     Alternatively, the method might include generating a gradient signal representing a phase-amplitude space gradient of the error signal of the power amplifier. In this case, the method desirably further includes: receiving a feedback signal corresponding to an output signal output from the power amplifier, receiving a reference signal corresponding to an input signal input to the power amplifier, modulating the phase of the reference signal through a bounded range of phase shift angles, modulating the amplitude of the reference signal through a bounded range of amplitude shift levels, subtracting the feedback signal from the modulated reference signal to produce a difference signal, and rectifying the difference signal to produce a rectified difference signal that corresponds to the magnitude and phase of the gradient signal. 
     Still alternatively, generating the gradient signal might include: receiving a feedback signal corresponding to an output signal output from the power amplifier, receiving a reference signal corresponding to an input signal input to the power amplifier, subtracting the magnitude of the feedback signal from the magnitude of the reference signal to produce an amplitude component of the gain error gradient, modulating the phase of the reference signal through a bounded range of phase shift angles, subtracting the feedback signal from the modulated reference signal to produce a difference signal, rectifying the difference signal to produce a rectified difference signal that corresponds to a phase component of the gradient signal. 
     It should be added that either generating a directional derivative signal or generating a gradient signal might further include; attenuating the output signal by the nominal gain of the amplifier, time delaying the input signal by the loop delay of the amplifier, demodulating the rectified difference signal to produce a demodulated difference signal, lowpass filtering the demodulated difference signal to produce a filtered difference signal, or normalizing the filtered difference signal with respect to the instantaneous power of the reference signal to produce a normalized signal. 
     According to another embodiment of the invention, there is provided means for linearizing the vector gain of a power amplifier including means for generating a directional derivative signal representing a phase-amplitude space directional derivative of an error signal of the power amplifier and means for modulating the gain of the power amplifier in response to the directional derivative signal. 
     According to yet another embodiment of the invention, there is provided an apparatus for linearizing the vector gain of a power amplifier having an input terminal, an output terminal, and a gain control terminal, the apparatus including a derivative signal generator having: a first input stage connected to the power amplifier input terminal to produce a reference signal corresponding to an input signal input to the power amplifier; a second input stage connected to the power amplifier output terminal to produce a feedback signal corresponding to an output signal output from the power amplifier; and an output stage connected to the power amplifier gain control terminal to provide a directional derivative signal representing a phase-amplitude space directional derivative of the power amplifier gain error. 
     While specific embodiments of the invention are described and illustrated, such embodiments should be considered illustrative of the invention only and not as limiting the invention as construed in accordance with the accompanying claims. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In drawings which illustrate embodiments of the invention, 
     FIG. 1 is a schematic diagram of a phase error estimator according to a first embodiment of the invention; 
     FIG. 2 is a schematic diagram of a directional derivative estimator according to a second embodiment of the invention; 
     FIG. 3 is a schematic diagram of a gain error estimator according to a third embodiment of the invention, the embodiment including a phase error estimator as shown in FIG. 1; 
     FIG. 4 is a schematic diagram of a gain error estimator according to a fourth embodiment of the invention, the embodiment including two of the directional derivative estimators shown in FIG. 2; 
     FIG. 5 is a schematic diagram of a gain error estimator according to a fifth embodiment of the invention, the embodiment including a frequency division multiplexed directional derivative estimator as shown in FIG. 2; and 
     FIG. 6 is a schematic diagram of a gain error estimator according to a sixth embodiment of the invention, the embodiment including a time division multiplexed directional derivative estimator as shown in FIG.  2 . 
    
    
     DETAILED DESCRIPTION 
     Overview 
     A power amplifier has a nominal complex gain G o , which includes both amplitude and phase components. In practice however, the amplifier&#39;s actual gain G differs from the nominal gain G o  and instead varies with the power of the amplifier&#39;s input signal X in . 
     The amplifier&#39;s complex gain error (G/G o )−1 is defined with reference to its nominal gain and its actual gain. To linearize the amplifier, one must converge the gain error to zero. Estimating the instantaneous gain error approximates the necessary gain adjustment for amplifier linearization. 
     The gain error&#39;s gradient is the most direct path to the error minimum. In this case, the gain error gradient is equivalent to the instantaneous gain error itself. Although the gain error gradient is the most direct convergence path, for circuit implementation reasons one may instead choose to converge along other directional derivatives. 
