METHOD AND APPARATUS FOR REDUCING INTERMODULAΗON

DISTORTION IN DIGITAL WIDEBAND

TRANSMISSION SYSTEMS

BACKGROUND OF THE INVENTION

The present invention relates to the field of signal processors and more specifically to methods and apparatus for minimizing the distortion caused by non-linear elements, such as amplifiers, in wireless communica¬ tions systems.

Distortion

Non-linear elements such as amplifiers in communications systems inherently provide distorted output signals that are not simply linear functions of the input signals. Distortion can be reduced by separating wanted in-band signal components from unwanted out-of-band signal components. Such separation is particularly effective for narrow band signals where filters can be used to remove unwanted signal components. For a communications system that operates only on a single (in-band) channel, the out-of-band signal components generated by an amplifier can be easily removed by filtering or can be ignored if they are non- interfering. However, in commu¬ nications systems that use signals that extend over a broad band of frequen¬ cies (broadband signals), the separation of in-band and out-of-band signals is more difficult. In communications systems that operate with a number of adjacent channels in adjacent frequency bands, the out-of-band components for one channel frequently exist in the same frequency spectrum as (and interfere with) the in-band signals of adjacent channels.

Cellular Systems

Present day cellular mobile telephone systems operate with many adjacent frequency channels and operate in an environment where elimination of signal distortion is important. Cellular mobile telephone systems "reuse" frequency within groups of cells to provide wireless two-way radio frequency (RF) communication to large numbers of users. Each cell covers one geographic area and collectively groups of adjacent cells can cover a larger geographic region. Each cell is allocated a fraction of the total amount of RF spectrum for cellular users located in the cell.

In cellular systems, typically each cell has a base station with radio frequency (RF) transmitters and RF receivers co-sited for transmitting and receiving communications with cellular users. The base station employs forward RF frequency bands (carriers) to transmit forward channel communi¬ cations to users and employs reverse RF carriers to receive reverse channel communications from users in the cell. Conventional forward channel communications employ fixed power, at fixed frequencies and have fixed sectors if sectorized antennas are used.

The forward and reverse channel communications use separate frequency bands to enable simultaneous transmissions in both directions using frequency domain duplex (FDD) signaling. Time domain duplex (TDD) signaling, in which the forward and reverse channels take turns using the same frequency band, is possible.

In cellular systems, particular channels are allocated to individual users. 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 allocated channels.

"Out-of-band" signals produced by intermodulation signal components from one channel can cause unacceptable interference with communications in other channels. Intermodulation products are produced through the interac-

tion of two or more wanted signals in non-linear system elements. For example, in a multiband 40-channel communications system, thousands of intermodulation products may exist. Cellular communications systems, therefore, place stringent restrictions on out-of-band signal emissions in order to minimize channel-to-channel interference.

Distortion Reduction

In order to reduce the out-of-band signal emissions from one channel into another, many different methods have been proposed. As one example, feedback systems have been proposed in which a portion of the amplifier output signal is processed and fed back to alter the input signal to the ampli- fier. Feedback systems, however, are expensive and have not achieved entirely satisfactory results. As another example, predistorting systems have been proposed to predistort the input signal to an amplifier with a predistortion transformation which is complementary to the distorting trans¬ formation of the amplifier. The predistorted input signal is intended to produce an undistorted output signal that is a linear function of the input signal, su Known predistortion techniques, however, have primarily operated on the wanted signal components and have not satisfactorily processed unwanted intermodulation signal components.

One prior art predistortion method is described in U.S. Patent 4,462,001 entitled BASEBAND UNEARIZER FOR WIDEBAND, HIGH

POWER, NONLINEAR AMPLIFIERS. In that patent, separate look-up tables containing amplitude and phase correction factors are provided contain¬ ing a multiplicity of entries which define predistortion transformation param¬ eters appropriate for use with a corresponding multiplicity of different input signals. In operation, the fluctuating power level of the input signal to be amplified is continuously measured and used to address the look-up tables to

obtain corresponding predistortion parameters to predistort the input signal before it is input to the amplifier.

Another prior art method is described in U.S. Patent 4,700, 151 entiϋed MODULAΗON SYSTEM CJAPJABLE OF IMPROVING A TRANS¬ MISSION SYSTEM. In that patent, the real and imaginary quadrature components of the input signal sample are used to index a look-up table containing more than a million entries of predistortion transformation param¬ eters. The look-up table entries are adaptively changed in response to varia¬ tions in the amplifier's distorting characteristics. If the channel is changed, as is common in cellular systems, every entry in the look-up table must be updated. This updating process is burdensome and often takes a longer time than is available. This technique is used to modify the wanted signal rather than the intermodulation components.

Still another prior art method is described in U.S. Patent 5,049,832 entitled AMPLIFIER LINEARIZATION BY ADAPTIVE PREDISTORTION. In that patent, stored table entries in rectangular coordi¬ nate format are provided to enable the subsequent predistortion operation to be performed more simply than in the 4,700,151 patent with a smaller look-up table. Again, this technique is used to modify the wanted signal, not the intermodulation components. The known methods, like those discussed above, primarily address the wanted signals, not the intermodulation components, so that the results obtained are not entirely satisfactory and particularly are not satisfactory for broadband systems having many adjacent frequency channels. In broad¬ band systems, even though the adverse effects of intermodulation distortion are known, amplifiers have not been able to adequately overcome the intermodulation distortion problems. To date, even the best amplifiers achieve only 20 to 40% efficiency for broadband applications. This low efficiency is in large part due to the third order intermodulation products.

The intermodulation products are described, for example, in the article entitled "Intermodulation Distortion in a Multi-Signal Environment, " by Michael Leffel, RF Design, pages 78-84, June 1995, which describes the intermodulation caused by multiple input tones. The article predicts the behavior of amplifiers having multiple input tones based upon two-tone intermodulation parameters.

Although predistortion has been proposed to improve the operation of amplifiers, in actual practice, there is much need for improved predistortion methods and apparatus for broadband applications such as cellular systems. Although intermodulation distortion products have been identified as the cause of problems, the known signal processors have not been satisfactory for reducing those problems in broadband systems.

In accordance with the above background, there is a need for im¬ proved signal processors for use in broadband communications systems.

