Least-Squares Phase Predistortion of a +30dBm Linköping University Post Print

Least-Squares Phase Predistortion of a +30dBm Linköping University Post Print
Least-Squares Phase Predistortion of a +30dBm
Class-D Outphasing RF PA in 65nm CMOS
Ylva Jung, Jonas Fritzin, Martin Enqvist and Atila Alvandpour
Linköping University Post Print
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Ylva Jung, Jonas Fritzin, Martin Enqvist and Atila Alvandpour, Least-Squares Phase
Predistortion of a +30dBm Class-D Outphasing RF PA in 65nm CMOS, 2013, IEEE
Transactions on Circuits and Systems Part 1: Regular Papers, (60), 7, 1915-1928.
http://dx.doi.org/10.1109/TCSI.2012.2230507
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-71862
1
Least-Squares Phase Predistortion of a +30dBm
Class-D Outphasing RF PA in 65nm CMOS
Ylva Jung, Jonas Fritzin, Member, IEEE, Martin Enqvist, Member, IEEE,
and Atila Alvandpour, Senior Member, IEEE
Abstract—This paper presents a model-based phase-only predistortion method suitable for outphasing radio frequency (RF)
power amplifiers (PA). The predistortion method is based on
a model of the amplifier with a constant gain factor and phase
rotation for each outphasing signal, and a predistorter with phase
rotation only. Exploring the structure of the outphasing PA, the
model estimation problem can be reformulated from a nonconvex
problem into a convex least-squares problem, and the predistorter
can be calculated analytically. The method has been evaluated for
5 MHz Wideband Code-Division Multiple Access (WCDMA) and
Long Term Evolution (LTE) uplink signals with Peak-to-Average
Power Ratio (PAPR) of 3.5 dB and 6.2 dB, respectively, applied
to one of the first fully integrated +30 dBm Class-D outphasing
RF PA in 65nm CMOS. At 1.95 GHz for a 5.5 V (6.0 V) supply
voltage, the measured output power of the PA was +29.7 dBm
(+30.5 dBm) with a power-added efficiency (PAE) of 27 %. For the
WCDMA signal with +26.0 dBm of channel power, the measured
Adjacent Channel Leakage Ratio (ACLR) at 5 MHz and 10 MHz
offsets were -46.3 dBc and -55.6 dBc with predistortion, compared
to -35.5 dBc and -48.1 dBc without predistortion. For the LTE
signal with +23.3 dBm of channel power, the measured ACLR
at 5 MHz offset was -43.5 dBc with predistortion, compared to
-34.1 dBc without predistortion.
Index Terms—Outphasing, amplifier, linearization, predistortion, complementary metal-oxide-semiconductor (CMOS).
I. I NTRODUCTION
P
OWER amplifiers (PA) for modern wireless communications standards typically employ non-constant amplitude
modulation schemes to use the limited frequency spectrum
more efficiently and larger bandwidths to meet the demand
for higher data rates. However, to meet the spectral and
modulation requirements, highly linear PAs are required.
With the improved speed of CMOS transistors, highly
efficient switched PAs, like Class-E and Class-D, have gained
increased interest in polar modulation, shown for Enhanced
Data rates for GSM Evolution (EDGE) [1], and outphasing [2],
[3]. In the outphasing PA, an input signal, s(t), containing
both amplitude and phase modulation, is divided into two
Manuscript received June 26, 2012; revised September 27, 2012; accepted
October 21, 2012. This work has been supported by the Excellence Center at
Linköping-Lund in Information Technology (ELLIIT), the Swedish Research
Council (VR), and Ericsson Research, Kista, Sweden.
Y. Jung, M. Enqvist, and A. Alvandpour are with the Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden, phone:
+46(0)13-284474, e-mail: ylvju@isy.liu.se, maren@isy.liu.se, atila@isy.liu.se.
J. Fritzin was with the Department of Electrical Engineering, Linköping
University, SE-581 83 Linköping, Sweden. He is now with Ericsson AB, SE164 80 Stockholm, Sweden, e-mail: jonas@jonasfritzin.com.
Copyright (c) 2012 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending an email to pubs-permissions@ieee.org.
constant-envelope phase-modulated signals, s1 (t) and s2 (t), as
in Fig. 1(a). Fig. 1(b) shows how the two signals are separately
amplified by efficient switched amplifiers, A1 and A2 , and
connected to a power combiner whose output, y(t), is an amplified replica of the input signal. The two constant amplitude
outputs can be combined using an isolating combiner, like
Wilkinson [4], or a non-isolating combiner, like Chireix [5] or
transformers [6]. While an isolating combiner can have good
linearity as the apparent load impedance for each amplifier is
fixed [4], [7], the efficiency is reduced at back-off. With a nonisolating combiner, the apparent load impedance depends on
the outphasing angle [8], degrading linearity if load-sensitive
amplifiers are used [9], [10], but can help to reduce the
power dissipation for small output amplitudes and improve the
back-off efficiency [6]. As shown in [11]–[20], Class-D and
Class-F amplifiers with non-isolating combiners can provide
acceptable linearity.
Unless the amplifier stages, and signal paths, are identical,
the amplifiers will experience gain and phase imbalances,
creating nonlinearities and spectral distortion [5], [21], [22].
Earlier predistortion methods of RF PAs include modelbased predistorters using model structures such as Volterra
series [23]–[25], parallel Hammerstein structures [26], or lookup tables (LUT) [27], which also can be made adaptive [28],
[29]. Compared to conventional linear PAs, the outphasing PA
takes two constant envelope signals to create an amplitude
and phase modulated output signal. Thus, there is no linearity
between the individual outphasing signals and the output, and
new predistortion methods are needed.
Phase-predistortion was evaluated for Chireix combiners in
simulations and by using signal generators in measurements
(no PA was used) in [5]. To compensate for the gain mismatch,
earlier proposed predistorters changed the input signal amplitudes when linear amplifiers were used [30]–[32], or included
adjustments of the voltage supplies in the output stage [8], [9].
In [16], an amplifier model and predistorter were proposed to
compensate for amplitude and phase mismatches by changing
only the phases of the two input outphasing signals, evaluated
for uplink signals with PAPR of ∼3.5 dB applied to a +10 dBm
amplifier. A similar approach was used in [33], where an
implementation of an all-digital Signal Component Separator
(SCS) with branch mismatch compensation was presented.
This paper presents a model-based phase-only predistortion
method suitable for outphasing RF PAs. The predistortion
method is based on a model of the amplifier with a constant
gain factor and phase rotation for each outphasing signal, and a
predistorter with phase rotation only. The predistorter proposed
2
in this paper compensates for both amplitude and phase distortion by changing only the phases of the two input outphasing
signals, and thus a second voltage supply is not needed to
compensate for the gain mismatch. By making use of the
specific structure of outphasing PAs, the model estimation
can be reformulated from a nonconvex problem into a convex
least-squares problem, and the predistorter can be calculated
analytically, thus making iterations like in [16] unnecessary.
The solution in [33] considers the gain mismatch between
the two branches and computes the ideal phase compensation
when the outputs are approximated as two constant amplitudes.
This is possible when there is no interaction between the
amplifier stages, e.g. by using a combiner with isolation.
In this paper, we also consider the outputs as two constant
amplitudes generating amplitude and phase distortion, (14)(16), but the amplitude dependent phase distortion that occurs
due to the interaction and signal combining of the amplifiers’
outputs is also considered. Thus, the approach in this paper
includes another degree of distortion and the nonlinearities are
used to create a linear least-squares predistortion method. A
similar approach to the one in [16] is presented in [34], with
suggestions on how to find good initial values for the nonlinear
optimization needed to extract the model and DPD parameters.
However, the basic problem of nonconvexity has not been
solved in [16], [34] and local minima still risk posing problems
in the optimization. For PVT variations or load impedance
variations (as in handheld applications) our method can also be
made adaptive, without significantly increasing the complexity
as our method is a linear least-squares DPD method with a
single optimal solution.
The proposed predistortion method has been used for
WCDMA and LTE uplink signals with PAPR of 3.5 dB and
6.2 dB, respectively, applied to a fully integrated +30 dBm
Class-D outphasing RF PA in 65nm CMOS [15], which is
also one of the first fully integrated +30 dBm outphasing
PAs [11], [14], [15]. The predistortion method was applied at
the baseband level and has not been implemented in hardware.
Implementing the proposed method involves many considerations, and as concluded in [25], “different methodologies or
implementation structures will lead to very different results in
terms of complexity and cost from the viewpoint of hardware
implementation”. Therefore, the purpose of the paper is to
theoretically and practically demonstrate the capability of the
proposed DPD method and describe it such that it can serve as
a basis for hardware implementation. The evaluation criterion
of the predistortion method is based on the measured ACLR,
as Error Vector Magnitude (EVM) requirements were easily
met without predistortion [15].
The outline of the paper is as follows. In Section II, the outphasing concept is explained mathematically. In Section III, an
ideal phase predistorter is described, and the implementation of
a DPD estimator is shown in Section IV, followed in Section V
by a discussion on benefits with the least-squares formulation
presented here. The impact of PVT variations and dynamic
range is discussed in Section VI and the PA used in the
measurements is described in Section VII. In Section VIII, the
measured RF performance and the performance for modulated
signals with and without phase-predistortion are presented. In
(a)
(b)
Fig. 1. (a) Outphasing concept and signal decomposition. (b) Ideal power
combining of the two constant-envelope signals, where the amplitude of the
output is normalized to 1.
Section IX, the conclusions are provided.
II. O UTPHASING C ONCEPT
The outphasing concept is shown in Fig. 1(a), where a nonconstant envelope-modulated signal
s(t) = r(t)ejα(t) = rmax cos(ϕ(t))ejα(t) , 0 ≤ r(t) ≤ rmax (1)
where rmax is a real-valued constant, is used to create two
constant-envelope signals, s1 (t) and s2 (t), in the SCS in
Fig. 1(b) as
s1 (t) = s(t) + e(t) = rmax ejα(t) ejϕ(t)
s2 (t) = s(t) − e(t) = rmax ejα(t) e−jϕ(t)
s
2
rmax
− 1.
e(t) = js(t)
2
r (t)
(2)
The signals s1 (t) and s2 (t), contain the original signal, s(t),
and a quadrature signal, e(t), and are suitably amplified by
switched amplifiers like Class-D/E. By separately amplifying
the two constant-envelope signals and combining the outputs
of the two individual amplifiers as in Fig. 1(b), the output
signal is an amplified replica of the input signal. In the sequel,
PA refers to the complete outphasing amplifier and amplifier
refers to the switched amplifiers A1 and A2 .
Letting g1 and g2 denote two positive real valued gain
factors, on s1 (t) and s2 (t), and δ denote a phase mismatch in
the path for s1 (t), it is clear from
y(t) = g1 ejδ s1 (t) + g2 s2 (t)
= [g1 ejδ + g2 ]s(t) + [g1 ejδ − g2 ]e(t),
(3)
that besides the amplified signal, a part of the quadrature
signal remains. As the bandwidth of the quadrature signal is
larger than the original signal, s(t), the ACLR is degraded
and the margins to the spectral mask are reduced, unless the
quadrature signals are canceled in the power combiner [5],
[21]. The phase and gain mismatches between s1 (t) and s2 (t)
3
Parts of the results in this section were presented in [16] but
are restated here for sake of completeness. In [16], the results
were merely used for motivation and evaluation purposes,
whereas they are here part of the algorithm to obtain the predistorter, as shown later in Section IV. The new contribution
is the reformulation of the nonconvex modeling problem to a
convex problem, thus guaranteeing that the global minimum
will be found in the optimization.
In this section, the setup is explained and the ideal predistorter is described. This will be followed by a description of
how a DPD can be estimated in practice, in the next section.
(a)
A. PA and DPD description
(b)
Let
∆ψ (s1 , s2 ) = arg(s1 ) − arg(s2 )
Fig. 2. A schematic picture of (a) the amplifier branches setup and (b) the
amplifiers with predistorters. Note that the functions fk and hk , k = 1, 2, are
not functions of the input to the block only but are used to show the general
functionality of the PA and the DPD with the separation of the two branches.
must be minimized in order not to allow a residual quadrature
component to distort the spectrum or limit the Dynamic Range
(DR)
max(|y(t)|)
|g1 + g2 |
cDR = 20 log10
= 20 log10
min(|y(t)|)
|g1 − g2 |
(4)
of the PA [5]. While the DR defines the ratio of the maximum
and minimum output amplitudes the PA can achieve, all
amplitudes and phases within the DR can be reached by
changing the phases of the outphasing signals. As the output
of a Class-D stage can be considered as an ideal voltage source
whose output voltage is independent of the load [17], i.e.
the output is connected to either VDD or GND, the constant
gain approximations g1 and g2 are appropriate and make
Class-D amplifiers suitable for non-isolating combiners like
transformers [11]–[16].
III. A MPLIFIER AND PREDISTORTER MODELING
A physical outphasing PA differs from the ideal PA in more
ways than the constant gain and phase mismatches described
in the previous section. Initial tests with the PA have shown
a phase distortion on the output that seems to depend on
the amplitude of the input s(t). Based on this observation,
a mathematical description of the physical outphasing PA has
been formulated and a digital predistorter (DPD) has been
proposed. Since it is desired that the predistorter should invert
all effects of the PA except for the gain, the signals are
assumed to be normalized such that
max |s(t)| = max |y(t)| = 1.
t
t
(5)
Despite the fact that the PA is analog and the baseband
model is time-discrete, the notation t is used for indicating
the dependency of time. Based on the context, t may thus
be a continuous or discrete quantity and denote the time or
the time indexation. For notational convenience, the explicit
dependency on time will be omitted in parts of this section
and the following one.
(6)
denote the phase difference of the outphasing signals s1 (t)
and s2 (t). Since the amplitude of the nondecomposed signal
in the outphasing system is determined by ∆ψ (s1 , s2 ), this
difference can be used instead of the actual amplitude in many
cases. For notational convenience, ∆ψ will be used instead of
∆ψ (s1 , s2 ), unless specified otherwise. Here, all phases are
assumed unwrapped.
A description of the amplitude-dependent phase distortion
in yk (t), k = 1, 2 (the two amplifier branches) can be done as
yk (t) = gk ej fk (∆ψ ) sk (t),
k = 1, 2,
y(t) = y1 (t) + y2 (t),
(7a)
(7b)
in case there is no DPD in the system as in Fig. 2(a). Here,
f1 and f2 are two real-valued functions describing the phase
distortion
arg(yk ) − arg(sk ) = fk (∆ψ ),
k = 1, 2,
(8)
in each signal path. Furthermore, g1 and g2 are the gain factors
in each amplifier branch. The normalization (5) implies that
g1 + g2 = 1. Hence, an ideal PA would have f1 = f2 = 0 and
g1 = g2 = g0 = 0.5 and any deviations from these values will
cause non-linearities in the output signal and spectral distortion
as previously concluded. In order to compensate for these
effects, a DPD can be used to modify the input outphasing
signals, i.e. s1 (t) and s2 (t), to the two amplifier branches.
Since the outputs of the Class-D stages have constant
envelopes, the DPD may only change the phase characteristics
of the two input outphasing signals. With this in mind, a DPD
that produces the predistorted signals
sk,P (t) = ej hk (∆ψ ) sk (t),
k = 1, 2,
(9)
to the two amplifier branches is proposed. Here, h1 and h2 are
two-real-valued functions that depend on the phase difference
between the two signal paths. By modifying the signals in
each branch using the DPD in (9) as shown in Fig. 2(b), the
PA output yP (t) can be written
yP = g1 ej f1 (∆ψ (s1,P ,s2,P )) s1,P + g2 ej f2 (∆ψ (s1,P ,s2,P )) s2,P .
{z
} |
{z
}
|
∆
=y1,P
∆
=y2,P
(10)
4
The phase difference between the two paths after the predistorters is described by
∆ψ (s1,P , s2,P ) = arg(s1,P ) − arg(s2,P )
= [arg(s1 ) + h1 (∆ψ )]
− [arg(s2 ) + h2 (∆ψ )]
∆
= ∆ψ + h1 (∆ψ ) − h2 (∆ψ ) = h̃(∆ψ ) (11)
and the phase difference between the two paths at the (predistorted) outputs by
∆ψ (y1,P , y2,P ) = arg(y1,P ) − arg(y2,P ) =
= [arg(s1,P ) + f1 (∆ψ (s1,P , s2,P ))]
− [arg(s2,P ) + f2 (∆ψ (s1,P , s2,P ))]
h
i
= arg(s1 ) + h1 (∆ψ ) + f1 (h̃(∆ψ ))
h
i
− arg(s2 ) + h2 (∆ψ ) + f2 (h̃(∆ψ ))
(b)
Fig. 3.
(a) Decomposition of the input signal s(t) into s1 (t) and s2 (t)
when g1 = g2 = g0 = 0.5 and into s̃1 (t) and s̃2 (t) when decomposed as
in (14) with nonidentical gain factors g1 and g2 . (b) Trigonometric view of
the decomposition of s(t) using nonidentical gain factors. Note that |s̃k | =
gk , k = 1, 2.
and
g12 = g22 + |s|2 − 2g2 |s| cos(b2 ).
(16)
The angles b1 and b2 that define s̃1 (t) and s̃2 (t) can be computed from these expressions and can be viewed as functions
of ∆ψ since |s| is a function of ∆ψ . This implies that the
angles
= ∆ψ + h1 (∆ψ ) − h2 (∆ψ )
+ f1 (h̃(∆ψ )) − f2 (h̃(∆ψ ))
= h̃(∆ψ ) + f1 (h̃(∆ψ )) − f2 (h̃(∆ψ ))
∆
= f˜(h̃(∆ψ )).
(a)
(12)
Thus, the absolute phase change in each branch is given by
1
∆
ξ1 (∆ψ ) = arg(s̃1 ) − arg(s1 ) = b1 − ∆ψ
2
(17)
1
∆ ψ − b2
2
(18)
and
arg(yk,P ) = arg(sk ) + hk (∆ψ ) + fk (∆ψ (s1,P , s2,P )) (13)
for k = 1, 2.
∆
ξ2 (∆ψ ) = arg(s̃2 ) − arg(s2 ) =
can also be viewed as functions of ∆ψ .
Assume that an ideal DPD (9) is used together with the
PA (7). In this case, the equalities
B. The ideal DPD
As mentioned above, the PA output signal y(t) is a distorted
version of the input signal. An ideal DPD should compensate
for this distortion and result in a normalized output signal
yP (t) = y1,P (t) + y2,P (t) that is equal to the input signal
s(t) = 0.5s1 (t) + 0.5s2 (t). In the ideal case when g1 = g2 =
g0 = 0.5, this is obtained when y1 (t) = 0.5s1 (t) and y2 (t) =
0.5s2 (t). However, this is not possible to achieve when gk 6=
0.5, k = 1, 2.
In this case, the ideal values for y1,P (t) and y2,P (t) are
instead s̃1 (t) and s̃2 (t), respectively. These signals define an
alternative decomposition of s(t) such that
s̃1 (t) + s̃2 (t) = s(t),
|s̃k | = gk ,
k = 1, 2,
(14a)
and
arg(s̃1 ) ≥ arg(s̃2 ).
(14b)
(14c)
Given g1 , g2 = 1 − g1 and s(t), the signals s̃1 (t) and s̃2 (t)
can be computed from (14). Let
(19)
y2,P (t) = s̃2 (t)
(20)
and
hold, which results in
yP (t) = y1,P (t) + y2,P (t) = s̃1 (t) + s̃2 (t) = s(t).
In particular, in order not to change the amplitude at the output,
the phase difference between y1,P (t) and y2,P (t) must be
equal to the one between s̃1 (t) and s̃2 (t), i.e.,
∆ψ (y1,P , y2,P ) = arg(s̃1 ) − arg(s̃2 ) =
= [arg(s1 ) + ξ1 (∆ψ )] − [arg(s2 ) + ξ2 (∆ψ )]
∆ ˜
= ∆ψ + ξ1 (∆ψ ) − ξ2 (∆ψ ) = ξ(∆
ψ ). (21)
Hence, inserting (21) into (12) gives
˜ ψ)
f˜(h̃(∆ψ )) = ξ(∆
b1 = arg(s̃1 ) − arg(s)
y1,P (t) = s̃1 (t)
⇔
˜ ψ )),
h̃(∆ψ ) = f˜−1 (ξ(∆
(22)
assuming that f˜ is invertible. Furthermore, for (19) and (20)
to hold, we require
and
b2 = arg(s) − arg(s̃2 )
denote the angles between the decomposed signals and s(t)
as shown in Fig. 3(a). Fig. 3(b) shows that the decomposition
can be viewed as a trigonometric problem and application of
the law of cosines gives
Combined with (11), (13) and (17) or (18), respectively, this
gives
g22 = g12 + |s|2 − 2g1 |s| cos(b1 )
arg(sk )+hk (∆ψ )+fk (h̃(∆ψ )) = arg(sk )+ξk (∆ψ ), k = 1, 2,
(15)
arg(yk,P ) = arg(s̃k ),
k = 1, 2.
5
which results in
hk (∆ψ ) = −fk (h̃(∆ψ )) + ξk (∆ψ )
˜ ψ ))) + ξk (∆ψ )
= −fk (f˜−1 (ξ(∆
(23)
for k = 1, 2. Here, (22) has been used in the last equality.
Hence, using the predistorters (23) as in (9), the output y(t)
will be an amplified replica of the input signal s(t), despite
the gain mismatch and nonlinear behavior of the amplifiers.
IV. A L EAST- SQUARES DPD ESTIMATOR
The ideal predistorter requires the unknown characteristics
g1 , g2 , f1 and f2 of the PA, which have to be estimated from
a sequence of input and output measurements referred to as
estimation data. In [16], a model of the PA was estimated by
minimizing a quadratic cost function measuring the difference
between the measured and predicted output signal. Based
on this model, the DPD was estimated in a second step by
minimizing a similar cost function measuring the difference
between s(t) and the predicted, precompensated output yP (t).
Both these estimation problems involve solving a nonconvex
optimization problem, which might be a challenging task due
to the presence of local minima. However, using the mathematical characterization of the DPD and PA derived here, there
is an alternative way which essentially only involves solving
standard least-squares problems.
Consider first the two gain factors g1 and g2 = 1−g1 , where
the relation between them comes from the normalization (5).
Let
g1 = g0 ± ∆g ,
g2 = g0 ∓ ∆g ,
(24)
where ∆g ≥ 0 represents the gain imbalance between the
amplifier stages and g0 = 0.5. Inserting (24) into (4) gives
g0
.
(25)
cDR = 20 log10
∆g
Hence, the imbalance term ∆g can be computed as
∆g = g0 · 10−cDR /20 ,
(26)
making it possible to find approximations of g1 and g2 from
the dynamic range of the output signal. The value of cDR
can be estimated from measurements as the ratio between the
maximum and minimum output amplitudes. This estimate is
noise sensitive, but this can be handled by averaging multiple
realizations. These approximations are valid for input signals
with large peak to minimum power ratios, like WCDMA and
LTE, where the PA generate an output signal including its peak
and minimum output amplitudes (i.e. its full dynamic range).
If this is not fulfilled or the noise influence is too large, an
alternative to this approach is to evaluate a range of values of
g1 and g2 = 1 − g1 and then solve the PA modeling problem
for each pair of gain factors.
Once the gain factors have been determined, s(t) can be decomposed into s̃1 (t) and s̃2 (t) using (14) to (16). Furthermore,
the standard outphasing decomposition of s(t) into s1 (t) and
s2 (t) as in (2) will be used in the sequel.
The distortions originate both from imperfect gain factors
and nonlinearities in the amplifiers. Once the gain factor
impact has been accounted for, the amplifier nonlinearities can
be modeled. Since the gain factors are now assumed known,
the output can be decomposed into y1 (t) and y2 (t) in the same
way as s(t) is decomposed into s̃1 (t) and s̃2 (t). The remaining
output distortion is due to the nonlinearities in the path from
s̃k (t) to yk (t), k = 1, 2.
The phase distortion in each signal path caused by the
amplifiers can thus be modeled from measurements of s(t)
and y(t). Here, polynomials
p(ηk , ∆ψ ) =
n
X
ηk,i ∆iψ ,
k = 1, 2,
(27)
i=0
have been used as parameterized versions of the functions fk ,
motivated by the Stone-Weierstrass theorem (Theorem 7.26
in [35]). Estimates η̂k,i of the model parameters ηk,i have
been computed by minimizing quadratic cost functions, i.e.,
η̂k = arg min Vk (ηk ),
k = 1, 2,
(28)
ηk
where
Vk (ηk )
=
N
X
(arg(yk (t)) − arg(sk (t)) − p(ηk , ∆ψ (s1 (t), s2 (t))))2 ,
t=1
(29)
and
ηk = ηk,0
ηk,1 . . .
ηk,n
T
.
The cost function (29) can be motivated by the fact that the
true functions fk satisfy (8) when the amplifier is described
by (7). Minimization of V1 and V2 are standard least-squares
problems, which guarantees that the global minimum will be
found [36].
Once the LS problem is solved for each setup of g1 and g2 ,
the problem of finding the best setup is now reduced to a one
dimensional (possibly nonconvex) optimization problem over
g1 (g2 = 1−g1 ), wich is much easier to solve than the original,
multidimensional problem. A problem this small can be solved
at a small computational cost by a global optimization method
such as parameter gridding.
The parameter estimates η̂k define function estimates
fˆk (z) = p(η̂k , z),
k = 1, 2,
from which an estimate
ˆ
f˜(z) = z + fˆ1 (z) − fˆ2 (z)
(30)
(31)
of the function f˜ from (12) can be computed. Provided that
this function can be inverted numerically, estimates ĥk of the
ideal phase correction functions can be computed as in (23),
i.e.,
ˆ ˜
ĥk (∆ψ ) = −fˆk (f˜−1 (ξ(∆
(32)
ψ ))) + ξk (∆ψ )
˜ ξ1 and ξ2
for k = 1, 2, where ∆ψ is given by (6) and ξ,
by (21), (17) and (18), respectively.
Hence, the complete DPD estimator consists of the selection
of gain factors g1 and g2 , the two least-squares estimators
6
given by (28), a numerical function inversion in order to obtain
ˆ
f˜−1 and the expressions for the phase correction functions
in (32). The DPD estimation can either be done at each point
in time, or (as has been done here) by evaluating the function
for the range of possible ∆ψ and saving this nonparametric,
piecewise constant function.
The DPD estimator will result in two functions ĥ1 and ĥ2
which take ∆ψ as input, and by using these as in (9), the
predistorted input signals s1,P (t) and s2,P (t) can be calculated
for arbitrary data. The measured results for a new data set,
validation data, not used during the modeling, are presented
in Section VIII.
The algorithm thus consists of two main parts, A – Estimation of PA model and B – Calculation of DPD functions. Part A
consists of three parts where the first, A.I, produces candidates
for the gain factors g1 and g2 by either using the DR or by
gridding the possible values. A.II produces LS estimates of
the nonlinear functions fˆ1 and fˆ2 for each pair of g1 and g2
and in A.III, the best performing model is chosen among all
the candidates. In Part B, the DPD functions ĥ1 and ĥ2 are
calculated. The different steps are described in more detail in
Algorithm 1.
V. B ENEFITS OF THE L EAST- SQUARES DPD ESTIMATOR
A. Convex and Nonconvex Formulations
The minimization of the cost function (9)-(10) in [16],
θ̂ = arg min
N
X
2
|y(t) − ŷ(t, θ)|
(33)
θ
ŷ(t, θ) = g1 e
t=1
j p(η1 ,∆ψ )
s1 (t) + g2 ej p(η2 ,∆ψ ) s2 (t)
where θ = [g1 g2 η1T η2T ]T , y(t) is the measured output data and ŷ(t) is the modeled output, is a nonconvex
optimization problem in 2n + 4 dimensions with possible
presence of local minima. Nonconvex optimization problems
can either be solved by a local optimization method or a
global one. A local optimization method minimizes the cost
function over points close to the initial point, and guarantees
convergence to a local minimum only. Global methods find
the global minimum, at the expense of efficiency [37]. Hence,
even under ideal conditions (noise-free data, true PA described
exactly by one model with the proposed structure), there is
no guarantee that the previous non-convex approach (33) will
produce an optimal model of the PA in finite time, whereas
the new least-squares approach does exactly this and results
in a closed-form expression for the parameter estimate. This
is a major advantage since it removes the need for error-prone
sub-optimality tests and possible time-consuming restarts of
the search algorithm. Additionally, the computation time for
the iterative, non-convex, and potentially sub-optimal solution
is significantly longer than for the proposed least-squares
method.
A two dimensional projection of the cost functions to be
minimized, (33) in the nonlinear formulation and (29) in the
LS reformulation, can be seen in Fig. 4. The WCDMA signal
used is described in more detail in Section VIII. All parameters
but two have been fixed at the optimum, and the linear term in
Algorithm 1 LS DPD method
Require: model order n, method for choice of g1 and g2 ,
precision of PA model (pM ) and inverse (pI ), estimation
data.
{A – Estimation of PA model}
y(t)
1: Normalize the output y(t) = max(|y(t)|)
2: Calculate ∆ψ ∀t according to (6).
{A.I – Estimation of gain factor candidates g1 and g2 }
3: if Use Dynamic Range to determine g1 and g2 then
4:
Calculate cDR using (4), and ∆g using (26).
5:
Calculate possible choices of g1 , g2 according to (24).
6: else {g1 and g2 over a range of values}
7:
Grid g1 ∈ [gmin , gmax ] with precision pM and let g2 =
1 − g1 .
8: end if
{A.II – Estimation of nonlinearity function candidates
fˆ1 and fˆ2 }
9: for all pairs of g1 , g2 do
10:
Create s̃k = gk ej arg(s̃k ) and yk = gk ej arg(yk ) , k =
1, 2 using (15) to (18).
11:
Find ηk using (28) and calculate fˆk , k = 1, 2 using (30).
12:
Simulate the output
ˆ
j fˆ2 (∆ψ )
ŷg1 ,g2 (t) = g1 ej f1 (∆ψ ) s1 (t) +
s2 (t).
Pg2 e
13:
Calculate error Vg (g1 , g2 ) = t |y(t) − ŷg1 ,g2 (t)|2 .
14: end for
{A.III – Choose best forward model, ĝ1 , ĝ2 , fˆ1 and fˆ2 }
15: Select ĝ1 = arg ming1 Vg (g1 , 1 − g1 ), ĝ2 = 1 − ĝ1 and
the corresponding fˆ1 and fˆ2 .
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:
26:
{B – Calculation of DPD functions ĥ1 and ĥ2 }
{Create a Look Up Table (LUT) for different values of
∆ψ by using an intermediate signal s.}
Grid ∆ψ ∈ [0, π] with precision pI .
for each value of ∆ψ do
Create s = cos(∆ψ /2) according to (1) assuming α = 0
∆
and rmax = 1 (ϕ = 2ψ ).
Create s1 and s2 according to (2) and s̃1 and s̃2
using (14) to (16).
Find ξ˜ using (21), (17) and (18).
ˆ˜
Calculate f˜(ξ)
using (31)
end for
ˆ˜
ˆ
Invert f˜(ξ)
numerically to get f˜−1 . This can e.g. be done
ˆ˜ ˜
by calculating f (ξ) for a number of values of ξ˜ ∈ [0, π],
ˆ˜
grid f˜(ξ)
and match with the ξ˜ that gives the closest value.
for each value of ∆ψ in line 16 do
Find estimate ĥk (∆ψ ) according to (32).
end for
7
number of different initial points need to be tested in order to
get a reasonable performance.
Linear Term in Amplifier Path 2
10
8
B. Recursive Least-Squares and Least Mean Squares
6
4
2
0
−2
−4
−6
−6
−4
−2
0
2
4
6
8
10
Linear Term in Amplifier Path 1
(a)
Linear Term in Amplifier Path 2
10
8
6
4
2
0
−2
−4
−6
−6
−4
−2
0
2
4
6
8
10
Linear Term in Amplifier Path 1
(b)
Fig. 4. Two dimensional projections of the cost functions of (a) the original
nonconvex optimization problem and (b) the least-squares reformulation. All
but two parameters in each amplifier branch have been fixed at the optimal
value, and the linear terms (ηk,1 in (27)) are varied. In (a), the visible local
minima are marked with 5 and the minimum obtained clearly depends on
the initial point of the local optimization. In the least-squares formulation
illustrated in (b), there is only one minimum (the global one) and convergence
is guaranteed. The + is the global minimum.
each amplifier branch (ηk,1 in (27)) has been varied. Clearly,
there is a risk of finding a local minimum in the nonconvex
formulation illustrated in (a) whereas there is only one (global)
optimum in the least-squares formulation in (b).
The local minima in themselves might not be a problem
if they are good enough to produce a wellperforming DPD,
but there are no guarantees that this is the case. Typically, a
In addition to the guaranteed convergence, the least-squares
formulations also have the advantage that there are many
efficient numerical methods to solve them, and they can be
solved recursively [38] making them suitable for an online
implementation. An even less complex parameter estimation
algorithm is the Least Mean Square (LMS) method, which can
make use of the linear regression structure of the optimization
problem, developed here in equations (8) and (27). LMS has
been used for RF PA linearization [39] and implemented in
Field Programmable Gate Array (FPGA) technology, as shown
in [40].
With a recursive implementation of the algorithm, it is even
more important that the algorithm can be proved to converge
to good values, as no monitoring of the performance should
be necessary in order to be useful in practice. On the other
hand, a nonconvex solution as in (33) is not suitable for online
implementation since it cannot guarantee convergence to good
enough minima. In an offline application, the possibility to
restart the optimization could be added but, together with the
lack of a bound on the number of iterations, this does not
seem like a good solution for an online version. Using well
explored methods like RLS or LMS would result in a lowcomplexity implementation, and though it is hard to judge the
exact complexity of the iterative implementation that would
be needed for the online version of nonconvex solution, it is
clear that it would be very hard to find an simpler one than
for the low-complexity LMS version of the new method.
One way to handle the PVT variations and changes in
the setting, such as aging, would be to use a method with
a forgetting factor, reducing the influence of older measurements [36]. The RLS and LMS solutions assume the changes
in the operating conditions to be slow.
VI. ACLR OVER DYNAMIC R ANGE AND THE I MPACT OF
PVT VARIATIONS
As previously explained, the result of the limited DR is
that all amplitude and phase errors occurring outside the DR
cannot be corrected for. The signal clipping in an outphasing
PA occurs at small amplitudes, while peak amplitudes in a
conventional linear PA are clipped. Thus, the DR in an outphasing PA limits the spectral performance when amplifying
modulated signals.
To investigate the performance limits of the predistorter,
simulations have been done using two amplifiers with a given
DR (no phase distortion), with and without DPD. In Fig. 5
the ACLR over DR at 5 MHz and 10 MHz for the WCDMA
signal are plotted with DPD (solid line) and without (dashed
line). In Fig. 6 the ACLR over DR at 5 MHz for the LTE
signal are plotted with DPD (solid line) and without (dashed).
In the figures, the phase error between the outphasing signals
is assumed to be zero as this can be compensated for in the
baseband. For a PA with a DR of 25 dB the differences in
ACLR between the nonpredistorted and predistorted output
8
(a)
(b)
Fig. 7.
(a) A Class-D stage with DC bias driven by a sinusoidal RF
waveform. (b) Signal combining of the outputs of the Class-D stages and
s1 (t) and s2 (t). The DC bias levels are adjusted to change the pulse width,
D1 and D2 , to change the amplitude of the fundamental tone.
Fig. 5. Simulated ACLR at 5 MHz and 10 MHz offset with DPD (solid line)
and without (dashed line) for the WCDMA signal.
Fig. 8.
Fig. 6. Simulated ACLR at 5 MHz offset with DPD (solid line) and without
(dashed line) for the 5 MHz LTE signal.
is 7.5-13 dB. When the DR is 25 dB the optimal theoretical
ACLR is achieved after DPD. For a PA with 45 dB of DR the
difference is negligible.
Changing the process corner (in simulations) from the
nominal corner to all other process corners, the DR is, at worst,
reduced by 2.5 dB. Similarly, varying the temperature from
25 to 125 degrees, the DR does vary by 2.5 dB. Combining
the choice of process corner and temperature, the DR can be
reduced up to 3.5 dB. As seen in Fig. 5 and Fig. 6, a variation
of 3.5 dB in DR, the ACLR is degraded by approximately
the same amount (here considering the case without DPD). If
the supply voltage is varied, the DR in the output stage may
vary as in Fig. 11(c) and may have an effect on how much
of the distortion can be compensated for and also how much
distortion can be expected without any DPD.
A. Dynamic Range Improvement by DC Bias Adjustment
In [41] a pulse-width and pulse-positioning method was
described for a Class-E PA. Similarly, the amplitudes of the
outphasing signals can be changed by changing the DC bias
levels, VBIAS , at the inputs of the Class-D stages as in Fig. 7(a),
which are driven by the sinusoidal RF signal, vRF . By changing
the DC bias levels, the pulse widths and the associated duty
Harmonic amplitude versus duty cycle, D.
cycles D1 and D2 of the output signal, Vout , of the ClassD stages are changed [42]. By changing the duty cycles, the
corresponding amplitudes of the fundamental tone, v1,fund and
v2,fund , can be changed as illustrated in Fig. 7(b) and Fig. 8
and by (34).
2VDD
sin(πD1 )ejϕ(t)
π
2VDD
= g2
sin(πD2 )e−jϕ(t)
π
v1,f und = g1
v2,f und
(34)
Thereby, the gain mismatch can be compensated for and the
DR in the output stage can be improved. The only drawback
of this method is that the peak output power is reduced as the
pulse width of strongest signal is made larger (or smaller) to
reduce the amplitude of the fundamental tone to compensate
for the smaller amplitude of the other signal. This principle
was used in measurements at 2.5 GHz for the measurement
setup in Fig. 12, where the bias levels were adjusted to improve
the DR from approximately 25 dB to 30 dB. The ACLR, before
and after bias adjustment without DPD, was approximately
-27 dBc and -32 dBc at an average output power of about
+ 21 dBm for the LTE signal.
VII. I MPLEMENTATION OF THE C LASS -D
O UTPHASING RF PA
Fig. 9 shows the implemented outphasing PA, based on a
Class-D stage utilizing a cascode configuration illustrated in
9
Fig. 9. The implemented Class-D outphasing RF PA using two transformers
to combine the outputs of four amplifier stages. Chip photo in [15].
(a)
(a)
(b)
Fig. 10. (a) The Class-D stage used in the outphasing PA [15]. C1 -C4 are
MIM capacitors. (b) Off-chip biasing resistors, R and Ri .
Fig. 10(a). The output stage is driven by an AC-coupled lowvoltage driver operating at 1.3 V, VDD1 , to allow a 5.5 V, VDD2 ,
supply without excessive device voltage stress as discussed
in [14], [15]. By driving all transistors in the cascode configuration it is possible to achieve a low on-resistance in the
on-state, and distribute the voltage stress on the devices in the
off-state, making sure that the root mean square (rms) electric
fields across the gate oxide is kept low to improve the lifetime of the transistors [44]. This enables the use of a high
supply voltage in the output stage to achieve an output power
of +30 dBm [14], [15].
The four amplifier outputs are combined using two trans−
−
formers, TR1 and TR2 , where s+
1 (t) and s2 (t), and s1 (t)
+
and s2 (t), are connected as described in [6]. In [6], the
−
s+
1 (t)/s1 (t) (and corresponding s2 (t)) signals were kept close
to minimize parasitic differential ground inductance. With
−
−
+
this connection, i.e. s+
1 (t) and s2 (t), and s1 (t) and s2 (t)
connected to the primary windings of the transformers also
used in this design, the matching network losses are reduced
−
at power back-off. At zero output amplitude (s+
1 (t) = s2 (t),
−
+
s1 (t) = s2 (t)), the matching network losses are zero as the
phase difference between the signals driving each individual
transformer are zero. Thus, the load impedance is increased
and the current through the load is reduced at power back-off
and may help to improve efficiency as in [12]. To minimize
AM-AM and AM-PM distortion, the left side of the layout
of the outphasing PA is a mirrored version of the right side.
Similarly, the PCB design is mirrored and off-chip baluns with
small phase errors have been used. As the load is connected
to one port of the equivalent secondary winding, consisting of
the two series-connected transformers, and the second port is
grounded, the signal combining is not perfectly symmetrical
(b)
(c)
Fig. 11. Measured Pout , DE and PAE for VDD1 = 1.3 V and VDD2 = 5.5 V [15]:
(a) over carrier frequency.
(b) over outphasing angle, ϕ, at 1.95 GHz.
(c) Measured Pout , DE and PAE over VDD2 for VDD1 = 1.3 V at 1.95 GHz.
resulting in a small gain mismatch between the amplifier stages
and AM-AM and AM-PM distortion.
On-chip resistors were used for an equivalent input
impedance of 50 Ω at the chip edge for each of the RF inputs
−
+
−
(s+
1 (t), s1 (t), s2 (t), and s2 (t)). The chip photo of the PA
implemented in a 65nm CMOS process is available in [15].
The chip was attached to an FR4 PCB and connected with
bond-wires.
10
TABLE I
C OMPARISON OF CMOS C LASS -D RF PA S - P EAK VALUES
Fig. 12. Measurement setup for IQ data with two Master-Slave-configured
SMBV signal generators [43].
VIII. M EASUREMENT R ESULTS
A. Measured RF Performance
Fig. 11(a) and Fig. 11(b) show the measured output power
(Pout ), drain efficiency (DE), and power-added efficiency
(PAE) over frequency and outphasing angle, ϕ in (2), for
VDD1 = 1.3 V and VDD2 = 5.5 V for the amplifier only. The
predistortion method has not been implemented in hardware.
At 1.95 GHz for a 5.5 V (6.0 V) supply voltage, the measured
output power of the PA was +29.7 dBm (+30.5 dBm) with a
power-added efficiency (PAE) of 27%. The PA had a peak to
minimum power ratio of ∼35 dB. The gain was 26 dB from
the drivers to the output. The DC power consumption of the
smallest drivers was considered as input power.
The performance of the PA used in the measurements,
denoted ‘This PA’, can be put in the context of published
Class-D PAs as listed in Table I, sorted with regard to their
output power.
Assuming that only the thin-gate oxide were driven by
the driver and that the thick-gate oxide is held at the same
bias level, the simulated DE and PAE, and output power
were reduced about 15 % and 5 % and 0.5 dB relative the
performance when all devices in the output stage are driven
by the buffer. However, the rms gate-drain electrical fields in
the oxide were increased by approximately 25 %, reducing the
life-time of the oxide [44]. To have the same electrical fields
as when all devices are driven by the buffer, the supply voltage
had to be reduced to about 4 V, resulting in an output power
reduction of ∼3 dB.
B. Measured Performance of Modulated Signals
The PAPR of the WCDMA and LTE uplink signals were
3.5 dB and 6.2 dB, respectively. The spectrum of the estimation
data sets are shown in Fig. 15(c) and Fig. 16(c). Two R&S
SMBV100A signal generators with phase-coherent RF outputs
and arbitrary waveform generators with maximum IQ sample
rate of 150 MHz, where s1 (t) and s2 (t) were stored, were
used in the measurements. Fig. 12 shows the measurement
setup. For the measurements without predistortion, the phase
offset between s1 (t) and s2 (t) in the baseband was adjusted to
minimize phase mismatch (ideally 0 ◦ between non-modulated
s1 (t) and s2 (t), i.e. maximum output power for a continuous
signal). Moreover, the IQ delay between the signal generators
was adjusted for optimal performance [43].
Measurements with two amplitude-matched signal generators, i.e. g1 = g2 = g0 = 0.5, show that phase errors of 12 ◦
Ref. Year
[45] 2008
[19] 2007
[20] 2010
[46], [47] 2010
[12] 2010
[48] 2011
[49] 2011
[6] 2011
[50] 2009
[15] 2011
This PA
[11] 2011
[14] 2011
Pout
[dBm]
+20.0
+20.0
+21.6
+21.8
+25.1
+25.2
+25.2
+25.3
+28.1
+29.7
VDD
[V]
2.4
1.8
1.9
1.0
2.0
2.5
3.0
2.0
2.4
5.5
DE
[%]
61.0
64.0
30.2
PAE
[%]
38.5
42.0
52.0
44.2
40.6
55.2
45.0
35.0
19.7
26.6
f
[GHz]
2.40
0.80
1.92
2.25
2.40
2.25
2.25
2.40
2.25
1.95
Tech.
[nm]
90
180
130
65
32
90
90
32
45
65
BW
[GHz]
0.3b
0.3a
1.0a
1.0a
1.0b
1.0b
1.0b
0.6b
1.6b
+31.5
+32.0
2.4
5.5
20.1
27.0
15.3
2.40
1.85
45
130
1.7b
0.9b
(a) 1 dB and (b) 3 dB bandwidth (BW)
TABLE II
DATA C OLLECTION
Standard
WCDMA
LTE
Nid
100 000
100 000
Nval
100 000
100 000
fs
92.1 MHz
92.1 MHz
fs,out
92.1 MHz
92.1 MHz
K
10
10
and 7.5 ◦ are acceptable for WCDMA and LTE to meet
the ACLR requirements. Thus, a predistortion implementation
would require a phase resolution of at least 6 bits, i.e. 360 ◦ /26
= 5.63 ◦ . For each bit of increased phase resolution, the ACLR
improves by ∼3 dB.
The estimation and validation data sets contain Nid and
Nval samples, respectively. The input and output sampling
frequencies are denoted fs and fs,out , respectively. To minimize
the influence of measurement noise, the signals were measured
K times, and a mean was calculated. The data collection
parameters are shown in Table II.
An evaluation of the predistorter estimation can be seen in
Fig. 13 for a model using the LTE input signal with polynomial
degree n = 5. Here and in all following experiments, the
DPD estimates ĥk , k = 1, 2, have been calculated for 3142
uniformly distributed points (pI = 0.001 in Alg. 1). Ideally,
the deviations from the ideal phase difference and the ideal
phases in each branch should be zero, and (providing the PA
model is good enough) it is clear that the predistorters reduce
these deviations. Fig. 14 shows the measured AM/AM and
AM/PM characteristics with DPD and without DPD for the
LTE signal. The figure shows that the amplitude and phase
errors are significantly reduced for small output amplitudes.
The performance was similar for the WCDMA measurements.
For the WCDMA signal at 1.95 GHz without predistortion,
the measured ACLR at 5 MHz and 10 MHz offsets were
-35.5 dBc and -48.1 dBc, respectively. The spectrum is shown
in Fig. 15(a). The estimation output data y(t) were used in the
predistortion method to extract the model parameters, with
n = 5. The predistorted input signals, s1,P (t) and s2,P (t),
were computed for the validation input signal, resulting in an
output spectrum as shown in Fig. 15(b). The power spectral
densities of the predistorted and the non-predistorted input outphasing signals are very similar and therefore not included in
11
(a)
(a)
(b)
(b)
Fig. 13. Simulated predistorter evaluation for a model with polynomial
degree n = 5 using the LTE input signal. The signals are generated using
the PA model and the DPD estimator. (a) The deviation from the ideal
phase difference without predistortion, y, and with predistortion, yP . (b) The
deviations from the ideal phase of the signals y1 and y2 without predistortion
and with predistortion, y1,P and y2,P .
Fig. 14. (a) Measured AM/AM characteristics with DPD (black) and without
DPD (grey) for LTE signal. (b) Measured AM/PM characteristics with DPD
(black) and without DPD (grey) for LTE signal.
the paper (similarly for the LTE signal). With predistortion, the
measured ACLR at 5 MHz and 10 MHz offsets were -46.3 dBc
and -55.6 dBc, respectively. Thus, the measured ACLR at
5 MHz and at 10 MHz offsets were improved by 10.8 dB
and 7.5 dB, respectively. The average power at 1.95 GHz was
+26.0 dBm with 16.5 % PAE.