     Six embodiments of a system for estimating the gain error of a power amplifier are disclosed. Each embodiment converges the gain error along at least one directional derivative, in some embodiments the gain error gradient. 
     The invention may be incorporated into adaptive predistortion and adaptive feedforward compensation networks. However, for simplicity, all embodiments discussed are directed to vector feedback networks connected to control a vector modulated power amplifier. 
     FIG.  1   
     Referring now to FIG. 1, a phase error estimator embodying a first aspect of the invention is generally illustrated at  10 . The phase error estimator  10  is connected to control a vector modulated power amplifier  11  having a nominal gain G o  and an actual gain G. 
     The phase error estimator  10  is connected to the amplifier  11  to receive both its input signal X in  and its output signal Y out . The phase error estimator  10  is also connected to feedback to the amplifier  11  a gain-modulating error signal δø that represents the phase component of the amplifier&#39;s complex gain error (G/G o )−1. The estimated phase error may be viewed as an estimated directional derivative oriented parallel to the phase coordinate axis. 
     In greater detail now, an attenuator  12  is connected to the amplifier  11  to sample its output signal Y out  and to attenuate it by the nominal gain G o  to produce an attenuated feedback signal y. 
     A splitter  14  is connected to the amplifier  11  to sample its input signal X in . A first time delay loop  16  is connected to the splitter  14  to receive the sampled input signal X in , and to produce in response a time-delayed reference signal x by introducing a time delay τ equal to the delay time through the amplifier  11  and around the feedback loop to the attenuator  12  output terminal. 
     A waveform source  18  is connected through a first scalar multiplier  20  to control a phase modulator  22 . The phase modulator  22  has an input terminal, an output terminal, and a control terminal, to which the scalar multiplier  20  is connected. The phase modulator  22  introduces a phase shift between its input and output terminals in response to the signal amplitude at its control terminal. 
     The waveform source  18  may produce any arbitrary periodic waveform; however, a constant amplitude, constant frequency sine wave having the form sin(ω ø t) is preferred. The first scalar multiplier  20  scales the amplitude of the waveform source signal by a predetermined value m ø  to yield a scaled waveform having the form Δø=m ø ·sin(ω ø t). The scaled waveform determines the phase-shifting range of the phase modulator  22 . 
     The input terminal of the phase modulator  22  is connected to the first time delay loop  16  to receive the time delayed reference signal x. The output terminal of the phase modulator produces a phase modulated reference signal x· e   jΔø . This modulation of the reference signal x enables a rolling range of phase angles to be searched to determine which yields the lowest gain error. 
     A summing junction  24  is connected to the attenuator  12  and the phase modulator  22  to subtract the attenuated feedback signal y from the phase modulated reference signal x·e jΔø  to produce a modulated error signal ε m =X·e jΔø −y. 
     A first square-law detector  26  is connected to the summing junction  24  to receive the modulated error signal ε m  and to producc at its output terminal a signal proportional to the power of the error signal |ε m|   2 . 
     The module formed by the interconnection of the first scalar multiplier  20 , the phase modulator  22 , the summing junction  24 , and the first square-law detector  26  will hereafter be termed a phase search module  27 . Essentially, the phase search module searches for the phase adjustment that yields the lowest error power signal |ε m | 2 . Thus the phase search module provides for searching for the best approximation of the directional derivative along a bounded one-dimensional path in the phase-amplitude space of the error signal, the path being parallel to the phase axis of the phase-amplitude space. 
     A demodulator  28  is connected at a first input terminal to the waveform source  18  and at a second input terminal to the output terminal of the first square-law detector  26 . The demodulator  28  removes the source waveform component from the error power signal |ε m | 2 . 
     A first lowpass filter  30  is connected to the demodulator  28  to receive the demodulated error power signal to remove any modulation harmonics and DC offset appearing at the output of the first square-law detector  26 . 
     A second time delay loop  32  is connected to the splitter  14  to receive the sampled amplifier input signal X in . The second time delay loop  32  introduces a time delay τ equal to the delay time through the amplifier  11  and around the feedback loop to the attenuator  12  output terminal, thereby producing a second instance of the time delayed reference signal x. 
     The second time delay loop  32  is connected to feed in series a second square-law detector  34 , a second lowpass filter  36 , and an inverter  38 , which in combination produce a normalizing signal 1/|x| 2 . 