SUMMARY OF THE INVENTION The present invention is an apparatus and method for signal process¬ ing to remove unwanted signal components and particularly for signal processing with non-linear elements such as RF amplifiers in broadband wireless communications systems that exhibit unwanted intermodulation products. The signal processor operates to process an information signal to form a processed output signal, V^. The signal processor includes a signal preprocessor that in turn includes a base modulator for modulating the information signal to form base modulated signals, a band modulator for modulating the base modulated signals to form an input signal, Vj, a distortion processor for providing a predistortion signal V _pd having selected predistortion terms of the Taylor series form [k,(V,)', k ₂ (V,) ² , k ₃ (V;) ³ , ...], and a broadband processor for combining the input signal Vj and the

predistortion signal V _pd to form a predistorted signal V _)d with the selected predistortion terms of the Taylor series form [k^V,) ¹ , k ₂ (V _; ) ² , k ₃ (Vi) ³ , ...] having reduced amplitude. The signal processor additionally includes a non-linear element having a non-linear transfer function for processing the processed predistorted signal V _id to form the output signal with the selected predistortion terms of the Taylor series form [k^V,) ¹ , k ₂ (V _; ) ² , k ₃ (V _; ) ³ , ...] having reduced amplitude.

This invention reduces the unwanted emissions caused by intermodulati on distortion that have been a major limitation in linear and wide band transmission systems. .Amplifiers using the present invention achieve substantial improvements in efficiency, an increase of 40-80% of usable power, due to the reduction of intermodulation products. In particu¬ lar, the third order products are reduced to improve the performance of transmission systems. This invention can be used to eliminate any selected intermodulation distortion desired. The present invention can operate with or without feedback and greatly simplifies the design and cost of wideband power amplifiers.

The foregoing and other objects, features and advantages of the invention will be apparent from the following detailed description in conjunc¬ tion with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a signal processor suitable for use in wireless commu¬ nications systems.

FIGs. 2a, 2b, 2c and 2d depict signals in a GSM system.

FIG. 3 depicts a block diagram of the modulator of the FIG. 1 circuitry.

FIG. 4 depicts a block diagram of the distortion processor of the FIG. 1 circuitry.

FIG. 5 depicts a block diagram of one particular distortion processor of FIG. 4 for a two carrier example. FIG. 6 depicts a graph representing the power levels of output waveforms of the FIG. 1 circuitry both without predistortion processing and with predistortion processing.

DETAILED DESCRIPTION

Signal Processor - FIG. 1 In FIG. 1, a signal processor 8 is provided suitable for use in wireless communications systems. The signal processor 8 includes a signal preprocessor 7, formed of a modulator 10 and a broadband processor 11 , and includes a non¬ linear element 12. Element 12 is typically a power amplifier. The modulator 10 receives one or more input information signals V(t) including information signals V,(t), V ₂ (t), ..., V _j (t), ...,Vj(t), receives one or more carriers ω(t) including carriers ω,(t), ω ₂ (t), ..., G) _j (t), ... , ω ₃ (t) and receives one or more sealers "a" including sealers a,, a^ ..., a., ..., a. _} . The modulator 10 includes a base modulator unit 21 for baseband modulation of the V(t) information signals to provide baseband modulated signals m(t), a band modulation unit 22 for modulation of the ω(t) carrier bands by the baseband signals m(t) to form modulated signals V„ and a distortion processor 13 that generates a predistortion signal, V _pd .

The output from the base modulator unit 21 is one or more modulated signals m(t) including modulated signals m,(t), m ₂ (t), ..., m t), ..., m^t) that each occupy a portion of a broadband spectrum. The distortion processor 13 in modulator 10 receives one or more of the modulated signals m(t), one or more of the carriers, ω(t), and one or more of the sealers, "a," to form the distortion signals, V _pd .

In FIG. 1 , broadband processor 11 operates to combine the modulated signals V, and the distortion signals, V^, to form a broadband signal, V _ld . The Vy output from the broadband processor 11 is the input signal to the non-linear element 12. Non-linear element 12, typically an RF power amplifier, has a non- linear transfer function, T, _* , which operates on the input signal, V,,,, to produce the broadband output signal, V ₀ . The distortion processor 13 of FIG. 1 operates to insert a pre-distortion signal, V _pd , into the broadband input signal, V _ld , as an input to the non-linear element 12 so that the output signal, V ₀ , from the non¬ linear element 12 has unwanted signal components minimized or eliminated. In particular, unwanted intermodulation distortion products are eliminated.

In FIG. 1, the broadband processor 11 typically includes a frequency division multiplexer 14 and a digital-to-analog (D/A) converter 15 and may include other conventional elements such as frequency shifters. The frequency division multiplexer 14 operates to multiplex each of the corresponding components of the Vj and V _pd signals into corresponding channels. Typically, the broadband signal has channels corresponding to the carriers ω,(t), ω ₂ (t), ... , ω/t). The frequency division multiplexer 14, in one embodiment, is an adder that concurrently adds each of the component signals of V ₍ and V _pd . The multiplexed broadband signal from multiplexer 14 is D/A converted in converter 15 to provide the digital broadband signal V _ld .

GSM Broadband Signals - FIG. 2

In FIG. 2a, the carrier frequencies ω,(t), ω ₂ (t), ..., ω _; (t) of FIG. 1 are the center frequencies for the channels CH ₀ , CH,, ... , CH _C , ... , CH _C . In a GSM embodiment, the channels are each 200KHz in bandwidth as shown for typical channel CH _C in FIG. 2b. In FIG. 2b, the channel CH,, typical of each of the channels, has a low frequency ω _cl and a high frequency ω _ch . Typically, ω _cl is 67.7KHz below the center frequency ω _c and ω _ch is 67.7KHz above the center frequency ω _c . For a GSM system, the information signal is in digital form and has logical 1 's and logical O's. The logical 1 's and logical O's of the information signal frequency shift key (FSK) modulate the carrier frequency of FIG. 2b such that data values of logical 1 cause the carrier ω _ch to be on and the carrier ω _cl to be off. Similarly, for data values of logical 0, the modulation causes the carrier ω _cl to be on and the carrier ω _ch to be off. Such modulation causes the distribution of the baseband data signal of the modulated signal ni _j (t) for the channel with center frequency carrier ω _c to be within the 200KHz. While FSK modulation is typical in GSM systems, other types of baseband modulation can be employed. For example, Δ modulation or FM modulation are other typical baseband modulations. In FIG. 2b, FM modulation is represented if ω _cl is the frequency for one cosine function and ω _ch is a frequency for another cosine function. In GSM systems, each of the frequency channels of FIG. 2b is typically further divided into eight additional subchannels by time division multiplexing as shown in FIG. 2c. In FIG. 2c, the eight time slots TS ₀₎ TS,, ... , TS ₇ are shown for a typical one of the channels CH _C of FIG. 2b. Each of the time slots of FIG. 2c in a GSM system is defined to include 156.25 bits spread over a duration of 577 x 10 ⁶ seconds. These bits in each time slot are selected as a logical 1 or a logical 0 for transmitting information including data and control information.

