PREDISTORTER DESIGN EMPLOYING PARALLEL PIECEWISE LINEAR STRUCTURE AND INVERSE COORDINATE MAPPING FOR BROADBAND COMMUNICATIONS

Transcription

1 4th European Signal rocessing Conference (EUSICO 26), Florence, Italy, September 4-8, 26, copyright by EURSI REDISTORTER DESIGN EMLOYING RLLEL IECEWISE LINER STRUCTURE ND INVERSE COORDINTE MING FOR BRODBND COMMUNICTIONS Mei Yen Cheong, Stefan Werner, Timo I. Laakso, Juan Cousseau 2 and Jose L. Figueroa 2 SMRD, Center of Excellence, Signal rocessing Laboratory Helsinki University of Technology, Finland 2 CONICET - Department of Electrical Engineering and Computer Science, Universidad Nacional del Sur, Bahía Blanca, rgentina BSTRCT This paper proposes a low-complexity predistorter (D) for compensation of both the M/M and the M/M conversions with memory. The nonlinear power amplifier () is modeled as a Wiener type nonlinearity. The quasi-static nonlinearities are modeled using a class of piecewise linear (WL) functions. The WL function facilitates an efficient D identification algorithm. The proposed algorithm involves a novel inverse coordinate mapping (ICM) method that maps the nonlinear characteristics of the to that of the D, and parameter estimations that do not require matrix inversion. The indirect learning architecture is used to provide an on-line compensation of the memory effect of the. Simulation results show that the D that compensates also the M/M distortion performs significantly better than one that considers only the M/M nonlinearity. The proposed D is also shown to outperform the orthogonal polynomial D in both adjacent channel interference suppression and inband distortion compensation.. INTRODUCTION In communications systems, the power amplifier () in the transmitter is an important component. It compensates the signal attenuation caused by path loss between the transmitter and the receiver. ower efficiency of the s in the base stations contributes directly to the network operation cost savings. Due to the intrinsic characteristics of existing designs, power efficient s are nonlinear. Nonlinear causes in-band and out-of-band distortion to the transmitted signals, which lead to bit-error-rate (BER) degradation and adjacent channel interference (CI), respectively. In broadband systems, the nonlinear s also exhibit memory effects. In practice, the operating point of the is chosen to optimize between power efficiency and linearity. For system transmitting constant envelope signals, the optimum operating point is easier to determine. However, the trend for the future mobile communications systems is to employ linear modulation scheme such as M-QM and multi-carrier systems such as orthogonal-frequency division multiplexing (OFDM) and multi-carrier code division multiple access (MC-CDM) to gain spectral efficiency and robustness towards interference and multipath effects. While linear modulation produces non-constant modulus constellation, multicarrier system produces high peak-to-average-power ratio (R) signals. Due to the high R, the optimum operating point for s are more difficult to determine without sacrificing power efficiency by means of large power backoff. The spectral mask requirements for the 3G WCDM system require an adjacent channel power ratio (CR) of more than 45 dbc and more than 6 dbc at the mobile terminals and base stations, respectively []. To comply with this requirement, linearization techniques are required when power efficient s are employed. Digital baseband predistorter (D) is a promising linearization technique due to its effectiveness and low-cost implementation. Volterra model based Ds have been proposed for compensation of nonlinear with memory in [2] [3] [4]. However, the designs are complex and render computationally demanding implementations. s alternatives to the Volterra model, the Wiener and Hammerstein models have been considered for the modeling and compensation of nonlinear system with memory. The most popular models considered for modeling the static nonlinear block of these systems are polynomial models [3] [5] [6]. The drawback of using a conventional polynomial for nonlinear system modeling is the numerical problem associated with matrix inversions in the parameter estimation. To alleviate the numerical problems, orthogonal polynomials have been proposed for D designs in [7] [8]. In this paper, the simplicial canonical piecewise linear (SCWL) function [9] is used to parameterize the static nonlinearities of the and D. The SCWL function was first proposed for D design in []. The D identification method in [] is limited to normalized M/M responses of the and does not compensate for M/M distortion. This paper extends the application of the SCWL function to model quasi-static nonlinearities, i.e., both the M/M and M/M functions. novel inverse coordinate mapping (ICM) method is introduced. The ICM method does not require the M/M response to be normalized as was the case in []. The D is also extended to a semi-adaptive algorithm, where the memory of the is compensated adaptively using the indirect learning architecture [2]. Finally, the performance of the SCWL D is compared to that of the orthogonal D proposed in [7]. In Section 2 the models for the to be linearized and the D are described. The development of the novel ICM method and the estimation of the SCWL coefficients are presented in Section 3. The SCWL function is briefly reviewed in this section. The D identification procedure is discussed in Section 4 followed by simulation examples in Section 5. Conclusions are drawn in Section 6.