     A mixer  40  is connected to the inverter  38  to receive the normalizing signal 1/|x| 2  and to the first lowpass filter  30  to receive the filtered and demodulated error power signal |ε m | 2 . The mixer  40  produces in response a normalized error power signal |ε m | 2 /|x| 2 . 
     A second scalar multiplier  42  is connected to the mixer  40  to receive the normalized error power signal |ε m | 2 /|x| 2  and scale it by the reciprocal of the first scalar multiplier  20 , k ø =1/m ø . This scaling operation compensates for the sensitivity increase introduced by the first scalar multiplier  20  and generates the estimated phase component of the gain error δø. The second scalar multiplier  42  is connected to the amplifier  11  to modulate its gain by the estimated phase component of the gain error δø. 
     Thus, the phase error estimator  10  provides for searching for the beat approximation of the directional derivative along a bounded one-dimensional path in the phase-amplitude space of the error signal, the path being parallel to the phase axis of the phase-amplitude space. 
     FIG.  2   
     Referring now to FIG. 2, a directional derivative estimator according to a second embodiment of the invention is generally illustrated at  50 . The directional derivative estimator  50  is connected to control a vector modulated power amplifier  51  having a nominal gain G o  and an actual gain G. 
     The directional derivative estimator  50  is connected to the amplifier  51  to receive both its input signal X in  and its output signal Y out . The directional derivative estimator  50  is also connected to feedback to the amplifier  51  a gain modulating error signal (m a ·δ a +m ø ·δ ø ) that represents the amplifier&#39;s complex gain error (G/G o )−1. 
     In greater detail now, an attenuator  52  is connected to the amplifier  51  to sample its output signal Y out  and to attenuate it by the nominal gain G o  to produce an attenuated feedback signal y. 
     A splitter  54  is connected to the amplifier  51  to sample its input signal X in . A first time delay loop  56  is connected to the splitter  54  to receive the sampled input signal X in  and to produce in response a time-delayed reference signal x by introducing a time delay τ equal to the delay time through the amplifier  51  and around the feedback loop to the attenuator  52  output terminal. 
     A waveform source  58  is connected through a first scalar multiplier  60  to control a phase modulator  62  and connected through a second scalar multiplier  61  to control an amplitude modulator  63 . 
     The phase modulator  62  has an input terminal, an output terminal, and a control terminal, to which the first scalar multiplier  60  is connected. The phase modulator  62  introduces a phase shift between its input and output terminals in response to the signal amplitude at its control terminal. Thus the phase modulator  62  provides for modulating the phase of the reference signal through a bounded range of phase shift angles. 
     The amplitude modulator  63  has an input terminal, an output terminal, and a control terminal, to which the second scalar multiplier  61  is connected. The amplitude modulator  63  scales amplitude between its input and output terminals in response to the signal amplitude at its control terminal. Thus the amplitude modulator  63  provides for modulating the amplitude of the reference signal through a bounded range of amplitude shift levels. 
     The waveform source  58  may produce any arbitrary periodic waveform; however, a constant amplitude, constant frequency sine wave having the form sin(ω 526  t) is preferred. 
     The first scalar multiplier  60  scales the amplitude of the waveform source signal by a predetermined value m ø  to yield a scaled waveform having the form Δø=m ø ·sin(ω ø t). This scaled waveform sets the phase-shifting range of the phase modulator  62 . 
     The second scalar multiplier  61  scales the amplitude of the waveform source signal by a predetermined value m a  to yield a scaled waveform having the form Δa=m a ·sin(ω ø t). This scaled waveform determines the amplitude scaling range of the amplitude modulator  63 , 
     The phase modulator  62  and the amplitude modulator  63  are connected in series to the output terminal of the first time delay loop  56  to receive the time delayed reference signal x. In combination, they produce a phase and amplitude modulated reference signal x·e Δa+jΔø . This modulation of the reference signal x enables a rolling range of amplitudes and phase angles to be searched to determine which yields the lowest gain error. The fixed ratio −(m a /m ø ) sets the trajectory of that search. 
     A summing junction  64  is connected to the attenuator  52  and the phase modulator  62  to subtract the attenuated feedback signal y from the modulated reference signal x·e Δa+jΔø  to produce a modulated error signal ε m =x·e Δa+jΔø −y. 
