Adaptive predistortion lineariser using polynomial functions

M. Ghaderi S. Kurnar, PhD D.E. Dodds

Indexing t e r m : Adaptiue predistortion lineariser, Linear modulation, Polynomial f i t i o n s

Abstract: Linear modulation methods are highly sensitive to high power amplifier nonlinearities. An adaptive predistortion lineariser using poly- nomial functions is described. An important problem faced in optimisation of nonliner circuits such as predistortion linearisers is that of con- vergence into a local minimum. This results from the nonquadratic shape of the objective functions. A solution to this problem is presented using a postdistorter with similar polynomial functions as the predistorter. Sample of the signals at the power amplifier output and predistorter input are demodulated. These demodulated signals are used for estimation of the postdistorter polynomial coefficients. The objective functions used in the estimation are quadratic functions of the coeffi- cients being estimated resulting in a rapid con- vergence to the global minimum. The coefficients of the predistorter polyniomials are set from those of the postdistorter. Computer simulation results for the proposed lineariser are presented. These results show 50 dB spectrum spreading improve- ment. Further, unlike previously reported linear- isers, the proposed lineariser is insensitive to the demodulator gain and phase imperfections. The performance of the proposed lineariser structured with fifth order polynomial functions is also com- pared with that for earlier polynomial type linear- isers.

1 Introduction

Linear modulation methods such as multilevel quadra- ture amplitude modulation (QAM) are of considerable interest for future mobile communication systems because of their high spectral efficiency. In QAM, the baseband information is contained in both the amplitude and phase of the carrier. This makes such a modulation method highly sensitive to the nonlinearities of the ampli- fication process. High power amplifier nonlinearities gen- erate intermodulation products and distortion resulting in adjacent channel interference and system performance degradation. The traditional method of using a heavily

0 IEE, 1994 Paper 10481 (E5, E8), first received 26th March and in revised form 13th October 1993 The authors are with the Department of Electrical Engineering, Uni- versity of Saskatchewan, Saskatoon, Saskatchewan, Canada STN ow0 IEE Proc.-Commun., Vol. 141, No. 2, April 1994

backed off class A amplifier is very inefficient, especially for mobile transmitters where cost and dissipation are important factors.

An effective solution is to linearise the power amplifier using predistortion [l-91. In this linearisation method, the input signal is predistorted such that the amplitude and phase nonlinearities of the power amplifier are com- pensated by the lineariser. Both the AM/AM and AM/PM characteristics of the predistorter have to be inverse of the power amplifier AM/AM and AM/PM characteristics. Predistortion could be implemented at the baseband, IF or RF frequency. In References 1-5 the predistorter amplitude and phase characteristics have been implemented using look up tables. In References 6-9 the nonlinear characteristics of the predistorter have been implemented using polynomial functions. The lin- eariser is required to be adaptive because of variations in the predistorter and power amplifier parameters with time, temperature and different operating channels.

An important problem in optimisation of nonlinear circuits, such as predistortion linearisers, is that of con- vergence into a local minimum [lo]. This results from the objective functions having a nonquadratic shape. Polynomial type linearisation methods reported in Refer- ences 7-9 estimate the predistorter polynomial coeffi- cients by minimisation of out-of-band intermodulation distortion (IMD). It has been shown [9] that the IMD components of 3rd, 5th and a part of 7th order are quad- ratic functions of the coefficients. However, this is not true for higher order IMD components. Thus, if the optimisation process is based on IMD minimisation, sta- bility and convergence to the global minimum are not guaranteed. In fact, minimisation of 3rd and 5th order IMD results in convergence into a local minimum only. Spectrum spreading improvement achieved by 3rd and 5th order IMD minimisation is limited. To get better per- formance, higher order IMD components must be con- sidered. The nonquadratic shape of IMD also limits the predistorter polynomials to be no higher than the fifth order.

