AMPLIFIER PREDISTORTION USING UNSCENTED ... - CiteSeerX

AMPLIFIER PREDISTORTION USING UNSCENTED KALMAN FILTERING Amaresh V. Malipatil and Yih-Fang Huang Department of Electrical Engineering University of Notre Dame {amalipat,huang}@nd.edu ABSTRACT Bandwidth efficient modulation schemes like M-ary quadrature amplitude modulation (M-QAM) suffer from amplifier nonlinearities when operated near saturation. The high power amplifier (HPA) characteristics could be memoryless or with memory, as experienced in wideband CDMA (WCDMA). Digital baseband predistortion is an efficient technique to compensate for the amplifier nonlinearities. In this paper, an Unscented Kalman Filter (UKF) is used to design an adaptive predistorter (PD) for the general case of nonlinear amplifier with memory. Simulation results are shown to verify the performance of the proposed method. 1. INTRODUCTION Modern wireless communication systems use highly spectral efficient modulation schemes like M-QAM to preserve bandwidth. Such modulation schemes have large peak-toaverage power ratio(PAPR); hence when the amplifier is operated near saturation for power efficiency, nonlinear distortion is introduced into the signal. The nonlinear distortion causes spectral regrowth resulting in Adjacent Channel Interference (ACI), and hence the spectrum may not conform to the standard specifications. It also deteriorates the performance of the system in terms of increased bit error rate (BER) due to constellation warping and nonlinear inter-symbol interference (ISI). To reduce the nonlinear effects, the amplifier output can be backed-off from the saturation point but it leads to very low power efficiency. Hence, power efficient but reliable communication requires that the amplifier be operated close to the saturation point and the signal be compensated for the nonlinear distortion. The nonlinear distortion can be suppressed either at the transmitter (predistortion) or at the receiver (nonlinear equalization). Equalization at the receiver can provide improved BER performance but the received signal still produces ACI to other users or systems operating in the adjacent frequency This work has been supported, in part, by the National Science Foundation under Grant EEC02-03366, by the U.S. Department of the Army under Contract DAAD 16-02-C-0057-P1, and by the Indiana 21st Century Fund for Research and Technology.

bands. Hence digital predistortion at the transmitter is considered more effective way of improving the BER performance along with the reduction in the ACI. The nonlinear amplifier is usually modeled as a frequency independent memoryless system with a given AM/AM and AM/PM characteristics [1]. Much research has been done on predistortion of memoryless nonlinear HPAs [2, 3, 4]. However memory could creep into the system in two ways. In high bandwidth communications like W-CDMA the HPAs exhibit frequency dependent characteristics, introducing memory into the system [5]. Otherwise, the HPA could be preceded by a pulse shaping filter which can be represented as a linear dynamical system [6]. In such cases, it is essential to design predistorters which compensate for the nonlinear ISI, since memoryless predistortion does not provide acceptable performance. The characteristics of the HPA may vary with time due to change in physical operating conditions like temperature. It thus becomes necessary to adapt the predistorter to match the time-varying amplifier characteristics. Recently, the UKF [7] has been gaining popularity in tracking applications involving nonlinear dynamical systems. Thanks to better convergence properties and simplicity of its design, the UKF is preferred over the traditional Extended Kalman Filter (EKF) [9] in nonlinear filtering and estimation problems. In this paper, the UKF is used to design a predistorter for a nonlinear HPA with memory. The nonlinear amplifier and the predistorter are modeled using memory polynomials [10]. The proposed method succeeds in suppressing the nonlinear distortion and the spectral regrowth. In section 2, the system model and the notation is presented. The UKF is applied to design the predistorter in section 3. In section 4, the simulation set up is discussed and the results are presented. Section 5 concludes the paper. 2. SYSTEM MODEL A block diagram of the system under consideration is shown in Fig. 1. In the figure, sk , uk and yk represent, respectively, the baseband input signal, the output of the predistorter and the output of the HPA. The vectors xk and vk are formed from sk and uk as given below and are input to the predis-

Delay

sk

xk

PD

uk

yk

HPA

wk

ek(w) vk

HPA Model

y^k

δk

hk

Fig. 1. Adaptive predistorter for nonlinear amplifier with memory torter and the HPA model, respectively. si