     A first square-law detector  66  is connected to the summing junction  64  to receive the modulated error signal ε m  and to produce at its output terminal a signal proportional to the power of the error signal |ε m | 2 . Thus the first square-law detector  66  provide for rectifying the difference signal to produce a rectified difference signal that corresponds to the magnitude of a directional derivative oriented parallel to the phase-amplitude space phase axis. 
     The module formed by the interconnection of the first scalar multiplier  60 , the phase modulator  62 , the second scalar multiplier  61 , the amplitude modulator  63 , the summing junction  64 , and the first square-law detector  66  will hereafter be termed a one-dimensional search module  67 . The one-dimensional search module  67  searches along a trajectory having slope −(m a /m ø ) for the amplitude and phase adjustments that yield the lowest error power signal |ε m | 2 . Thus the one-dimensional search module  67  provides for locating in the phase-amplitude space of the error signal a vector that is substantially equal to the directional derivative. Furthermore, the one-dimensional search module  67  provides for searching for the best approximation of the directional derivative along a bounded one-dimensional path in the phase-amplitude space of the error signal. 
     A demodulator  68  is connected at a first input terminal to the waveform source  58  and at a second input terminal to the output terminal of the first square-law detector  66 . The demodulator  68  removes the source waveform component from the error power signal |ε m | 2 . 
     A first lowpass filter  70  is connected to the demodulator  68  to receive the demodulated error power signal to remove any modulation harmonics and DC offset appearing at the output of the first square-law detector  66 . 
     A second time delay loop  72  is connected to the splitter  54  to receive the sampled amplifier input signal X in . The second time delay loop  72  introduces a time delay τ equal to the delay time through the amplifier  51  and around the feedback loop to the attenuator  52  output terminal, thereby producing a second instance of the time delayed reference signal x. 
     The second time delay loop  72  is connected to teed in series a second square-law detector  74 , a second lowpass filter  76 , and an inverter  78 , which in combination produce a normalizing signal 1/|x| 2 . 
     A mixer  80  is connected to the inverter  78  to receive the normalizing signal 1/|x| 2  and to the first lowpass filter  70  to receive the filtered and demodulated error power signal |ε m | 2 . The mixer  80  produces in response a normalized error power signal |ε m | 2 /|x| 2 , which is a directional derivative of the gain error. The mixer  80  is connected to the amplifier  51  to modulate its gain by the estimated directional derivative m a ·δ a +m ø ·ø ø  of the gain error. Thus the directional derivative estimator  50  provides for generating a directional derivative signal representing a phase-amplitude space directional derivative of an error signal of the power amplifier and modulating the gain of the power amplifier in response to the directional derivative signal. 
     FIG.  3   
     Referring now to FIG. 3, an error gradient estimator according to a third embodiment of the invention is generally illustrated at  90 . The error gradient estimator  90  is connected to a vector modulated power amplifier  92  having a nominal gain G o , an actual gain G, and being connected to receive an input signal X in  and transmit an output signal Y out . 
     It may be observed that the error gradient estimator  90  includes a phase error estimator as illustrated at  10  in FIG.  1 . 
     The error gradient estimator  90  is connected to the amplifier  92  through appropriate splitters, time delay loops, and attenuators as described in FIGS. 1 and 2 to receive a time delayed reference signal x(t)=X in (t−τ) and an attenuated feedback signal y=G o   −1 ·Y out . The error gradient estimator  90  is also connected to feedback to the amplifier  132  a gain-modulating error signal (δa, δø) that represents the gradient of the amplifier&#39;s complex gain error (G/G o )−1. 
     In greater detail now, a first three-way splitter  94  is connected to receive from the amplifier  92  the time delayed input signal x. A second three-way splitter  96  is connected to receive from the amplifier  92  the attenuated feedback signal y. 
     A phase search module  98 , which was shown in greater detail at  27  in FIG. 1, is connected to the first and second three-way splitters  94 ,  96  to receive the time-delayed reference signal x and the attenuated feedback signal y. A square-law detector  100  is similarly connected to the first three-way splitter to receive the time-delayed reference signal x. 
     The phase search module  98  and the square-law detector  100  are assembled according to the embodiment of FIG. 1 along with a waveform source  102 , a demodulator  104 , a first lowpass filter  106 , a second lowpass filter  108 , a first normalizing divider  110 , and a scalar multiplier  112  to form a phase error estimator that yields an estimated phase error δø. 
