Abstract:
In one embodiment, a signal processor for linearizing a non-linear circuit through pre-distortion of an input signal is provided that includes: a first coupler for extracting a version of the input signal, wherein a remaining portion of the input signal not extracted by the first coupler is provided to a first node; a mixer for multiplying the extracted version of the input signal with a pre-distortion signal to produce an additive signal, the pre-distortion signal having a relatively small or zero constant component such that the additive signal includes either no linear version of the input signal or a linear version of the input signal that has a lower power than the remaining portion of the input signal; and a second coupler to add the additive signal to the remaining portion of the input signal at the first node to form a pre-distorted input signal, whereby if the non-linear circuit processes the pre-distorted input signal to form an output signal, the output signal is a substantially linear function of the input signal.

Description:
RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 60/992,227 filed Dec. 4, 2007. 
    
    
     TECHNICAL FIELD 
     The present invention relates generally to signal processing, and more particularly to additive pre-distortion for linearizing non-linear circuits. 
     BACKGROUND 
     All amplifiers will have some non-linearity across their dynamic range. Although there are contexts in which the resulting distortion in the amplified signal is desirable (such as an electric guitar amplifier), most applications do not benefit from non-linear distortion. For example, modern wireless telecommunication protocols such as 2.5G and 3G use non-constant amplitude (envelope modulation) signals. These non-constant envelope signals are sensitive to non-linear-power-amplifier-induced distortion. Given the general desire for linear power amplification, a number of techniques have been developed to enhance the linearity of power amplifiers. For example, feedforward power amplifiers have good linearity but generally do not improve efficiency and greatly increase cost and complexity. 
     An attractive alternative to feedforward linearization techniques is to pre-distort the input signal in an inverse fashion with regard to the non-linearity of the power amplifier. This pre-distortion may be performed on the input signal in the digital domain prior to a digital-to-analog conversion. Alternatively, the input signal may be pre-distorted in the analog RF domain. To pre-distort the input signal in the RF domain, the input signal is typically multiplied with the pre-distortion signal. For example, an RF input signal that will be multiplied by the pre-distortion signal may be represented by the real part of {R(t)*exp(jω c t)}, where R(t) is the complex envelope, j is the imaginary unit, ω c  is the angular frequency of the RF carrier bearing the complex envelop modulation, and t is time. The desired pre-distortion of the complex envelope may be better understood with reference to  FIG. 1 , which illustrates the non-linear dependence of an output signal amplitude  101  with respect to input signal amplitude for a conventional amplifier. As known in the arts, a conventional amplifier has a linear region of operation and a saturation region of operation (these regions being separated by dashed line  106  in  FIG. 1 ). For relatively small input signal amplitudes, a real-world amplifier will amplify such small signal amplitudes into corresponding output signal amplitudes according to the gain of the amplifier in a close-to-perfectly-linear fashion. However, as the amplifier approaches saturation, output signal amplitude  101  progressively distorts away from an ideal linear response for the amplifier (the ideal linear response for a perfect amplifier being represented by dashed line  100 ). Given this non-linearity, it may be seen that if the input signal amplitudes to the amplifier were pre-distorted in a reciprocal fashion to the distortion seen in output signal amplitude  101  with respect to ideal response  100 , the amplifier would provide an output signal that would mirror ideal response  100 . As seen by an output signal amplitude  105  (which would be produced by an ideal amplifier amplifying the pre-distorted signal), the pre-distortion mirrors the distortion in output signal amplitude  101  with respect to ideal response  100 . By multiplying such a pre-distortion signal with the input signal, the amplifier is thereby linearized, to the limit of saturation for the amplifier in question. 