2 4th European Signal rocessing Conference (EUSICO 26), Florence, Italy, September 4-8, 26, copyright by EURSI M/M functions, respectively. The quasi-static nonlinearities of the D are identified using the ICM method that is presented in the following section. Figure : Wiener model Figure 2: Hammerstein model D 2. SYSTEM MODEL This section introduces the model and the D model used in our simulation examples. The input, output and intermediate signals of these block models are defined in this section. 2. ower amplifier model Fig. illustrates the Wiener model to be compensated. It is composed of a linear dynamic block followed by a quasistatic nonlinear block. The input, output and intermediate signals of the model are denoted by x, y and w, respectively. The linear dynamic block is modeled with an FIR filter h = [h h h Lh ] T of length L h. The quasistatic block is modeled with a Saleh model [] given by y( w ) = ( w )exp [ j(θ + ( w ) ], () where θ is the phase of w. The functions ( w ) and ( w ) are respectively the M/M function and M/M function of the given by and ( w ) = α a w +ξ a w 2 (2) ( w ) = α p w 2 +ξ p w 2. (3) where α a is the small signal amplitude gain factor, ξ a is the amplitude compression factor, α p is the phase gain factor, and ξ p is the phase compression factor. 2.2 redistorter model The Hammerstein model D which has the same functional blocks as the but cascaded in reverse order, is shown in Fig. 2. The quasi-static block is modeled with two SCWL functions, one for the M/M function and the other for M/M function (see Section 3 for details). n FIR filter g = [ g g g Lg ] T is used to model the linear dynamic block. The signals of the D are given by v = f (D) ( u )exp ( j(φ + f (D) ( u )) ), x = g H v, where φ is the phase of the input signal u. The functions f (D) ( u ) and f (D) ( u ) are the SCWL M/M and (4) 3. THE D IDENTIFICTION LGORITHM This section develops the ICM method and the estimation of the SCWL coefficients that are used to identify the D. First, the SCWL function is briefly reviewed. Thereafter, the ICM method is developed by exploiting the linear affine property of the SCWL function. Finally, the estimation method for the SCWL coefficients is discussed. 3. SCWL function The SCWL function introduced in [9] uses σ predefined partition points to divide the input space to (σ ) linear affine regions. The output of the function is given by f [ s(k) ] = Λ T[ s(k) ] c, (5) where s is the amplitude of the input signal to the function, and Λ [ s(k) ] = [ λ [s(k)] λ σ [s(k)]] T is the basis function vector, and c = [c c σ ] T is the SCWL coefficient vector. The basis function [9] is given by λ i [ s(k) ] = { 2 ( s(k) βi + s(k) β i ), s(k) β σ 2( βσ β i + β σ β i ), s(k) > β σ, where β i, (i =,...,σ), is the ith predefined partition point. We define the basis function matrix as a collection of basis function vectors evaluated over a block of input data s = [s() s(k)] in a matrix as (6) L(s) = [Λ[s()] Λ[s(k)]] T. (7) Then the block of output data can be expressed in matrix form as f(s) = [ f[s()] f[s(k)]] T = L(s)c. (8) 3.2 Inverse coordinate mapping (ICM) When a static nonlinearity of a is modeled using a WL function, each subregion of the function is represented by a linear affine function. If the D s nonlinearity is modeled using another WL function, the subregions of the D are also linear affine functions. linear affine region can be solely represented by the two boundary coordinates that define the region. Based on this property, the ICM method is developed by finding the relationship between the partitions points of the and that of the D. This relationship is represented in the proposed mapping matrix that translates the WL partitions of the to that of the D. The ICM matrix that maps the s nonlinearity to the D s nonlinearity for any arbitrary linearized gain, is developed based on the following conditions. C The output of the D- cascade is a linear amplified version of the input signal with linearized gain G. C2 The output space of the D must coincide with the input space of the. In order to fulfill C, let the gain of the in an arbitrary WL region be ρ. Then the gain of the D, denoted K, resulting in a D- output gain G is K = G ρ. (9)

