Information criteria based optimal structure identification of RF power amplifier models

Abstract

This article presents a novel information criterion based optimal model parameter selection algorithm for behavioral modeling of Radio Frequency Power Amplifiers (RF PAs). The proposed approach uses Particle Swarm Optimization (PSO) along with the Information Criterion (IC) based cost functions for determining the most parsimonious model from all the available combinatorial models. The proposed technique thereby helps in deriving complexity reduced models without compromising modeling accuracy. The validation of the proposed approach was carried out by modeling a GaAs based PA driven by a 20-MHz generic random input signal. It was shown that, the model performance was maintained while its complexity in terms of number of coefficients was reduced by around 35% in the considered cases. In addition, the proposed PSO based approach helps in deriving the most parsimonious PA model in a very short amount of time compared to the conventional sweep technique.

Keywords

RF Power amplifiers particle swarm optimization (PSO)information criterion (IC)maximum entropy (ME)

1. Introduction

The Radio Frequency Power Amplifier (RF PA) is one of the major component in the transmitter of any modern wireless communication system. However, they are inherently non-linear and exhibit memory effects when excited by wideband or multi carrier signals such as Code Division Multiple Access (CDMA) and Orthogonal Frequency Division Multiplexing (OFDM). These unwanted characteristics of RF PAs distort the modulated signal, causing degradation in the performance of bit error rate (BER) in the in-band region and spectral regrowth in the adjacent side bands. Therefore, accurate behavioral modeling of RF PAs is essential for evaluating the impact of the signal distortions caused due to their adverse characteristics on the overall performance of wireless systems. Moreover, inverse behavioral modeling of PAs finds application in digital pre-distortion (DPD) [1] techniques which compensate the impairments caused due to non-linear PAs.

Numerous PA models based on Volterra series [2], Artificial Neural Network (ANN) [3, 4], Wiener [5, 6] and Hammerstein [7 –9] models are available in the literature. Although the Volterra model is efficient in capturing the behavior of PA more accurately, deriving its kernels is computationally intensive and also consume lot of resources. Hence, simplified versions of Volterra such as Memory Polynomial (MP) [10] and Generalized Memory Polynomial (GMP) [11] models are widely used. The Volterra and its truncated versions (MP and GMP models) are parametric in nature and these parameters depends on the non linear order and memory length of the PA. Hence, identifying the required parameters for accurate PA modeling is a challenging task.

Nowadays, certain applications require RF integrated circuits with multiple PAs which together support multiple frequency bands. The chip may also support multiple protocols such as WLAN, Bluetooth, Zigbee or proprietary schemes. Depending on the characteristics of PA, various aspects of transmission (for example, in WLAN, the bandwidth/FFT size which effects the Peak-to-Average-Power-Ratio (PAPR) and/or the MCS (modulation and coding selection)) are chosen. Hence, it may serve well to save a model of the PA on chip. And a single model even for a single PA for all frequency bands or even powers may not suffice. Instead multiple models of the PA are required. In order to save this information, precious memory is used. By reducing the parameters of the PA behavioral model without significant degradation in modeling accuracy, a substantial reduction in the memory to save this information is possible. Hence, there is a need to derive PA models with reduced model dimension [12 –14].

The conventional method of identifying the accurate PA model structures, for instance [11], is based on sweeping the parameters over an initialized range, then quantify the accuracy of each model. Finally the model parameters corresponding to the highest accuracy (minimum modeling error) are selected. However, this requires enormous amount of computation time. Hence, several optimization algorithms like genetic (GA) [15], hill-climbing (HC) [16] and particle swarm optimization (PSO) [17 –19] are proposed. All these methods perform a guided deterministic or random search to identify the best model and hence reduce the model estimation time. In addition, applying Information criteria (IC) [20] which considers both model performance and complexity can help in identifying the PA model with reduced dimension without compromising the model performance [18]. In this work, we propose a modified PSO along with the Maximum Entropy (ME) based Information Criteria [24] to identify the most parsimonious PA model. Compared to the conventional PSO [18], our proposed approach assigns specific range for the individual parameters of the considered PA model and hence helps in reducing the search space for the PSO algorithm. This in-turn helps in deriving accurate PA models with substantial reduction in CPU time compared to the conventional sweep based method.

This paper is organized as follows: Section 2 discusses the state of the art PA behavioral models, Section 3 discusses the algorithms used for model parameter identification with different search criteria (conventional and the proposed information criteria based), Section 4 presents the simulation results and Section 5 marks the conclusion.

2. Behavioral Modeling of RF Power Amplifiers

Some of the widely used models for predicting the behavior of PA are: Volterra Series, Memory Polynomial (MP) Model and Generalized Memory Polynomial (GMP) model.

2.1. Volterra Series

Volterra series is the most popular mathematical tool used to model the output of RF PAs that are characterized by non linear systems with memory. The discrete baseband equivalent form of Volterra consists of a sum of multi dimensional convolutions and can be written as [2], $\begin{matrix} y_{v} (n) = & \sum_{k = 1}^{K} \sum_{q_{1} = 0}^{Q} \sum_{q_{2} = 0}^{Q} . . \sum_{q_{k} = 0}^{Q} h_{k} (q_{1}, q_{2} . . q_{k}) \prod_{j = 1}^{k} x (n - q_{j}) \end{matrix}$ (1) where x (n) and y_v (n) are the complex baseband input and Volterra based output samples respectively. K represents the degree of non linearity, Q is the memory depth of the PA and h_k (q₁, q₂ . . . q_k) represents the kernels or basis of the Volterra model. For a given K and Q, the size of the Volterra kernels (model dimension)is given as: $P = (Q + 1)^{1} + (Q + 1)^{2} + . . . (Q + 1)^{K}$ (2)

From Eq. 2, it can be inferred that, the Volterra model is highly complex and its practical usage is limited due to the large kernel size and estimation time. Hence, the applications of Volterra based models are limited to weakly non linear systems with less memory effects.

2.2. Memory Polynomial (MP) Model

A simplified version of the Volterra series that consists of only its diagonal terms is the memory polynomial model. The MP modeled output [10], for any complex baseband input x (n) is given by: $y_{MP} (n) = \sum_{k = 1}^{K_{a}} \sum_{q = 0}^{Q_{a}} a_{k, q} x (n - q) | x (n - q) |^{k - 1}$ (3) where K _a and Q _a are the non linear degree and memory length of the PA corresponding to the diagonal terms respectively and a _k,q represents the model coefficients. The model dimension for the MP model is given by, $P = K_{a} (Q_{a} + 1)$ (4)

2.3. Generalized Memory Polynomial (GMP) model

The GMP model is an enhanced version of the MP model and it consists of both the diagonal and cross terms (leading and lagging envelope terms) from the expanded Volterra series. The GMP model performs better than the MP model and the improvement in performance comes at the cost of higher order model dimensions. The GMP based model [11] output for any baseband sampled input data x (n) can be given as:

$\begin{matrix} y_{GMP} (n) & = \sum_{k = 1}^{K_{a}} \sum_{q = 0}^{Q_{a}} a_{k, q} x (n - q) | x (n - q) |^{k - 1} \\ + \sum_{k = 1}^{K_{b}} \sum_{q = 0}^{Q_{b}} \sum_{l = 1}^{L_{b}} b_{k, q, l} x (n - q) | x (n - q - l) |^{k} \\ + \sum_{k = 1}^{K_{c}} \sum_{q = 0}^{Q_{c}} \sum_{l = 1}^{L_{c}} c_{k, q, l} x (n - q) | x (n - q + l) |^{k} \end{matrix}$ (5) where K_a, Q_a are the index arrays for the diagonal terms as in MP model, K_b, Q_b, L_b are the index arrays for the lagging terms, K_c, Q_c, L_c are the index arrays for the leading terms of the signal envelope and a_k,q, b_k,q,l, c_k,q,l are the model coefficients of the GMP model. The model dimension for the GMP model is given by $P = K_{a} (Q_{a} + 1) + K_{b} (Q_{b} + 1) L_{b} + K_{c} (Q_{c} + 1) L_{c}$ (6)

3. Structure Identification Algorithms

Sweep method is one of the widely used approach for model parameter selection where the range of all parameters are swept from a minimum to a maximum value and the model which gives minimum Normalized Mean Square Error (NMSE, a model error metric) is chosen as the best model. However, due to the enormous time that it takes to determine the best model, sweep technique is not a preferred choice for large parametric models such as GMP model. Hence there is a need to apply optimization techniques to determine the best model in a short amount of time. Literature presents various optimization procedures including GA, HC and PSO. PSO is a kind of stochastic optimization method inspired by the social behavior of animals or birds and can search the best solution from a large search space. Although there are several alternatives with in the domain of stochastic optimization, the PSO has special features like faster rate of convergence and higher success rate compared to other optimization algorithms. Hence, it is widely used in Global optimization problems. Every model order selection algorithm requires a range of initial values to be defined for the model parameters first and then the algorithm chooses the best model based on some predefined criterion, which is defined in terms of its cost function. In PA behavioral modeling, the parameter range for any model is governed by several parameters including modulation scheme, PAPR, operating power and bandwidth of the input signal.

3.1. Sweep Method

In this approach, an exhaustive search is performed over the initialized model parameter range according to the defined cost function to determine the best model. The algorithm performs the model validation step for all the possible parameter combinations within the defined range and then selects the model with minimum cost function as the best. The sequence of steps involved in the sweep method are given by the pseudo code shown in Table 1.

Table 1
Pseudo code for sweep algorithm

Start Sweep

Initialize $x_{j} = [\min_{j} : \max_{j}]$ // Input

where j=1,2…d, d is the number of model parameters;

Define the Cost Function (f);

Form x = [x₁, x₂, . . . . x_d]; // Form all possible parameter combinations

T = Number of all possible combinations for x; // Total combinations

fort = 1 : T

Evaluate the Cost Function (f(x)) for all possible T sets; // Evaluation

end

Best cost = Min(f(x=x^)); // Identify the best cost

Return x^ and Best cost; // Output

End Sweep

3.2. Particle Swarm Optimization (PSO)

Unlike the full-scale search procedure used in sweep method, PSO performs a guided random search to select the parameters of the best model. The conventional PSO considers a swarm of particles with certain position and velocity, which are d-dimensional vectors. The variable d represents the number of model parameters in our case (like d=2 for MP model and d=8 for GMP model). The position of each particle represents a potential solution in the provided search space and particle velocity control the convergence speed of the algorithm. The conventional PSO used for PA modeling [18] assigns the same range for all the parameters of the PA model. This makes the search mechanism highly complex and also consumes lot of resources and time to determine the best model. Our proposed approach overcomes this drawback by introducing a variant of PSO, where different ranges are assigned for each individual parameters of the considered PA model and thereby reduces the search space. The range of each parameters are defined based on the input signal parameters like, modulation scheme, PAPR, operating power and bandwidth of the baseband signal used for PA modeling. In the proposed approach, each potential solution is represented by a particle and each particle has d-different positions (x_i,j, j = 1, 2 . . . d) and velocities (v_i,j, j = 1, 2 . . . d), where i and j represents the index of the particle and dimension respectively and d is the number of model parameters. The positions and velocities of the particles are updated at each iteration according to Eqs. 7 and 8. The optimization goal is to find the parameter set (x^*) that minimizes the required cost function. At t + 1^th iteration, the updated individual velocity of the particle is: $\begin{matrix} v_{i, j}^{t + 1} = w v_{i, j}^{t} + c_{1} rand (1, 1) (l_{best, i, d}^{t} - x_{i, j}^{t}) + \\ c_{2} rand (1, 1) (g_{best, d}^{t} - x_{i, j}^{t}) \end{matrix}$ (7) where $v_{i, j}^{t}$ and $x_{i, j}^{t}$ are the j^th velocity and position of the i^th particle respectively at the t^th iteration. w is the inertia coefficient, c₁ (cognitive component) and c₂ (social component) are the acceleration coefficients, rand (1, 1) is the 1-dimensional random variable uniformly distributed between [0,1]. $l_{best, i, d}^{t}$ is the local best of the i^th particle at t^th iteration and $g_{best, d}^{t}$ is the global best of all the particles at t^th iteration. The corresponding new updated position is given by $x_{i, j}^{t + 1} = x_{i, j}^{t} + v_{i, j}^{t + 1}$ (8)

The flow of steps involved in the proposed variant of PSO are illustrated in the pseudo code given in Table 2. The algorithm returns x^*, the optimal model order set and the best cost (minimum cost function), where x^* = [K_a, Q_a] for the MP model and x^* = [K_a, Q_a, K_b, Q_b, L_b, K_c, Q_c, L_c] for the GMP model respectively.