     Referring back to the complex envelope representation of the RF input signal, it may be seen that the pre-distortion signal is a baseband signal in that the pre-distortion signal is a function of the complex envelope R(t) and not of the RF carrier. In that regard, a pre-distortion signal that will be multiplied by the complex envelope may be represented as a Taylor series expression: α 1 +α 2 *R(t)+α 3 *R(t) 2 +α 4 *R(t) 3 + . . . , where the alpha symbols represent complex series coefficients. Upon multiplication with the RF input signal (the real part of {R(t)*exp(jω c t)}), the resulting pre-distorted RF signal becomes the real part of {[α 1 *R(t)+α 2 *R(t) 2 +α 3 *R(t) 3 +α 4 *R(t) 4 + . . . ]*exp(j ω c t)} that will then form an input signal for the amplifier. The alpha coefficients are controlled so as to pre-distort the input signal so as to produce a linear response in the downstream amplifier. 
     Turning now to  FIG. 2 , an RF signal processing (RFSP) circuit  200  that addresses the non-linear distortion discussed with regard to  FIG. 1  is illustrated. An amplifier  205  amplifies an RF input signal  201  (designated as the complex signal (R(t)exp(jω c t)) after it has been properly pre-distorted such that a resulting output signal  210  from the amplifier is amplified in a substantially linear fashion. To generate an appropriate pre distortion pre-distorted signal  265 , the degree of non-linearity in this output signal should be determined so that the degree of pre-distortion necessary to linearize amplifier  205  may in turn be determined. The non-linearity of amplifier  205  may be determined in a number of fashions. For example, a version of output RF signal  210  may be suitably scaled in an attenuator  215  and have its sign reversed through a 180 degree phase-shifter  220  so that it may be subtracted from a version of the RF input signal in an adder  225  to produce an error signal e(t)  226 . Each version of the RF input signal and the RF output signal is supplied through, for example, couplers  230 . Based upon the non-linearity as exhibited in error signal  226 , a signal generator  235  may then generate an appropriate pre-distortion signal  236  such as the complex Taylor series discussed above: α 1 +α 2 *R(t)+α 3 *R(t) 2 +α 4 *R(t) 3  and so on up until some final power of R(t). This final power depends upon the complexity of the design and desired precision. For example, suppose the final power in the series expression is five, corresponding to R(t) 5 . In such an embodiment, it may be seen that signal generator must then solve for six coefficients in the Taylor series, ranging from α 1  to α 6 . The envelope function associated with each coefficient may be designated as the corresponding “basis” function. Thus the monomial basis function associated with coefficient α 1  is R(t) 0 , the basis function associated with coefficient α 2  is R(t) 1 , the basis function associated with coefficient α 3  is R(t) 2 , and so on. These coefficients may be determined in a variety of fashions. In an example analytical approach, signal generator  235  may include a correlator for each coefficient. Each coefficient&#39;s correlator correlates error signal  226  with the basis function corresponding to the coefficient. For example, coefficient α 2  may be produced responsive to a correlation of the error signal and the envelope R(t), coefficient α 3  may be produced responsive to a correlation of the error signal and the squared envelope R(t) 2 , and so on. It may be shown that the preceding selection of monomial basis functions will not typically provide desirable real-world results because numerous calculation cycles are necessary to converge to a solution. To enhance the convergence speed, each basis function may be an orthonormal polynomial formed from the above-discussed monomial powers of R(t) such as discussed in U.S. application Ser. No. 11/484,008, filed Jul. 7, 2006, now U.S. Pat. No. 7,844,014, the contents of which are incorporated by reference. The correlation of the basis functions and the error functions may be performed in an analog domain or in a digital domain. In alternative embodiments, signal generator  235  may simply use a brute force approach or non-linear optimization techniques to select an appropriate value for the coefficients such that the error signal is minimized. 
     Regardless of how signal generator  235  processes the error signal, signal generator  235  will determine values for the coefficients in the series representation of pre-distorted RF input signal  265  as discussed above. The number of coefficients depends upon the highest power of the complex envelope R(t) that will be generated for pre-distorted RF input signal  265 . For example, signal generator  235  may generate up to a sixth power of the complex envelope R(t) in a complex pre-distortion signal  236  represented as (α 1 +α 2 *R(t) 1 +α 3 *R(t) 2 +α 4 (t) 3 +α 5 *R(t) 4 +α 6 *R(t) 5 +α 7 *R(t) 6 ). Depending upon the resulting non-linearity produced in output RF signal  210  for a given set of coefficients, the signal generator may then drive the coefficients (from α 1  to α 7 ) until the non-linearity reaches a minimal value. 