This paper describes an adaptive polynomial type pre- distortion lineariser for which a new method for poly- nomial coefficients estimation is used. Using this method of estimation, a solution to the problem of local con- vergence is proposed. The coefficient estimator in the

Science and Engineering Research Council of Canada under the University-Industry chair program.

49

proposed lineariser uses a postdistorter which distorts the demodulated power amplifier output using polynomial functions of the same type as the predistorter. This results in objective functions used for the estimation of the pre- distorter polynomial coefficients being of quadratic shape, which guarantees stability and convergence to the global minimum. The recursive least square (RLS) algo- rithm is employed for the optimisation process. The pre- distorter functions in effect simulate a nonlinearity which is the inverse of the power amplifier nonlinearity. The coefficient of the predistorter polynomial functions are set using the postdistorter polynomial coefficients.

It is shown that a predistorter with 7th order poly- nomials provides considerable improvement compared to a 5th order predistorter. Because of quadratic cost func- tions higher orders of predistorter polynomials present no problem.

2 Lineariser structure

The lineariser structure is shown in Fig. 1. As shown in this Figure, the proposed predistortion circuit is imple- mented at IF. A sample of the high power amplifier (HPA) output is downconverted to IF and this signal is demodulated using the IF signal at the predistorter input. The envelope of the predistorter input IF signal is also detected. Samples of the demodulator and envelope detector outputs form the required inputs for the estima-

analog predistorter - - - - - - - -

tor. A more detailed block diagram of the predistorter and estimator is shown in Fig. 2. The HPA introduces amplitude and phase nonlinearities, and the signal at the

sigml P Fig. 1 Block scl predistorter

predist

~~

w t i c of a QAM system transmit chain with adaptive

output of the HPA may be written as

v d t ) = cos cw, t + W) + Wml = g,(r(t)) cos CO, t + Wl

+ gp(r(t)) sin CO, t + Wl (1) where is the complex envelope of the modulated QAM signal and w, is the carrier frequency. A(.) and @(.)

. -

digital estimator * Fig. 2

M

Schematic diagram of adaptive predistorterlestimator

I E E Proc.-Commun., Vol. 141, No. 2, April 1994

denote the amplitude and phase transfer functions of the amplifier. Functions g,(.) and sa(.) are termed as the inphase and quadrature amplitude distortion functions of the HPA.

As shown in Fig. 2, the predistorter is a complex quad- rature modulator whose inphase and quadrature ports are adjusted by the nonlinear gain functions G; "(rdt)) and Q,,"(rdt)), where ri(t) is the amplitude of the'QAM signal at the predistorter input. These functions are an analog approximation of the estimator polynomial func- tions GI, &it)) and G,, "(rdt)), which may be implemented by using nested multipliers and summers. The functions G;,.Jrdt)) and Gb, "(ri(t)) differ from their corresponding estimator polynomial functions G,, "(rdt)) and G , "(rdt)), because of imperfections in the circuits used for imple- mentation.

As shown, the signal at the output of the power ampli- fier is downconverted and then demodulated using a quadrature demodulator (QDM), where the LO port of the QDM is the IF signal from the predistorter input. The signals at the inphase and quadrature ports of the QDM are

rdt)ro(t) cos (eo@) - W ) )

- rdt)ro(t) sin (OO(t) -

and

respectively. ro(t) L Oo(t) and rdt) L O,(t) are the power amplifier output and the predistorter input signals, respectively, in polar form. The operation of the rec- tangular to polar converter is to find ro(t) and (Oo(t) - e&)) using the signals at the QDM and envelope

detector outputs. As shown, the postdistorter has two polynomial functions at the inphase and quadrature ports. The inputs to the postdistorter are found using samples of ro(t), (Oo(t) - e&)) and a constant phase shift denoted by 4. The terms XI, and X,, are obtained as shown in Fig. 2 as

X,O(t) = ro(t) cos (Oo(t) - at) + 4) (2)

xQO(t) = rO(t) sin - Oi(t) + 4) (3) From eqns. 2 and 3 it can be seen that if the QAM signal is predistorted such that

X,O(t) = rdt) cos (4) and

x Q O ( t ) = rdt) sin ('$1 then ro(t) = rAt) and Oo(t) = Odt) and the amplifier is lin- earised. In the following Section it will be shown that X,,(t) and X,,(t) take these values when the errors e, and e, in Fig. 2 are converged to their minimum values. The adaptation algorithm used to minimise the error functions is described in the following Section.