= [si si |si | · · · si |si |Nn −1 ]T

xk

= [sTk sTk−1 · · · sTk−Nl +1 ]T

uk

= wkT xk

ui vk

= [ui ui |ui | · · · ui |ui |Mn −1 ]T = [uTk uTk−1 · · · uTk−Ml +1 ]T

yˆk

= hTk vk

(w)

ek

δk

is then propagated analytically through the “first-order” linearization of the nonlinear system. Hence EKF requires deriving complicated analytical derivatives (Jacobians or Hessians). Moreover, in highly nonlinear systems, error due to linearization could lead to divergence. To overcome these problems a new extension of Kalman Filter called as Unscented Kalman filter (UKF) was proposed by Julier et al [7]. UKF is based on the concept of Unscented Transformation (UT), which is a deterministic sampling method to determine the statistics of a random variable undergoing nonlinear transformation. It is founded on the intuition that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function or transformation. In employing the UKF, the mean and covariance of the state estimates are accurate at least to the second order (Taylor series approximation) as opposed to the first order accuracy of the EKF estimates. The greater accuracy of UKF leads to better recursive estimates and robustness with respect to divergence compared to the EKF. Motivated by these advantages, the UKF is now applied to linearize a nonlinear HPA. The state space equations representing the evolution of the predistorter coefficients w for the nonlinear system is given as follows [12].

= sk−∆ − yk

wk+1

= yk − yˆk

where T represents the transpose operation. Nl and Nn represent the memory and the highest order of the nonlinearity of the predistorter, respectively. Ml and Mn are defined similarly for the HPA model. wk and hk represent the coefficients of the polynomial predistorter and the HPA model (w) respectively, at instant k. ek and δk are, respectively, the errors/innovations of the predistorter and the HPA estimator. The HPA introduces predominantly odd-order harmonics, therefore only the odd orders could be used in forming the vector inputs (the magnitude terms | · | will have even powers). However adding the even orders gives improved performance [10]. The delay (∆) is due to the causality of the predistorter. The adaptive estimation of the polynomial predistorter and the HPA model is presented in the next section.

yk

(1)

= wk + r k = G(xk , wk ) +

(w) ek

(2)

where G(·) represents the cascade of the predistorter followed the HPA, and rk is the process noise. The covariance matrix of the process noise determines the convergence and tracking performance. When the input-desired output pair {xk , yk } is available for training, the aim is to recursively estimate the parameter vector w minimizing the MSE of the (w) error ek = yk − G(xk , wk ). This requires the knowledge of the first and second order statistics of yk . To estimate the mean and covariance of yk , the UKF forms a matrix Wk with a set of 2L + 1 vectors Wk,i (w is a L × 1 vector, where L = (Nn + 1)/2 × Nl if only odd orders are used and L = Nn × Nl if both even and odd orders are used ) called as sigma points and propagates them through the actual nonlinear system.

3. PREDISTORTER DESIGN USING UKF

Wk,0

=

ˆ k− w

Kalman filter has been very popular in tracking applications. Formulated with a linear state-space model, the Kalman filter a linear minimum mean square error (MMSE) filter that recursively estimates the state of the dynamical system [11]. When the underlying system is nonlinear, the EKF has been applied for state estimation [9]. In EKF, the state distribution is approximated by a Gaussian random vector, which

Wk,i

=

ˆ k− + ( w

Wk,i

=

q

− (L + λ)Pw k ) , i = 1, · · · , L, i q − − ˆ k − ( (L + λ)Pwk ) , i = L + 1, · · · , 2L, w i

− ˆ k− and Pw where w are, respectively, the a priori mean and k covariance of w at instant k. λ = α2 (L + κ) − L is a scaling parameter. The constant α determines the spread of

the sigma points around wk , and is usually set to (10−4 ≤ α ≤ 1) [8].q The constant κ is a scaling parameter usually set

− to 3 − L. ( (L + λ)Pw k ) is the ith column of the matrix i − square root of (L + λ)Pwk . The algorithm is initialized as follows.

ˆ0 w P w0

(3)

= E[w]

(4)

H

ˆ 0 )(w − w ˆ 0) ] = E[(w − w

where H represents the complex-conjugate transpose operation. The time update equations including sigma point propagation and prediction of yk are given below. ˆ k− w − Pw k

=

(5)

ˆ k−1 w

(6)

r Rk−1

Yk

= Pwk−1 + = G(xk , Wk )

yˆk

=

2L X

(m)

Wi

(7)

Yk,i

i=0

The measurement-update equations are as follows. P yk yk

=

2L X

Wi (Yk,i − yˆk )(Yk,i − yˆk )H + Rke(8)

2L X

ˆ k− )(Yk,i − yˆk )H Wi (Wk,i − w

(c)

i=0

P w k dk

=

(c)