     In this fashion, signal generator  235  functions to cancel the non-linear components in RF output signal  210 . For example, suppose the amplifier has a non-linearity such that it produces a component proportional to R(t) 2  having a certain phase relationship to the baseband envelope R(t). Signal generator  235  must then generate the coefficients such that this R(t) 2  component is cancelled in the RF output signal. It may thus be seen that each coefficient may require a unique and independent phase relationship to the baseband signal so that the corresponding non-linear component in the RF output signal may be cancelled. To enable such independent phasing, the multiplication of RF input signal  201  and pre-distortion signal  236  should be performed in the in-phase (I) and quadrature (Q) domain. Thus, the RF input signal R(t) may be decomposed into its I and Q components after passing through a buffer  240  and a quadrature phase-shifter (QPS)  245 . Signal generator  235  generates its coefficients in corresponding I and Q forms (designated in  FIG. 2  as the real (Re) and imaginary (Im) parts of pre-distortion signal  236 , respectively). The resulting I components of the RF input signal and the pre-distortion signal are multiplied in a mixer  250 . Similarly, the resulting Q components of the RF input signal and the pre-distortion signal are multiplied in a mixer  255 . The mixer output signals may be combined in a combiner  260  to provide pre-distorted RF input signal  265  to the amplifier. 
     But note that the generation of the analog non-linear components R(t) 2 , R(t) 3 , etc., in the signal generator is an inherently noisy process. The noise in the pre-distortion signal may then dominate the resulting pre-distorted RF input signal  265  that is to be amplified as shown by the following analysis: Let the input signal to be pre-distorted be represented by X such that its signal-power-to-noise-power ratio (SNR X ) is X 2 /(nx) 2 , where nx represents the rms noise “n” in the input signal X. Similarly, the pre-distortion signal may be represented by Y such that its signal-power-to-noise-power ratio (SNR Y ) is Y 2 /(ny) 2 . The multiplied signal (corresponding to pre-distorted RF input signal  265 ) is thus represented by Y*X. It may then be shown that the SNR for the signal YX is 1/((1/SNR X )+(1/SNR Y )). This expression for output SNR indicates that the output SNR is lower than the lowest SNR of the two inputs. This is a worst case scenario for the output SNR because the pre-distortion signal Y is typically noisy as compared to the input signal X. For example, suppose SNR X  is 100,000 and SNR Y  is 10,000 such that the RF input signal X is 10 times less noisy than the pre-distortion signal Y. However, the output SNR will be 9,091, slightly less than the pre-distortion signal&#39;s SNR Y  because of its SNR dependence discussed above. 
     Because pre-distortion in the RF domain is noisy, linearization using pre-distortion is typically performed in the digital domain. However, digital pre-distortion has its own problems because of the sampling noise introduced by the required conversions of the pre-distortion signal into the digital domain and then back into the analog domain. Moreover, these conversions use large amounts of power (often as much as a low-power power amplifier) and require complex circuitry. Accordingly, there is a need in the art for more robust pre-distortion techniques. 
     SUMMARY 
     Disclosed embodiments exploit the weakly non-linear nature of non-linear circuits such as power amplifiers. In other words, amplifiers are designed to be predominately linear such that a linear portion of an amplifier output signal is more powerful than a non-linear portion of the output signal. Appropriate pre-distortion of an amplifier input signal will thus mirror this imbalance between linearity and non-linearity—the linear portion in the pre-distorted amplifier input signal is more powerful than the non-linear portion. Although the non-linear portion of the pre-distorted input signal is relatively weak, it is inherently noisier than the linear portion. To prevent the domination of the signal-to-noise ratio in the pre-distorted input signal by the noisy non-linear components, the non-linear components in the pre-distorted input signal are formed separately from the linear term such that the pre-distorted input signal is formed by the addition of the non-linear and linear terms. This additive pre-distortion is very advantageous because the signal-to-noise ratio in the pre-distorted input signal is not polluted by the relatively noisy nature of the non-linear terms. 