3

The recursive least square (RLS) algorithm was employed for the optimisation process [ll]. This algorithm was used to minimise the errors e, and e, for all samples of data leading up to the nth iteration. At each iteration the predistorter coefficients are set based on the postdistorter coefficients of the previous iteration. The predistorter gain polynomials at the Ith iteration and the postdistor- ter gain polynomials at the nth iteration may be written

IEE Proc.-Commun., Vol. 141, No. 2, Apn'l 1994

Lineariser operation and adaptation algorithm

in terms of powers of the input or output signal as M

G I , dri(I)) = 1 ak, I '(I) (4) k = l

U

k = l

M

(5 )

for I = 1,2, . . . , n. Here I is the discrete time sample index and rXI), X,,(I), and XQo(I) are values of rdt), XI,@) and XQO(t) at the Ith iteration. ak, ! and f i t , I are the coefficients of predistorter gain polynomials at the Ith iteration, and ak," and bk," are the coefficients of postdistorter gain polynomials being estimated at the nth iteration. Note that, in implementing the recursive least square (RLS) algorithm the postdistorter coefficients are assumed to be constant for calculation of its polynomial functions for all samples of data leading up to the nth iteration.

At each iteration (i.e. nth iteration) the circuit sets the predistorter polynomial coefficients from the postdistor- ter polynomial coefficients which were estimated at the previous iteration (i.e., (n - 1)th iteration). This is calcu- lated using the following relations

GI. k d 4 ) = &, "- drAn) cos 4) GQ, k i n ) ) = Yp. "- Ar,(n) sin 4)

(8)

(9) The relationship between the predistorter and postdistor- ter polynomial coefficients is obtained by substituting for GI," and G , , from eqns. 4 and 5 and for &.n-l and Y,, - l, from eqns. 6 and 7 in eqns. 8 and 9. Equating the coefficients on both sides gives the following relation between the predistorter and postdistorter polynomial coefficients at the nth iteration

a k . n =(cos ( 4 ) l k - l a k , n - l (10)

p k , n = (sin (4)lk- bk, II - 1 (1 1)

At the nth iteration, the error functions for the data sampled at the time of the Ith iteration may be defined as

eQ(I) = 'dQGQ, krkO) - 'O(0 'Q. n ( x Q O ( 0 ) (13) where I = 1, 2, . . . , n and ro(l) = (XX(I) + X~o(l))l'* is the value of the output envelope ro(t) at the Zth iteration. The cost functions for the RLS algorithm are defined as a residual sum of the weighted squares. At the nth iteration these are given by

(14)

(15)

where I is a positive number equal to or less than one. Since the predistorter polynomial coefiicients are set using ah, and bk, where I = 1, 2, . . ., n, therefore GI. XrdN GQ, krdl)), X,O(O and XpdI), I = 1, 2, . . . , n, are independent of the coeffiaents ah, ,, and bk, ". As a result of this, the cost functions J,(n) and J&) are quadratic functions of the coefficients being estimated at the nth iteration, i.e. ak." and bk," . The derivatives of the cost

51

functions aJ,(n)/aa,,, and aJQ(n)/ab, are derived in Appendix 8. Let a:, I and b:, denote ihe optimum value of the kth coefficients for which the derivatives of the cost functions are zero at the nth iteration. Then

m=l I =1

fork = 1, 2, ..., M. The solution of these two sets of M linear simultan-

eous equations yields the coefficients of the inphase and quadrature postdistorter polynomials, at the nth iter- ation. The recursive least square (RLS) algorithm is set up using eqns. 16 and 17. As the equations are linear, the system yields a unique solution for the polynomial coeffi- cients and convergence to the global minimum is assured.

