FIXED POINT IMPLEMENTATION FOR PARAMETERS EXTRACTION IN A DIGITALPREDISTORTER USING ADAPTIVE ALGORITHMS

Martin Garcia-Hernandez, Alfonso Prieto-Guerrero, Gerardo Abel Laguna-Sanchez, Paulino Mendoza-Valencia.

Autonomous Metropolitan University - Iztapalapa,Department of Electric Engineering

C.P. 09340, Mexico D.F.

ABSTRACT

In this paper, the parameters extraction from the Volterraseries to analyze the performance of a Digital Predistorter(DPD), for the Power Amplifier (PA) with memory, is in-troduced in two different ways: (1) different numericalmethods for the parameters extraction and (2) the fixedpoint numerical format implementation for this numericalmethod. The parameters in the Volterra Model are typi-cally calculated based on the mean square error criteria.In this paper, we present some alternatives to reduce thecomplexity, number of operations, and a PA linearizationtime ,with DPD dealing with OFDM signals. The simu-lation results show that with the Volterra model, both theLMS and the VSS algorithms are faster and more effectiveto calculate the parameters and mantain their convergenceproperties for a 32-bits implementation.

1. INTRODUCTION

Digital predistortion techniques are used to reduce the non-linearities and to compensate the spectral distortion causedby the PA due to the high peak to average power ratio(PAPR), introduced by the nonconstant envelope modula-tion techniques like the modulation based on orthogonalfrequency division multiplexing (OFDM). Some authorspropose several ways to linearize the PA and reduce thememory effects, and a detailed study can be found in [1].All these proposals need a non iterative stage for param-eter extraction and therefore are difficult to implement inreal-time. We focus our attention on the recently proposedDPD cited in [2], using dynamic deviation reduction -based Volterra Series but giving emphasis to parametersextraction using numerical methods such as the adaptativealgorithms Least Mean Square (LMS), Normalized LMS(NLMS), and Variable Step Size (VSS) implemented infixed point.

If we implement a DPD on a dedicated hardware withfixed point number representation, such as a sampled sig-nal or a very large-scale integration chip (VLSI), the fixedwordlength constrains resolution of the Volterra coeffi-cient rounds may produce multiplications and additionsresults overflow. In this paper we study three methods forparameters extraction implemented with fixed point andthey are compared with the corresponding float point ver-sions.

This paper is organized as follows. In Section II weintroduce the DPD model for this work. Section III pres-ents an overview about the parameters extraction from theVolterra series using and comparing some adaptive algo-rithms. In Section IV we present the fixed-point imple-mentation for the extracted parameters. Simulation resultsare given in Section V and finally the conclusion is pre-sented in Section VI.

2. CREATING A DIGITAL PREDISTORTER

The Digital Predistorter (DPD) is a device with a nonlin-ear predistortion function that must be the inverse of thedistortion function exhibited by the amplifier. This non-linear predistortion function is built up adapting the char-acteristics of the power amplifier [3]. Figure 1 shows thispredistortion system. At the beginning, the DPD is by-passed while the adaptive algorithm calculates the opti-mum kernel for Volterra series and the DPD is built. Afterthis stage, the DPD is connected and the compensation iscompleted.

To describe the relationship between the input and out-put of a DPD, in this work we use the following Volterrapruned series, called dynamic deviation reduction, origi-nally proposed in [2]:

u(n) =

P−12∑

l=0

M∑i=0

k2l+1,1(i)|x(n)|2lx(n− i)

+

P−12∑

l=1

M∑i=1

k2l+1,2(i)|x(n)|2(l−1)x2(n)x∗(n− i) (1)

where x(n) and u(n) are respectively the original inputand output of the DPD, and k2l+1,j , with j = {1, 2}, isthe complex Volterra kernel of the system. (.)∗ representsthe complex conjugate operation and |.| returns the mag-nitude. P and M are respectively, the nonlinearity orderand the memory effect. The parameters extraction used todetermine the values for the complex Volterra kernel, isgiven by:

k = (Y HY )−1Y HU (2)

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Fig. 1. Predistortion system. The predistortion functionin the DPD is the inverse from the PA distortion function.The cascade of the DPD followed by the PA must result ina linear system.

where (.)H represents the Hermitian transpose. In thismodel it is necessary to capture and save the observa-tions Y (received samples) and the desired response U(transmitted symbols). This way of extracting the parame-ters minimizes the model error criterion (MMSE) . In thiswork the parameters k are determined by an adaptive al-gorithm avoiding the computation of the matrix inverse inequation (2).

3. PARAMETERS EXTRACTION USINGADAPTIVE ALGORITHMS

Because the Volterra model is linear with respect to theparameters, we can use it as a numerical method to give agood approximation to the MMSE solution. The iterativeextraction of parameters from the Volterra kernel to buildthe DPD is realized through adaptative algorithms. Thesealgorithms are widely used to solve linear optimizationproblems.

We select the commonly used adaptive algorithms, brie-fly outlined below. Least Mean Square or LMS algorithm[4], updates the coefficients according to

K(n+ 1) = K(n) + µe(n)Y (n), (3)

where Y (n) is the input signal and K(n) is the vector ofthe filter coefficients and the e(n) is the error for the nth

sample expressed by

e(n) = U(n)− Y T (n)K(n), (4)

here U(n) is the desired output and Y T (n)K(n) is thesignal estimation. Finally, µ is the parameter that controlsthe step size and stability that has some dependence on theinput signal and must accomplish certain requirements toensure their convergence [5]. In order to eliminate somerequirements, there is a normalized version of the LMSalgorithm called NLMS (Normalized-LMS) algorithm ex-plained in detail in[6]. For the normalized algorithm, theupdate coefficients are estimated by

K(n+ 1) = K(n) + βe(n)Y (n) (5)

where the parameter β is

β =α

Y T (n)Y (n). (6)

In this case, α is a constant that is possible to set withindependence on the characteristics from the input signal.In these two algorithms the step is fixed. When the stepsize increases, the speed of convergence also increases,but with a bigger error. On the other hand, with a smallstep size, the algorithm needs more iterations and takeslonger to adapt the coefficients. To remove this depen-dence and decrease the error and the convergence time,the Variable Step Size (VSS) algorithm was proposed in[7], where the adjustment for the new step is giving by

µ(n) = αµ(n− 1) + γe2(n− 1), (7)

where the parameter α works as a memory factor of theprevious step and the parameter γ is the memory factor ofthe previous iteration. The step is bounded by

µ(n+ 1) =

µmax if µ(n) > µmax,µmin if µ(n) < µmin,µ(n) otherwise.

(8)

The coefficients vector is computed with the equation (3).Several proposed modifications to the VSS algorithm

can be found in literature [8]-[9] but in all of them thestep size iterative equation can be written as the equation(7). We do not consider these algorithms because theyincrease the number of parameters to update the step andthis complicates a real implementation.

4. FIXED POINT IMPLEMENTATION

To iteratively evaluate the series of equation (1), the x val-ues are bounded to 1.3, to get a good approximation of kcoefficients. Assuming 5th order powers, the algorithmproduces extremely small numbers. In each iteration thealgorithm needs to update the coefficients with very smallvalues, and with less than 32 bits accuracy it would not bepossible to achieve good results compared with the float-ing point case. In our fixed point implementation, wordlengths are of 32 bits, of which 25 bits are for the frac-tional part, 6 bits for the integer part and 1 the sign bit.The round mode type floor and the overflow mode typewrap are employed. These parameters were chosen ac-cording to the dynamic range of the predistorted signal.Table 1 shows the parameters used in the fixed-point im-plementation.

The signal used to stimulate the adaptations of the al-gorithms is an OFDM signal defined and described in theIEEE802.11a standard [10] for WiFi wireless networks.In this case, the OFDM signal is sampled to 20 Msps(Mega samples per second) and quantized to 16 bits withsign and 13 bits for the fractional part.

5. SIMULATION RESULTS

To evaluate the performance of the proposed algorithmsdescribed in Section 3, we will take the amplitude to am-

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Fig. 2. AM/AM response simulated with an OFDM sig-nal and the DPD built by the LMS algorithm. Real PA’swork close to nonlinear region resulting in a compressionof the output signal. The DPD tries to compensate thiscompression and to restore the original signal.

plitude conversion or effect AM/AM curve to observe thelinearization of the PA with the fixed point and the float-ing point implementations. We observe the Power Spec-tral Density (PSD) of the system and how the nonlinearPA’s response causes an unwanted spread of the spectrum.Finally, we show the convergence of the kernel values ofthe Volterra series.

We tested the three adaptive algorithms with specificparameters. Table 2 presents the parameters for the LMSand NLMS algorithms (fixed-step type) while Table 3 pres-ents the parameters for the VSS algorithm (variable-steptype). These values are the same as those reported in theoriginal papers. The parameters of the Volterra series ex-pressed in equation (1) were estimated with polynomialorder of the 5 and a memory length effect set to 2 (P = 5and M = 2). With these parameters we have a kernel ofsize k : 13x1. In order to compare the convergence be-haviors, we made simulations with 70,000 samples, cor-responding to approximately 4 OFDM symbols.

Figure 2 shows the AM/AM curve with the DPD andwithout the DPD for the LMS algorithm implemented infixed-point and floating-point. The simulated PA is com-pressed with respect to the ideal curve, however the digitalpredistortion achieves a very good compensation for thecompression introduced by the nonlinear PA. The othertwo algorithms reach a similar linearization performance,except that more iterations are required to reach the sameresult. This is mainly because the linearization itself is notthe result of the adaptive algorithm but it is made by the

Table 1. Parameters for Fixed-Point ImplementationWordLength Integer Fraction Sign32 bits 6 bits 25 bits 1bit

Fig. 3. Spectral plots for the system with the DPD builtby the VSS algorithm for both fixed-point and float pointimplementations, and a zoom window for them.

Volterra series, which is responsible for the DPD building.Here the good performance of the built DPD is observedas a result of the pruned Volterra Series and the parame-ters extraction by adaptative algorithms implemented withfixed point. Figure 3 presents the Power Spectral Densityresults for the VSS algorithm. In the same graph the be-haviors for both implemented cases are plotted. In this fig-ure it is easy to observe the good performance in restoringthe original spectra with the fixed-point implementation.The variation with respect to floating point performanceis minimal and it can be seen in the zoom window practi-cally the same result despite the loss due to rounding andoverflow implicit in this type of implementation.

After simulations, the results for the convergence dur-ing the extraction of the parameters for the Volterra series

Table 2. Parameters for fixed-step algorithmsAlgorithm StepLMS µ = 4x10−1

NLMS α = 6x10−1

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Fig. 4. Convergence for the LMS algorithm during theparameters extraction for the Volterra series kernel. Fixed-point and floating point implementations are plotted. Themost significant coefficients for the Volterra series kernel,in terms of value, are k1, k2 and k3. Coefficients kn>3 aresmall.

kernel are presented in Figures 4, 5, and 6. In dotted linesthe optimal solutions given by MMSE are plotted, and insolid lines the convergence of each algorithm to the opti-mum is plotted. The step size for the fixed-step algorithmsand that bounded for variable-step algorithms are chosenin such a way the convergence presents the minimum stan-dard deviation. Any of the three algorithms presented hereis faster than that often used for the parameters extractionrepresented in equation (2), and they are also very easyand practical to implement in terms of number of opera-tions.

Figure 4 shows the convergence for the LMS algo-

Table 3. Parameters for variable-step algorithmAlgorithm Update step BoundedVSS α = 0.9995 µmax = 75x10−2

γ = 4.8x10−4 µmin = 65x10−2

Fig. 5. Convergence for NLMS algorithm fixed-point andfloating point implementations.

rithm with fixed point and floating point implementations.The achieved performance for the fixed-point implemen-tation is very similar to that of the floating-point, which isbest seen in the zoom window. The convergence is closeto the MMSE solution, in terms of iterations (4x104), italready shows a good approximation to the MMSE solu-tion.

The NLMS algorithm is the slowest because it requiresmore than 7x104 iterations, but makes a smoother conver-gence than any other, and the standard deviation is smallcompared with the other algorithms. Particularly, this al-gorithm shows some problems during the evaluation ofequation 6, since in the guard interval between OFDMsymbols the value in the observed samples is zero, Y (n) =0. This phenomenon is corrected in the floating-point im-plementation replacing the zeros by a small value, to pre-vent the algorithm divergence but this correction causes areset in the convergence for this algorithm. This is shownin figure 5, the horizontal lines after guard time of OFDMsymbols is a small reset in the convergence. In the fixed-point implementation, the problem is solved more easily,and only requires a shift to avoid division by zero and nonpresent the reset in the convergence problem. This advan-tage is clearly noticed in Figure 5, where it can be seenthat the fixed-point implementation converges faster thanthe respective floating point implementation. In the dot-ted circle of figure 5 we see the smooth convergence offixed-point implementation and the reset phenomenon inthe convergence between OFDM symbols of floating pointimplementation.

The LMS algorithm is almost two times faster thanthe NLMS algorithm, although NLMS algorithm presentsbenefits in terms of convergence and stability [6], easy so-lution to avoid the division by zero, and smooth conver-gence with the fixed point implenentation. Finally, TheVSS algorithm has a convergence similar to the LMS al-gorithm, and it requires about 4x104 iterations to approx-imate to the MMSE solution. As in the LMS algorithm,

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Fig. 6. Convergence for VSS algorithm fixed-point imple-mented and floating point.

convergence for fixed point and floating point implemen-tations are very similar. In the zoom window of figure 6we can see the closed convergence between them.

6. CONCLUSIONS

In this article we showed the convergence properties forthree adaptive algorithms during the extraction of param-eters for the DPD with the Volterra Pruned Series. Thefixed-point numeric format with constant word length isimplementable in VLSI applications of sampled signals.

The distortions in the power amplifier, nonlinear re-sponse and unwanted spread of spectrum, are properlycompensated by the DPD with the Volterra Pruned series.We tested three adaptive algorithms implemented in fixedpoint and compared them with the floating-point versionsand the MMSE solution during the parameters extractionfor the Volterra series. We evaluated: a) the performancein the linearization and spectral compensation, and b) theconvergence speed during the extraction of parameters forthe kernel. We have shown that the three proposed al-gorithms are faster than the MMSE. Both the VSS andthe LMS are better in terms of convergence speed. With

respect to stability and smoothness of convergence, theNLMS algorithm showed the best performance but withlower convergence speed and the risk of divisions by zero.

AcknowledgmentThe authors wish to acknowledge the support of the IntelTecnologıas de Mexico and the Department of ElectricEngineering of Autonomous Metropolitan University Iz-tapalapa.

7. REFERENCES

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[8] Aboulnasr T., Mayyas K., “A robust variable step-sizeLMS-type algorithm: analysis and simulations,” Sig-nal Processing, IEEE Transactions on, Vol. 45, No. 3,pp. 631–639, March 1997.

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