Irrespective of the different forms of modulation which are employed in any system, the resultant baseband modulation from the base modulator unit 21 of FIG. 1 is typically spread over a particular bandwidth for a particular channel.

Modulator - FIG. 3 In FIG. 3, further details of the modulator 10 of FIG. 1 are shown. The base modulator unit 21 of FIG. 1 includes the base modulators 21-1, ..., 21-J which in turn receive the information signals V,(t), ..., V _j (t) and provide the baseband modulated output signals m(t), specifically the modulated baseband signals m,(t), ..., m,(t). In FIG. 3, the band modulator unit 22 of FIG. 1 includes the band modulators 22-1, ..., 22-J. The modulators 22-1, ..., 22-J receive the baseband modulated signals m,(t), ..., mj(t), the carrier frequencies ω,(t), ... , ω,(t), and the sealers a,, ..., a _j5 respectively, and produce the carrier frequency outputs, Vj, which include the carrier outputs V _i{1) , ..., V _(l)(J) , respectively. In FIG. 3, the distortion processor 13 receives the baseband modulated signals πiι(t), ... , m/t), receives the carrier frequencies ω,(t), ... , ω/t), and receives the sealers a,, ..., a _} and produces the predistortion output signal, V _pd , including the predistortion terms V _pd(1) , ...,V _pd(5) , ... , V^^, where S is some integer.

Distortion Processor- FIG. 4

In FIG. 4, further details of the distortion processor 13 of FIG. 1 and FIG. 3 are shown. The distortion processor receives the baseband signals m(t) and the carrier frequencies ω(t) as inputs to a term generator 41. The term generator 41 generates various ones of the terms to form the predistortion signal. The terms from term generator 41 are input to a phase shifter 42 which properly aligns the phase of the terms so that they will be out of phase by 180° from the

phase of corresponding terms in the V _; signal input to the broadband processor of FIG. 11. The phase shifter 42 provides the phase shifted signals to the periodic signal generator 43 which, for example, is a cosine generator that generates the cosine of the outputs from the phase shifter 42. The periodic function signals are then scaled in sealer 44 and are provided as the components of the predistortion output signal V _pd . While the phase shifter 42 is shown in

FIG. 4 as providing a phase shift angle to the signal input to the periodic signal generator 43, the same result can be achieved by changing the sign of the output after the cosine generator.

Two Carrier Distortion Processor - FIG. 5 In FIG. 5, further details of the distortion processor of FIG. 4 are shown for a two-carrier embodiment. The term generator 41 receives the carrier terms ω(t) and specifically, for the two carrier example, receives the ωj(t) and ω ₂ (t) carriers. Similarly, the term generator 41 receives the baseband modulated information signals m(t) and, for the two-carrier example, receives the information modulated baseband signals m,(t) and m ₂ (t). The multipliers 45-1 and 45-2 generate the one times, two times, and three times signals from the ω,(t) and m,(t) signals. The multipliers 46-1 and 46-2 generate the one times, two times, and three times signals from the α> ₂ (t) and m ₂ (t) signals.

In FIG. 5, the adders 47-1 , 47-2 and 47-3 add the one time signals ω,(t) and m,(t) to form add the two times signals to form 2ω, (t)+2m, (t), and add the three times signals to form 3(0,(1) +3m,(t). Similarly, the adders 48-1, 48-2 and 48-3 add the one, two and three times signals to form lω ₂ (t)+m ₂ (t), 2ω ₂ (t)-l-m ₂ (t), 3ω ₂ (t)+3m ₂ (t). The adders 49-1 , 49-2 have complementary inputs and provide the difference signals [2ω ₁ (t)+2m,(t)]- [lω ₂ (t)+ lm ₂ (t)] and [3ω ₁ (t)+3m ₁ (t)]-[2ω ₂ (t)+2m ₂ (t)]. Similarly, the adders

49-3 and 49-4 have complementary inputs and provide the difference signals [2ω ₂ (t)+2m ₂ (t)]-[lω _l (t) + lm ₁ (t)] and [3 _W2 (t)+3m ₂ (t)]-[2 _Wl (t)+2 _mi (t)],

respectively. The difference signals from the adders 49-1 , 49-2, 49-3 and 49-4 are input to the adders 52-1, 52-2, 52-3 and 52-4, respectively. The adders 52-1, 52-2, 52-3 and 52-4 receive inputs φ,, φ ₂ , φ ₃ , φ ₃ from the phase shifters 51-1 , 51-2, 51-3 and 51-4, respectively, to provide the phase shifted difference signals to the cosine units 43-1, 43-2, 43-3 and 43-4, respectively. The cosine signals from the cosine units 43-1, 43-2, 43-3 and 43-4 are as follows:

{[2ω ₁ (t)+2m ₁ (t)]-[lω ₂ (t) + lm ₂ (t)3+φ ₁ } {[2ω ₂ (t) +2m ₂ (t)]-[l ω, (t) + 1 m,(t)] + φ ₂ } {[3ω ₁ (t)+3m ₁ (t)]-[2ω ₂ (t)+2m ₂ (t)]-r-φ ₃ } {[3ω ₂ (t) -3m ₂ (t)]-[2ω,(t)+2m ₁ (t)]+φ ₄ } . The cosine signals from the cosine units 43-1 , 43-2, 43-3 and 43-4 are input to the multipliers 54-1 , 54-2, 54-3 and 54-4, respectively, which are multiplied by scaled values S _f(1) , S _f(2) , S _f(3) and S _f(4) determined by the a, and a ₂ sealers input to the lookup tables 53-1, 53-2, 53-3 and 53-4, respectively. The output from the multipliers 54-1, 54-2, 54-3 and 54-4 are the V _p ,, signals V _pdl (t), _p .β ), V _pd3 (t) and V _pd4 (t), respectively, having the scaled harmonic terms that are to be inserted into the non-linear element 12 of FIG. 1 as follows:

V _pdl (t) = S _f()) cos{[2ω _I (t)+2m ₁ (t)]-[lω ₂ (t)-r-lm ₂ (t)]+φ ₁ } _p azC = S _f(2) cos{[2ω ₂ (t)+2m ₂ (t)]-[lω ₁ (t)-l- lm ₁ (t)]+φ ₂ }

V _pd3 (t) = S _f(3) cos{[3ω ₁ (t)+3m _I (t)]-[2ω ₂ (t)+2m ₂ (t)]+φ ₃ } V _pd4 (t) = S _f(4) cos{[3ω ₂ (t)+3m ₂ (t)]-[2ω ₁ (t)-l-2m _I (t)]-r-φ ₄ }

Operation

Referring to FIG. 1, modulator 10 receives the input information signals, V _j (t), the carriers, ω^t), and the sealers, z. _t , to produce the modulated input signal, Vj, and a predistortion signal, V _pd . The broadband processor 11 receives the modulated input signal, Vj, and the predistortion signal, V _pd , and forms the broadband output, V _id , as follows:

The non-linear element 12 receives the broadband signal, V _jd , and has a transfer function, T _pa , so that the output signal, V ₀ , is formed as follows:

V ₀ =V T _pa ) Eq. ₍₂ )

Using Eq. (1) in Eq. (2), the output signal, V ₀ , is given as follows:

Vo = V, ^{+ V} _P ^T _P a Eq. (3)

In Eq. (3), the output signal, V ₀ , is a function of both the normal input signal, V„ that is present in the absence of any predistortion processing and the predisdortion signal, V _pd . If the distortion processor 13 is inactivated and does not perform distortion processing, then V _pd in Eq. (3) is equal to zero and therefore Eq. (3) reduces to the normal relationship that exists for any non-linear element without predistortion processing where the distorted output, V^, is given as follows:

Vo, = W _pa ) Eq. (4)

In a typical example, the non-linear element 12 of FIG. 1 is a power amplifier used for wireless transmission and the non-linearities of the power amplifier cause unwanted signal components to be generated in the transmitted signal. Government regulations and operational requirements dictate that such unwanted signal components not be transmitted at power levels that will cause interference. One way to achieve non-interference of unwanted signal

components is to reduce the power output of the power amplifier, but such reduction also reduces the output power of the wanted signal components and reduces, therefore, the efficiency of the amplifier. For broadband signals that have multiple input frequencies, this efficiency problem is particularly severe. Any distorted output signal such as V^ of Eq. (4) can be expressed in terms of the input signal, Vj, using a Taylor series expansion as follows:

^V « = W >' ⁺ W ^{)2 +} W ^{)3 +} W ^{)4 +} W ⁾⁵ »• Eq. (5)

Also, it is known that the distorted output signal, V _^ ,, of a non-linear power amplifier can be represented as a function of the input voltage, V _; , as follows:

^• od G e (VilVtr)

1 ) Eq. (6)

where,

threshold voltage (value useful in connec¬ tion with relationship between V _f and G). (R)J(l _Wa J, the product of the load resistance and the quiescent current which are properties that are dependent upon the physics of the amplifier circuit design.

When Eq. (6) is used to represent the power amplifier, the Taylor series expansion of Eq. (5) has the values of k as follows:

k _χ GIV _tr

* ₂ =σ/(4v;>

In general the input signal, V„ is formed as the sum of any number of carrier terms, C. where if N is the number of carrier terms, V, is given as follows:

Each of the carrier terms C. is represented by a modulated periodic function, for example, a cosine function, as follows:

C = a _} cos [ ω _; (f)+m r) ] Eq. (9)

where: a. = carrier amplitude ω _; (t) = carrier frequency m.(t) = carrier modulation

When the input signal, V„ includes terms as represented by Eq. (9), the distorted output signal, V^, of Eqs. (4), (5) and (6) includes many unwanted signal components. For example, for any two carriers C, and C _J+1 (including the

carrier frequencies ω-(t) and ω _j+ ,(t), the distorted output includes not only ω-(t) and ω _j+1 (t) but also components uft(t) ± v _1+l αμ, (t) for each of q and v _)+l equal to all positive integers. More generally, for N carriers, the distorted output includes products of the terms «,ω.(t), « ₂ ω ₂ (t), ..., 1^(1), w _j+] ω _j+1 , ..., w _N ω _N for each of the values u u ₂ , ... , u ₃ , w _J+ ,, ... , u _N equal to all positive integers. Clearly, the number of terms that are possible, even for a simple two carrier example, is computationally large. To form V^ with full precision would be extraordinarily complex and can be represented as follows:

Vpd = ^["ι ^ω P u ₂ ω ₂ , « _y ω, ^w r«°V ι » ■WN] Eq. (10)

Most of the terms in Eq. (10) need not be selected in formation of V _pd .

The terms in Eq. (10) are partitioned into groups including the selected group of terms, H, ₀₎ , that are utilized in forming the predistortion value V _p ,, and the non-selected group, H,^,, not utilized in forming the predistortion value V _pd so that V _pd as used for predistortion is given as follows:

^V _P -i = ^H 4 _f t Eq. (11)

The selection of terms for H, ₀₎ is based on recognition that some of the components in Eq. (10) are more significant than others. For example, the third harmonic components 2ωj(t) and 2ω _J+1 (t) are particularly significant. The third harmonic components 2ω _; (t) and 2ω _j+I (t) which appear in the form 2ω(t) - ω _j+1 (t) and 2(t)ω _J+1 - ω,(t) for ; equal 1 , 2, ..., N where N is equal to the number of carriers in the input signal so that H _sΘ is given as follows:

H *.V) [2ω ₂ (r) - ω,(/)] + [2ω.(/) - ω,(f)] +

[2ω _Λ/ (t) - ω _w _,( ]

While the third harmonic components of Eq. (12) are particularly significant, the present invention can be used to remove these or any other unwanted components that cause unwanted intermodulation distortion. The unwanted componets to be removed are selected as part of Η _>(j) in the formation of V _pd . The formation and selection of V _pd is considered in connection with some examples.