     An amplitude comparator  114  is connected to the first and second three-way splitters  94 ,  96  to receive the time-delayed reference signal x and the attenuated feedback signal y and to produce in response a difference signal. A second normalizing divider  116  is connected to receive the difference signal from the amplitude comparator  114  and the reference power signal from the square-law detector  100  in order to divide the former by the latter. 
     A scalar multiplier  118  is connected to receive the normalized signal from the second normalizing divider  116  and to multiply it by a scalar constant k a =0.5 to yield an estimated amplitude error δa. 
     Thus the error gradient estimator  90  provides a gradient signal generator, a first input stage connected to the power amplifier input terminal to produce a reference signal corresponding to an input signal input to the power amplifier, a second input stage connected to the power amplifier output terminal to produce a feedback signal corresponding to an output signal output from the power amplifier, an output stage connected to the power amplifier gain control terminal to provide a directional derivative signal representing a phase-amplitude space directional derivative of the power amplifier gain error, a first subtracting junction connected to the first and second input stages to subtract the magnitude of the feedback signal from the magnitude of the reference signal to produce an amplitude component of the gain error gradient, a phase modulator connected to the first input stage to modulate the phase of the reference signal through a bounded range of phase shift angles, a second subtracting junction connected to the phase modulator and the second input stage to subtract the feedback signal from the modulated reference signal to produce a difference signal, and a rectifier connected to the subtracting junction to rectify the difference signal to produce a rectified difference signal that corresponds to a phase component of the gradient signal. In this manner, the error gradient estimator  90  generates a signal representing an estimated error gradient. 
     FIG.  4   
     Referring now to FIG. 4, an error gradient estimator according to a fourth embodiment of the invention is generally illustrated at  130 . The error gradient estimator  130  is connected to a vector modulated power amplifier  132  having a nominal gain G o , an actual gain G, and being connected to receive an input signal X in  and transmit an output signal Y out . 
     It may be observed that the error gradient estimator  130  includes two interdependent directional derivative estimators as illustrated at  50  in FIG.  2 . 
     The error gradient estimator  130  is connected to the amplifier  132  through appropriate splitters, time delay loops, and attenuators as described in FIGS. 1 and 2 to receive a time delayed reference signal x(t)=X in (t−τ) and an attenuated feedback signal y=G o   −1  ·Y out . The error gradient estimator  130  is also connected to feedback to the amplifier  92  a gain modulating error signal (δa,δø) that represents the gradient of the amplifier&#39;s complex gain error (G/G o )−1. 
     In greater detail now, a first three-way splitter  134  is connected to receive from the amplifier  132  the time delayed input signal x. A second three-way splitter  136  is connected to receive from the amplifier  132  the attenuated feedback signal y. 
     A first one-dimensional search module  138  is connected to the first and second three-way splitters  134 ,  136  to receive the time-delayed reference signal x and the attenuated feedback signal y. The first one-dimensional search module  138  has preset search-trajectory constants m ø1  and m a1 . 
     A square-law detector  140  is also connected to the first three-way splitter to receive the time-delayed reference signal x. 
     The first one-dimensional search module  138  and the square-law detector  140  are assembled according to the embodiment of FIG. 2 with a waveform source  142 , a first demodulator  144 , a first lowpass filter  146 , a second lowpass filter  148 , and a first normalizing divider  150 , to form a first directional derivative estimator. second one-dimensional search module  152  is connected to he first and second three-way splitters  134 ,  136  to receive the time-delayed reference signal x and the attenuated feedback signal y. The second one-dimensional search module  152  has preset search-trajectory constants m ø2  and m a2 . Preferably, the search-trajectory constants m ø1  and m a1  and m ø2  and m a2  are selected such that the search-trajectories of the first and second one-dimensional search modules  138 ,  152  are orthogonal. 
     The second one-dimensional search module  152  and the square-law detector  140  are assembled according to the embodiment of FIG. 2 with the waveform source  142 , a second demodulator  154 , a third lowpass filter  156 , the second lowpass filter  148 , and a second normalizing divider  158 , to form a second directional derivative estimator. 
