On the system level convergence of ILA and DLA

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 present the results for systemlevel convergence of indirect learning architecture (ILA) anddirect learning architecture (DLA) for digital predistortion. Weshow that best performance with ILA and DLA can only beobtained if the system level identification of the power amplifier

and predistorter is done iteratively. Results are demonstrated interms of improvement in adjacent channel power ratio (ACPR)and error vector magnitude (EVM) at the output of poweramplifier (PA) with each system level iteration for both thearchitectures when a Long Term Evolution-Advanced (LTE-Advanced) signal is applied at the input. We also show thatpredistorter identification with DLA is more robust compared toILA in presence of additive white Gaussian noise (AWGN).

Index Terms—Digital predistortion, nonlinear distortion, non-linear filters, high power amplifiers

I. INTRODUCTION

Power amplifiers (PAs) usually exhibit nonlinear character-

istics when driven towards high efficiency saturation region

[1]. This nonlinear behavior of the PA tends to distort the

transmitted signal resulting in out-of-band distortion (spectral

regrowth beyond the signal bandwidth) and inband distortion

(Error Vector Magnitude degradation). The problem is fur-

ther exacerbated with the use of varying-envelope waveforms

(EDGE, WCDMA, LTE) or multiple carrier signals (e.g.,

4xWCDMA) which occupy wide bandwidth and have high

peak-to-average power ratio (PAPR) [2].

Plethora of techniques for linearizing power amplifiers has

been proposed in literature [1]: feedback, feedforward and

predistortion. However digital predistortion (DPD) has been

gaining widespread popularity as an effective and cost efficient

method [3]. The basic principle of a baseband DPD is to add

a digital predistorter (PD), which has the inverse nonlinear

characteristics of that of the PA. This way the cascaded digital

predistorter-PA system becomes linear and the output of the

system is constant gain (linearly) amplified version of the

input.

DPD can be further classified into indirect learning ar-

chitectures (ILA) [3], [4] and direct learning architectures

(DLA) [5], [6]. In the DLA approach, a model for the PA

is first identified. The predistorter is then obtained based on

the extracted PA model and a reference error between the input

to the predistorter and output of the PA. Depending upon the

extracted PA model and the reference error, various algorithms

have been proposed to identify a PD in DLA. Prominent

among them are the Analytical Method (AM) proposed by

Kim et al. [6] which will be denoted in rest of the paper as

DLA-AM and nonlinear filtered algorithms proposed in [5],

[7] which will be denoted in rest of the paper as DLA-NF. In

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

and then used as a predistorter [4]. The post-inverse can be

identified by using either least mean square (LMS), recursive

least square (RLS) or least squares (LS) approach.

Although much research has been done to investigate the

performance of different DLA and ILA digital predistortion

systems, to the best of our knowledge system level conver-

gence of these architectures has not been investigated. For

instance, the authors in [8] and [9] have restricted themselves

to general comparison between the ILA and DLA, with the

results being demonstrated in terms of power spectral density

[8] and, AM/AM and AM/PM performance [9]. Moreover

the authors have only considered a single instance of DLA

without giving due considerations to algorithms proposed in

[7]. Specifically only DLA-NF based on LMS algorithm and

DLA-AM were considered in [8] and [9], respectively. Apart

from above, most of the research on DLA has been on the

identification of the predistorter with an assumption that the

parameters of predefined nonlinear model for PA has been

extracted a priori [7].

In this paper, we try to put forward a comprehensive

overview by taking into consideration the system level con-

vergence of these architectures. For instance, in DLA opti-

mal performance can only be obtained if the system level

identification of both PD and PA is done iteratively. In other

words, for each identification of predistorter, the parameters

for PA need to be “re-extracted” and this process needs to

be repeated till the cascaded PD-PA system converges to the

best possible solution. Similarly, for ILA a new post-inverse

should be identified with each system level iteration until the

cascaded PD-PA system converges to the best possible solution

[3]. We demonstrate the results by showing the improvement

in ACPR and EVM at the output of PA with each system

level iteration for both architectures for a LTE-Advanced

input signal. Moreover, we will also investigate the impact

of noisy measurements on the identification of the PD in both

architectures.

978-1-4673-0762-8/12/$31.00 ©2012 IEEE 870

The remainder of this paper is organized as follows. Section

II discusses the modeling of a nonlinear systems by Volterra

series. Section III presents the theoretical background related

to DLA and ILA. In Section IV simulation results for the

system level convergence of DLA and ILA are presented and

discussed. Finally Section V concludes the paper. In the fol-

lowing, the vectors and matrices are denoted by bold lowercase

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

The superscripts (.)∗, (.)T and (.)H denote the conjugate, the

transpose and the conjugate transpose, respectively.

II. MODELS FOR NONLINEAR SYSTEMS

Before proceeding to give a comprehensive overview of

DLA and ILA it is important to discuss Volterra series which

is generally used to model a nonlinear system with memory.

A low-pass equivalent finite memory Volterra model relating

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

memory to the baseband input signal x(n) can be written as

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 and

L represents the memory depth. The main problem associ-

ated with the Volterra series is the computational complexity

involved in the identification of the Volterra kernels and

hence several simplified memory model structures have been

proposed. The most studied among them is the memory

polynomial (MP) model [4] in which only the diagonal terms

of (1) are considered, i.e. l1 = l2 = · · · = l2k+1 = l. The MP

model can be written as (2)

y(n) =

K−1∑

k=0

L∑

l=0

aklΦkl[x(n)], (2)

where Φkl[x(n)] = x(n − l)|x(n − l)|k, K is the maximum

order of nonlinearity, L is memory depth and the total number

of terms is K× (L+1). For the rest of the study, the MP will

be used for the identification of PA and PD. Considering an

input of N samples, (2) can be written in a matrix form as

y =Xa, (3)

where y= [y(1), . . . , y(N)]T , a=[a00, · · ·, akl, · · · , aK−1L]T ,

X=[Φ00[x], · · · ,Φlk[x], · · · ,ΦK−1L[x]], Φlk[x]=x(n −l)|x(n − l)|k, and x= [x(1), . . ., x(N)]

T. The least square

method can thus be used to extract the coefficients

a =(

XHX)

−1

XHy. (4)

Parametersestimation

PAMod.PDMod.

Linearized PA

PAPDy(n)

d(n) = gu(n)

x(n)

e(n)

u(n)

H

Fig. 1. Direct Learning Architecture - DLA

III. PREDISTORTION LEARNING ARCHITECTURES

In this section we describe in detail the direct and indirect

learning architecture for digital predistortion.

A. Direct learning architecture

As mentioned before, the identification of the PD based on

a direct learning architecture is done in two steps. First, the

parameters of a predefined nonlinear model for the PA are

extracted, and then in the second step, the identified model of

the PA is used for the estimation of the PD. Fig. 1 illustrates

the block diagram of DLA. In the following, we describe two

different ways to identify the PD in DLA.

1) DLA based on analytical method (DLA-AM): In DLA-

AM, an analytical method is used to compute the output of

the predistorter using the extracted MP model of the power

amplifier [6]. Therefore predefined model for PD is not needed

and the output, x(n), can be deduced directly from (2)

x(n) =1

∑K−1k=0 αk0|x(n)|k

(

y(n)−K−1∑

k=0

L∑

l=1

aklΦkl[x(n)]

)

(5)

where x(n) and y(n) are the input and output of the PA

respectively as shown in Fig. 1. In an ideal PD-PA chain the

output y(n) = d(n) = gu(n) where g is the small signal gain

of the power amplifier, u(n) is the input to PD and d(n) is

the desired output. Thus in a linearized system, y(n) in (5)

must be substituted by gu(n). The only constraint in (5) is that

|x(n)| is needed to calculate x(n). Hence, initially |u(n)| is

used as an estimate of |x(n)| to obtain x(n) an approximation

of x(n). Then in the next iteration |x(n)| is used as an estimate

of |x(n)|. It has been stated in [6] that (5) converges after few

iterations.

2) DLA based on nonlinear filter architectures (DLA-NF):

Unlike DLA-AM, in DLA-NF [5], [7] an adaptive algorithm is

used to identify a model for PD from the extracted parameters

of PA. As stated before, an MP is used as a model for PD.

The output x(n) can thus be written as

x(n) =

P−1∑

p=0

M∑

m=0

wpmΦpm[u(n)] (6)

871

where u(n) and x(n) are the input and output sequences of

the predistorter as shown in Fig. 1, P is the maximum order of

nonlinearity, M is memory depth and wpm, p = 0, . . . , P − 1and m = 0, . . . ,M are the the complex coefficients of the

predistorter. (6) can be written in vector form as

x(n) = uT (n)w(n) (7)

where u(n) = [Φ00[u(n)] · · · Φpm[u(n)] · · · ΦP−1M [u(n)]]T

and w(n) = [w00 · · · wpm · · · wP−1M ]T .

The output x(n) of the predistorter is fed into the PA. We

wish to have the output of the PA, y(n) as close as possible to

the desired signal d(n). The error at the output can be defined

as e(n) = d(n) − y(n). In the following, we present two

iterative identification methods, the Nonlinear Filtered-x LMS

(NFxLMS) [5] and Nonlinear Filtered-x RLS (NFxRLS) [7],

to estimate the coefficients of the PD, w(n).a) Nonlinear Filtered-x LMS algorithm (NFxLMS): For

NFxLMS algorithm the coefficients for the predistorter can

be obtained by minimizing the mean square error (MSE),

E{|e(n)|2} = E

{

|d(n) − y(n)|2}

. The weight update equa-

tion for the predistorter based on gradient method is

w(n+ 1) = w(n)−1

2µ∇(n) (8)

where µ is the step size and ∇(n) represents an instanta-

neous estimate of the gradient of E{|e(n)|2} with respect to

predistorter coefficients w(n).

∇(n) = −2e∗(n)ψ(n) (9)

where ψ(n) = ∂y(n)∂w(n) . Hence (8) can be rewritten as

w(n+ 1) = w(n) + µe∗(n)ψ(n). (10)

Since the output of PA, y(n) is a function of x(n) where

x(n) = [x(n), x(n − 1), · · · , x(n − L)]T , which is in turn a

function of w(n), ψ(n) can be written as

ψ(n) =∂x(n)

∂w(n)

∂y(n)

∂x(n)=

L∑

l=0

∂y(n)

∂x(n− l)

∂x(n− l)

∂w(n). (11)

From (2) we can obtain

∂y(n)

∂x(n− l)=

K∑

k=0

k + 2

2akl∣

∣x(n− l)∣

∣

k(12)

Using (7) and assuming that w(n) ≈ w(n − l), l ∈{1, 2, · · · , L} 1, we obtain

∂x(n− l)

∂w(n)=

∂uT (n− l)w(n− l)

∂w(n)≈ u(n− l). (13)

Using (12) and (13) in (11), we get

ψ(n) =

L∑

l=0

K∑

k=0

k + 2

2akl∣

∣x(n− l)∣

∣

ku(n− l) (14)

1Note that, this assumption is also needed to derive (11).

And finally, substituting (14) in (10) we obtain,

w(n+ 1) =w(n) + µe∗(n)L∑

l=0

K∑

k=0

k + 2

2akl∣

∣x(n− l)∣

∣

ku(n− l). (15)

Note that in (15), u(n− l) involves the delayed input up to a

delay equal to L+M .

b) Nonlinear Filtered-x RLS algorithm(NFxRLS): For

NFxRLS algorithm the cost function to be minimized is

defined by the exponentially weighted sum of error squares

as follows

ξ(n) =

n∑

i=1

λn−1|e(i)|2 (16)

where λ is the exponential forgetting factor taking values

between 0 and 1. The minimum can be obtained by differ-

entiating (16) with respect to the coefficient vector w(n) and

equating the result to zero,∂ξ(n)∂w(n) = 0. Hence

∂ξ(n)

∂w(n)= −

n∑

i=1

2λn−1ψ(i)e∗(i) (17)

where ψ(i) is given in (14). Similarly e(i) can be derived as

[7]

e(i) ≈ d(i)− w(n)ψ(i) (18)

for 1 ≤ i ≤ n, with an assumption that the parameters of

the predistorter are constant during the time interval (i =1, · · · , n+M). Combining (17) and (18),

∂ξ(n)

∂w(n)= −

n∑

i=1

2λn−1ψ(i)d∗(i) +n∑

i=1

2λn−1ψ(i)w∗(n)ψH(i).