For a two carrier example, N = 2, Eq. (8) becomes:

2

V, = ∑ α »s(ω (f)+m (ø) Eq. (13)

7 = 1

Eq. (13) is expanded as follows:

V _t = + α ₂ cos[ω ₂ (t) + m ₂ (ή] Eq. (14)

Using the input signal, V„ of Eq. (14) and using an expansion to the third and fifth harmonics, the Taylor series expansion for the output signal, V _^ ,, in the Eq. (5) format is given as Eq.(15) as follows:

^V a* '

^a \ ^k 2 a ₂ k ₂ 3fl,* ₄ 3 , a ₂ k ₄ 3a ₂ k ₄

2 2 8 2 8

, _r , _rt , _Λ1 3α ^3; ₃ Co ^j [« (/) + ω,(/)]

₃ 2

15α, a ₂ k _s Cos [m _χ (ή + ω,(/)] 15o, a ₂ k ₅ Cos [m _χ (t) + ω,( ]

4 8

+ 2ω,(/)] [2m, (r) + 2ω,(/)]

3 _x a ₂ k _Λ Cos [2m _χ (ή + 2ω,( ] a^hfos [3/w.(/) + 3ω,( ]

2 ⁺ 4 ⁺

5a _x k ₅ Cos [3/M,( + 3ω _j (/)3 5a _x a ₂ k ₅ Cos [3m,( + 3ω,( ]

_ _

a _x a ₂ k ₅ Cos [2m,(r) - 3w ₂ ( + 2ω,(/) - 3ω ₂ (/)] 3a _χ a ₂ k ₃ Cos [m _χ (ή - 2m ₂ (t) + ω,( - 2ω ₂ (]

_ _

[m _x (t) - 2m ₂ (t) + ω,(/) - 2ω ₂ (/)] 5a _χ a ₂ k _s Cos [m _χ (t) - 2m ₂ (t) + ω _χ (t) - 2ω ₂ (ή]

2 2 3 2, a _x a ₂ %Cos [2m _χ (ή - 2m ₂ (t) + 2ω,(/) - 2ω ₂ (/)] 5a _{x 2} k _$ Cos [3 _χ (l) - 2m ₂ (ή + 3ω,( - 2ω ₂ ( J

3a _x a ₂ k ₄ Cos [m _χ (t) - m ₂ (f) + ω,(/) - ω ₂ (/)] aa ₂ k ₂ Cos [m _χ (ή - m ₂ (t) + ω _χ {t) - ω ₂ (/)] +

3a _x a ₂ k _j Cos [2m _χ (t) - m ₂ (t) + 2ω,(/) - ω ₂ (/)]

a _x a ₂ k ₄ Cos [3m(ή - m ₂ (l) + 3ω,(?) - ω ₂ (/)] 5a _x a ₂ k ₅ Cos [4m _χ (ή - m ₂ (l) + 4ω,( - ω ₂ ( ] 2 16

3a.aJ^Cos [m ή ^■ + ω t)] a ₂ k _χ Cos [m ₂ (l) + ω ₂ ( ] + ^{2 3} ϊ- ² — ² — - +

3a ₂ k^Cos [m ₂ (ή + ω ₂ (/)] l5 _x a ₂ k _$ Cos [ ₂ (ή + ω ₂ (/)]

\5a _x ² ₂ k ₅ Cos [m ₂ (ή + ω ₂ (ή] 5a ₂ k^Cos [m ₂ t) + ω ₂ (/)]

4 8

3tf, ³ α ₂ * ₄ C<« [m,( + m ₂ (t) + ω,( + α> ₂ (/)] a _χ aJ ₂ Cos [m _χ (t) + w ₂ (/) + ω _x (t) + ω ₂ ( ] + — ^! ■ — +

3a _χ a ₂ k ₄ Cos [m _x (ή + m ₂ (ή + ω,(Q + ω ₂ (/)j

3a _l a ₂ k _i Cos [2m _χ (l) + m ₂ (t) + 2ω,(/) + ω ₂ ( ]

a ³ a ₂ k ₄ Cos [3m,( + m ₂ (t) + 3ω,( + ω ₂ (/)] 5a _x a ₂ k _$ Cos [4m,(/) + m(t) + 4ω,( + ω ₂ (/)] 2 16 a ₂ k ₂ Cos [2m ₂ (t) + 2ω ₂ ( ] 3a a k ₄ Cos [2m ₂ (t) + 2ω ₂ (/)]

- + _ +

a ₂ kfos [2m ₂ ( + 2ω ₂ ( ] [m _χ (l) + 2m ₂ (t) + ω,(/) + 2ω ₂ (/)]

15α, a ₂ k ₅ Cos \mΛt) + 2m t) + ω.(t) + 2ω,( ] -7,02 'fc _j Co.S' [ifi.(r) + 2m ₂ ( + ω.( + 2ωJή]

a ₂ k ₃ Cos [3m ₂ (l) + 3ω ₂ ( ] 5a _x a ₂ k ₅ Cos [3m ₂ (ή + 3ω ₂ ( ]

_ + _ +

5a ₂ k ₅ Cos [3m ₂ (i') + 3ω ₂ (/)] a _χ a ₂ k ₄ Cos [m _χ (l) + 3m ₂ (t) + ω,(c) + 3ω ₂ (/)]

_ + _

5a _x a ₂ k ₅ Cos [2m,( + 3 ₂ (l) + 2ω.( + 3ω ₂ (/)] a ₂ k ₄ Cos [4m ₂ ( + 4ω ₂ (/)]

_ _ +

5a _{χ 2} k ₅ Cos [m,( + 4w ₂ (/) + ω,(/) + 4ω ₂ ( ] ₂ k _s Cos [5m ₂ (t) + 5ω ₂ ( ] _ _

The distorted output, V^, of Eq. (15) includes the unwanted third harmonic component terms of the following form [m,(t) - 2m ₂ (t) + ω,(t) - 2ω ₂ (t)] and [2m,(t) - m ₂ (t) + 2ω,(t) - ω ₂ (t)]. More particularly, Eq. (15) includes the terms:

[m _x (ή - 2m ₂ (t) + ω,(/) - 2ω ₂ ( ]

+ Exp. (16)

Exp. (17)

The terms of Exp. (16) and Exp. (17) are the selected terms of Eq. (12) used to form the predistortion signal, V _pd , for the third harmonic. In order to be used, however, the Exp. (16) and Exp. (17) terms are scaled. Referring to Eq. (3), the magmtude of the scaling factor required is determined by the transfer function of the power amplifier.