After convergence, i.e. n,th iteration, the errors e,(n) and e&) can be assumed to be zero for n z n, , and the coefficients are converged, i.e. G,. "( .) = G I , .+ .) = . . . , GQ..(.)= G Q , ~ + I ( . ) = . . . , Yr..(.)= ? , . + I ( . ) = . . . and YQ, "( .) = YQ, + .) = . . . . From eqns. 12 and 13

(18)

(19) for n > n, .

Substituting from eqns. 8 and 9 for G1,m(rdn)) and G , "(rdn)) and also using the fact that at convergence the coefficients of the Dostdistorter are the same as for the

rdnF1, .(rdn)) = r0 (4 Yr. n(XIo(n))

rdn)GQ, ~ ( ~ d ~ ) ) = rO(n)YQ, n(xQO(n))

previous iteration, i.e. Yr."(.) = YQ.n-l(.), we have

and YQ..(.) =

rXn)r,* Ardn) cos 4) = (X?o(n) + X60(n))"2%, n(xlo(n))

= (x:O(n) + X60(n))1'2 yQ. d x Q O ( n ) )

(20)

(21) for n > n, .

As HPA inphase and quadrature distortion functions, gI and gQ are single valued functions, eqns. 20 and 21 have only one solution for X,,(n) and XQo(n), which is X,&) = rAn) cos 4, and XQo(n) = rdn) sin 4 for n > n, . As stated at the end of Section 2, this implies that the amplifier is linearised, i.e. ro(t) = rdt) and f?,(t) = edt). In the following Section the effect of demodulator imperfec- tions on lineariser performance is discussed.

rdn)YQ, n(rdn) sin 4)

4

A problem in predistortion linearisers reported before [2-91 has been the effect of demodulator gain and phase imperfections on the lineariser performance. It has been shown [5] that the demodulator gain and phase imper- fections degrade the lineariser spectrum spreading per- formance by at least 10 dB. It can be proved that, in the proposed method demodulator gain and phase imperfec- tions result in a convergence which introduces a small phase offset equal to the output demodulator phase error. Even with imperfect demodulators the output amplitude

Effect of demodulator and other imperfections

52

is proportional to the input amplitude and no IMD is generated.

Predistorter I and Q modulator imperfections as well as imperfections in the circuit used to generate the control signals corresponding to gain functions G,, and GQ,.,, can be treated as a part of the amplifier nonlin- earities. As the reference signals rAoG,, I(r#)) and rdI)GQ, ,+-XI), 1 = 1,2, . . . , n, in the estimator are independ- ent of these errors, these nonlinearities are automatically compensated by the lineariser.

As shown in Fig. 2, suitable delay elements are required to compensate for the delay in the various signal paths. For example the modulated IF signal at the pre- distorter input is delayed before being used to demodu- late the HPA output sample. An accurate estimation of the delay is required to implement delay compensation. Generally an accuracy of k1/64 of a symbol period is recommended [3]. The lineariser performance was evalu- ated using computer simulation. Results of this simula- tion are presented in the next Section.

5 Lineariser performance

A power amplifier with the AM/AM and AM/PM char- acteristics shown in Fig. 3 was used in the computer simulation. In order to compare performance of the pro- posed lineariser with the earlier reported [7-91 method of IMD minimisation, a predistorter with 5th order poly- nomial functions was considered first. The gain poly- nomials for such a lineariser may be written as

GI, .(rdt)) = al. + a3, ,, r,2(t) + as.