3 4th European Signal rocessing Conference (EUSICO 26), Florence, Italy, September 4-8, 26, copyright by EURSI.9 Coordinate projections Output amplitude y Nonlinearity b i b i+ Inverse nonlinearity Desired linear gain line Input amplitude u b i b i+ Figure 3: The Inverse-Coordinate Mapping method To fulfill C2, we search for the input u i to the D that results in the output that coincides with the ith partition point β i of the. It follows that G ρ u i = β i, u i = G (ρβ i). () Notice that ρβ i is the s response f(β i ) = ρβ i. Thus, we deduce from (), that the D s input partition points can be mapped from that of the s output as β i = G f(β i). () Condition C2 also indicates that the signal range and the partition points of the D s output are identical to that of the s input. Thus, the mapping is simply a copy of the s input partition points. s a result, we can define an ICM matrix for any arbitrary desired linear gain G as [ ] Q = G. (2) The ICM matrix maps the s M/M coordinates to those of the D s as b i = Qb i, as illustrated in Fig Estimation of the SCWL coefficients When (8) is evaluated over all the partition points of the SCWL function, the equation can be rewritten as f(β) = L(β)c, (3) where L(β) is a nonsingular matrix. The coefficient vector can then be estimated as c = L (β) f(β). (4) s shown in [], L (β) is a tridiagonal matrix with elements given by the the known user defined partition sizes. Thus, no matrix inversion is needed for the estimation of c in (4). Figure 4: Identification procedure of the M/M SCWL- D s can be seen from (4), the M/M and M/M are functions of the input amplitude only. Thus, the SCWL functions f ( ) ( ) (β) and f (β) share a same basis function matrix L(β). The simultaneous estimation of the SCWL coefficient vectors c a and c p for the M/M and M/M, respectively, can be written in matrix form as [c a c p ] = L (β) [ f ( ) ( ) (β) f (β)]. (5) 4. THE HMMERSTEIN MODEL D IDENTIFICTION The Hammerstein D is identified in two steps. First, the quasi-static model of the D is identified in a training session. Thereafter, the linear dynamics of the D is identified adaptively using the indirect learning architecture [2] [7]. The following subsections provide the details of the two steps. 4. D s nonlinear block The quasi-static block of the is identified as two parallel SCWL functions by exciting the with a powerswept single-tone signal u s. The output amplitude and output phase shift caused by the M/M and M/M conversions are fitted to the SCWL functions f () ( u s ) and f () ( u s ), respectively. Then the coordinates corresponding to the responses of the M/M and M/M at the predefined SCWL partition points β = [β β σ ] are determined. The coefficient vectors are then identified using (5). Thereafter, the ICM matrix (2) is used to map the s M/M coordinates to that of the D s as shown in Fig. 3. The partition points of the D and the corresponding M/M responses b = [ β (β )] T are obtained. The SCWL coefficient vector κ a for the M/M D is identified using (4) as f D κ a = L (β ) f (D) (β ). (6) Note that the D s M/M function has to take into account the total effect of the M/M conversion caused by the and the D itself. Therefore, the M/M model of the f () ( ) is fitted to the phase shift at the output of the M/M predistorted as shown in Fig. 4. The coefficient vector c p is obtained using (4). Since the desired linearized output phase shift of the D- cascade is zero the M/M coefficient of the D is obtained as κ p = c p. (7)