The power amplifier operates on both the input signal V _j and the predistortion signal, V^ and Eq. (3) is rewritten as follows:

^V _o = (V,)(T _po ) ₊ Eq. (18)

The terms of Exp. (16) and Exp. (17) are derived from the output (V _j CT _p and hence must be scaled by l/(T _pa ) in order to form V _pd . Eq. (6) can be rewritten as follows:

^y od _e (Vι/Vtr) Eq. (19)

G

Using Eq. (20) to calculate the scaling factors for the third harmonic terms of Exp. (16) and Eq. (17), assuming V^ is the only input to the non-linear element 12, produces the following:

where V _Q .,, ₃ is given by the sum of the multipliers for the Cos functions of Exp. (16) as follows:

3«!«2*3 I5a _x a ₂ k _s 5a _x a ₂ k ₅

Odl3 Eq. (22)

where V^ is given by the sum of the multipliers for the Cos functions of Exp. (17) as follows:

3fl _j α ₂ & ₃ 5α, a ₂ k^ I5a _x a ₂ k ₅

Od23 Eq. (23)

The scaling factor, S _f , using Eq. (22) and Eq. (23) in Eq. (21) is given as follows:

The predistortion signal, V _pd , determined for the third harmonic terms is given as follows:

^v _pd = V _pdl3 + V _pd23 Eq _{, (} 25 ₎

where V _p jju and V _p ^ are the scaled component signals derived from the third harmonic terms of Exp. (16) and Exp. (17), respectively, given as follows:

pdli (S _f + 2m I) - ω ) - m t) + φ βq. ₍₂ 6 ₎

^V _pd23 = (S _f )(V _od23 )Cos[2ω ₂ (t) + 2m β) - ω,(t) - m,(t) * φ ₂ ] βq. ₍₂ 7)

While the above description explained the derivation of the third harmonic terms and the resulting predistortion signal, V _p „, any unwanted signal components can be eliminated in the same manner.

By way of further explanation, the following PROGRAM LISTING determines the components for the fundamental (1") and for the 3 ^rd and 5 ^th harmonics, produces the predistortion components for the 3 ^rd and 5 ^th harmonics, and produces the outputs for the 3 ^rd , 5 ^th , and 7 harmonics as shown in the graph of FIG. 6.

PROGRAM LISTING COPYRIGHT ©CELLULAR TELCOM LTD. 1996

% This program specifies the operation of a non-linear amplifier and compares the % intermodulation performance of two FM modulated carriers under three conditions: % No predistortion (v*l*) % Predistortion to reduce the 3rd order term (v*2*)

% Predistortion to reduce the 3rd and 5th order terms (v*3*)

%

% Sampling Parameters % frequency = 64e6; numsamps = 2 ^A 16; x = 1 /frequency : 1 /frequency : ((1 /frequency) *(numsamps));

% Transistor model %

V„ = 10; % this is the threshold Voltage only important relative to Vin and G

G = 100; % this is Voutmax = R(load)*I(quiescent)*((exp(Vs/V„)-l) G=R]*Iq

% NOT GAIN this is circuit dependent PhaseShift = -ϊ ; % cancelling terms 180 degrees out of phase kl = G/V _lr ; % Constants for Taylor Expansion of the function k2 = G/(4*V„ ^Λ 2);% Vout = G*(exp(Vs/V, _r ) -1) k3 = G/(6*V _lr ^* 3); k5 = G/(120*V„ ^A 5);

% Plotting Parameters

% plot strt = 6000; plot stop = 14000;

% % frequency parameters frequency_deviation = 30e3; % 30 KHz k = (frequency_deviation/frequency)*2*pi; fl_Hz = 10e6; f2_Hz = 10.4e6; tll Hz = 800; tl2_Hz = 900; t21_Hz = 1000; t22_Hz = 1100; wl = fi_Hz*2*pi; w2 = f2_Hz*2*pi; ti l = tl l_Hz*2*pi: tl2 = tl2_Hz*2*pi;

t21 = t21_Hz*2*pi; t22 = t22_Hz*2*pi;

% Loop Settings % loop stop =V _tr ; loop step = .5; loop cnt = 1;

% Calcutate where to look for power fl center = ceil(fl_Hz*(numsamps/frequency)); fl_3_center = ceiI((2*fl_Hz-f2_Hz)*(numsamps/frequency)); fl_5_center = ceil((3*fl_Hz-2*f2_Hz)*(numsamps/frequency)); fl_7_center = ceil((4*fl_Hz-3*f2_Hz)*(numsamps/frequency)); bw = frequency_deviation*(numsamps/frequency); Samρle ide

% % Initialize result vectors

sig in = zeros(l ,loop_stop/loop_step); sig_in2 = zeros(l ,loop_stop/loop_step); sig_in3 = zeros(l,loop_stop/loop_step); imd3_l = zeros(l,loop_stop/loop_step); imd3 2 = zeros(l,loop_stop/loop_step); imd3_3 = zeros(l,loop_stop/loop_step); sig ol = zeros(l,loop_stop/loop_step); sig_o2 = zeros(l,loop_stop/loop_step); sig_o3 = zeros(l ,loop_stop/loop_step); imd5_l = zeros(l ,loop_stop/looρ_step); imd5_2 = zeros(l ,loop_stop/loop_step); imd5_3 = zeros(l ,looρ_stop/loop_step); imd7_l = zeros(l,loop_stop/loop_steρ); imd7_2 = zeros(l ,loop_stop/loop_steρ); imd7_3 = zeros(l ,loop_stop/loop_step);

%

% Modulate

infol = .5 .* (cos(tl l*x) + cos(tl2*x)); %Information Signal info2 = .5 .* (cos(t21*x) + cos(t22*x)); ml = k*cumsum(infol); % Modulating functions m2 = k*cumsum(info2);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Main Loop

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% for iii = Ioop_step : loop step :loop_stop

% FM Modulated Carriers and Modulating functions % al = iii; Peak input amplitude a2 = iii; cl = al*cos(wl*x + ml); % Modulated Carriers c2 = a2*cos(w2*x + m2);

% Input Signal Creation

Vs = cl+c2;

Vs aft = abs(fft(Vs))/numsamps;

Vs_fft = 20*loglO(Vs_aft);

% Measure sig in(loop cnt) = max(Vs_fft(fl_center-bw:fl_center+bw-l));

%

% Amplifier without Predistortion - Generate FFT of output

voutl = G*((exp(Vs/V _lr )-l)); voutl aft = abs(fft(voutl))/numsamps; voutl fft = 20*logl0(voutl_aft);

% Measure terms imd3_l(loop_cnt) = max(voutl_fft(fl_3_center-bw:fI_3 center+bw-1)) imd5_ 1 (loop cnt) = max(vout l_fft(f 1 _5_center-bw: f 1 S center + bw- 1 )) imd7_l(loop_cnt) = max(voutl_fft(fl_7_center-bw:fl_7_center+bw-l)) sig ol(loop cnt) = max(voutl_fft(fl_center-bw:fl_center-r-bw-l));

% Amplifier WITH PREDISTORTION

% - predistortion scaling based on measuring gain of vout including % estimated cancelling terms to Vin, they reduce the gain

% Compute Cancelation terms (third order only)

% First collect the terms for the desired signal to be cancelled

% These include terms from fifth order expansion

Voutl3 = (3/4)*al ^A 2*a2*k3 + ...