     To separate the linear and non-linear formation of the pre-distorted input signal, the non-linear signal portion of the pre-distorted input signal may be formed at a mixer from a version of the input signal and a pre-distorting signal. A first coupler may be used to extract the version of the input signal provided to the mixer such that the input signal is divided into a remaining input signal portion and the extracted version. A second coupler may be used to add the non-linear signal portion from the mixer with the remaining input signal portion to form the pre-distorted input signal. A variable gain amplifier may be used to amplify the remaining input signal portion prior to addition with the non-linear signal portion. 
     A method to separately form the linear and non-linear portions of the pre-distorted input signal comprises: multiplying a version of the input signal with a pre-distortion signal to produce a multiplied signal, the pre-distortion signal having no constant component such that the multiplication produces no linear versions of the input signal; adding the multiplied signal to the input signal to provide a pre-distorted input signal; and processing the pre-distorted input signal through the non-linear circuit to provide an output signal that is a substantially linear function of the input signal. 
     The invention will be more fully understood upon consideration of the following detailed description, taken together with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates the non-linear properties of a conventional power amplifier. 
         FIG. 2  illustrates an RF signal processing circuit that pre-distorts an input signal to a non-linear circuit component such as an amplifier such that the non-linear circuit produces an output signal that is linearly related to the input signal. 
         FIG. 3  illustrates an improvement of the RF signal processing circuit of  FIG. 2  wherein the non-linear portion of the pre-distorted RF input signal provided to the amplifier is produced in a different path as compared to the production of the linear portion of the pre-distorted RF input signal. 
         FIG. 4  illustrates an improvement of the RF signal processing circuit of  FIG. 3  such that no mixers are required in the signal path that produces the linear term in the pre-distorted RF input signal. 
         FIG. 5  illustrates a further improvement to the RF signal processing (RFSP) circuit of  FIG. 3  such that no signal modification need be performed on the RF input signal to form the linear term in the pre-distorted RF input signal. 
         FIG. 6  is a simplified representation of the RFSP of  FIG. 5 . 
         FIG. 7  illustrates an amplifier serial cascade that may be linearized by the RFSP of  FIG. 5 . 
     
    
    
     Embodiments of the present invention and their advantages are best understood by referring to the detailed description that follows. It should be appreciated that like reference numerals are used to identify like elements illustrated in one or more of the figures. 
     DETAILED DESCRIPTION 
     Reference will now be made in detail to one or more embodiments of the invention. While the invention will be described with respect to these embodiments, it should be understood that the invention is not limited to any particular embodiment. On the contrary, the invention includes alternatives, modifications, and equivalents as may come within the spirit and scope of the appended claims. Furthermore, in the following description, numerous specific details are set forth to provide a thorough understanding of the invention. The invention may be practiced without some or all of these specific details. In other instances, well-known structures and principles of operation have not been described in detail to avoid obscuring the invention. 
     The following pre-distortion linearization techniques exploit certain characteristics of the pre-distorted input signal that is processed through a non-linear circuit such as an amplifier. As discussed above, the pre-distorted input signal represents an inversion of the non-linearity introduced by the non-linear circuit. Examination of  FIG. 1  shows that the linear component in the desired pre-distorted RF input signal will dominate over the non-linear components when the non-linearity is merely moderately non-linear. Turning now to  FIG. 3 , this domination of the linear component may be exploited by providing parallel paths for the I and Q signal from QPS  245 . A pair of I and Q mixers plus a combiner analogous to mixers  250  and  255  and combiner  260  discussed with regard to  FIG. 2  are provided in each path. A first pair of mixers and a corresponding combiner  300  mix only the I and Q components for the DC coefficient in pre-distorted signal  236  (generated as discussed with regard to FIG.  2 —for illustration clarity, the circuitry to generate signal  236  is not shown in  FIG. 3 ) with I and Q components of the RF input signal. The output from mixers and combiner  300  is thus the linear term in the pre-distorted signal (a scaled and phase-shifted version of an RF input signal  201 ). Similarly, a second pair of mixers and a corresponding combiner  305  mix the remaining I (Re) and Q (Im) components for the pre-distortion signal with corresponding I and Q components of the RF input signal. Thus, the second pair of mixers  305  do not mix any DC portion of pre-distortion signal  236  with the I and Q components such that an output signal from mixers and combiner  305  contains only non-linear versions (powers) of the complex envelope R(t). The output signals from the respective combiners  300  and  305  are combined in a combiner  310  to produce pre-distorted RF input signal  265 . For illustration clarity, related components discussed with regard to  FIG. 2  such as signal generator  235  are not shown in  FIG. 3 . Note the improvement with regard to FIG.  2 —the production of the noisy (and low-power) non-linear components α 2 *R(t) 2 +α 3 *R(t) 3 +α 4 *R(t) 4 + . . . in the envelope for pre-distorted RF input signal  265  is decoupled from the production of the relatively less noisy (and higher power) linear term α 1 *R(t). As will be explained further herein, this decoupling provides a significant improvement in the signal-to-noise-ratio (SNR) for pre-distorted RF input signal  265  and hence for the SNR in an output signal resulting from the amplification of pre-distorted RF input signal  265 . It will be appreciated that any real-world signal generator such as signal generator  235  of  FIG. 2  cannot produce an infinite series of powers of the complex envelope. In other words, the series must end at some finite power. For example, it is believed that generating a series ending at the sixth power of R(t) (such that the output of combiner  305  may include a seventh power of R(t)) is sufficient to substantially linearize a power amplifier such as those used in cellular base stations. It will be appreciated that is some embodiments, a linear component may be present in pre-distortion signal  236  in that some signal-to-noise ratio (SNR) improvement will be realized in the downstream amplifier so long as the DC component of pre-distortion signal  236  is of lower power than the power in the remaining portion of the signal. 
     Turning now to  FIG. 4 , dramatic improvements in dynamic range may also be achieved through an appropriate decoupling in the production of the linear and non-linear terms in the pre-distorted RF input signal in an RF signal processing (RFSP) circuit  400 . In circuit  400 , the linear term in a pre-distorted RF input signal  405  is produced by an appropriate attenuation (or amplification) of the RF input signal in a variable amplifier  410 . Because there is no I/Q channel formation with regard to the linear term, it may be seen that this linear term cannot be phased according to the complex value of the corresponding coefficient α 1  (assuming that the signal generator, which is not shown for illustration clarity but corresponds to generator  235  of  FIG. 1 , determines that α 1  should be complex). The remaining non-linear terms in the pre-distorted RF input signal are generated analogously as discussed with regard to  FIG. 3  using a buffer  240 , QPS  245 , and I/Q mixers and combiner circuit  410 . But note that the I portion (real portion) of the pre-distortion signal that mixes with the corresponding I portion of the RF input signal and also the Q portion of the pre-distortion signal that mixes with the corresponding Q portion of the RF input signal are not necessarily the same as discussed with regard to  FIG. 3 . This is because the I and Q portions of the pre-distortion signal may need to be re-phased since the linear portion of the pre-distorted RF input signal has been produced without an I/Q mixing. In other words, the phase relationship between α 1 *R(t) and the remaining non-linear terms may change because the linear envelope term is no longer re-phased as it would be if it were multiplied by a complex coefficient α 1  in an I/Q fashion. To account for this phase relationship loss, the I and Q portions of the pre-distortion signal may be re-phased according to coefficients that may differ from those discussed with regard to  FIG. 3 . Thus, the in-phase portion I and the quadrature-phase portion Q of the pre-distortion signal are denoted as the real and imaginary parts of [α 2 ′*R(t) 2 α 3 ′*R(t) 3 +α 4 ′*R(t) 4 + . . . ], respectively, where the alpha coefficients are given a prime signal to signify that these coefficients may differ from those discussed earlier. Note the advantages of RFSP  400  over the corresponding RFSP discussed with regard to  FIG. 3 . While RFSP  400  enjoys the same SNR decoupling from the noisy non-linear term formation, it also has a much wider dynamic range because the I and Q signals from QPS  245  of  FIG. 3  must be mixed to produce both the linear term and the non-linear terms in the resulting pre-distorted RF input signal. The linear term is considerably larger such that the dynamic range in the mixers in circuit  305  will be wasted. In contrast, the I and Q signals from QPS  245  in RFSP  400  are mixed to just provide the non-linear terms in the pre-distorted RF input signal. The mixers in circuit  410  may then be configured to use their full dynamic ranges whereas the mixers in circuit  305  cannot be so configured. 