4 4th European Signal rocessing Conference (EUSICO 26), Florence, Italy, September 4-8, 26, copyright by EURSI u redistorter Quasi-static NL-D (fixed) v Linear filter g x y /G y G 4 ˆx Linear filter ĝ ˆv Training branch Quasi-static NL-D (fixed) MSE after convergence 5 Figure 5: The indirect learning architecture for memory compensation 4.2 D linear dynamic block fter the training session an indirect learning architecture as shown in Fig. 5 is used to compensate the s memory iteratively. broadband multisine signal is used to emulate a general multicarrier signal transmitted by the system. The static nonlinear D blocks in the architecture is fixed while the LMS algorithm updates the linear filter coefficients in the training branch. The filter coefficients are then copied to the D. The LMS update equation for the linear filter is ĝ(k) = ĝ(k )+ µε ˆv(k), (8) where µ is the step size that controls the convergence of the algorithm, and ˆv(k) is a vector with the L g most recent output samples. 5. SIMULTIONS The described in Sec. 2 is the device under test (DUT) in our simulations. The static nonlinear block parameters are α a = 2, ξ a =, α p = 3.5 and ξ p = 25. The FIR filter parameters of the linear dynamic part is h = [ ] T [3]. The Hammerstein model SCWL-D, identified as described in Section 4 is used to compensate the DUT. The number of WL partitions examined for the modeling of the M/M and M/M functions are σ = 6, 8, 2, 5 and 8. The length of the FIR filter that models the linear dynamic block is L h = 3 and L h = 5. The influence of the number of WL partitions on the accuracy of the model is shown in Fig. 6. The mean-squared error (MSE) between the training branch and the D output after the FIR filter has converged is used as a performance metric. It is observed that by increasing the number of partition points from 6 to 2, the MSE dropped by almost 2 orders of magnitude. However, increasing the number of partition points beyond 5 does not provide significant improvements. In the following, the SCWL-D with σ = 2 and σ = 5 are used in the performance evaluation of out-of-band and in-band distortion, respectively. The CR improvement using the SCWL-D with σ = 2 that compensates nonlinearity with memory is compared to an uncompensated in Fig. 7. The input back off (IBO) of the multisine excitation signal is 9 db. The IBO is defined as ( i,sat ) IBO = log, (9) i where i and i,sat are the mean input signal power and the input signal power at saturation, respectively. In order to The number of SCWL partiton points σ Figure 6: Influence of the number of partition points σ on model accuracy SD db only with IBO = 5 db With SCWL FIR D Frequency Hz x 8 Figure 7: The CR improvement using a SCWL-D with σ = 2 compared to an uncompensated. match the in-band signal attenuation caused by the D, the input signal to the is further backed off by 5 db, i.e., a total of IBO = 4 db, for a fair comparison of CRs. The average CR improvement achieved is 2 db, resulting in an average CR of 5 dbc. Fig. 8 shows the CI suppression obtained by an SCWL-D that compensates only the M/M nonlinearity and an SCWL-D that compensates both M/M and M/M nonlineairities. It is observed that the quasi-static D performs db better than the M/M D. Next, the performance of the SCWL-D is compared to that of the orthogonal polynomial D (O-D) proposed in [7]. Fig. 9 shows the average out-of-band power of the output compensated by the SCWL-D and the O-D. The out-of-band power is calculated by averaging the adjacent channel power over the third-order and fifth-order intermodulation distortion (IMD) zones. The results show that the SCWL-D outperforms the 3rd-order, 5th-order and 7thorder O-D by approximately db when the corresponding SCWL-D uses σ = 6, 8 and 2 respectively. It is also observed that a 5-partition point SCWL-D results in a further CI suppression of approximately 3 db. To illustrate the compensation of in-band distortion, Fig. shows the bit-error rate (BER) performance generated with 2 OFDM symbols, each with 28 subcarriers employing 6-QM modulation received over an WGN channel. The IBO of the OFDM signal used in the BER evalution is 8 db. The quasi-static SCWL-D and the O- D achieved similar performance in compensation of in-band distortion. Both achieved BER in the range of 3 and 4 for SNR above 5 db and 8 db, respectively. Further improvement in BER performance is observed when

5 4th European Signal rocessing Conference (EUSICO 26), Florence, Italy, September 4-8, 26, copyright by EURSI Normalized SD db QS D M/M D Frequency Hz x 8 Figure 8: erformance comparison of an SCWL M/M D and an SCWL quasi-static D. Mean out of band power σ=6 3rd order 5th order σ=8 σ=2 Model order 7th order SCWL D O D Figure 9: erformance comparison of the SCWL-D and the O-D. [Note: The x-axis scale does not indicate equal complexity model order of the two Ds.] the SCWL-D that compensates nonlinearity with memory is employed. pproximately db gain is obtained at SNR = 3 db, and gradually increase to a gain of 2 db at SNR = 8 db. The BER performance of the dynamic SCWL-D at IBO = 8 db attained the performance of the system with a linear in an WGN channel. 6. CONCLUSIONS low-complexity predistorter (D) designed using the simplicial canonical piecewise linear (SCWL) function that compensates for nonlinear power amplifiers () with memory is proposed. The properties of the SCWL function is exploited to develop a simple and efficient D identification algorithm. The low-complexity D identification algorithm BER σ=5 5 SCWL D, quasi static SCWL D, dynamic O D WGN channel SNR db Figure : Bit-error rate performance comparison of the SCWL-D and the O-D. includes a novel inverse coordinate mapping method, and an efficient method for the estimation of the SCWL coefficients that does not require matrix inversions. Unlike polynomial models, the SCWL parameter identification does not impose numerical problems. In addition, the modeling capability of the SCWL function is highly accurate even with reasonably few number of partitions. Simulations verified that the SCWL-D outperforms the O-D in adjacent channel interference suppression as well as in-band distortion compensation. We also shown that compensation of memory effects improves both CR as well as BER performance more significantly as compared to Ds that compensates only static nonlinearities. REFERENCES [] 3G, TS 25.4 V , Specifications, 25. [2] C. Eun and E. J. owers, new Volterra predistorter based on the indirect learning architecture, IEEE Trans. Signal rocessing, vol. 45, no., pp , Jan [3], predistorter design for a memoryless nonlinearity preceded by a dynamic linear system, in roc. IEEE Global Telecommunication Conf. (GLOBECOM 95), vol., 995, pp [4] I.-S. ark and E. J. owers, new predistorter design technique for nonlinear digital communications channels, in roc. The International Symposium on Signal rocessing and Its pplication ISS, Gold Coast, ustralia, 996, pp [5] L. Ding, G. T. Zhou, D. R. Morgan, Z. Ma, J. S. Kenney, J. Kim, and C. R. Giardina, robust digital baseband predistorter constructed using memory polynomials, IEEE Trans. Commun., vol. 52, no., pp , Jan. 24. [6] H. W. Kang, Y. S. Cho, and D. H. Youn, On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter, IEEE Trans. Commun., vol. 47, no. 4, pp , pr [7] R. Raich, H. Qian, and G. T. Zhou, Orthogonal polynomials for power amplifier modeling and predistorter design, IEEE Trans. Veh. Technol., vol. 53, no. 5, pp , Sep. 24. [8]. Midya, olynomial predistortion linearizing device, method, phone and base station, United States atent, No. 6,236,837, July 998. [9]. Julián,. Desages, and O. gamennoni, High-level canonical piecewise linear representation using a simplicial partition, IEEE Trans on Circuits and Systems I: Fundamental Theory and pplications, vol. 46, pp , pr [] M. Y. Cheong, S. Werner, J. Cousseau, and T. I. Laakso, redistorter identification using the simplicial canonical piecewise linear function, in roc. 2th International Conference on Telecommunications, ICT 25, Cape Town, South frica, May 25. [].. M. Saleh, Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers, IEEE Trans. Commun., vol. 29, pp , Nov. 98.