     Thus, together the first and second one-dimensional search modules  138 ,  152  provide for searching for the best approximation of the directional derivative along two bounded one-dimensional paths in the phase-amplitude space of the error signal. Furthermore, they provide for searching f or the beat approximation of the directional derivative along two substantially orthogonal bounded one dimensional paths in the phase-amplitude space of the error signal. In effect, the first and second one-dimensional search modules  138 ,  152  function as sweep circuits, each sweeping its one-dimensional path in phase-amplitude space for the best approximation of the directional derivative. 
     A coordinate transform network  160  is connected to receive the signals output from the first and second normalizing dividers  150 ,  158  in order to extract the gain error gradient from the two directional derivatives. The coordinate transform network  160  includes first, second, third and fourth scalar multipliers  162 ,  164 ,  166 ,  168 . The first scalar multiplier  162  is connected to receive the signal output from the first normalizing divider  150  and to multiply it by m ø2 . The second scalar multiplier  164  is connected to receive the signal output from the first normalizing divider  150  and to multiply it by m a2 . The third scalar multiplier  166  is connected to receive the signal output from the second normalizing divider  158  and to multiply it by −-m ø1 . The fourth scalar multiplier  168  is connected to receive the signal output from the second normalizing divider  158  and to multiply it by m a1 . 
     The coordinate transform network  160  also includes first and second summing junctions  170 ,  172 . The first summing junction  170  is connected to receive the signals output from the first and third scalar multipliers  162 ,  166 . The second summing junction  172  is connected to receive the signals output from the second and fourth scalar multipliers  164 ,  168 . 
     Thus together the first and second one-dimensional search modules  138 ,  152  and the coordinate transform network  160  provide for searching for the best approximation of the directional derivative simultaneously along each of two substantially orthogonal bounded one-dimensional paths in the phase-amplitude space of the error signal and combining into a weighted average the best approximation of the directional derivative along each of the two substantially orthogonal bounded one-dimensional paths. 
     The coordinate transform network  160  further includes fifth and sixth scalar multipliers  174 ,  176 . The fifth scalar multiplier  174  is connected to receive the signal output from the first summing junction  170  and to multiply it by the scalar k Δ =[m a1 ·m ø2 −m ø1 ·m a2 ] −1  thereby yielding an estimate of the amplitude component δa of the gain error gradient. The sixth scalar multiplier  176  is connected to receive the signal output from the second summing junction  172  and to multiply it by the k Δ  thereby yielding an estimate of the phase component δø of the gain error gradient. 
     FIG.  5   
     Referring now to FIG. 5, an error gradient estimator according to a fifth embodiment of the invention is generally illustrated at  190 . The error gradient estimator  190  is connected to a vector modulated power amplifier  192  having a nominal gain G o , an actual gain G, and being connected to receive an input signal X in  and transmit an output signal Y out . 
     It may be observed that the error gradient estimator  190  includes one directional derivative estimator as illustrated at  50  in FIG.  2 . It may also be observed that waveform superposition permits the single directional derivative estimator to be frequency division multiplexed and to thereby simultaneously search two trajectories. 
     The error gradient estimator  190  is connected to the amplifier  192  through appropriate splitters, time delay loops, and attenuators as described in FIGS. 1 and 2 to receive a time delayed reference signal x(t)=X in (t−τ) and an attenuated feedback signal y=G o   −1 ·Y out . The error gradient estimator  190  is also connected to feedback to the amplifier  192  a gain modulating error signal (δa,δø) that represents the gradient of the amplifier&#39;s complex gain error (G/G o )−1. 
     In greater detail now, a two-way splitter  194  is connected to receive from the amplifier  192  the time delayed input signal x. 
     A one-dimensional search module  196  is connected to receive the time-delayed reference signal x from the two-way splitter  194  and the attenuated feedback signal y from the amplifier  192 . The one-dimensional search module  196  is similar but not identical to the one illustrated at  67  in FIG.  2 : it contains no internal scalar multipliers. 
     A square-law detector  198  is connected to the two-way splitter  194  to receive the time-delayed reference signal x. 
     The one-dimensional search module  196  and the square-law detector  198  are shared between two directional derivative estimators assembled according to the embodiment of FIG.  2 . 
     The first directional derivative estimator includes, besides the one-dimensional search module  196  and the square-law detector  198 , a first waveform source  200 , an amplitude scalar multiplier  202 , a first demodulator  204 , a first lowpass filter  206 , a second lowpass filter  208 , and a first normalizing divider  210 , and a second scalar multiplier  212  to form a first directional derivative estimate δa aligned parallel to the amplitude axis. 
