Generalized Rational Functions for Reduced-Complexity Behavioral ...

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 2, FEBRUARY 2014

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Generalized Rational Functions for Reduced-Complexity Behavioral Modeling and Digital Predistortion of Broadband Wireless Transmitters Meenakshi Rawat, Member, IEEE, Karun Rawat, Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE, Shubhrajit Bhattacharjee, and Henry Leung, Member, IEEE Abstract— In this paper, we present and analyze rationalfunction-based digital predistortion (DPD) of transmitters for broadband applications where system noise and prominent memory effects contribute to the overall nonlinearity of the system. The performance is reported for simulation and measured results for gallium nitride (GaN)-based class-AB and laterally diffused MOS (LDMOS)-based Doherty power amplifiers (PAs) using three different wideband code division multiple access signals with peak-to-average-power ratios of around 10 dB. The performance of the proposed model, in terms of normalized meansquare error, adjacent channel power ratio, matrix condition number, and coefficient dispersion, is compared against those of a memory polynomial (MP) model and a previously proposed rational-function-based model. It is shown by simulation and measurement that the previously proposed absolute-term denominator rational functions have limitations in the inverse modeling needed for DPD. A new variation of the rational function is proposed to alleviate this limitation. Depending on the type of PA and signals, a floating-point operation reduction of 8%–38% is reported as compared with a low-complexity MP model. Index Terms— Adaptive filters, digital signal processing, modeling, nonlinear dynamical systems, predistortion, 3G mobile communication, transmitters.

I. I NTRODUCTION

T

HE power amplifier (PA) is the most nonlinear component in a wireless transmitter, which is responsible for most of the out-of-band distortion in the presence of envelopevarying third- and fourth-generation broadband signals [1]. Accurate estimation of PA nonlinearity has been attempted using subsampled temporal data in order to avoid inherent modulator imperfections during modeling [2]. Moreover,

Manuscript received February 16, 2013; revised May 17, 2013; accepted May 22, 2013. Date of publication September 4, 2013; date of current version January 2, 2014. This work was supported in part by the iRadio Laboratory team, in part by the sponsors of the laboratory, in part by the Alberta Innovate Technology Futures, in part by the Chair Program, in part by the Canada Research Chair Program, in part by the Canada Foundation of Innovation, and in part by the National Science and Engineering Council of Canada. The Associate Editor coordinating the review process was Dr. John Lataire. M. Rawat, F. M. Ghannouchi, and H. Leung are with the University of Calgary, AB T2N1N4, Canada (e-mail: [email protected]; [email protected]; [email protected]). K. Rawat is with the Indian Institute of Technology, Delhi 110 001, India (e-mail: [email protected]). S. Bhattacharjee is with SNC Lavalin T&D, Calgary, AB T2B 3G4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2013.2278598

frequency-domain approaches have also been reported by taking the best linear approximations [3], [4]. A rather simple approach requires the measurement of time-domain input/output voltage waveforms and behavioral modeling in the digital domain [1], [5]. Digital-domain behavioral modeling has a more practical and popular application as indirect digital predistortion (DPD) to suppress spectral regrowth and signal distortion, where an inverse digital model based on input/output waveforms is synthesized and applied in the baseband domain before the PA, to compensate for the PA nonlinearity. At present, digital-domain compensation is favored in many fields [6], [7] due to its compatibility with softwareenabled transmitters and the simplicity of its implementation through the utilization of high-speed digital signal processors and digital-to-analog converters [8]. Moreover, software-based implementation offers high reconfigurability for any change in transmitter hardware and configuration, and also supports the concept of software-defined radio at the transmitter end for effective communication [9]. In this paper, we propose a novel rational-function behavioral model for digital-domain baseband modeling of PAs along with the DPD application. Section II provides a brief introduction to PA behavioral modeling, the indirect DPD concept, the need for inverse modeling, and the metrics used for PA behavioral modeling and DPD in this paper. Section III describes the experimental setup, signals, and devices used for the measurements and indirect DPD implementation. Section IV describes the rational-function-based modeling and DPD, and presents the proposed model. Section V shows various modeling results for DPD. It is proven that the model offers certain advantages over memory polynomial (MP) models, such as reduced number of coefficients and low complexity, while achieving comparable or better modeling performance for practical PAs. Section VI reports the experimental results achieved for the proposed model compared with a state-of-the-art low-complexity MP model. II. PA B EHAVIORAL M ODELING AND I NVERSE M ODELING FOR I NDIRECT D IGITAL P REDISTORTION A PA should provide a linearly amplified version of an input signal at a higher power. A typical application of a PA is

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Fig. 1.

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 2, FEBRUARY 2014

Block diagram of a transmitter, and data extraction for behavioral modeling in the digital domain.

found in a wireless communication transmitter, where the PA is the last active component in the transmitter chain before the antenna. Fig. 1 shows a schematic of a transmitter system [10]. Data bits are encoded into digitally modulated waveforms. In the baseband, the input signal x(n) = I (n) + j Q(n) is comprised of complex data, as shown in Fig. 1. Digital data are converted to an analog signal and up-converted to the required carrier frequency, which is then amplified using the PA and transmitted to the receiver. The PA contributes the most distortion to the transmitter output, due to its nonlinearity at high input power levels; therefore, modeling of the PA is an important technique, which is used in the simulation of the physical layer for a complete communication system. Behavioral modeling of the PA captures the nonlinear relation between the complex input data x(n) and the complex output data y(n) available in the baseband, as shown in Fig. 1. Any variation in the signal envelope affects the amplitude and phase of the output signal, which are referred to as the amplitude modulation/amplitude modulation (AM/AM) and the amplitude modulation/phase modulation (AM/PM) characteristics of the PA. An effective behavioral model should imitate the AM/AM and AM/PM characteristics accurately. The normalized meansquare error (NMSE) is the most popular metric for determining the modeling performance of a PA, which is given by ⎛ N ⎞  2 |e(n)| ⎜ ⎟ ⎜ k=0 ⎟ (1) NMSEdB = 10 log10 ⎜ ⎟ N ⎝ ⎠ 2 |ymeas. (n)| k=0

where N is the total number of samples, and e(n) = ymeas. (n) − yest. (n)

(2)

is the complex error between the measured output ymeas. (n) and the estimated model output yest. (n), for any sample n. The NMSE is considered a measure of in-band performance [11]. The adjacent channel error power ratio (ACEPR) is used as a metric to assess the out-of-band modeling performance [11].

1 G

Fig. 2.

Principle of indirect learning architecture for DPD.

Its expression is given as follows [11]: ⎛ ⎞ f 1 −+ BW f 2 ++ BW 2 2 ⎜ |E( f )|2 d f + |E( f )|2 d f⎟ ⎜ ⎟ BW BW ⎜ ⎟ f −− f +− 1 1 2 2 2 ⎟ ACEPR = ⎜ ⎜ ⎟ 2⎜

f2

⎟

Ymeas. ( f )2 d f ⎝ ⎠ f1

(3) where E( f ) and Ymeas. ( f ) are the discrete Fourier transforms of the error signal e(n) and ymeas. (n), respectively; f1 and f2 define the limits of the output signal band in the frequency domain; and BW defines the bandwidth of the adjacent channel at  frequency offset from the carrier frequency. Digital behavioral models can also be used for DPD to compensate for the distortion from the PA nonlinearity. Fig. 2 shows the indirect learning architecture (ILA) for DPD. The input signal is passed through predistorter K with parameters α, creating the predistorted signal u(n). This predistorted signal u(n) is passed through the PA to provide output y(n), which is passed through predistorter K to create u. ˜ ILA works on the principle that, when u(n) = u(n), ˜ x(n) = y(n)/G or y(n) = Gx(n), which represents the required linear power amplification. For the first iteration, u(n) = x(n), and DPD is a model that maps x(n) = K (y(n)/G), where K represents the predistorter nonlinear function. The modeling of the complex input, with

RAWAT et al.: GENERALIZED RATIONAL FUNCTIONS FOR REDUCED-COMPLEXITY BEHAVIORAL MODELING

respect to the normalized PA output, is the inverse of the PA behavioral model; therefore, DPD modeling is also known as inverse modeling. This inverse modeling is the foremost requirement for the successful application of a DPD model. Note that the AM/AM characteristics of a PA show compression at high power levels, while the AM/AM characteristics of an inverse PA model should have expansion characteristics at high power levels; therefore, the requirements for PA modeling and PA inverse modeling are quite different. Apart from PA nonlinearity, prominent memory effects due to the use of wider band signals further contribute to signal deterioration at the output of the transmitter [12]. Several behavioral models and inverse models for nonlinear PAs with memory effects have been proposed in the literature, offering a good inverse model for indirect DPD in most cases [13]–[22]. Among these models, the lookup table (LUT)-based models are generally assumed to be simple to implement; however, optimal spacing and proper bin selection are required for best results [14], [15]. Moreover, when memory effects are considered, a large number of variables need to be stored for cascaded LUTs [16], [17]. Among the parametric models, the Wiener and Hammerstein models assume that memory effects are linear and, hence, can be separated from the memoryless nonlinear behavior of the PA [18]. However, it has been established that this is not a valid assumption for all PAs [19]. Volterra models [20] and neural network models [21], [22], with theoretical support for their modeling properties, can be considered the most successful models, but their good performance is shadowed by their computational complexity. The Volterra series [9] is given as follows: y(t) = H1 [x(t)] + H2 [x(t)] + · · · Hn [x(t)] + · · · (4)  ∞  ∞ Hn [x(t)] = ··· h n (τ1 , · · · τn )x(t − τ1 ) . . . −∞

−∞

× x(t − τn )dτ1 . . . dτn

(5)

where integral Hn is called an nth-order Volterra operator. Although theoretically justified, the Volterra model, including all nonlinear kernels, is too complex in practice to be identified correctly. Some pruning approaches have, therefore, been proposed in order to keep only the required kernels, including the generalized MP [19] and pruned Volterra models [20], [23], where off-diagonal terms are pruned to eliminate ineffective terms and keep only the effective terms. However, such pruning approaches have several practical shortcomings. For instance, the Volterra model given by (4) and (5) can extend to infinite values, and many selections may provide similar solutions, and all combinations need to be considered. Hence, a general direction for complexity reduction is hard to define. A pruned model can, therefore, create large errors, even with a small variation of the PA characteristics, due to the mutual dependencies of cross terms and the numerical instabilities arising from the large size of the matrices. These reductions of Volterra models may prove to be excellent, from a behavior modeling perspective, for a model that only needs to be identified once. However, DPD applications

487

Fig. 3. Measurement setup for data acquisition, preprocessing, and DPD evaluation.

may require frequent updating of the model coefficients and processing of the input signals when the PA characteristics change, due to environmental and self-heating effects. Therefore, in addition to good modeling capability, simplicity of the model and a low number of coefficients are also desirable requirements [24]. Due to these limitations, the simplified Volterra model with only diagonal terms, also known as the MP model, seems to be an attractive choice, due to its reasonable accuracy, compact parametric representation, simplicity, availability of noniterative least squares (LS) solutions, and the ability to model nonlinear and linear memory effects [25]. The MP model is given as [19] y(n) =

M P

a p,m x(n − m) |x(n − m)| p

(6)

p=0 m=0

where P and M are the nonlinearity order and memory depth, respectively. In this paper, we propose a rational-function-based model that aims to achieve a similar modeling performance as the MP model with fewer coefficients. This model is presented in Section IV, along with previous rational-function-based attempts for PA modeling. III. D EVICE U NDER T EST AND E XPERIMENTAL S ETUP Fig. 3 shows the PA characterization setup used for data extraction in forward and inverse PA modeling. A baseband signal is uploaded into the vector signal generator (Agilent E4438C), where the signal modulation and frequency upconversion to the required carrier frequency are carried out. The radio frequency (RF) modulated signal, which is to be transmitted using an antenna, is amplified via the PA. For modeling and DPD, this PA output is down-converted, demodulated, and captured in the digital domain using a vector signal analyzer (Agilent 4440). The captured data are sent to the digital processing unit (laptop/handheld computer), using a general-purpose interface bus (GPIB). More details on each component of experimental setup can be found in [26]. The baseband output data are time aligned with the input data by using cross correlations between sampled input and the output data. Normalized cross correlation ρx y between two

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waveforms x and y is calculated as follows: ρx y =

L 1 (y(n) − μ y )(x(n+τ ) − μx ) L σ y σx

(7)

n=0

where μx and μ y represent the mean values of the input (x) and output (y) waveforms, respectively; σx and σ y represent the standard deviations of the x and y waveforms, respectively; L is the length of waveforms x and y and τ is the sample delay between the two waveforms. ρx y is observed with the increase in τ values, and the τ value for which ρx y is the maximum represents the sample delay between x and y. The ratio of this sample delay and sampling rate provides time delay between two baseband waveforms. Therefore, the accuracy of the delay estimation is limited by the sampling rate of the device. A detailed discussion on the fine-tuning of the delay alignment can be found in [27]. This time-aligned input–output data can be used for PA modeling. Digital signal processing for time alignment and model identification are carried out in MATLAB software. In DPD, the output digital data are normalized with the small-signal gain of the PA. The normalized output data are used as the model input, and the original input data are used as the desired output to achieve an inverse model. The coefficients of each layer are copied to the predistorter, as shown in Fig. 3. Using this experimental setup, the modeling performance of the proposed model was assessed for two different PAs. 1) A commercial laterally diffused MOS (LDMOS) Doherty PA (Powerwave Technologies) with a saturation power of 300 W at a carrier frequency of 2.14 GHz. The PA had a gain compression of 1.7 dB [with an overall gain variation of 3.5 dB as shown in Fig. 4(a)] and a phase compression of 33° when driven by a signal of peak-to-power ratio (PAPR) of approximately 10.5 dB. The drain voltage for both the peaking and carrier amplifiers was 28 V and the total drain current was 1.7 A. 2) A 10-W gallium nitride (GaN) class-AB PA biased at a drain voltage and current of 28 V and 780 mA, respectively, at a center frequency of 3.5 GHz. The PA had a gain compression of 4.3 dB and a phase compression of 30°. The AM/AM and AM/PM characteristics of these amplifiers are shown in Fig. 4. Clearly, the Doherty PA had more nonlinearity with a slight expansion and then compression, in terms of AM/AM; whereas the class-AB PA had a simpler yet highly compressed AM/AM, leading to high distortion. Both PAs had a phase compression of approximately 30°. Three different wideband code division multiple access (WCDMA) signals with 10 and 15 MHz of bandwidth and a PAPR of 10.5 dB were used in the experiments. All the WCDMA signals had a chip rate of 3.84 MHz, and the chips/slot length was 2560. Each chip had 24 samples. The sampling frequency was 92.16 MHz, and the signal was captured for 2-ms time frame for data acquisition.

(a)

(b)

Fig. 4. AM/AM and AM/PM characteristics. (a) Doherty PA. (b) ClassAB PA.

IV. R ATIONAL -F UNCTION -BASED A PPROACHES FOR PA M ODELING A rational function is defined as the ratio of two power polynomials given by J 

x J (n)

a0 + a1 x(n) + . . . + a J i=0 = y(n) = K b0 + b1 x(n) + . . . + b K x K (n)  j =0

ai x i (n) (8) bj

x j (n)

where x(n) is the input and y(n) is the output of the rational function at instance n. In [28], the AM/AM and AM/PM were captured separately using (8), where x(n) represented the input signal amplitude for a traveling wave tube (TWT) amplifier and a solid-state PA (SSPA). The amplifiers were considered static nonlinear systems; therefore, memory effects were not considered. Predistortion was reported only for simulations, approximating the SSPA with a cubic spline and the TWT amplifier with Saleh’s model [29], [30]; however, the effects of measurement errors and noise were neglected, which are necessary to validate a model’s usefulness in a real practical scenario. Saleh’s model [31] was used to represent the TWT amplifier itself, which is a form of a rational function; and, it is expected that the use of another rational function should easily be able to model DPD. A modified Saleh’s model was also proposed for SSPAs, but it did not compensate for memory effects and it targeted the behavioral modeling of the PA [32]. Considering the importance of memory effects, a behavioral model based on a rational function was proposed in [33], where memory effects were included as time delay taps in a complex-term numerator and absolute-term denominator. The absolute-termdenominator rational function (ADRF) proposed in [33] is given by  K n  Mn 2i m n =0 ai,m o x(n − m) |x(n − m)| i=0 (9) y(n) =  K d  Md 2 j +1 1 + j =0 m d =0 b j,m d |x(n − m)| where x(n) and y(n) are the input and output baseband signals, respectively; K n and K d are the nonlinearity orders for the numerator and denominator, respectively; and Mn and Md are the memory depths used in the numerator and denominator, respectively. It was proposed in [33] that (9) converges to a finite value for large x(n) only when K n = K d = K and Mn = Md = M. Using a power series expansion, (9) can be rewritten as

RAWAT et al.: GENERALIZED RATIONAL FUNCTIONS FOR REDUCED-COMPLEXITY BEHAVIORAL MODELING

(a)

489

(b)

Fig. 5. NMSE performances of the MP and ADRF models of a Doherty PA. (a) PA modeling. (b) PA inverse modeling. X-axis represents nonlinearity order P for MP model and K for ADRF model.

follows:

⎛

y(n) = ⎝1 −

K M

⎞ b j, m d |x(n − m d )|2 j +1 + · · ·⎠

j =0 m d =0

×

M K

ai,m n x(n − m n ) |x(n − m n )|2i .

(10)

i=0 m n =0

Thus, (10) contains odd- and even-order monomials, such as ai,m n b j,m d x(n − m n ) |x(n − m n )|2i |x(n − m d )|2 j +1 , depending on the values of i and j . Theoretically, the use of only odd-order terms should provide good baseband modeling performance; however, it has been proven that the inclusion of even terms is essential for PA modeling and DPD, providing a richer basis function set and leading to much better in-band and out-of-band performances with an overall reduced complexity [34], [35]. The performance of the ADRF model has been reported for PA behavioral modeling [33], which captures the AM/AM compression characteristics; however, indirect DPD (explained in Section II) for transmitters needs inverse modeling of the PA, which requires modeling of expansion characteristics. To the best of the authors’ knowledge, the use of a rational function for inverse modeling (modeling of expansion characteristics) and DPD application has never been reported for practical SSPAs containing measurement noise and memory effects. Fig. 5 shows PA modeling and PA inverse modeling performances, in terms of NMSE, for the ADRF model proposed in [33] and the MP model. The x-axis of the graph represents nonlinearity order P for the MP model

and K n = K d = K for the ADRF model. The PA under test is the Doherty PA [AM/AM and AM/PM characteristics shown in Fig. 4(a)] driven with a WCDMA11 signal (twocarrier WCDMA). We can perceive the following points from Fig. 5. 1) Convergence curves and the best performance achieved for both models were different for the PA model and the PA’s inverse model. 2) Inclusion of delay taps M led to lower NMSE performances for both models, but as memory length (i.e., number of delay taps) increased, the performance improvement diminished. No performance improvement was observed for M > 2. 3) The best NMSE performances of the ADRF and MP models for PA behavioral modeling were almost similar. 4) For PA inverse modeling, the ADRF model provided better NMSE performance for low nonlinearity order than that of the MP model. However, the best NMSE performance of the ADRF model was ultimately not as good as the MP model for inverse modeling performance. 5) The MP model tended to converge to a constant value with the increase in the nonlinearity order; however, the curve for the ADRF model started diverging after converging to a best performance. Indeed, the accuracy of the inverse model is of utmost importance for a successful DPD operation. Moreover, it is also desired that the number of coefficients be as low as possible, in order to reduce the number of floating-point operations (FLOPs) per each input signal datum, leading to overall savings on computation during transmission. In this

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paper, we present a novel and modified rational-function model for application in DPD as a compromise between the ADRF model and the MP model, which provides better performance than the ADRF model with lower number of coefficients than the MP model. A. Rational Function Denominator Model

With

Memoryless

Flexible-Order

It can be perceived from (10) that, for any selected nonlinearity order K , the ADRF model attempts approximation of a power series with a nonlinearity order of 2 × (2K +1), which provides the capability of modeling a highly nonlinear system using a low-power-term rational function. However, the ADRF model proposed in [33] has the following constraints. 1) It includes only odd-order power terms in the numerator and denominator. 2) The numerator and denominator have the same nonlinearity orders (K n = K d ). 3) The numerator and denominator terms have the same memory depths (Mn = Md ). 4) The model denominator is an absolute term that does not include signal phase information. Due to the rigidity of the ADRF model, the best PA inverse modeling performance achieved with the ADRF model is not as good as that of the MP model. Therefore, we propose a dynamic rational function with memoryless flexible-order denominator (DRF-MFOD) model, which has memory effects in the numerator polynomial, but has memoryless flexible nonlinearity order in the denominator polynomial. Both the numerator and denominator have even- and odd-order term polynomials, resulting in a richer basis function domain to be utilized in modeling. The proposed DRF-MFOD model is given as Nn  M 

y(n) =

j =0 m=0

a j,m x(n − m) |x(n − m)|

1+

Nd 

j

(11) bi x(n)

|x(n)|i

i=0

where x(n) and y(n) denote the nth sample of the PA input and output complex signals, respectively; M is the memory depth; Nn and Nd are the nonlinearity orders in the numerator and denominator polynomials, respectively; and a j,m and bi denote complex coefficients for the numerator and denominator, respectively. Equation (11) represents the multiplications of static (memoryless) and dynamic (with memory) parts, after the power series expansion of the denominator can be rewritten as ⎞ ⎛ Nn M j a j,m x(n −m) |x(n −m)| ⎠ y(n)=⎝ j =0 m=0

⎞ ⎞2 ⎛ N N d d ⎟ ⎜ ×⎝1− bi x(n) |x(n)|i + ⎝ bi x(n) |x(n)|i⎠+. . .⎠. ⎛

i=0

i=0

From (12), we find that the proposed model approximates a case that has an Nn -order nonlinearity for memory-effect terms and an (Nd + Nn )-order nonlinearity term for the static branch of the MP. Using this representation, there is flexibility to adjust the nonlinearity orders for the static and memory-tap branches, which can be searched iteratively. The absence of memory terms in the denominator reduces the total number of coefficients to identify for the model. An iterative search is part of the model selection for the device under test. This iterative search is carried out in a digital signal processor in an offline mode, which requires a 3-D search for the denominator and numerator orders and the memory depth. This is achieved by the creation of a model for different values of Nn , Nd , and M and the comparison of their NMSE performances. An initial value of 0 is assumed for each parameter, which is then incremented in steps of 1; and, performance improvement is observed. The dimension values for which performance improvement becomes constant or deteriorates with increases in the dimensions are selected to be appropriate model dimensions. Once the model dimensions are selected, they are kept constant for any particular PA. Any fluctuations in PA behavior due to environmental changes can be adjusted by updating model coefficients without changing model dimensions. One can argue that, unlike the ADRF model, the proposed DRF-MFOD model is not bounded for large x(n) values. However, the power series convergence criterion with an infinite limit is not applicable for PA modeling and inverse PA modeling, due to the following practical constraints. 1) Practical PAs have a drive limit, which is the input power saturation point (Pin,sat ). After this limit, PAs are overdriven and may get damaged; therefore, input signals always have finite values bounded by the Pin,sat value and can never tend to infinite/very large values for nonlinear nonclipped PA modeling applications. 2) For DPD applications, output power is normalized with the gain of PA; therefore, the normalized output signal is a distorted version of the input signal within the power ranges of the actual input signal. Due to the PA input power drive limitation, as described in the first point of this list, the input for the inverse model can also only have finite values. 3) It has been established that input values in either PA or inverse PA modeling are finite values; therefore, the radius of convergence is a finite value depending on the values of x(n), as well as on the complex coefficients.

B. Rational-Function Parameter Extraction The DRF-MFOD model given by (11) can also be written in a recursive form y(n) = −y(n)

bi x(n) |x(n)|i

i=0

+ (12)

Nd

Nn M j =0 m=0

j

a j, m x(n − m) |x(n − m)| .

(13)

RAWAT et al.: GENERALIZED RATIONAL FUNCTIONS FOR REDUCED-COMPLEXITY BEHAVIORAL MODELING

⎡

−y(n)x(n) ⎢ −y(n − 1)x(n − 1) ⎢ ⎢ . U1 (n) = ⎢ ⎢ . ⎢ ⎣ . −y(n − L)x(n − L) ⎡

x(n) ⎢ x(n − 1) U2 (n) = ⎢ ⎣ ... x(n − L)

... −y(n)x(n) |x(n)| Nd ... −y(n − 1)x(n − 1) |x(n − 1)| Nd . . . . . . ... −y(n − L)x(n − L) |x(n − L)| Nd

... x(n) |x(n)| Nn ... x(n − M) ... x(n − 1) |x(n − 1)| Nn ... x(n − 1 − M) ... ... ... ... x(n − L) |x(n − L)| Nn ... x(n − L − M)

Moreover, parameter extraction is possible by rewriting (13) in a matrix form y = AφDRF−MFOD

(14)

where y is the output vector with the dimension of L × 1 and L is the length of the training data; and, φDRF−MFOD is the coefficient vector, which is defined as φDRF−MFOD = [b0 , . . . , b Nd , a00, . . . , a Nn 0, a01, . . . , a Nn 1, . . . , a Nn M ]T. (15) In (14), the observation matrix is denoted by A = [U1 U2 ], where U1 is a Vandermonde matrix containing recursive terms and is given as (16), shown at the top of the next page. The Vandermonde matrix U2 is similar to the MP model and is given by (17), shown at the top of the next page. From (14)–(17), it is clear that the model identification equation (16) is linear in its parameters and φDRF−MFOD is obtained using the LS solution of φˆ DRF−MFOD = (AT A)−1 AT y, which is implemented according to the singular value decomposition method. Based on an individual PA’s data, the denominator complex coefficient automatically takes small values to keep the denominator stable. This is possible because the PA model is created within the data range that falls in the PA power handling range. Once the coefficients are extracted, the output is calculated using (11). It has been reported in [36] that recursive models are closer to the physical analogy of the PA and provide robust modeling. In recursive models, the coefficients are extracted using the stored values of x(n) and y(n). However, for practical applications such as DPD, y(n) at the present instance is not known beforehand, which is a limitation when applying any recursive model. However, for a rational-function-based model, the output is calculated using (11). Therefore, even with recursive model extraction, DPD application does not require the current output; and, the predistorted inputs can be synthesized independently using only the input signals. V. I NVERSE M ODELING R EQUIREMENT FOR I NDIRECT L EARNING D IGITAL P REDISTORTION A. In-Band and Out-of-Band Modeling Performances Figs. 6 and 7 show the NMSE performances of the DRFMFOD model in comparison to the MP and ADRF models

491

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ... x(n − M) |x(n − M)| Nn ... x(n − 1 − M) |x(n − 1 − M)| Nn ⎥ ⎥ ⎦ ... ... ... x(n − L − M) |x(n − L − M)| Nn

(16)

(17)

for the two PAs (characteristics shown in Fig. 4) and two signals. Iteratively, the best memory length is found to be 3 for all models. This is achieved by increasing the memory length (delay taps) and observing the corresponding decrease in the NMSE performance until no performance improvement is observed for a fixed nonlinearity order. For a memoryless case, the MP model is represented as a function of the nonlinearity order (P), and the ADRF model is represented as a function of K . However, the DRF-MFOD model requires a two-dimensional search for the best values of Nn and Nd . Once the model dimensions are known for all models, we introduce delay taps for the memory effects and observe the decrease in the NMSE with the increase in the number of the delay taps. The best memory-depth length/number of delay taps is achieved when no further performance improvement is observed. Fig. 6 shows the NMSE performances of the MP, ADRF, and DRF-MFOD models for different Nn and Nd values for the memory length of 3. The general trend of the NMSE for both PAs shows that the ADRF converges for a very small nonlinearity order, followed by the DRF-MFOD and MP models. The DRF-MFOD model converges faster than the MP model, even for Nd = 0, due to the recursive nature of the model. For the class-AB PA and WCDMA101 signal, as shown in Fig. 6(a), any increase in Nd in the lower Nn region leads to a better performance. As the denominator does not contain delay taps, it leads to fewer coefficients. As an example, the MP model converges at N = 8 (8 × 4 = 32 coefficients), while a similar performance for the DRF-MFOD model can be achieved with Nn = 4 and Nd = 5 (4×4+5 = 21 coefficients). However, for the same PA when the WCDMA111 signal is used [Fig. 7(a)], the denominator terms in the DRF-MFOD model have a negligible effect. It is also noted that, for the class-AB PA with a WCDMA111 signal [Fig. 7(a)], the ADRF model also performs poorly. One can observe from Figs. 6 and 7, that a maximum improvement of 7 dB is achieved at Nn = 3 for the class-AB PA with the DRF-MFOD model compared with the MP model; and, a maximum improvement of 6 dB was achieved at Nn = 5 for the Doherty PA for the DRF-MFOD model compared with the MP model for both the WCDMA101 and WCDMA111 signals. Clearly, due to the rational function properties, the maximum improvement is achieved for smaller nonlinearity

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(a)

(b)

Fig. 6. NMSE performances for a WCDMA101 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

(a)

(b)

Fig. 7. NMSE performances for a WCDMA111 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

orders, as with the ADRF model; however, the transition is much smoother with the DRF-MFOD model. Although the ADRF model also seems to provide a comparable NMSE for the WCDMA101 signal, this only validates the in-band performance, as the NMSE is

more favorable to high-power data and is considered as an in-band performance figure of merit [11]. Therefore, as a metric to assess the out-of-band modeling performance, the ACEPR has much more significance for DPD applications [11].

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(a)

493

(b)

Fig. 8. ACEPR performance for a WCDMA101 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

(a)

(b)

Fig. 9. ACEPR performance for a WCDMA111 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

Figs. 8 and 9 show the ACEPR performances for the two different PAs and signals, and it can be observed that the DRF-MFOD and MP models have an improvement in the

ACEPR of 7 dB over that of the ADRF model for the class-AB PA and a WCDMA101 signal. It is interesting to note that, although each model converges to a different final

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TABLE I C OMPLEXITY AND F LOPS P ER D ATUM

10

10 MP ADRF D RF -M FO D (N =0) d

Coefficient Dispersion

10

D RF -M FO D (N =1)

8

d

D RF -M FO D (N =2) d

D RF -M FO D (N =3) d

10

6

10

4

D RF -M FO D (N =4) d

D RF -M FO D (N =5) d

100

1

0

2

4 6 8 N onlinearity O rder

10

12

Fig. 11. Coefficient dispersion for the class-AB PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves. Fig. 10. Matrix condition number for the class-AB PA. X-axis represents nonlinearity order P for MP model, K for ADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

performance, there are similar convergence trends for the NMSE and ACEPR. B. Coefficient Dispersion and Matrix Conditioning The MP and the proposed DRF-MFOD models require an LS solution for model coefficient identification. Matrix conditioning is an important factor in the determination of the accuracy of the inversion operation in the LS algorithm, which is defined as [37] Condition Number =

λmax λmin

(18)

where λmax and λmin are the maximum and minimum singular values of the matrix to be inverted.

A poorly conditioned matrix makes the pseudoinverse calculation very sensitive to slight disturbances. It can also be shown that the condition number is an indicator of the transfer of error from the matrix to the solutions. As a rule of thumb, if a condition number is 10n , then one can expect to lose at least n digits of precision in solving the system [38]. For example, the numerical precision is around 10−7 for a single-precision floating-point calculation. Any condition number greater than 102 leads to an approximate precision of 10−5 and any condition number greater than 104 leads to an approximate precision of 10−3 . Therefore, the condition number should be as low as possible. Another factor considered in this paper is the coefficient dispersion, which is obtained as a ratio of maximum to minimum absolute coefficient value. If the coefficients are much dispersed, the accuracy level of coefficients in FPGA is compromised to cover the whole range of coefficients [39].

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(a)

(b) Fig. 12. Characteristics of the linearized class-AB PA. (a) AM/AM. (b) AM/PM.

495

Fig. 14. Predistorter performance with ACPR correction for the Doherty PA. (a) WCDMA101 signal with P = 12 for MP model and Nn = 10 and Nd = 0 for DRF-MFOD model. (b) WCDMA111 signal with P = 12 for MP model and Nn = 10, Nd = 0 for DRF-MFOD model.

coefficient dispersion with respect to the nonlinearity order (P, Nn , and K for MP, DRF-MFOD, and ADRF models, respectively) for the class-AB PA with a WCDMA101 signal. It can be observed that both the matrix condition numbers were similar, or slightly lower for the DRF-MFOD model (Nn = 4 and Nd = 5) than for the MP model (P = 8), for the best NMSE and ACEPR performance. It is interesting to note that the ADRF model matrix condition number trends are very similar to those of the MP model; however, the value of the condition number is larger by a factor of 100 compared with the MP model. The proposed DRF-MFOD model has a unique pattern where the matrix condition number is almost constant for Nn < (Nd + 1). C. Complexity Comparison

Fig. 13. Predistorter performance with ACPR correction for the class-AB PA. (a) WCDMA11 signal with P = 8 for MP model and Nn = 4 and Nd = 6 for DRF-MFOD model. (b) WCDMA111 signal with P = 8 for MP model and Nn = 6 and Nd = 6 for DRF-MFOD model.

The Vandermonde-type matrices of (16) and (17) are notoriously ill-conditioned for larger matrix sizes [40]. It has been claimed that the conditioning of a Vandermonde matrix increases exponentially with its order [41]. It may seem that the DRF-MFOD model matrix, which contains two Vandermonde-like matrices, may lead to a large condition number; however, as the model converges for a smaller numerator order, the resultant matrix properties are similar to or better than that of the MP model. Fig. 10 shows the matrix condition number for three models while Fig. 11 shows the

Table I shows the total number of FLOPs required for calculating the output of the ADRF, MP, and the proposed DRFMFOD models. The equivalent numbers of FLOPs for each operation are based on [42]. The ADRF model has the lowest number of FLOPS however; the out-of-band performance of the ADRF model is unacceptable for DPD applications. The proposed DRF-MFOD model needs fewer FLOPs for similar performance, and therefore, can be substituted for the MP model for providing similar performance with lower complexity. It can be observed from Table I that the DRFMFOD model requires 182 fewer FLOPs than the MP model for a class-AB PA. Similarly, in the case of a Doherty PA, the DRF-MFOD model requires 44 fewer FLOPs than the MP model. Note that these calculations are done only for single-input sample; however, for continuous transmission, each input needs to be processed through the predistorter, and the saving in the number of FLOPs will be much significant cumulatively. In the above case, we assumed that the system is under controlled environmental conditions and that the inverse model coefficients need to be extracted only once to identify the predistorter digital model. However, where frequent processing is required due to changes in the environmental conditions, established linear adaptive algorithms such as the recursive LS (RLS) and the least mean squares can be applied to (13)

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TABLE II M EASURED ACPR A CHIEVED W ITH L INEARIZED PAS

instead of LS. The details of these algorithms can be found in [27]. VI. A PPLICATION IN D IGITAL P REDISTORTION Fig. 12 shows the AM/AM and AM/PM characteristics of the class-AB PA, the DPD (inverse PA characteristics) response using the proposed model, and the resulting linearized response. The excess scattering in the gain and phase characteristics is due to memory effects, which give multiple values of gain and phase for a single-input value. As can be seen from the figure, the DPD derived using the proposed model is an accurate inverse of the PA characteristics; and, the resulting DPD and PA characteristics are linear. The performance of the proposed DPD scheme is verified for the above-mentioned PAs and signals, in terms of adjacent channel power ratio (ACPR) performance at ± 5, ± 10, and ± 15 MHz, according to industrial norms [43]. The power spectral density curves are shown in Figs. 13 and 14, which illustrate the performance of the proposed DRF-MFOD model for the class-AB and Doherty PAs with various WCDMA signals. The WCDMA signals used in the experiment have a PAPR of approximately 10.5 dB, and the output power for each PA is maintained at an almost constant level. These figures also show curves without DPD for comparison. It can be seen that the static model (i.e., memoryless model) fails to compensate for memory effects in each case, whereas the MP and DRF-MFOD models provide reasonable and comparable performances, with the DRF-MFOD model performing slightly better than the MP model. Table II shows the ACPR performance for the different PAs and signals. The ACPR is a frequency-domain evaluation metric used to assess the DPD performance. It is defined as the ratio of the power of the output signal in an adjacent channel to the power of one of the in-band carriers and is given as  ω2  2 ω1 |YDPD ( f )| d f (19) ACPR dBc = 10 log10 ω4 2 ω3 |Yin ( f )| d f where YPSD denotes the power spectrum density of the linearized output signal; w1 and w2 denote the lower and upper