(19)

Equating (19) to 0, we obtain,

n∑

i=1

λn−1ψ(i)w∗(n)ψ(i)H =

n∑

i=1

λn−1ψ(i)d∗(i) (20)

which can be rewritten as

Rψ(n)w(n) = rψ(n) (21)

where, Rψ(n) =∑ni=0 λ

n−1ψ(i)ψH(i) and rψ(n) =∑n

i=0 λn−1ψ∗(i)d(i). Hence, the Kalman gain can be ob-

tained as

k(n) =λ−1P(n− 1)ψ∗(n)

1 + λ−1ψH(n)P(n− 1)ψ(n)(22)

where P(n) = λ−1P(n−1)−λ−1k(n)ψH(n)P(n−1). Hence

the predistorter update based on NFxRLS algorithm can be

given as

w(n+ 1) = w(n) + µe∗(n)k(n) (23)

872

estimation

parameters

Parameters

Copying

PA

Post−inverse

P’

Pre−inverse: PD

P

x(n) y(n)

zp(n)

u(n)

z(n)

1g

Fig. 2. Indirect Learning Architecture - ILA

B. Indirect learning architectures

The PD identification in ILA is done in a single step as

shown in Fig. 2. A post-inverse of the PA is identified and

used as a PD. If the post-inverse is modeled as a MP, then its

output can be written as

zp(n) =

P−1∑

p=0

M∑

m=0

cpmΦpm[z(n)] (24)

where z(n) = y(n)G

is the input to the post-inverse block as

shown in Fig. 2, cpm, p = 0, . . . , P−1 and m = 0, . . . ,M are

the complex coefficients and Φpm[z(n)] = z(n − m)|z(n −m)|p. After convergence, we should have zp(n) = x(n). For

a total number of samples equal to N , we can write

zp = Zc (25)

where zp= [zp(1), . . . , zp(N)]T

and c=[c00, · · ·, cpm, · · · ,cP−1M ]T , Z=[Φ00[z], · · ·,Φpm[z], · · ·,ΦP−1M [z]],

Φpm[z]=z(n−m)|z(n−m)|p and z= [z(1), . . ., z(N)]T

.

The least square solution for (25) will be

c = (ZHZ)−1ZHzp. (26)

C. System level convergence

In most of the studies on DLA, it is assumed that parameters

of PA model are extracted apriori [5], [7]. Using these param-

eters the coefficients of PD are obtained by applying any one

of the DLA methods discussed above. Often the parameters

of PA are usually extracted from one particular set of input

and output data. However, after the 1st PD identification the

characteristics of the input signal will change substantially. In

fact the PD itself being a nonlinear system, after the 1st system

level identification the spectrum of the PD output signal i.e.

the input of PA will have wider bandwidth. Therefore, the

behavior of the PA will likely change and a new model should

be “re-extracted” and used for the identification of the PD.

This process should be repeated until the complete PD-PA

system converges to the best possible solution. Similarly for

ILA, more than one system level iteration is needed for PD-

PA system to converge [3], [4]. Depending upon the model

of PA, the number of system level iterations needed for the

convergence of DLA and ILA might vary.

−60 −40 −20 0 20 40 60−120

−100

−80

−60

−40

−20

0

Frequency (MHz)

Norm

aliz

ed M

agnitude (

dB

/Hz)

Input

Output w/o PD

20dB−ILA

w/o Noise−DLA−NF

w/o Noise−ILA

w/o Noise−DLA−AM20dB−DLA−NF

20dB−DLA−AM

Fig. 3. PSD Comparison for DLA and ILA

IV. SIMULATION RESULTS

In this section, we present and discuss the simulation results

for system level convergence of ILA and DLA. For this

purpose, we use a Wiener model given in [10] as a PA. The PA

is driven by an LTE-Advanced signal with bandwidth 20 MHz,

sampling frequency 122.8 MHz and PAPR of approximately

8dB. The output back-off is approximately equal to 8.5dB.

An MP model (2) with K = 9 and L = 5 (only odd orders

terms are considered) has been used to extract the parameters

of PA in DLA using LS method (4) at each system level itera-

tion. This model gives best modeling performance in terms

of Normalized Mean-Square Error (NMSE) and Adjacent

Channel Error Power Ration (ACEPR) [11]. From simulations,

it was also observed that this model gives the best performance

when used as a post-inverse in ILA and PD in DLA. To

observe the impact of noise on the PD identification in both

architectures, we also add white Gaussian noise at the input

of PA or post-inverse identification blocks. The performance

with noise is demonstrated for SNR of 10dB, 20dB and 30dB.