; % These are the peak voltages Vout23 = (3/4)*al*a2"2*k3 + ...

(15*ar3*a2"2*k5)/8 + ... (5*al*a2"4*k5)/4;

% First calculate the magnitude of the predistortion signals % by collecting terms of the desired frequency from the series expansion % and Vs = V _tr *log( (Vout/G) + 1 )

Vpd in = V, _r *log(l + (Voutl3/G)) + V _tr *log(l + (Vout23/G));

%This is used to calculate a prescale term based on the gain of the amplifier

PreScale = ((2*(al +a2+ Vpd_in))/(G* ( exp((al +a2)/V _lr ) - exp(-l*(al +a2)/V _lr ) ) ) )

%Now the actual Vpdl3 = PhaseShift*PreScale * Voutl3 * cos((2*wl *x) + (2*ml) - (w2*x) - m2);

Vpd23 = PhaseShift*PreScale * Vout23 * cos((2*w2*x) + (2*m2) - (wl*x) - ml);

% Generate output with predistorted input

Vs = cl + c2 + Vpdl3 + Vpd23; Vs aft = abs(fft(Vs))/numsamps; Vs fft = 20*logl0(Vs_aft);

% Measure sig _in2(loop_cnt) = max(Vs_fft(fl_center-bw:fl_center+bw-l)); vout2 = G*((exp(Vs/V _lf )-l)); vout2_aft = abs(fft(vout2))/numsamps; vout2_fft = 20*logl0(vout2_aft);

% Measure imd3_2(loop_cnt) = max(vout2_fft(fl_3_center-bw:fl_3_center+bw-l)) imd5_2(loop_cnt) = max(vout2_fft(fl_5_center-bw:fl_5_center+bw-l)) imd7_2(loop_cnt) = max(vout2_fft(fl_7_center-bw:fl_7_center+bw-l)) sig_o2(loop_cnt) = max(vout2_fft(fl_center-bw:fl_center+bw-l));

% Amplifier with predistortion 3RD AND 5TH

% - predistortion scaling based on measuring gain of vout including

% estimated cancelling terms to Vin, they reduce the gain

% Compute Cancelation terms (third order only)

% First estimate the magnitude of the predistortion inputs Vsl and Vs2

Vpd_in = V _lr *log(l +(Voutl3/G)) + V _lr *log( 1 + (Vout23/G) );

PreScale = ((2*(al +a2+ Vpd_in))/(G* ( exp((al +a2)/V _lr ) - exp(-l *(al +a2)/VJ ) ) ); al3 = PhaseShift*PreScale*Voutl3 ; a23 = PhaseShift*PreScale*Vout23 ; Vpdl3 = PhaseShift*PreScale * Voutl3 * cos((2*wl*x) + (2*ml) - (w2*x) - m2);

Vpd23 = PhaseShift*PreScale * Vout23 * cos((2*w2*x) + (2*m2) - (wl*x) - ml);

% Now estimate the magnitude of the 5th order term

Voutl5 = (3*al*al3 ^Λ 2*k3)/4+ ...

(3*al*al3*a2*k3)/2+ ...

(3*al ^Λ 2*a23*k3)/4+...

(3*al3*a2*a23*k3)/2+...

(15*ar3*al3 ^A 2*k5)/8+ ...

(5*al*al3"4*k5)/4+ ...

(15*ar3*al3*a2*k5)/4+ ...

(15*al*al3 ^* 3*a2*k5)/4+ ...

(5*ar3*a2 2*k5)/8 + ...

(15*al ""al 3 ^A 2*a2~2*k5)/4 + ...

(15*al*al3*a2 ^A 3*k5)/4+ ...

(15*al ~2*al3"2*a23*k5)/4 + ...

(15*ar2*al3*a2*a23*k5)/2+ ...

(15*al3 ^* 3*a2*a23*k5)/4+ ...

(15*al "2*a2 ^* 2*a23*k5)/4 + ...

(l5*al3 ^Λ 2*a2'2*a23*k5)/8 + ...

(15*al3*a2 ^Λ 3*a23*k5)/4+ ...

(15*al *aI3 ^Λ 2*a23 "2*k5)/4+ ...

(15*al*al3*a2*a23 ^A 2*k5)/2+ ...

(15*al*a2"2*a23"2*k5)/8 + ...

(15*al ^Λ 2*a23"3*k5)/8 + ...

(15*al3*a2*a23"3+k5)/4;

Vout25 = (3*al3*a2'2*k3)/4+ .

(3*al*al3*a23*k3)/2+ ...

(3*al*a2*a23*k3)/2+ ...

(3*a2*a23 ^A 2*k3)/4+ ... (15*ar2*al3'2*a2*k5)/8+ ...

(15*al ^* 2*al3*a2 ^Λ 2*k5)/4+ ...

(15*al3"3*a2"2*k5)/8+ ...

(5*ar2*a2 ^Λ 3*k5)/8 + ...

(5*al3*a2~4*k5)/4+ ... (15*ar3*al3*a23*k5)/4 + ...

(15*al*al3~3*a23*k5)/4+ ... (15*al ^Λ 3*a2*a23*k5)/4+ ... (15*al*al3"2*a2*a23*k5)/2+ ... (15*al*al3*a2 ^* 2*a23*k5)/2+ ... (15*al*a2 ^Λ 3*a23*k5)/4+ ... (15*al ^Λ 2*al3*a23"2*k5)/8+ ...