     In addition to these dynamic range improvements, RFSP  400  is also less costly to build because a set of mixers has been eliminated. The linear signal integrity has also been improved due to the signal no longer being passed through the QPS, which can be quite noisy and lossy. Indeed, further circuit simplification may be achieved as seen with regard to an RFSP  500  of  FIG. 5 . In RFSP  500 , the linear envelope term in the pre-distorted RF input signal  265  is not attenuated or amplified according to any coefficients. As discussed with regard to  FIG. 4 , the corresponding coefficients in the pre-distortion signal must then be altered from those discussed with regard to  FIG. 3 . For example, analysis of the RF output signal from a circuit such as amplifier  205  of  FIG. 2  by a signal generator may indicate that a non-linearity may be cured by pre-distorting the complex envelope R(t) of the RF input signal to the amplifier such that the complex envelope becomes [α 1 *R(t)+α 2 *R(t) 2 +α 3 *R(t) 3 +α 4 *R(t) 4 + . . . ] as discussed previously. In such a pre-distorted signal, the linear term has a certain phase relationship to the quadratic term, a certain phase relationship to the cubed term, and so on. If the linear term is then changed to just R(t), it may be seen that these phase relationships are disturbed. But the remaining coefficients may adjust their phase such that the overall phase difference between the linear term and the remaining non-linear terms is maintained the same as in the original pre-distorted RF input signal. Because such a preservation of phase relationships may require different coefficients than those discussed with regard to  FIG. 4  (unless the variable amplifier of  FIG. 4  was implementing a unity gain and had zero delay), the in-phase and quadrature portions of the pre-distortion signal may be designated as the real and imaginary parts, respectively, of [α 2 ″*R(t) 2 +α 3 ″*R(t) 3 +α 4 ″*R(t) 4 + . . . ], where the double prime signal for the alpha coefficients is used to signify that these coefficients may differ from those discussed earlier. RFSP  500  includes a coupler  505  to provide a version of the RF input signal to buffer  240  and QPS  245 . In this embodiment, buffer  240  and QPS  245  are configured to process double-ended signals (for illustration clarity, these components are illustrated in a single-ended configuration) such that a transformer  510  may be used to transform the RF input signal into a differential (double-ended) signal. A circuit  515  functions analogously to circuit  410  of  FIG. 4  to mix the I and Q portions of the RF input signal version obtained from transformer  510  with the corresponding in-phase and quadrature portions of the pre-distortion signal (for illustration clarity, circuit  515  is also illustrated in a single-ended configuration). The resulting pre-distorted signal is transformed back into a single-ended form in a transformer  520  so it can couple with the RF input signal at a coupler  525  to form pre-distorted RF input signal  265 . 
     Regardless of whether an RFSP enjoys the simplicity and enhanced dynamic range discussed with regard to  FIGS. 4 and 5 , so long as the formation of the non-linear terms in the pre-distorted RF input signal is decoupled from the formation of the corresponding linear term, the resulting RFSP will advantageously not suffer from a reduction in SNR due to noise from the non-linear term formation as discussed with regard to  FIG. 1 . 
     Turning now to  FIG. 6 , a simplified representation of RFSP  500  is illustrated. However, the ensuing SNR analysis will apply to any implementation in which the non-linear term formation is decoupled from the linear term formation. The RF input signal is represented by signal X having an SNR designated as SNR X . Similarly, the powers of the RF input signal&#39;s envelope R(t) are represented by signal Y having an SNR designated as SNR Y . The coefficients that scale signal Y (representing the scaling of the powers of R(t)) are represented by a scaling factor α. Couplers  505  and  525  each introduce an attenuation modeled by a coefficient β. Thus, a version of the input signal X is scaled by an attenuation β, multiplied with the pre-distortion signal αY in a multiplier  600  and coupled back with the input signal X to form a pre-distorted signal  605  that equals X (minus the βX extracted by coupler  505 )+αβ 2 XY. It can be shown that an output SNR (SNR out ) for pre-distorted signal  605  equals 
               SNR   out     =       1             (     αβ   2     )     2     ⁢     Y   2         SNR   Y       +         (       αβ   2     ⁢   Y     )     2       SNR   X       +     1     SNR   X           +       1       1     SNR   Y       +     1     SNR   X       +     1       SNR   X     ⋆         (     αβ   2     )     2     ⁢     Y   2               .             
It may thus be observed that the second term in this expression will be dominated by the noise in the Y signal such as is the case for the output signal for RFSP  200  of  FIG. 2 . However, the entire second term is dominated by the first term where the effect of SNR Y  is tempered by the (αβ 2 ) 2 Y 2  factor. Thus, the output SNR is substantially equal to SNR X  rather than being dominated by SNR Y  (assuming that SNR Y  is lower than SNR X  and further assuming that the predistortion term, (αβ 2 ) 2 Y 2 , that is added is lower power than the original linear team, as is the case for a weakly non-linear system).
 
     Referring again to  FIG. 5 , it may be seen that RFSP  500  offers numerous advantages over RFSP  200  of  FIG. 2 . For example, consider the serial cascade of amplifiers shown in  FIG. 7 , which is representative of a typical amplifier train for power amplification, for example, in cellular base stations. A digital-to-analog converter  700  receives a digital signal so as to produce an analog RF input signal. This RF input signal is then amplified across the serial cascade of amplifiers such that each amplifier in the cascade produces a successively more powerful output signal. For example, a first amplifier  705  produces a −18 dBm output signal, a second amplifier  710  amplifies this output signal into a −5 dbm output signal, and so on until a final amplifier  715  produces a 47 dBm output signal that may be transmitted through an antenna  720 . Should RFSP  200  of  FIG. 2  form an integrated circuit produced using conventional CMOS processes, the maximum signal power that may be processed may be no higher than −5 dBm, depending upon the particular CMOS process used. Thus, RFSP  200  could be inserted in this amplifier cascade (or chain) no higher than an output terminal of amplifier  705 . However, because RFSP  500  perfoims its pre-distortion in an additive fashion (rather than a multiplicative fashion as for RFSP  200 ) and because the system need only inject a relatively small pre-distortion signal due to the system being weakly nonlinear, RFSP may inject its pre-distortion signal in multiple locations in the cascade of  FIG. 7 . For example, if it is assumed that coupler  505  introduces −20 dBm of loss and RFSP  500  can process no more than a −5 dBm signal, RFSP  500  could be inserted after any amplifier whose output does not exceed 15 dbM (amplifier  710  or  705 ). 
     It will be obvious to those skilled in the art that various changes and modifications may be made without departing from this invention in its broader aspects. For example, although the linearization techniques and circuits discussed above have used an amplifier as the non-linear circuit to linearized, it will be appreciated that the resulting linearization advantages may be enjoyed by any non-linear circuit one wishes to linearize. For example, mixers and phase-shifters may be linearized by the techniques and circuits disclosed herein. The appended claims encompass all such changes and modifications as fall within the true spirit and scope of this invention.