Table 2

Pseudo code for the modified PSO based algorithm

Start PSO
Initialize population size(Pop.Size), number of iterations (Max.Iter),	// Inputs
positions (d), $x_{\min_{j}}$ , $x_{\max_{j}}$ , where (j = 1, 2, . . . d)
and d is the number of model parameters, w, c₁,c₂
Define Cost Function (f);
fori = 1: Pop.Size	// Initialization
Initialize x_i,j = $rand [x_{\min_{j}} : x_{\max_{j}}]$ for j = 1, 2, . . . d and;
Form x_i,d = [x_i,j=1, x_i,j=2 . . . x_i,j=d];
Initialize v_i,j = $rand [x_{\min_{j}} : x_{\max_{j}}]$ for j = 1, 2, . . . d and;
Form v_i,d=[v_i,j=1, v_1,j=2 . . . v_i,j=d];
l_best,i,d = x_i,d;
end
Evaluate fitness function f(x_i, d) for all Pop.Size particles;	// Evaluation
g_best,d = Min(f);
fort = 1: Max.Iter	// Start the Iteration
fori = 1: Pop.Size
update each particle velocity v_i,j using Eq. 7 and form v_i,d;	// Update velocity
update each particle position x_i,j using Eq. 8 and form x_i,d;	// Update position
evaluate fitness value for f(x_i, d);
iff(x_i, d) < f(l_best, i, dt)
$l_{best, i, d}^{t} = x_{i, d}$ ;	// Update local best
end
iff(g_best, dt) < f(l_best, i, dt);
$g_{best, d}^{t}$ = $l_{best, i, d}^{t}$ ;	// Update global best
end
end for
x^* = $g_{best, d}^{t}$ ;
end for	// End Iteration
Return x^* and $g_{best, d}^{t}$ ;	// Output
End PSO

3.3. Cost Function (CF)

The Cost Function defines the search criteria for the parameter identification algorithms of RF PA models. The algorithms should work to optimize (either minimize or maximize) the cost function to get the best validated model structures. Following are the different cost functions considered in this work:

3.3.1. Normalized Mean Square Error (NMSE)

The metric NMSE measures the error between the modeled output (y_modeled) and the actual measured output (y_actual) of the PA and it can be given as: ${NMSE}_{dB} = 10 \log_{10} \sum_{n = 1}^{N} \frac{| y_{actual} (n) - y_{modeled} (n) |^{2}}{| (y_{actual} (n) |^{2}}$ (9) where N is the size of data set used for model validation. For model order selection algorithms, defining the model error (NMSE) alone as the cost function will select the model parameters that will give the best validation. However, this leads to larger model dimension as there is no constraint on model size and model error is a descending function of model size. Hence, there is a need for defining new cost functions that could minimize both the model error and complexity. So different cost functions based on information criterion are introduced by aiming at selecting the smallest possible model structures without significant degradation in the performance. The parameter NMSE is also used to validate the model performance.

3.3.2. Information Criterion (IC)

Information Criterion is widely used for model order selection in statistics and linear regressions, as it considers both the model error and complexity (size) [20]. In general, the Information Criterion as given by [21] can be represented as: $IC = Nln (σ^{2}) + f (N, P) P$ (10) where N represents the observation sample size used for validating the model, σ is the standard deviation of the model error, f (N, P) is the penalty function and P is the number of estimated model coefficients. The first term in Eq. 10 represents the model fitness and the second term is used to penalize the model performance in terms of its size. Increasing the penalty function (f (N, P)) gives smaller dimension model structures at the cost of reduced model performance and decreasing the penalty function gives better modeling accuracy. Hence, the penalty function provides a trade-off between model performance and its complexity. Different types of Information Criterion based cost functions can be defined based on the penalty functions as given below:

Akaike Information Criterion (AIC) For a given model, the AIC [22] based cost function is described as, $AIC = Nln (σ^{2}) + 2 P$ (11)

Bayesian Information Criterion (BIC) Similarly, for BIC [23] the penalty is considered in terms of the number of samples used for validating the model and is given by, $BIC = Nln (σ^{2}) + \ln (N) P$ (12)

Two other criteria are introduced in [24], using Maximum Entropy (ME) method and information criteria. In these criteria, a tolerance factor is introduced in a way that it controls the model accuracy by fixing the penalty function and is described as below.

Maximum Entropy-Akaike Information Criterion (ME-AIC) It is derived from AIC and Maximum Entropy theory. For a given model, the ME-AIC is given by:

$ME - AIC = 2 Cln (σ) + P$ (13)

Maximum Entropy-Bayesian Information Criterion (ME-BIC) This is derived from BIC and Maximum Entropy theory and is given by: $ME - BIC = 2 Cln (σ) + \frac{\ln (N)}{3} P$ (14) where C is a constant criterion which controls the modeling accuracy. Higher the value of C better will be the modeling accuracy.

4. Results

The validation of the proposed work was carried out by modeling a GaAs based PA driven by a single carrier 20-MHz generic random input signal having an average power of 10 dBm, simulated using Keysight ADS tool. The conventional sweep method and the proposed information criteria based sweep and PSO algorithms are applied on the MP and GMP models to determine the most parsimonious model. The model extraction was done on an Intel CORE i7-4790 processor running at 3.60 GHz. Sampled complex baseband waveforms of the input and output of the PA are used for deriving the PA model. A set of 12,000 data samples are taken for model estimation and another set of 12,000 samples are used for model validation. The model extraction step was repeated for 50 times when PSO is used, to ensure consistency. To find the best population size for the PSO, swarms of various sizes (like 50,100,150,200 and 250) were tested with a fixed iteration order of 10 using AIC as the cost function, as shown in Figure 1. From Figure 1, it can be inferred that increasing the swarm size beyond 200 particles does not help in increasing the model performance and involves huge estimation time, hence 200 is taken as the best swarm size. The search space for the model order selection algorithms is defined by initializing a predefined range for each of the parameters in the MP and GMP models as shown in Table 3. Similarly the parameters chosen for the PSO algorithm are listed in Table 4.

Fig.1

Study of PSO Convergence with Different Population Sizes.

Table 3

Parameter Range Initialization for MP and GMP models

Model name	K _a	Q _a	K _b	Q _b	L _b	K _c	Q _c	L _c	Total possible combinations
MP model	1 : 20	0 : 7	-	-	-	-	-	-	160
GMP model	1 : 15	0 : 4	1 : 5	0 : 8	1 : 2	1 : 5	0 : 3	1 : 2	270000

Table 4

Parameter Initialization for PSO based Algorithm

Population size	Number of Iterations	w	c ₁	c ₂	Total possible combinations
200	10	0.8	5	5	2000

The sweep based algorithm is initially applied to derive the optimal PA model based on MP and GMP models with different cost functions and the results are tabulated in Table 5 and Table 6 respectively. These results illustrate that, compared to the NMSE, AIC and BIC based cost functions, the maximum entropy based information criteria such as ME-AIC and ME-BIC significantly reduces the model complexity without compromising the performance. For MP model, the best model is having an NMSE of -47.6974 with a size of 45, whereas the most parsimonious model has an equivalent performance of -47.2136 with 30 coefficients. Similarly for the GMP model, a model with -52.5613 NMSE and size of 75 is the best and the most parsimonious model has an NMSE of -52.3244 with only 49 coefficients are determined with ME based criteria. Hence, for both the MP and GMP models the complexity is reduced around 35% using the proposed ME based IC. Figure 2 and Figure 3 illustrates the variation of model performance (NMSE) with the constant criterion (C) of ME-AIC and ME-BIC for the sweep based MP and GMP models respectively. From Figure 2 and Figure 3, it can be observed that, the NMSE initially reduces and then becomes almost constant after a certain value of C. Based on that, for MP model the value of C for which the algorithm will give the most parsimonious model is chosen as 60 and 160 for the ME-AIC and ME-BIC cost functions respectively. Similarly for the GMP model, the C values are chosen as 60 and 180 for ME-AIC and ME-BIC respectively.

Table 5

Comparison Between Conventional Sweep Method and Information Criterion Based Sweep Method for MP Model

Method	NMSE(dB)	Best Model Size
Sweep-NMSE	-47.6974	45
Sweep-AIC	-47.3753	32
Sweep-BIC	-47.3753	32
Sweep-ME AIC (C=60)	-47.2136	30
Sweep-ME BIC (C=160)	-47.2136	30

Table 6

Comparison Between Conventional Sweep Method and Information Criterion Based Sweep Method for GMP Model

Method	NMSE(dB)	Best Model Size	CPU Time (Mins)
Sweep-NMSE	-52.5613	75	1242.70
Sweep-AIC	-52.5243	64	1275.23
Sweep-BIC	-52.5193	61	1271.38
Sweep-ME AIC (C=60)	-52.3244	49	1268.09
Sweep-ME BIC (C=180)	-52.3244	49	1267.99

Fig.2

Variation of NMSE with constant criteria (C) for ME-AIC and ME-BIC with sweep based MP model.

Fig.3

Variation of NMSE with constant criteria (C) for ME-AIC and ME-BIC with sweep based GMP model.

The sweep based approach determines the best solution with the required cost function, since it performs a full-scale search over the provided search space. However, this method takes enormous amount of time to determine the best solution. The CPU time required for estimating the MP model parameters is small, since its optimal structure is determined by only two parameters. On the other hand, the GMP model size is determined by eight different integer parameters and hence it consumes lot of estimation time as shown in Table 6. Hence, to reduce the time for estimating the GMP model parameters, the PSO algorithm is used. The PSO performs a guided random search over 2000 parameter combinations to find the best solution. For the GMP model, the PSO algorithm along with all the introduced cost functions are used to determine the best model and the results are summarized in Table 7. Since the PSO algorithm performs a random search, the solution provided by it is not same all the time. Hence, to ensure consistency the experiment was repeated for 50 times and the results are averaged. Table 7 illustrates that, with PSO based approach there is a significant reduction in time required to determine the best solution compared to the sweep based method. In addition, the information criterion based cost functions helps in substantially reducing the model complexity. It is to be noted that, the minor loss in performance of less than 1 dB comes with a significant reduction in the model size.

Table 7

Comparison of PSO performance with different Cost Functions

Method	NMSE(dB)	Best Model Size	CPU Time (Mins)
PSO-NMSE	-52.5538	78	14.86
PSO-AIC	-52.4721	63	12.07
PSO-BIC	-52.4132	61	11.67
PSO-ME AIC (C=60)	-52.0032	51	10.06
PSO-ME BIC (C=180)	-51.8051	48	9.84

Figure 4 shows the normalized power spectral density (PSD) of the actual input, output and the GMP modeled output waveforms of the PA based on the conventional sweep-NMSE and PSO-ME BIC methods. From Figure 4 it can be observed that, even though ME-BIC based PSO (48) requires only two third of the coefficients compared to the conventional approach (75), there is no significant difference between the modeled waveforms using the two approaches. This clearly illustrates that, applying information criterion based cost function to determine the PA model coefficients helps in deriving complexity reduced PA models without compromising the modeling accuracy.

Fig.4

Normalized PSD for GMP based model.

5. Conclusions

In this paper, different information criterion based cost functions are used in model parameter selection algorithms to derive complexity reduced PA models without significant degradation in model performance. The results clearly indicate the superior performance of the introduced maximum entropy based approaches compared to the traditional approaches in deriving the most parsimonious model. It was found that, information criterion based sweep method reduces the model complexity by 35% without significant degradation in its performance. In addition, the proposed variant of PSO helps in reducing the execution time by almost 1/100 to determine the optimal model parameters. Hence, the proposed approach helps in deriving complexity reduced models at faster time.

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Start Sweep
Initialize $x_{j} = [\min_{j} : \max_{j}]$	// Input
where j=1,2…d, d is the number of model parameters;
Define the Cost Function (f);
Form x = [x₁, x₂, . . . . x_d];	// Form all possible parameter combinations
T = Number of all possible combinations for x;	// Total combinations
fort = 1 : T
Evaluate the Cost Function (f(x)) for all possible T sets;	// Evaluation
end
Best cost = Min(f(x=x^)*);	// Identify the best cost
Return x^* and Best cost;	// Output
End Sweep