(15*al ^Λ 2*a2*a23 ^Λ 2*k5)/4+ ... (15*al3"2*a2*a23'2*k5)/4+ ... (15*al3*a2 ^A 2*a23 ^* 2*k5)/4+ ... (15*a2 ^Λ 3*a23 ^Λ 2*k5)/8 + ... (15*al*al3*a23"3*k5)/4+ ...

(15*al*a2*a23 ^Λ 3*k5)/4+ ... (5*a2*a23"4*k5)/4;

% Now recalculate Vpd in using Voutl 5 and Vout25

Vpd in = V, _r *log(l +(Voutl3/G)) + V,,*log(l +(Vout23/G)) + ... V _lr *log(l +(abs(Voutl5)/G)) + V _lr *log(l +(abs(Vout25)/G));

Prescale = (2*(al +a2 + Vpd_in)) / (G* (exp((al +a2)/V, _r ) - exp(-l*(al +a2)/V _tf )) ); %Prescale = ((al +a2+ Vpd in)) / (G* (exp((al +a2)/V„) - exp(-l*(al +a2)/VJ) );

Vpdl5 = PhaseShift*Prescale*Voutl5*cos((3*wl*x) + (3*ml) - (2*w2*x) - (2*m2));

Vpd25 = PhaseShift*Prescale*Vout25*cos((3' ^l 'w2*x) + (3*m2) - (2*wl*x) - (2*ml)); % Vpd - voltage with predistortion 3 and 5

Vs = cl + c2 + Vpdl3 + Vpd23 + Vpdl5 + Vpd25; Vs_aft = abs(fft(Vs))/numsamps; Vs fft = 20' ^i, loglO(Vs_aft);

% Measure sig_in30oop_cnt) = max(Vs_fft(fl_center-bw:fl_center+bw-l)); vout3 = G*((exp(Vs/V _lr )-l)); vout3_aft = abs(fft(vout3))/numsamps; vout3_fft = 20*logl0(vout3_aft);

% Measure imd3_3(loop_cnt) = max(vout3_fft(fl_3_center-bw:fl_3_center+bw-l)) imd5_3(looρ_cnt) = max(vout3_fft(fl_5_center-bw:fl_5_center+bw-l)) imd7_3 (loop cnt) = max(vout3_fft(f 1 _7_center-bw: f 1 7 center + bw- 1 )) sig_o3(loop_cnt) = max(vout3_fft(fl_center-bw:fl_center+bw-l)); loop cnt = loop cnt + 1 ; end plot(sig_in,'w') hold on

%plot(sig_in2, 'g') %plot(sig_in3, 'b ^* )

plot(sig_ol,V); plot(sig_o2,'g'); plo sig rf.'b*); plot(imd3_l,V); plot(imd3_2,'g'); plot(imd3_3,'b'); plot(imd5_l,V); plot(imd5_2,*g'); plot(imd5_3,'b'); plot(imd7_l,Y); _P lot(imd7_2 ₎ ^, g'); plot(imd7_3,'b'); hold off cil = sig ol - imd3_l; ci2 = sig_o2 - imd3_2; %plot(cil _> ^, g')

%ho!d on %ρlot(ci2,'r'); %hold off

Amplifier Performance Graph - FIG. 6

The effect of the insertion of the V _pd predistortion terms into a non¬ linear amplifier is shown by the traces in FIG. 6.

In FIG. 6, the output power is graphed as a function of the input power using units of dB _w where dB _w = lOlog {power (watts)/ 1 (watt)}. For cellular communications systems, the input power range is typically 10 to 20 milliwatts and the output power range is typically from 3 to 100 watts.

In FIG. 6, the V _out trace 6-1 represents the output without any predistortion and the V _0UlPD TRACE 6-2 represents the output with predistortion. The V _out and the V _0UtPD traces are in substance the same.

In FIG. 6, the IMD3 trace 6-3 represents the contribution of the intermodulation third harmonic terms to V^ in the absence of any predistortion. It is apparent that a substantial portion of the power of V^ is represented by third harmonic intermodulation products of IMD3. In FIG. 6, the IMD3 _PD trace 6-4 represents the third harmonic component of V _0UtPD when third harmonic predistortion, in accordance with the present invention, has been inserted into V _ld . As can be seen from a comparison of IMD3 and IMD3 _PD , the third harmonic component of V^ is substantially reduced (IMD3 _PD is much less than IMD3) so that the efficiency of the amplifier for a given level of distortion is substantially increased.

In FIG. 6, the IMD5 trace 6-5 represents the contribution of the fifth harmonic intermodulation products to V _out in the absence of any predistortion. The IMD5 _PD 6-6 trace indicates the contribution of the fifth harmonic products when the third harmonic and fifth harmonic predistortion components are inserted in accordance with the present invention. Note that even with the third and fifth predistortion, the third harmonic component IMD3 _PD is still the most significant unwanted signal component.

In FIG. 6, the IMD7 trace 6-7 represents the contribution of the seventh harmonic intermodulation products to V _out in the absence of any predistortion.

Also in FIG. 6, the IMD7 _PD . trace 6-8 indicates the effect of the seventh harmonic intermodulation distortion products when the third and fifth harmonic products are inserted into the predistortion input V _a . No seventh harmonic predistortion terms are inserted in the predistortion input V _id . The seventh harmonic intermodulation terms as represented by IMD7 _PD * are increased in

V _out p _o in the absence of the insertion of any seventh harmonic predistortion due to the shift of some energy in the third and fifth harmonics into the seventh harmonic. However, the seventh harmonic distortion IMD3 _PD * in V^ still is less than the third harmonic distortion IMD3 _PD and hence the seventh harmonic distortion has not interfered with the important improvements gained by reducing the third harmonic components.

In summary, the traces of FIG. 6 demonstrate that the selection of particular predistortion harmonic terms advantageously reduces unwanted intermodulation products in the output of non-linear elements without creating more adverse consequences in other unwanted intermodulation products. While seventh harmonic terms were not used as predistortion terms in the FIG. 6 example, seventh harmonic terms can be included thereby further reducing the effects of the seventh harmonic terms. While the effect of selecting additional harmonic terms has the effect of transferring some of the power in the selected predistortion terms to other non-selected predistortion terms, such a transfer is not concentrated in any particular one of the unselected harmonic terms but is generally distributed across many intermodulation terms so that the net effect of this transfer of energy does not detract from the beneficial and significant gains for the selected harmonic components. While the invention has been particularly shown and described with reference to preferred embodiments thereof it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention.