Provided by the author(s) and University College Dublin Library in accordance with publisher
policies. Please cite the published version when available.
Title Band-Limited Volterra Series-Based Digital Predistortion for Wideband RF Power Amplifiers
Authors(s) Yu, Chao; Guan, Lei; Zhu, Erni; Zhu, Anding
Publication date 2012-12
Publication information IEEE Transactions on Microwave Theory and Techniques, 60 (12): 4198-4208
Publisher IEEE
Item record/more information http://hdl.handle.net/10197/8643
Publisher's statement © 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be
obtained for all other uses, in any current or future media, including reprinting/republishing
this material for advertising or promotional purposes, creating new collective works, for
resale or redistribution to servers or lists, or reuse of any copyrighted component of this
work in other works.
Publisher's version (DOI) 10.1109/TMTT.2012.2222658
Downloaded 2022-08-09T23:50:33Z
The UCD community has made this article openly available. Please share how this access
benefits you. Your story matters! (@ucd_oa)
© Some rights reserved. For more information, please see the item record link above.
1
Abstract— The continuously increasing demand for wide
bandwidth creates great difficulties in employing digital
predistortion (DPD) for radio frequency (RF) power amplifiers
(PAs) in future ultra wideband systems because the existing DPD
system requires multiple times the input signal bandwidth in the
transmitter and receiver chain, which is sometimes almost
impossible to implement in practice. In this paper, we present a
novel band-limited digital predistortion technique in which a
band-limiting function is inserted into the general Volterra
operators in the DPD model to control the signal bandwidth under
modeling, which logically transforms the general Volterra
series-based model into a band-limited version. This new
approach eliminates the system bandwidth constraints of the
conventional DPD techniques, and it allows users to arbitrarily
choose the bandwidth to be linearized in the PA output according
to the system requirement without sacrificing performance, which
makes the DPD system design much more flexible and feasible. In
order to validate this idea, a high power LDMOS Doherty PA
excited by various wideband signals, including 100 MHz
LTE-Advanced signals, was tested. Experimental results showed
that excellent linearization performance can be obtained by
employing the proposed approach. Furthermore, this technique
can be applied to other linear-in-parameter models. In future
ultra wideband systems, this new technique can significantly
improve system performance and reduce DPD implementation
cost.
Index Terms—Behavioral model, digital predistortion,
linearization, LTE-Advanced, power amplifiers, Volterra series.
I. I
NTRODUCTION
In order to satisfy growing demands for high data rates and
large capacity, transmit signal bandwidth is continuously
increasing in modern wireless communication systems. For
example, 100 MHz instantaneous modulation bandwidth will
be required in the forthcoming Long Term Evolution Advanced
(LTE-Advanced) systems [1]. This demanding bandwidth
requirement creates great difficulties in designing radio
frequency (RF) power amplifiers (PAs) to meet efficiency
specifications while simultaneously conforming to spectral
This is an expanded paper from the IEEE MTT-S Int. Microwave
Symposium held on June 17-22, 2012 in Montreal, Canada. This work was
supported by the Science Foundation Ireland under the Principal Investigator
Award scheme, and in part by Huawei Technologies Co. Ltd.
C. Yu, L. Guan and A. Zhu are with the School of Electrical, Electronic and
Communications Engineering, University College Dublin, Dublin 4, Ireland
(e-mail: chao.yu@ucdconnect.ie; lei.guan@ucd.ie; anding.zhu@ucd.ie).
E. Zhu is with Huawei Technologies Co. Ltd., Shanghai, China (e-mail:
erni_zhu@huawei.com).
mask and in-band distortion requirements. Digital predistortion
(DPD) is one of the advanced linearization techniques that
compensates for nonlinear distortion in RF PAs by inverting
their nonlinear behavior using digital circuits [2]. DPD allows
PAs to be operated at higher drive levels for higher efficiency
without sacrificing linearity, which is nowadays one of the
essential units in high power wireless base stations [3].
It is well known that the RF power amplifier is inherently
nonlinear, and it produces in-band distortion and
inter-modulation products that cause spectral regrowth in the
adjacent channels. For a modulated signal, fifth-order products
appear within a range of five times the bandwidth of the input
signal. In order to accurately model and effectively linearize the
RF PA, an output signal occupying multiple times the input
bandwidth, usually five times, is required to be captured. In the
past decades, many DPD models have been developed [4]-[15].
Among them, truncated Volterra series-based models, such as
memory polynomial (MP) [5], generalized memory polynomial
(GMP) [6], and dynamic deviation reduction-based Volterra
series (DDR) [7], are very popular. These models are based on
polynomial-type of functions, of which the output bandwidth
increases proportionally with the nonlinear orders. This
property can effectively match the nonlinear behavior of the RF
PAs, and hence these models work very well in the existing
systems.
However, in the forthcoming wideband system, e.g.,
LTE-Advanced, 100 MHz modulation bandwidth is required,
which means that 500 MHz linearization bandwidth will be
required if the existing DPD techniques are employed. Such a
wide bandwidth requirement will remarkably increase the
difficulties in system design. It requires not only very high
speed data converters but also ultra wideband transmitter and
receiver chains, which make DPD sometimes become
infeasible. On the other hand, in practice, it may not be
necessary to linearize the PA up to such a wideband bandwidth
because the signal bandwidth is expanded too wide. We may
Band-Limited Volterra Series-Based Digital
Predistortion for Wideband RF Power Amplifiers
Chao Yu, Student Member, IEEE, Lei Guan, Student Member, IEEE, Erni Zhu,
and Anding Zhu, Member, IEEE
Fig. 1. Band-limited input and output.
Behavioral
Model
Band-limited Input
Band-limited output
2
only need to remove the distortion near the input center
frequency band, e.g., within 200 to 300 MHz range. The
distortion beyond that band can be filtered by using a bandpass
filter at the PA output, or it may have already been taken care of
by the cavity filters in the duplexer. Therefore, in the future
system, we may face a scenario that only the distortion within a
limited bandwidth is captured, as illustrated in Fig. 1, and we
only need to linearize the system within that limited bandwidth.
In this new scenario, however, the conventional DPD models
can no longer be employed. This is because almost all of the
existing models are constructed in the time domain with very
little or no control in the frequency domain. For instance, in the
MP model, the bandwidth of the frequency domain signal
solely depends on the nonlinear order selected and the
distortion resulted from each nonlinear operation are spread
over the corresponding bandwidth. However, in the new
system, as shown in Fig. 1, only distortion within a limited
bandwidth is captured and the rest of distortion is missing,
which leads that the signal bandwidth produced by the model
does not match the bandwidth of the system under modeling,
and thus the model accuracy may be significantly decreased, so
that the linearization performance can be deteriorated.
In [16], a band-limited Volterra series-based behavioral
modeling approach was proposed, which allows us to
accurately model a PA with band-limited input and output. This
is achieved by inserting a band-limiting function into each
Volterra operator before multiplying with its coefficients. By
controlling the bandwidth of the band-limiting function, we can
arbitrarily choose the bandwidth under modeling. It naturally
transforms the general Volterra series-based models into
band-limited ones. This approach eliminates the inherent
bandwidth requirement of the general Volterra series but still
keeps the same model structure, such as the linear-in-parameter
property. Experimental results showed that the proposed
approach significantly improves the model accuracy.
In this paper, we introduce a band-limited DPD system by
employing the proposed modeling technique. In this new
system, the DPD bandwidth can be arbitrarily chosen according
to the system requirement, which removes the system
bandwidth constrains of the conventional system. It provides an
extra freedom for DPD designers to make tradeoffs between
linearization bandwidth and system cost. Detailed theoretical
analysis and rigorous experimental validations are presented in
the paper.
The paper is organized as follows. In Section II, we briefly
re-introduce the band-limited behavioral modeling
methodology proposed in [16]. A band-limited DPD system is
then proposed in Section III while the model implementation is
presented in Section IV. Experimental results are given in
Section V with a conclusion in Section VI.
II. B
AND-LIMITED BEHAVIORAL MODELING
A. General Volterra Series
In the discrete time domain, a general Volterra series [17] can
be written as
1
1
10 0
() (, , ) [()]
p
ppp
pi i
yn h i i D xn
∞∞ ∞
== =
=
∑∑ ∑
(1)
where x(n) and y(n) represents the input and output,
respectively,
1
(, , )
pp
hi i
is the pth-order Volterra kernel and
1
[()] ( )
p
p
j
j
D
xn xn i
=
=
−
∏
(2)
where D
p
is the pth-order Volterra operator. In a real
application, the general Volterra series is normally simplified to
a certain format. For instance, only limited dynamic orders are
considered in the DDR-Volterra model [7]. One of the main
advantages of the Volterra models is that the output of the
model is linear with respect to its coefficients, meaning that it is
possible to extract a nonlinear Volterra model in a direct way by
using linear system identification algorithms, such as least
squares (LS).
However, one common feature of these models is that the
signal bandwidth in the frequency domain increases with the
nonlinear orders involved in the model. As shown in Fig. 2,
when the input signal passes each nonlinear Volterra operator,
the signal bandwidth proportionally increases with the order of
nonlinearity selected. For example, the 3
rd
-order operator will
expand the signal bandwidth 3 times, and the final output
bandwidth depends on the highest order of nonlinearity chosen.
Sufficient accuracy can be achieved when the nonlinear order
in the model matches the bandwidth of the system under
modeling. For instance, a 5
th
-order model can be used when the
output signal within five times the input signal bandwidth is
captured. However, in the wideband scenario described earlier,
the output signal may be filtered before it is captured. In that
case, although the high order nonlinear distortion is spread over
a wide bandwidth, only part of distortion within the filter
passband is obtained while the other part is missing. If we still
use the Volterra model to map the input to the output, the model
accuracy will decrease because the signal bandwidth of the
model does not match the signal bandwidth under modeling.
B. Proposed Band-limited Volterra Series
To resolve the bandwidth mismatch problem, a new model
called band-limited Volterra series was proposed [16], in which
a band-limiting function is introduced to control the bandwidth
expansion when the signal passes each nonlinear Volterra
operator. As shown in Fig. 3, a band-limiting function,
()w
⋅
, is
inserted into the Volterra operator. This band-limiting function
can be a linear filter, and it can be pre-designed in the frequency
domain with an effective bandwidth chosen according to the
bandwidth requirement of the system output. It is then
Fig. 2. General Volterra series model.
3
converted into the time domain and represented by a finite
impulse response. The new pth-order band-limited Volterra
operator T
p
can be represented by
[()] [()] ()
pp
Txn Dxn wn=∗
(3)
where * represents the convolution operation. The general
Volterra series can thus be transformed into a band-limited
version as
{}
1
1
1
,1
10 0
,1
10 0
,1
10 0 0
1
() (, , ) [()]
(, , ) [()] ()
(, , ) ( )()
p
p
p
pBL p p
pi i
pBL p p
pi i
p
K
pBL p j
pi i k
j
yn h i i T xn
hi iDxnwn
hi i xnikwk
∞∞ ∞
== =
∞∞ ∞
== =
∞∞ ∞
== = =
=
=
=∗
⎧⎫
⎡⎤
⎪⎪
=−−
⎨⎬
⎢⎥
⎪⎪
⎣⎦
⎩⎭
∑∑ ∑
∑∑ ∑
∑∑ ∑ ∑
∏
(4)
where T
p
is the pth-order band-limited Volterra operator,
()w
⋅
is
the band-limiting function with length K,
,1
(, , )
pBL p
hi i
is the
pth-order band-limited Volterra kernel, and x(n) and y(n)
represents the input and output, respectively. As shown in Fig.
3, because the signal is filtered by the band-limiting function
after it passes each Volterra operator, the bandwidth of the
output from each Volterra operator is limited within a certain
frequency range, e.g., BW. After linearly scaled by the
coefficients and recombined together, the final output y(n) is
logically band-limited to BW. It allows the bandwidth of the
signal produced by the model to perfectly match that of the
actual PA output if BW is chosen to be equal to the bandwidth
of the PA output. The accuracy of the model is therefore
significantly increased. Although linear convolution is
required, the model structure is still the same as that of the
general Volterra series, e.g., the output is still linear with
respect to the model parameters. The coefficients can therefore
also be extracted by using linear system identification
algorithms.
By inserting a band-limiting function into each Volterra
operator, the general Volterra series is naturally transformed
into the band-limited version. This intervention to the model
completely changes the model behavior and significantly
enhances its modeling capability. In the conventional model,
there is no control in the frequency domain and the bandwidth
of the output is solely decided by the nonlinear order chosen. In
the new model, an extra control or freedom is obtained, namely,
we can arbitrarily control the signal bandwidth produced by the
model in the frequency domain by selecting different
bandwidths of the band-limiting function, despite the model is
still operated in the time domain. In this case, no matter how
narrow the bandwidth of the PA output is, the relationship
between the input and output can always be accurately
represented by the new model if the bandwidth of the
band-limiting function matches that of the PA output. If we
consider integrating the band-limiting function with the
original Volterra operator together, we can treat the new
nonlinear operation (Volterra + filtering) as a new basis
function. The model based on the new band-limited basis
function can be used to represent a wide range of nonlinear
systems, including the general Volterra series if the filter
bandwidth is set to infinite. In this sense, the general Volterra
series can be treated as one of special cases of the new model.
III.
BAND-LIMITED DIGITAL PREDISTORTION
The band-limited modeling idea proposed in Section II looks
very simple. However, it not only just improves the model
accuracy, but also significantly enhances the modeling
capability. If employed in digital predistortion, it dramatically
changes the way the DPD system is operated, as we will discuss
below.
A.
Conventional DPD
In the conventional systems, behavioral models developed
for RF PAs can be directly employed in digital predistortion
and the indirect learning approach [18][19] is normally used for
model extraction. It is based on the theory of the pth-order
inverse described in the classical Volterra series book [17],
which states that the pth-order post-inverse is the same as its
pth-order pre-inverse when a nonlinear system is linearized up
to the pth-order nonlinearity. In these systems, an identical
model is normally used for both the pre-inverse in predistortion
and the post-inverse in model extraction. In the model
extraction, the output of the PA y(n) is used as input of the
model while the input of the PA x(n) is used as the expected
output. The extracted coefficients are then directly copied to the
pre-inverse model, i.e., the predistortion block, as illustrated in
Fig. 4. If z(n) approaches u(n), y(n) is then close to x(n).
As mentioned earlier, a signal passing a nonlinear PA will
generate spectrum regrowth, and the nonlinear operators in the
DPD model also expand the signal bandwidth. In the frequency
domain, we can see the signal bandwidth changes when the
signal passes each nonlinear block, as illustrated in Fig. 4: (i) in
the post-inverse case, when the original input signal passes the
PA, the bandwidth of the signal is expanded 5 times, and then
after the post-inverse, the signal bandwidth returns to the
original bandwidth; (ii) in the pre-inverse case, the signal
bandwidth is expanded 5 times when the signal passes the
predistorter block, and then returns to the original after passing
the PA. In a real system, DPD is normally operated in digital
baseband. The block diagram of a general DPD system is
shown in Fig. 5. In order to accommodate the frequency
domain characteristics and achieve high linearization
performance, a general rule of thumb is that five times the input
signal bandwidth would be required in both the feedback path
and the transmitter chain.