(9)

i=0

Kk

= Pwk dk Py−1 k yk

ˆk w

=

P wk

=

ˆ k− + Kk (yk − yˆk ) w − Pw − Kk Pyk yk KkT k

(10) (11) (12)

where Rr is the process noise covariance and Re is the mea(m) (c) surement noise covariance. The weights Wi and Wi are calculated as follows. (m) W0 (c)

W0

(m)

Wi

λ = , L+λ λ + 1 − α2 + β, = L+λ 1 (c) = Wi = , 2(L + λ)

adaptive algorithm for system identification can be used. A recursive least squares (RLS) algorithm could be employed for this purpose, but it has high computational complexity. Another alternative could be the least mean squares (LMS) algorithm. It has less complexity, but the convergence and the mean square error performance of the filter depend critically on the choice of the step size. Here, a setmembership framework is used as an alternative to RLS and LMS, to estimate the nonlinear characteristics of the HPA. The advantages of the SM-NLMS algorithm include simplicity of the update procedure, data-dependent optimized step-size and selective update of the parameter estimates. The selective update feature can be exploited to reduce the implementation complexity by sharing updating processors [15]. This property is particularly well-suited for the identification of the HPA model in parallel radio-frequency (RF) chains, since the HPA characteristics vary slowly with time and hence the updations are performed only when necessary. The complexity of the SM-NLMS algorithm is comparable to the normalized least mean squares (NLMS) algorithm. In set-membership filtering, the objective is to find a set of feasible filter coefficients such that the resulting estimation errors are bounded in magnitude by a fixed threshold ν, for all the input vector-desired output pairs (vk , yk ). The SM-NLMS algorithm is a supervised learning algorithm, belonging to the set-membership adaptive recursive techniques (SMART) family [14]. The SM-NLMS is formulated by the following update equation [14]:

where β represents the a priori knowledge of the distribution of w (for Gaussian distribution β = 2 is optimal). The complexity of the UKF algorithm presented above for parameter estimation is O(L3 ). Finding the square root of the − matrix Pw is the most complex operation in the algorithm. k The complexity can be reduced to O(L2 ) using the square root implementation of the UKF proposed in [13]. The propagation of the sigma points through the actual nonlinear system necessitates the identification of the nonlinear HPA. Since identification of a nonlinear system is comparably simpler than determining its pre-inverse, any

(13)

where ∗ denotes the complex conjugation and the gain γk are given by γk

i = 1, · · · , 2L

(h)

= hk−1 + γk ek vk∗ /vkH vk

hk

=



1 − ν/|δk |, if |δk | > ν 0, otherwise.

(14)

Initially, only the HPA estimator is trained, till a moderately accurate estimate of the true HPA has been obtained. This model is then incorporated in the estimation of the predistorter and can be later adaptively trained along with the predistorter as the training progresses. Thus, of the 2L + 1 sigma points {Wk,i }2L+1 i=0 , the a priori mean Wk,0 is passed through the actual HPA and the remaining points {Wk,i }2L+1 i=1 are propagated through the estimate of the HPA. 4. SIMULATION RESULTS In this section, simulation results are presented for predistortion of nonlinear HPA with and without memory using the algorithm presented in the previous section.

1

1

Real part Imaginary part

0.8 0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −1 0

Restored sinusoidal waveform and the negligible imaginary part in the PD output plotted in Fig. 2(b) implies excellent compensation of the AM and PM distortions of the nonlinear HPA. The spectrum of the nonlinear HPA output signal plotted in Fig. 3(a) supports the claim that the HPA nonlinearity introduces predominantly odd-order distortions in the spectrum. The spectrum of the output signal for the PDHPA combination is as shown in Fig. 3(b). This demonstrates the validity of the proposed PD scheme which mitigates the spectral regrowth, thereby avoiding the ACI. The most dominant harmonic which is of the third order is suppressed by almost 40 dB. This is about 10 dB better than the suppression achieved by the method proposed in [16], where an EKF is used to design the predistorter.

Real part Imaginary part

0.8

−0.8 20

40

60

80

100

120

140

−1 0

20

40

60

80

100

120

140

time

time

(a)

(b)

Fig. 2. Output signal amplitudes (a) without predistorter, (b) with predistorter. 4.1. Predistortion of Memoryless nonlinear HPA

−20

−20

In narrow-band communications the HPAs exhibit frequencyindependent characteristics, which can be characterized by their AM/AM A(·) and AM/PM Φ(·) effects. The AM/AM function indicates the relationship between the input and the output signal amplitudes. Likewise, the AM/PM function represents the relationship between the input amplitude and the output phase. The model proposed by Saleh [1] is used to simulate the AM/AM and AM/PM effects. The Saleh model is widely used because of its greater accuracy and simplicity compared to other available models for the memoryless HPA. The amplitude and phase transfer functions are given as follows.