GQ. n(rdt)) = 83, n r?(t) + 8 5 . n r?(t)

(22) (23)

normalised input amplitude

Fig. 3 amplijier

0 (----)phase

AMIAM and AMIPM characteristics of a class A B powm

(-) amplitude

Phase shift 4 used for the rotation of the demodulated output was assumed to be -30". An output back-off of 2 dB was also assumed. The convergence rates of the nor- malised amplitude and phase of the predistorter/power amplifier system to the desired linear system are shown in Fig. 4. As may be seem from this figure, amplitude and phase errors of less than 2% were obtained after 100 iter- ations. After convergence, the coefficients of the predis- torter gain polynomials were found to be as follows

al,,=0.50165 a,,,=0.10442 a,..=0.10334

8 3 , " = -0.111 57 85. . = -0.14089 I E E Proc.-Commwt., Vol. 141,No. 2, April I994

To compare the proposed method with that using IMD minimisation, predistorter polynomial coefficients were also computed using IMD minimisation. The 3rd,

0 1 - - \ -e--

6 io io $0 do 20 iteration numbers, n

Fig. 4 Error performance of fii/th order predistorter estimator (proposed method)

(-) amplitude 0 (----)phase

5th and a part of the 7th order IMD components which are quadratic functions of coefficients were equated to zero. The resulting coefficients are

u1," = 0.5000 a3," = 0.12500 = 0.043 15 83," = -0.1300 85," = -0.07025

As may be seen, the values of these coefficients are differ- ent from those found by the proposed method. The power spectrum at the output of the linearised power amplifier is shown in Fig. 5, for the two methods. A

frequency.kHz Fig. 5 Power spectrum of filtered 16QAM data signul with and without linearisation, 5th order P D ( IMD based method and proposed

~ output without PD input

. - -. . . . output: proposed method output: IMD method

method)

_ _ _ _

16 QAM signal with 25% raised-cosine filtered pulse was used as the input signal. Using the proposed method, spectrum spreading reduction of 35 dB was achieved. This is 15 dB better than that for the IMD based method.

From a series of simulations it was seen that, in a pre- distorter with 7th order polynomials, the 5th order terms do not have significant effect on the performance. Next, a predistorter with the seventh order gain polynomials shown below was simulated.

I E E Proc.-Commun., Vol . 141, No. 2, April 1994

G,,.(rXt)) = al. I + ~ 3 , I r 3 t ) + ~ 7 . II r 3 )

Gp, "(CW = 81," + 83, I ri" + 87. " rP(4

(24)

(25) No circuit imperfections were considered. As before, 4 was assumed to be - 30" and an output back-off of 2 dB were assumed. After convergence the coefficients were found to be

u1,.=0.49952 ~ ~ , ~ = 0 . 1 3 3 9 5 ~ , , ~ = 0 . 0 9 9 3 1

B1,. = 0.00109 83," = -0.14545 1 7 , " = -0.14417

The AM/AM and AM/PM characteristics of the predis- torter are shown in Fig. 6. A comparison of Figs. 3 and 6

f- 30 I m I

0 01 02 03 0 4 05 0 6 07 0 8 09 normalised input amplitude

AMIAM and AMIPM characteristics ofpredistorter Fig. 6 (-) amplitude

0 (----)phase

shows that the predistorter is quite effective in realising the inverse nonlinearity for both amplitude and phase. The highly linear AM/AM and very small AM/PM result from this precise realisation of inverse nonlinearity. The convergence of amplitude and phase errors was seen to be somewhat slower because of the higher order of the predistorter polynomials. The power spectrum at the amplifier output was also computed. A further 15dB improvement in spectrum spreading was obtained com- pared to fifth order lineariser results.