frequency limits of the adjacent channel, respectively; and w3 and w4 designate the lower and upper frequency limits for the in-band channel, respectively. For comparison purposes, the ACPR values of the memoryless, MP, and the proposed models are reported in Table II. The ADRF model’s values are not included, as the model failed to provide acceptable predistortion performance. This has already been established for out-of-band performance in terms of ACEPR in Figs. 8 and 9. The output power back-off, with respect to the saturation point, is selected to achieve the desired linearization performance. This is achieved by adding the gain expansion of the predistorter to the PAPR of the signal, while ensuring the linearized PA is not overdriven. It can be seen from Table II that, in the case of the class-AB amplifier driven by a WCDMA11 signal, an improvement of 5 dB in the ACPR is achieved compared with that of the memoryless model at an offset of 5 MHz. It can also be noticed that in all cases, the ACPR values at ± 5, ± 10, and ± 15 MHz remain well below the spectrum mask specified for WCDMA signals [43]. VII. C ONCLUSION In this paper, we proposed a novel dynamic rationalfunction-based PA model, taking memory effects of the PA into consideration. The modeling and predistortion performances were evaluated using class-AB and Doherty PAs with two- and three-carrier WCDMA signals as excitation signals. A study of the proposed model and comparisons with a previously proposed rational-function and memorypolynomial techniques showed that the proposed model could achieve comparable performance in terms of NMSE and ACEPR, but with a reduced number of coefficients in comparison with the established MP model. The matrix conditioning and coefficient dispersion performance of the new model were also maintained, similar to that of the MP model. ACKNOWLEDGMENT The authors would like to thank the reviewers, who enhanced the quality of this manuscript immensely with their thoughtful detailed comments.

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Meenakshi Rawat (S’09–M’13) received the B.Tech. degree in electrical engineering from the Govind Ballabh Pant University of Agriculture and Technology, Pantnagar, India, in 2006, and the Ph.D. degree from the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, in 2012. She is currently a Post-Doctoral Research Fellow with the iRadio Laboratory, University of Calgary. She was with Telco Construction Equipment Co. Ltd., Jamshedpur, India, from 2006 to 2007, and Hindustan Petroleum Corporation Ltd., Mumbai, India, from 2007 to 2008. She is currently a Reviewer for several international transactions and journals. Her current research interests include signal processing for software defined radios, communications, and microwave active and passive circuit modeling using nonlinear models and neural networks.

Karun Rawat (M’08–S’09–M’13) received the B.E. degree in electronics and communication engineering from Meerut University, Meerut, India, in 2002, and the Ph.D. degree from the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, in 2012. He is currently an Assistant Professor with the Indian Institute of Technology, Delhi, India. He was a Scientist with the Indian Space Research Organization from 2003 to 2007. He was a PostDoctoral Research Fellow with the iRadio Laboratory, Schulich School of Engineering, University of Calgary, from April 2012 to April 2013. He is a reviewer of several well known journals. His current research interests include microwave active and passive circuit design and advanced transmitter and receiver architecture for software defined radio applications. Dr. Rawat was the leader of a University of Calgary team that won first prize and the Best Design Award in the Third Annual Smart Radio Challenge in 2010, conducted by the Wireless Innovation Forum.

Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07) is currently a Professor and iCORE/CRC Chair with the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, and the Director of the Intelligent RF Radio Laboratory. He has held numerous invited positions at several academic and research institutions in Europe, North America, and Japan. He has provided consulting services to a number of microwave and wireless communications companies. His research activities have led to over 500 publications, two books, and ten U.S. patents (three pending). His current research interests include microwave instrumentation and measurements, nonlinear modeling of microwave devices and communications systems, design of power and spectrum efficient microwave amplification systems and design of intelligent RF transceivers, and SDR Radio systems for wireless and satellite communications.

Shubhrajit Bhattacharjee received the M.Sc. degree in electrical and computer engineering from the University of Calgary, Calgary, AB, Canada, in 2011, and the B.Eng. degree from the University of Pune, Pune, India. He is currently a Project Engineer with SNC Lavalin T&D, Calgary. His current research interests include RF transceiver impairment mitigation, software defined radio, transmitter linearization, digital signal processing, and channel modeling.

Henry Leung (M’90) received the Ph.D. degree in electrical and computer engineering from McMaster University, Hamilton, ON, Canada. He is a Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. He was with the Defence Research Establishment Ottawa, Ottawa, Canada, where he was involved in the design of automated systems for air and maritime multisensor surveillance. His current research interests include chaos, computational intelligence, data mining, information fusion, nonlinear signal processing, multimedia, sensor networks, and wireless communications.