Moreover, our simulations showed that NFxLMS algorithm

has slow convergence and poor performance as compared to

NFxRLS algorithm. This has also been validated in [7], [12].

Hence only NFxRLS will be considered here to demonstrate

the performance of DLA-NF.

Fig.3 shows the power spectral density (PSD) performance

of ILA and DLA for system level convergence at 3rd system

level iteration with noise (SNR=20dB) and without noise.

For DLA-AM, three iterations per sample were used for

each system level iteration in absence of noise. However,

in presence of noise, we were restricted to two iterations

per sample due to errors accumulation leading to divergence.

Moreover, as observed, for DLA-AM the performance is

limited by the approximation |x(n)| ≈ |u(n)|, which does

not hold in the compression region [6]. This approximation

limits the performance of DLA-AM.

Another point to note is that in absence of noise ILA

performs better than DLA-NF. This is due to the fact that

in DLA the error is induced while estimating the parameters

of PA in the 1st step. However, in presence of noise DLA-

NF outperforms ILA. This is quite obvious as in ILA the

measurement noise appears at the input of the post-inverse

block and thus matrix inversion in (25) seems to be more

sensitive to noise than in (4).

873

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−100

−90

−80

−70

−60

−50

−40

−30

−20

Iteration

AC

PR

(dB

)

ILA

DLA−NF

30 dB

w/o noise

20 dB

10 dB

Fig. 4. ACPR performance for ILA and DLA: System level Iterations

Fig.4 and Fig.5 shows the ACPR and EVM performance

respectively of ILA and DLA-NF for different system level

iterations at different SNR values. The 1st iteration denotes

the performance without PD. As observed both DLA-NF and

ILA converges after three system level iterations. Like Fig.

3, in Fig.4 and Fig.5 as well, DLA-NF outperforms ILA

in presence of noise. From Fig. 4 it can also be observed

that at the 3rd system level iteration with SNR of 30dB and

20dB, DLA-NF can achieve a performance improvement of

approximately 30dB and 18dB respectively whereas ILA can

achieve a performance improvement of approximately 20dB

and 12dB respectively. Even for SNR of 10dB DLA-NF can

achieve a performance improvement of approximately 8dB

whereas no significant performance improvement is observed

in ILA.

From Fig.5 it is also obvious that noise significantly dete-

riorates the EVM performance of ILA. ILA has an EVM of

approximately 8% for SNR of 10dB. For the same value of

SNR, DLA has an EVM of approximately 2%. At 3rd system

level iteration for SNR of 30dB and 20dB, DLA-NF has an

EVM of approximately 0.2% and 0.5% respectively whereas

ILA has an EVM of 0.5% and 2% respectively. Although

not shown in Fig.4 and Fig.5, DLA-AM performance was

observed to be fluctuating and highly unstable at system level

iterations for low SNRs. Hence it was performing worst among

all three algorithms. As mentioned before, this is mainly due

to the accumulation of errors at each sample level iteration.

V. CONCLUSION

The system level convergence of direct and indirect learning

architectures (DLA and ILA) for DPD is investigated. The MP

model have been used for the identification of both PD and PA.

Two different algorithms, one based on an analytical method

and another based on the adaptive nonlinear filter have been

presented and used for the identification of PD in DLA. For

ILA, the LS algorithm is used for the identification of the

post-inverse of the PA. The results demonstrated that at least

three system level iterations are needed to converge to the best

possible solution. The performances were evaluated in terms of

ACPR and EVM at the output of the PA for an LTE-Advanced

input signal. It was also shown that DLA is more robust than

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

2

4

6

8

10

Iteration

EV

M (

%)

DLA−NF − w/o Noise

ILA − w/o Noise

DLA−NF − SNR=30dB

ILA − SNR=30dB

DLA−NF − SNR=20dB

ILA − SNR=20dB

DLA−NF − SNR=10dB

ILA − SNR=10dB

Fig. 5. EVM performance for ILA and DLA: System level Iterations

ILA when noisy measurements are considered instead of the

ideal ones.

ACKNOWLEDGMENT

The research leading to these results has received fund-

ing from the Seventh Framework Programme under grant

agreement n° 230688 and from the European Catrene Project:

CA101 PANAMA.

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