To study the effect of predistorter and gain function implementations, the I-Q modulator in the predistorter were assumed to have 4% gain and 5" phase quadrature error. The multipliers in the gain function circuits were, similarly, assumed to have 4% gain error and 4% DC offset. Once again 4 was assumed to be - 30". After con- vergence, the coefficients for the seventh order predistor- ter were found to be

= 0.44037 = 0.071 24 u7," = 0.071 67

P I , . = -0.03896 /?s,.= -0.18424 87,"' -0.13147

As may be seen, the estimated coefficients are different from the case of no imperfections. The predistorter is quite effective in compensating for the imperfections. This is evident from the gain function plots in Fig. 7. Note that functions G;,"(.) and Gh,.(.), are different from the estimated functions G,. "( .) and G , "( .) as the nonlinearity of the gain implementation circuits i s compensated through this difference. Similarly, the effective gain func- tions for the predistorter are different from G:, &&)) and Cb, &it)), to account for predistorter imperfections. The AM/AM and AM/PM characteristics and the power spectrum of the linearised amplifier was also computed in

53

the presence of circuit imperfections. These results were found to be quite close to the no imperfection case which showed that the proposed method was effective in com- pensating for these imperfections.

0 00

e

09 normalised input cmplitude

Fig. 7 PD effective gain

0 estimated gain implementation gain

Predistorter inphase and quad. gainfunctions

The demodulator imperfections were not considered in the above simulations. To investigate the effect of these imperfections on lineariser performance, the demodulator was assumed to have 4% gain and 4" phase errors. A 4% gain error was also assumed in the envelope detector. After convergence the coefficients were found to be

a,.m=0.43240 a3,.=0.06840 ~ , , ~ = 0 . 0 6 9 7 8

/!I,,.= -0.06916 P 3 , . = -0.20434 87 ," ' -0.151 16

The AM/AM and AM/PM characteristics of the predis- torter and the AM/AM and AM/PM characteristics of a class AB power amplifier linearised with this predistorter are shown in Figs. 8 and 9. The convergence rates and

O 6 1

R

0

0 01 02 03 0 4 05 0 6 07 08 I

-30

-20 U U U

0 c

-10; 3

0

nomlised input amplitude Fig. 8 demodulator imperfections (all circuit imperfections considered)

(-) amplitude 0 (----)phase

AMjAM and AMjPM characteristics of predistorter with

power spectral density of the QAM signal at the output of the power amplifier with and without predistortion are shown in Figs. 10 and 11 . As before, an output back-off of 2 dB was assumed. As seen from Fig. 11, the spectrum spreading was improved by greater than 50 dB with lin- earisation. This high spectrum spreading improvement is achieved because the imperfections in the demodulators were compensated for. A comparison of Figs. 6 and 8

54

shows that a phase offset of -4" has been included in the phase characteristic. As mentioned before, this is due to the 4" quadrature error in the demodulator. The RLS

'1

0 01 02 03 0 4 05 06 Cl7 08 09 rnrmlised lnput amplitude

Fig. 9 amplifier with predistortion (all circuit imperfections considered)

(-) amplitude 0 (----)phase

AMJAM and AMjPM characteristics of a class A B power

'1

0 100 200 300 400 iteration nurnbers.n

Error performance of seventh order predistorter estimator (all Fig. 10 circuit imperfections considered)

(-) amplitude 0 (- ---)phase

5 -50

-70

-9 0 -50 -40 -30 -20 -10

, I , I I

10 20 30 40 50 frequency, kHz

Power spectrum offiltered 16QAM data with and without lin- Fig. 11 earisation (all circuit imperfections considered) W (-) output without PD 0 (- - - -) output with PD

algorithm requires 3N(N + 3)/2 multiplications for each polynomial, where N is the number of coefficients. Using a DSP chips with multiplication time of 1 p , each iter-

I E E Proc.-Commun., Vol. 141, No. 2, April 1994

ation takes about 0.08 ms. Therefore, an improvement of greater than 50 dB is obtained after 50 ms.