−30

−30

A(r)

=

Φ(r)

=

2r (1 + r2 ) π 2r2 6 (1 + r2 )

where r is the amplitude of the input signal to the amplifier. The simulation results for a sinusoidal input signal of normalized frequency 0.04, with its amplitude scaled to 1, are shown in Fig. 2 and Fig. 3 . The schemes described in Section 3 are employed to estimate the coefficients of the predistorter and the HPA model. The HPA polynomial is modeled using both the even and the odd orders while the PD is modeled using only the odd orders. The simulation parameters for the predistorter and the HPA model were set as Nn = 5, Nl = 1, Mn = 5 and Ml = 1. The initial ˆ 0 is chosen randomly as a Gaussian parameter estimate w random vector with zero mean and covariance Pw0 = I. The measurement noise covariance matrix in (8) was fixed to Rke = 10−6 . The process noise covariance is updated as Rkr = (λ−1 RLS − 1)Pwk , where λRLS = 0.999 is the forgetting factor. The amplifier introduces amplitude(distorted waveform) and phase distortions (non-zero imaginary part). The real and imaginary parts of the HPA output without (Fig. 2(a)) and with (Fig. 2(b)) predistortion are plotted.

Spectrum (dB)

0 −10

Spectrum (dB)

0 −10

−40 −50 −60

−40 −50 −60

−70

−70

−80

−80

−90 −100 0

−90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normalized frequency

(a)

0.4

0.45

−100 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Normalized frequency

(b)

Fig. 3. Output Spectrum (a) without predistorter, (b) with predistorter. 4.2. Predistortion of nonlinear HPA with memory In this subsection, the memory effects are examined with a Weiner system which consists of a the nonlinear HPA preceded by a linear dynamical system [6]. The Saleh model is used for the memoryless nonlinear HPA and the linear filter considered is a 3-tap FIR filter with the coefficients [0.7692 0.1538 0.0769]. All the simulation parameters are initialized as in the previous subsection, except that in this case Nl = 4 and Ml = 4 to deal with the nonlinear ISI. To verify the performance of the proposed predistortion scheme, a 16-QAM signal, with equally probable symbols, is generated. It is then passed through the above mentioned linear FIR filter and the Saleh modeled HPA in cascade. The output back-off (OBO), defined as the ratio of the amplifier output saturation power to the average transmitted signal power, is set to 6 dB. The channel is assumed to be additive white gaussian noise (AWGN). The transmitted signal constellations without and with predistorter are shown in Fig. 4(a) and Fig. 4(b), respectively. The predistorter successfully compensates for the constellation warping and restores the signal constellation. The BER curves shown in Fig. 4(c) further confirm that the predistorter designed using the UKF is able to effectively linearize the nonlinear HPA with memory. Without predistortion, the uncompen-

6. REFERENCES

4

4

3

3

2

2

1

1

Im

Im

sated nonlinear ISI results in an “error floor”, whereas with predistortion the BER performance is close to the ideal case of linear amplifier.

0

0

−1

−1

−2

−2 −3

−3 −4 −4

−3

−2

−1

0

1

Re

2

3

4

−4 −4

−3

−2

−1

0

1

2

3

Re

(a)

(b)

Fig. 4. Transmitted signal constellations (a) without predistorter, (b) with predistorter. OBO=6 dB.

5. CONCLUSION In this paper, a new digital baseband predistortion structure using the UKF was proposed to linearize time-varying nonlinear HPAs with memory. The SM-NLMS algorithm was employed to estimate the HPA model, which was essential for the propagation of sigma points in the UKF. The predistorter and the HPA model were designed as coefficients of memory polynomials. The UKF has greater accuracy and robustness; this contributes to better convergence properties of the predistorter. The performance of the proposed predistorter was demonstrated from simulation results. The new predistortion method performs considerably well in either case of memoryless amplifier or amplifier with memory, resulting in close to linear operation of the PD-HPA combination. This enables the amplifier to be operated at a small output back-off which leads to improved power efficiency. 0

10

Nonlinear HPA No PD PD−HPA combination Ideal amplifier

−1

10

−2

10

−3

BER

10

−4

10

−5

10

−6

10

−7

10

6

7

8

9

10 11 Eb/No (dB)

12

13

14

15

Fig. 5. BER curves demonstrating the performance of the proposed PD. OBO = 6 dB

4

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