     The second directional derivative estimator includes, besides the one-dimensional search module  196  and the square-law detector  198 , a second waveform source  214 , a phase scalar multiplier  216 , a second demodulator  218 , a third lowpass filter  220 , the second lowpass filter  208 , a second normalizing divider  222 , and a fourth scalar multiplier  224  to form a second directional derivative estimate δø aligned parallel to the phase axis. 
     Thus the error gradient estimator  190  provides for searching at a first frequency for a first component of the best approximation of the directional derivative along a first bounded one-dimensional path in the phase amplitude space of the error signal and searching at a second frequency for a second component of the beat approximation of the directional derivative along a second bounded one-dimensional path in the phase amplitude space of the error signal. 
     FIG.  6   
     Referring now to FIG. 6, an error gradient estimator according to a sixth embodiment of the invention is generally illustrated at  240 . The error gradient estimator  240  is connected to a vector modulated power amplifier  242  having a nominal gain G o , an actual gain G, and being connected to receive an input signal X in  and transmit an output signal Y out . 
     It may be observed that the error gradient estimator  240  includes two directional derivative estimators as illustrated at  50  in FIG. 2, although the two estimators share a single processing path, including a single one-dimensional search module. It may also be observed that time division multiplexing permits the single one-dimensional search module alternately search each of the two trajectories. 
     The error gradient estimator  240  is connected to the amplifier  242  through appropriate splitters, time delay loops, and attenuators as described in FIGS. 1 and 2 to receive a time delayed reference signal x(t)−X in (t−τ) and an attenuated feedback signal y=G o   −1 ·Y out . The error gradient estimator  240  is also connected to feedback to the amplifier  242  a gain modulating error signal (δa,δø) that represents the gradient of the amplifier&#39;s complex gain error (G/G o )−1. 
     In greater detail now, a two-way splitter  244  is connected to receive from the amplifier  242  the time delayed input signal x. 
     A one-dimensional search module  246  is connected to receive the time-delayed reference signal x from the two-way splitter  244  and the attenuated feedback signal y from the amplifier  242 . The one-dimensional search module  246  is similar but not identical to the one illustrated at  67  in FIG.  2 : it contains no internal scalar multipliers 
     A square-law detector  248  is connected to the two-way splitter  244  to receive the time-delayed reference signal x. 
     The one-dimensional search module  246  is connected to a waveform generator  250  through a first single pole, double throw switch  252  which alternately connects the one-dimensional search module  246  to either a first pair of external scalar multipliers  254  having preset search-trajectory constants m ø1  and m a1  or a second pair of external scalar multipliers  256  having preset search-trajectory constants m ø2  and m a2 . 
     The error gradient estimator  240  further includes second and third single pole, double throw switches  258 ,  260 . The three switches  252 ,  258 ,  260  are ganged together so that they throw simultaneously. 
     With the three switches  252 ,  258 ,  260  in a first position, a first directional derivative estimator is assembled according to the embodiment of FIG. 2 from the one-dimensional search module  246 , the square-law detector  248 , the waveform source  250 , the first pair of scalar multipliers  254 , a demodulator  262 , a first lowpass filter  264 , a second lowpass filter  266 , and a first normalizing divider  268 . 
     With the three switches  252 ,  258 ,  260  in a second position, a second directional derivative estimator is assembled according to the embodiment of FIG. 2 from the one-dimensional search module  246 , the square-law detector  248 , the waveform source  250 , the second pair of scalar multipliers  256 , the demodulator  262 , a third lowpass filter  270 , a fourth lowpass filter  272 , and a second normalizing divider  274 . 
     As in FIG. 4, a coordinate transform network  276  is connected to receive the signals output from the first and second normalizing dividers  274  and to generate the amplitude δa and phase δø components of the gain error gradient from the directional derivative values. 
     Thus the error gradient estimator  240  provides for searching for the best approximation of the directional derivative alternately along each of two substantially orthogonal bounded one-dimensional paths in the phase-amplitude space of the error signal. 
     While specific embodiments of the invention have been described and illustrated, such embodiments should be considered illustrative of the invention only and not as limiting the invention as construed in accordance with the accompanying claims.