Two-Dimensional Memory Selective Polynomial

Model for Digital Predistortion

Mazen Abi Hussein #∗1, Vivek Ashok Bohara ξ2, Olivier Venard #ξ3

#LaMIPS, Laboratoire commun NXP-CRISMAT, UMR 6508 CNRS ENSICAEN UCBN Caen

ESIEE Paris, ∗Electronic systems Dept., ξ Telecommunication Dept.,

Cite Descartes, BP99, 93162 Noisy-Le-Grand, France1m.abihussein@esiee.fr,

2v.bohara@esiee.fr,

3o.venard@esiee.fr

Abstract—In this paper, we propose a Two-Dimensional Mem-ory Selective Polynomial (2-D MSP) model for digital predis-tortion (DPD). The proposed model has been demonstrated toachieve a comparable performance to the well known traditionalpredistorter models such as generalized memory polynomial(GMP) but with significantly less number of parameters. Theproposed model has been validated by evaluating the DPDperformance on a class AB power amplifier in terms of adjacentchannel power ratio (ACPR).

Index Terms—Digital predistortion, memory polynomial, gen-eralized memory polynomial

I. INTRODUCTION

Power amplifiers (PAs) are indispensable components of

modern wireless transceivers [1]. However to achieve maxi-

mum efficiency the power amplifier should be driven near the

saturation region which severely compromises the linearity of

the transmitted signal resulting in out-of-band spurious emis-

sion along with in-band distortions. To minimize the nonlinear

distortions, it is desirable to operate PA in the linear region i.e.

with huge back off, but in this case, efficiency has to be traded

off [1]. Moreover, on operating with multicarrier transmissions

such as wideband code division multiple access (WCDMA)

or orthogonal frequency division multiplexing (OFDM) which

has wide bandwidth and high peak-to-average power ratio

(PAPR), the high power PA exhibits memory effects [2]. These

memory effects further deteriorate the performance of the PA

and can not be modeled or compensated by conventional tech-

niques that solely depend upon the instantaneous amplitude of

the input signal.

With its ease of implementation, low operating cost and con-

siderable high performance improvement digital predistortion

(DPD) has been gaining widespread popularity as one of the

most efficient technique for linearization of power amplifiers

with memory effects [1]. Consequently, DPD has been a

frontrunner in solution for linearization of PAs in future multi-

standard multi-band cognitive radios where reconfigurability

and flexibility will be of paramount importance [1]. As per

recent development [3], depending upon the PA, DPD can

achieve adjacent channel leakage ratio (ACLR) improvements

in excess of 20dB as well as combination of PAPR reduction

and DPD can achieve efficiencies greater than 25% which

is considerably higher than what can be achieved by other

linearization methods.

The various DPD techniques available in literature can be

broadly classified into indirect learning architectures (ILA) and

direct learning architectures (DLA). In the DLA, a model for

the PA is first identified [4]. The predistorter is then obtained

based on the identified model and a reference error between the

input to the predistorter and output from the nonlinear system.

In the ILA approach, a post-inverse of the PA is first identified

and then used as a predistorter [5], [6]. Hence, the ILA

approach removes the requirement of identifying the model

of the PA which substantially reduces the computationally

complexity [7].

In this paper, we would discuss some of the widely accepted

models that are used for DPD applications. Some of the

models considered here are Generalized memory polynomial

(GMP) [6], memory polynomial (MP) [5], Wiener [8], Ham-

merstein [8] and Wiener Hammerstein (WH) [5] models.

These traditional models are shown to use large number

of coefficients to achieve desirable DPD performance with

ILA, that in turn complicates the extraction procedure for

the predistorter and increases the computational complexity.

Hence we propose a Two-Dimensional Memory Selective

Polynomial (2-D MSP) model for digital predistortion. This

model significantly reduces the computational complexity by

introducing two-dimensional memory “selectivity” over non-

linearity. The performance of the proposed 2-D MSP model

is evaluated with an ILA DPD system using a 10W class AB

power amplifier.

The remainder of this paper is organized as follows. Section

II discusses the various models for a digital predistorter and

tries to establish a correlation between them. Section III

presents the description and simulation results of the proposed

2-D MSP model. Finally, Section IV concludes the paper.

In the following, the arrays and matrices are denoted by bold

lowercase letters (eg. a) and bold uppercase letters (eg. A)

respectively.

II. MODELS FOR NONLINEAR SYSTEMS WITH MEMORY

Volterra series and its variants are predominantly used to

model a nonlinear system with memory [6]. A low-pass

equivalent finite memory Volterra model relating the baseband

output signal y(n) of a nonlinear system with memory to the

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469

baseband input signal x(n) can be written as1

y(n) =

K∑

k=0

y2k+1(n) (1)

where

y2k+1(n) =L∑

l1=0

· · ·L∑

lk+1=lk

L∑

lk+2=0

· · ·L∑

l2k+1=l2k

al1l2...l2k+1

k+1∏

m=1

x(n− lm)

2k+1∏

m=k+2

x∗(n− lm),

al1l2...l2k+1denotes the (2k + 1)th order Volterra kernel, L

represents the memory depth and (.)∗ denotes the conjugate

operation. The main problem associated with the Volterra

series is the computational complexity involved in the mea-

surement of the Volterra kernels. Hence, most of the sim-

plified structures have been obtained by imposing particular

conditions on kernel extraction and/or by forcing some of

the kernels to zero. For example, leaving only diagonal terms

in (1) and forcing all other coefficients to zero will lead to

the memory polynomial (MP) model [5], also called parallel

Hammerstein (PH) model as illustrated in (2)

y(n) =K−1∑

k=0

L∑

l=0

aklx(n− l)|x(n− l)|k, (2)

where we have generalized (2) by including odd as well as

even order terms [9]. If special conditions are imposed on the

extraction procedure of akl in (2), while maintaining all its

terms (diagonal terms), another structure commonly known as

Hammerstein model can be obtained. Every coefficient of the

Hammerstein model is restricted to be the product of two other

coefficients, splitting the 2-D array {akl} to two 1-D arrays

{ak} and {bl}. The Hammerstein model can be expressed as

y(n) =

K−1∑

k=0

L∑

l=0

akblx(n− l)|x(n− l)|k. (3)

Basically, this restriction on Hammerstein model parameters

results from separating the static nonlinearity in the PH

model from linear filtering. Doing so, reduces the number of

coefficients considerably (from K × (L + 1) to K + L + 1),

which may compromise behavioral modeling capability [10].

Another special case of the general Volterra model is the

Wiener model. The Wiener model is a cascade of a linear filter

followed by a static nonlinearity. Unlike the Hammerstein and

PH models, Wiener models include much more terms from

(1) in their expressions. If static nonlinearity is modeled by

a simple polynomial function, and a finite impulse response

(FIR) filter is used, the expression of Wiener model can be

expressed as follows [10]

y(n) =

K−1∑

k=0

ak

L∑

l=0

blx(n− l)

∣

∣

∣

∣

∣

L∑

l=0

blx(n− l)

∣

∣

∣

∣

∣

k

. (4)

1Note that for baseband applications, we are only interested in termsgenerating signals around the center frequency, hence only odd order termsare considered.

If we expand (4), we can see that the Wiener model contain

diagonal terms similar to MP model, cross terms similar to

generalized memory polynomial model (GMP) which will

be discussed later as well as some terms from the general

Volterra series. However, the output in (4) depends nonlinearly

on coefficients bm making its extraction more complicated

and thus limiting its use. If we add a filter in cascade with

the Wiener model, we obatin a model that is commonly

known as Wiener-Hammerstein (W-H) model, which includes,

exactly the same terms as the Wiener model but with other

restrictions imposed on parameters extraction. Recently in

[11], a special Wiener model with cross terms of the form

x(n)∏k

i=1 |x(n − li)| with li = 0, 1, 2, . . . L, was added in

parallel branch to the MP model, thus generating what is

known as Memory-polynomial/Wiener model. The Memory-

polynomial/Wiener model can be expressed as

y(n) =K−1∑

k=0

L∑

l=0

aklx(n− l)|x(n− l)|k

+

K−1∑

k=1

akx(n)

[

L∑

l=1

bl |x(n− l)|

]k

. (5)

This model has shown considerable improvement over MP

model due to the addition of cross terms. However it still

suffers from complicated extraction procedure as the output

depends nonlinearly on the coefficients bl.

In [6] a more direct way of adding cross terms to an MP

model is introduced without significantly complicating the

extraction procedure. The model in [6], known as generalized

memory polynomial model (GMP). The output of the GMP

model is linearly dependent on its coefficients and it has been

shown to outperform the MP model when used as a predis-

torter. The GMP model includes, in addition to the diagonal

terms, cross-terms having the form x(n− l)|x(n−m)|k where

l = 0, · · · , L, m = −M, · · · , 0 · · ·M and m 6= l. This model

is equivalent to the one proposed in [12], where an algorithm

was used to select the terms at a finite distance |l −m| from

the diagonal. The GMP model is expressed as

y(n) =∑

k∈Ka

∑

l∈La

aklx(n− l)|x(n− l)|k

+∑

k∈Kb

∑

l∈Lb

∑

m∈Mb

bklmx(n− l)|x(n− l −m)|k

+∑

k∈Kc

∑

l∈Lc

∑

m∈Mc

cklmx(n− l)|x(n− l +m)|k.

(6)

where Ka,Kb and Kc are index arrays for nonlinearity, and

La, Lb, Lc,Mb and Mc are the index arrays for memory.

The total number of coefficients is equal to ϑ(Ka)ϑ(La) +ϑ(Kb)ϑ(Lb)ϑ(Mb) + ϑ(Kc)ϑ(Lc)ϑ(Mc) with ϑ(X) denotes

the number of elements in X .

In general, it is perceived that, when the number of terms

included in the predistorter model increases, the correction

capability of the model increases, although the amount of

470

contribution might vary across the terms [6]. However, addi-

tion of each new term increases the computational complexity,

implementation cost and complicates the extraction procedure.

Moreover, its very difficult to predict a priori which terms to

be kept, which ones to be discarded and what is the contribu-

tion of each term as it depends heavily on the characteristics

of the power amplifier to be linearized. As a result there

is a significant trade-off in the desired performance of the

predistorter and the number of terms that can be included in

the predistorter.

III. TWO-DIMENSIONAL MEMORY SELECTIVE

POLYNOMIAL MODEL

As observed from all the models discussed above, the terms

that have a significant impact on modeling performance are

of this form: x(n − l)|x(n − m)|k, where l = 0, · · · , L,

m = −M, · · · , 0, · · ·M . However, the arrays of nonlinearity

are the same for all memory delays. For instance, in the GMP

model (6) the index array Kb is applied for any two pairs

of delays (l, l+m). A memory selectivity can be achieved if

nonlinearity index arrays are selected judiciously for each pair

of delays (l,m). For delays that do not contribute significantly

to the correction capability of the predistorter, it might be

reasonable to omit some of the terms by reducing the size

of array index of nonlinearity. As shown in later part of the

paper, such selectivity will achieve a performance comparable

to the traditional models but with number of terms reduced

significantly.

Moreover, instead of restricting the terms x(n − l)|x(n −m)|k for l = 0, · · · , L, m = −M, · · · , 0, · · ·M as in the

case of GMP, we would also like to incorporate other cross

terms like x(n− l)|x(n−m)|k , where l = −L, · · · ,−1, m =−M, · · · , 0, · · · ,M , on the performance of a predistorter.

Henceforth, we propose a more generic model for predistorter

application called as Two-dimensional memory selective poly-

nomial (2-D MSP) model that encompasses all the terms of

the form x(n − l)|x(n −m)|k, where l = −L, · · · , 0, · · · , Land m = −M, · · · , 0, · · · ,M . Fig. 1 shows a graphical

representation of the proposed 2-D MSP model. GMP and

MP models are shown as special cases of this model, and

an example of memory selectivity over the MP model is also

illustrated. The expression for 2-D MSP can be written as

y(n) =

L∑

l=−L

M∑

m=−M

A[i,j]=1

i=l+L+1j=m+M+1

K−1∑

k=0

Bk[i,j]=1

aklmx(n− l)|x(n−m)|k (7)

where

A =

A3 mn A4

ln c lp

A2 mp A1

. (8)

Elements in A, A[i, j], take values equal to 1 to select the

corresponding terms in the two-dimensional memory space

and 0 otherwise, as shown in Fig. 1, with A1, A2, A3 and

A4 L×M matrices controlling the terms in the four quadrants

(l 6= 0, m 6= 0). mn and mp are 1 × M column vectors

−3 −2 −1 0 1 2 3−4 4−L L

0

4

−4

−3

3

2

−2

1

−1

M

−M

A1

Two-Dimension Memory space

Terms of the form:

x(n− q1)|x(n− q2)|k

x(n)|x(n− q2)|k

x(n+ q1)|x(n− q2)|k

x(n+ q1)|x(n)|k

x(n)|x(n)|k

x(n+ q1)|x(n+ q2)|k

x(n)|x(n+ q2)|k

x(n− q1)|x(n+ q2)|k

x(n− q1)|x(n)|k

q1 = 1, . . . , Lq2 = 1, . . . ,M

MP Model

MSP Model (example)

GMP Model

K = 4, L = 4

La = [0, 1, 2, 3]Lb = [0, 1, 2],Mb = [1, 2]Lc = [0, 1, 2],Mc = [1, 2]

l = m = 0,K = [0, ..., 4]l = m = 1,K = [0, ..., 3]l = m = 2,K = [0, ..., 3]l = m = 3,K = [0, 1, 3]

A4A3

A2

l

m

k

Fig. 1. Illustration of 2-D MSM model

controlling terms with delays l = 0,m = −1 · · ·−M and l =0,m = 1 · · ·M , respectively. Similarly, ln and lp are L×1 row

vectors controlling terms with delays l = −1 · · · − L,m = 0and l = 1 · · ·L,m = 0, respectively. The value in c controls

the terms of the form x(n)|x(n)|k , i.e., l = m = 0.

Bk has the same structure as A. However, for particular value

of k it introduces the selectivity on the elements of A. The

structure of Bk can be written as

Bk =

B3 st B4

ut d uv

B2 sv B1

. (9)

For better illustration let us represent MP model in terms of

2-D MSP model. The 2-D MSP model will reduce to an MP

model if the following cases are true.

• A1 is a diagonal matrix with 1 on the diagonal and 0elsewhere.

• c = 1• A3 = A2 = A4 = 0L×M

• mn = mp = 0M×1

• lp = lm = 01×L

• Bk = A, ∀k

The above illustration is also given in Figure 1. Now suppose,

for k = 2 we want to select only the non delay element,

x(n)|x(n)|2, then the last case gets modified as

Bk =A, ∀k, k 6= 2

B2[i, j] =

{

1, if i = L+ 1, j = M + 1

0, otherwise.(10)

In the following, we show the results for our proposed 2-D

471

15 20 25 30 350

4

8

12

16

20

24

28

32

36

Input power (dBm)

Ga

in (

dB

)

15 20 25 30 35−90

−72

−54

−36

−18

0

18

36

54

72

90

Ph

ase

Sh

ift

(°)

Fig. 2. Gain (AM/AM) and AM/PM characteristics of the selected class ABPower Amplifier.

MSP model using a 10W class AB power amplifier. Figure 2

shows the measured AM/AM and AM/PM characteristics of

this power amplifier. A 3G modulated signal with 3.84 MHz

bandwidth and center frequency of 0.9 GHz was used to iden-

tify the predistorter. The ILA DPD system was implemented

using MP, GMP and the proposed 2-D MSP models. As shown

in figure 3, the MP model with 42 coefficients has relatively

good performance with improvements in terms of ACPR of

13.5 dB, 5.7 dB, 8.22 dB and 6.73 dB in the first lower channel

(L1), first upper channel (U1), second lower channel (L2) and

second upper channel (U2), respectively. With 74 coefficients

the GMP model improves the performance by approximately

3.5 dB in L1, 2.5 dB in U1, 3.3 dB in L2 and 3.2 dB

in U2. No further improvement was observed by increasing

the number of coefficients in the GMP model. For the 2-D

MSP model, we selected the best combination of arrays of

nonlinearities for the different delays considered for the GMP

model. As observed from Figure 3, the ACPR performance

of 2-D MSP model are comparable to the GMP model, but

with substantial reduction in the number of coefficients from

74 to 51, i.e. approximately by 30%. Thus, the selectivity

over memory allows to reduce considerably the complexity

of implementation while maintaining the same correction

capability as an equivalent GMP model. An important point

to note is that the proposed reduction in coefficients depends

significantly on the characteristics of the power amplifier.

−30 −20 −10 0 10 20 30

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

Norm

aliz

ed M

agnitude (

dB

/Hz)

InputOutputMP: 42 coefficientsGMP:74 coefficientsMSP: 51 coefficients

MP2D MSP

GMP

Fig. 3. The spectral regrowth suppression performance with MP, 2-D MSPand GMP.

IV. CONCLUSION

A Two-Dimensional Memory Selective Polynomial (2-D

MSP) model was proposed for digital predistortion. This

model generalizes the expressions of the traditional MP and

GMP models by introducing memory selectivity over a two-

dimensional memory space. As a result, for a given nonlinear

PA, 2D MSP model was able to achieve equivalent perfor-

mance as that of GMP but with significantly less number

of parameters. The performance of the proposed model was

evaluated on a 10W class AB PA in terms of spectral regrowth

suppression.

ACKNOWLEDGMENT

The research leading to these results has received funding

from the Seventh Framework Programme under grant agree-

ment n° 230688 and from Panama european project of the

EUREKA program CATRENE funded by the French Research

Ministry. The authors would like to thank R. Montesinos for

providing the power amplifier.

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