6 Conclusions

An adaptive polynomial predistortion lineariser which employs a new estimation method for the polynomial coefficients was presented. The predistorter is a quadra- ture modulator with inphase and quadrature polynomial functions. A solution to the problem of local convergence was described. A fast convergence of the coefficients was obtained using the recursive least square algorithm and objective functions of quadratic shape in the estimator. The method proposed in this paper has the following advantages compared to other polynomial type linear- isers reported before [7-91

(i) The quadratic shape of the objective functions result in stability and convergence to the global minimum

(ii) Higher order IMD products generated by the com- posite predistorter/HPA are also considered in the estimation of coefficients. These IMD products are, there- fore, compensated

(iii) The order of the predistorter nonlinear functions is not limited to the 5th. It was shown that a predistorter with 7th order polynomial results in 15 dB greater spec- trum spreading improvement than that with 5th order polynomials

(iv) The demodulator gain and phase imperfections do not generate IMD

(v) Any nonlinearities introduced by the imperfections of the circuits used to implement the predistorter poly- nomials can be compensated. IMD products generated by these nonlinearities may not be functions of the poly- nomial coefficients, and will not be compensated by an IMD based method.

Using the proposed lineariser, the spectrum spreading by a nonlinear HPA of a 16QAM 25% raised-cosine filtered signal was improved by greater than 50dB. This per- formance was achieved using predistorter modulator with 4% gain and 5" phase errors, a demodulator with 4% gain and 4" phase errors and gain functions circuits multipliers with 4% gain error and 4% DC offset.

7 References

1 GIRARD, H., and FEHER, K.: 'A new baseband linearizer for more efficient utilization of earth station amplifiers used for QPSK trans- mission', IEEE J. Sel. Arem Commun., Jan. 1983, SAC-1, (1). pp. 46-56

2 SALEH, A.A.M., and SALZ, J.: 'Adaptive linearization of power amplifiers in digital radio systems', Bell Syst . Tech. J . , Apr. 1983,62, (4). pp. 1019-1033

3 NAGATA, Y.: 'Linear amplification technique for digital mobile communications'. Proc. IEEE Vehicle technology conference 1984, pp. 159-164

4 CAVERS, J.: 'Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements', IEEE Trans. Veh. Technol., Nov. 1990,39, (4), pp. 374-382

5 WRIGHT, AS., and DURTLER, W.G.: 'Experimental performance of an adaptive digital linearized power amplifier', IEEE Trans. Veh. Technol., Nov. 1992,41, (4), pp. 395-400

6 NAMIKI, J.: 'An automatically controlled predistorter for multi- level quadrature amplitude modulation', IEEE Trans. Commun., May 1983, COM-31, pp. 707-712

7 STAPLETON, S., and COSTESCU, F.: 'Amplifier linearization using adaptive analog predistortion', IEEE Trans. Veh. Technol., Feb. 1992,41, ( I ) , pp. 49-56

8 HILBORN, D., STAPLETON, S.P., and CAVERS, J.K.: 'An adapt- ive direct conversion transmitter'. Proc. IEEE Vehicle technology conference, 1992

9 STAPLETON, S., KANDALA, G.K., and CAVERS, J.: 'Simulation and analysis of an adaptive predistorter utilizing a complex spectral convolution', l E E E Trans. Veh. Technol., Nov. 1992, 41, (4). pp. 387-394

10 MASSARA, R.E.: 'Optimization methods in electronic circuit design' (John Wiley & Sons, New York, 1991)

11 HAYKIN, S.S.: 'Introduction to adaptive filters' (Collier Macmillan, New York, 1984)

8 Appendix

From eqns. 6, 8, 12 and 14 the inphase cost function is given by

(26)

Note that, &,lFl(r#)) , X!o(r) and XQo(l) are functions of coefficients estimated using (I - 1) data at the (I - 1)th iteration where I = 1, 2, . . . , n, and are independent of the coefficients ak, and bk, ". Therefore, J,(n) is a quadratic function of the coefficients. The set of coefficients which minimises J,(n) must be found. Hence, differentiation of eqn. 26 with respect to ak, I results in

A similar analysis can be done for the quadrature poly- nomial coefficients, resulting in

IEE Pr0c.-Commun., Vol. 141, No. 2, April 1994 55