Fig. 3. Band-limited Volterra series model.
4
The configuration in Fig. 5 works very well in a relatively
narrow band system. However, for a wideband system, e.g., the
LTE-Advanced system, 100 MHz input bandwidth will be
required. If the existing DPD system is used, 500 MHz
linearization bandwidth will be needed. It creates enormous
difficulties in real implementation. In the feedback path, a
wideband down-conversion chain is required and a 500 MSPS
(Mega-samples per second) sampling rate will be required for
the analog-to-digital converters (ADCs) for I/Q signals
1
. If
digital intermediate frequency (IF) signal is used, it will require
1 GSPS (Giga-samples per second) sampling. In order to
produce the 500 MHz predistorted signal and allow it passing
through the transmitter chain before feeding into the PA, very
high speed digital-to-analog converters (DACs) and a
wideband transmitter chain will also be required. This is very
difficult and sometimes impossible to implement in practice.
On the other hand, it may not be necessary to clean up all the
distortions in such a wide bandwidth by using DPD. Because
when the distortion is spread into such a wide bandwidth, some
distortions in the sideband can be easily filtered by using a
bandpass filter in the output of the RF PA and this bandpass
filter may have already existed in the transmitter, e.g., the
cavity filter in the duplexer. Therefore, in practice, it would be
desirable to have a DPD system that only requires limited
bandwidth and only linearizes the PA up to a certain bandwidth.
However, this will require necessary changes in the existing
DPD models; otherwise the system performance will
deteriorate quickly. It is because that the model accuracy is
1
The ADC sampling rate may be reduced by employing the under-sampling
approach proposed in [8], but the anti-aliasing filter in the ADC must be
re-designed to preserve the full sideband information, and it still requires a
wideband down-conversion chain in the feedback path.
significantly reduced, when the bandwidth of the system under
modeling does not match the signal bandwidth of the model, as
discussed in Section II.
B.
Proposed Band-limited DPD
If the band-limited model proposed in Section II is
employed, the bandwidth mismatch problem can be easily
resolved. Assuming that the output of the PA is filtered, the
system under modeling can be represented by a nonlinear PA
with a bandpass filter, as represented by T shown in Fig. 6. If
we let the bandwidth of the band-limiting function equal to the
bandwidth of the bandpass filter, the relationship between the
PA input and output can be accurately modeled by employing
the band-limited model. Because part of the frequency
information is missing, invertibility of the model must be
investigated before employing it in DPD. First, let’s look at
how the signal is handled by the model. In the PA modeling
(the forward model), because we changed the basis function of
the model, the frequency bandwidth of the model can now
perfectly match the bandwidth of the system under modeling.
With correct model extraction, the input and output relationship
can be accurately represented by a pth-order band-limited
Volterra model. Despite each of the new Volterra operator only
takes into account the information within the selected
bandwidth, there must be one to one mapping from the input to
the output in the transfer function, the same as that in the
general Volterra series. For instance, the 3
rd
-order basis
function exactly represents the mapping from the original input
to the band-limited 3
rd
-order distortion. Therefore, in an inverse
order, if the same band-limited model structure is used and with
the same signal bandwidth, the one to one mapping can of
course be inversed back in the same way. The model hence
indeed is invertible.
Secondly, we should discuss whether or not the pth-order
inverse is still applicable. Let’s re-derive the equations in the
pth-order inverse theory [17]. As illustrated in Fig. 6, we simply
cascade the three systems together, similar to that in Fig. 4. We
first construct a Pth-order post-inverse,
1
()
P
post
T
−
,with a
band-limited model, and then copy it into the pre-inverse,
1
()
P
pre
T
−
. Let’s assume
11 1
() () ()
.
Ppre Ppost P
TT T
−− −
==
(5)
In the system Q with the post-inverse,
Fig. 6. Band-limited cascaded nonlinear system.
Fig. 5. Conventional DPD.
Fig. 4. Cascaded nonlinear system.