The receiver noise floor before and after predistortion,
assuming a 45 MHz offset, was -127 dBc/Hz. Using two amplitude and phase matched signal generators with the outphasing
signals, the noise level is the same. Thus, the noise level is
limited by the signal generators, not the outphasing amplifier.
The phase noise of a single signal generator was -140 dBc/Hz
for a sinousoidal signal.
For the LTE signal at 1.95 GHz without predistortion, the
measured ACLR at 5 MHz offset was -34.1 dBc. The spectrum
is shown in Fig. 16(a). The estimation output data y(t)
were used in the predistortion method to extract the model
parameters with n = 5. The predistorted input signals, s1,P (t)
and s2,P (t), were computed for the validation input signal,
resulting in an output spectrum as shown in Fig. 16(b). With
the predistorted spectrum in Fig. 16(b), a small assymetry can
TABLE III
M EASURED S PECTRAL P ERFORMANCE AT 1.95 GH Z FOR WCDMA AND
LTE U PLINK S IGNALS W ITH P REDISTORTION ( USING n = 5) AND
W ITHOUT
Standard
WCDMA
LTE
Measured Parameter
ACLR @ 5 MHz [dBc]
ACLR @ 10 MHz [dBc]
ACLR @ 5 MHz [dBc]
W DPD
-46.3
-55.6
-43.5
W/o DPD
-35.5
-48.1
-34.1
Req
-33
-43
-30
EVM requirements were easily met without predistortion for the WCDMA
signal and an LTE signal (20 MHz, 16-QAM) without predistortion [15].
be observed, which was expected due to the non-symmetrical
frequency spectrum of the reference signal. With predistortion,
the measured ACLR at 5 MHz offset was -43.5 dBc. Thus, the
measured ACLR at 5 MHz offset was improved by 9.4 dB. The
average power at 1.95 GHz was +23.3 dBm with 8.0 % PAE.
Fig. 17 shows the measured ACLR over the polynomial
degree n of the PA model for the WCDMA and LTE signals.
For each increase of order, the ACLR is improved by on
average ∼2 dB. However, making the order larger than 5 does
not significantly improve ACLR any further.
The measured performance of the PA for modulated signals
is summarized in Table III. The table shows measured ACLR
with DPD (W DPD), without DPD (W/o DPD), and the
required (Req) ACLR for the WCDMA [51] and the LTE [52]
12
Fig. 15. Measured WCDMA spectrum at 1.95 GHz.
(a) Measured WCDMA spectrum without DPD. The measured ACLR is
printed in grey.
(b) When DPD is applied to (a). The measured ACLR is printed in black.
(c) Spectrum of estimation signal. Spectrum of validation signal was similar.
Fig. 16. Measured LTE spectrum at 1.95 GHz.
(a) Measured LTE spectrum without DPD. The measured ACLR is printed in
grey.
(b) When DPD is applied to (a). The measured ACLR is printed in black.
(c) Spectrum of estimation signal. Spectrum of validation signal was similar.
Fig. 17. Measured ACLR depending on the polynomial degree n of the PA
model. Degree = 0 represents the performance without predistortion.
is investigated in [55] and [56], where it was concluded that
to obtain an optimal signal/distortion ratio over the entire
bandwidth, a compromise between the sampling frequency
and the filter characteristics has to be made. In this paper, we
have evaluated the required bandwidth/sampling rate based on
measurements with two signal generators and one combiner
(no PA). Increasing the sampling frequency from the original
15.36 MHz to 30.72 MHz and 61.44 MHz, the ACLR at 5 MHz
for the WCDMA signal improves from -44 dBc to -50 dBc and
-52 dBc, respectively. Similarly, the ACLR at 10 MHz offset
improves from -48 dBc to -52 dBc and -56 dBc, respectively.
The corresponding improvements for the LTE signal at 5 MHz
are -34 dBc to -43 dBc and -46 dBc. Thus, for the specific tests
performed here, the ACLR at 5 and 10 MHz can improve by
6-9 dB and 4-8 dB, respectively, when increasing the sampling
rate up to four times the original sampling rate of 15.36 MHz.
Further increasing the sampling frequency, up to 92.16 MHz,
shows no significant change.
IX. C ONCLUSIONS
standards. In measurements at 1.95 GHz, the DPD proved to
be successful and improved the WCDMA ACLR at 5 MHz
and 10 MHz offsets by 10.8 dB and 7.5 dB, respectively. The
LTE ACLR at 5 MHz offset was improved by 9.4 dB. Thus, the
predistortion method improves the measured ACLR to have at
least 12.6 dB of margin to the requirements [51], [52]. The
measured ACLR at 5 MHz is comparable to state-of-the-art
WCDMA [53], [54] transceivers.
Assuming a PA with 35 dB of dynamic range (neglecting
phase distortions), i.e. assuming g1 = 0.509 and g2 = 0.491,
and a polynomial degree of n = 5, the computed achievable
ACLR at 5 MHz and 10 MHz is ∼3 dB better compared to the
measurements with the WCDMA signal. Similarly, the computed achievable ACLR at 5 MHz is ∼2 dB better compared
to the measurements with the LTE signal.
The sampling rate was 92.16 MHz in the measurements,
six times the original sampling frequency of the signal, and
was chosen as high as possible while still being able to
practically map the measured IQ data with the estimation
signal. The impact of baseband filtering and limited bandwidth
This paper presents a model-based phase-only predistortion
method suitable for outphasing RF PAs. The model estimation
has here been reformulated from a nonconvex problem into
a convex least-squares problem, and the predistorter can be
calculated analytically. The predistortion method has been
applied at the signal generation level in the baseband, and
it has been used for WCDMA and LTE uplink signals applied
to a fully integrated +30 dBm Class-D outphasing RF PA
in 65nm CMOS, which is one of the first fully integrated
+30 dBm outphasing PAs. In measurements at 1.95 GHz, the
DPD proved to be successful and improved the WCDMA
ACLR at 5 MHz and 10 MHz offsets by 10.8 dB and 7.5 dB,
respectively. The LTE ACLR at 5 MHz offset was improved
by 9.4 dB.
ACKNOWLEDGMENT
The authors would like to thank Dr. Peter Olanders, Ericsson
Research, Kista, Sweden, for useful discussions.
13
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Ylva Jung was born in Stockholm, Sweden, in
1984 and recieved her M.Sc. degree in Applied
Physics and Electrical Engineering from Linköping
University, Linköping, Sweden, in 2010. At present,
she is working towards a Ph.D. degree at the at the
Division of Automatic Control within the Department of Electrical Engineering, Linköping University. Her main research interest is system inversion
of nonlinear dynamics with focus on power amplifier
predistortion.
Jonas Fritzin (S07-M12) received his M.Sc. degree
in electrical engineering from Chalmers University
of Technology, Göteborg, Sweden, in 2004 and the
Ph.D. degree from Linköping University, Linköping,
Sweden, in 2011. Since 2012 he is working with
research and development of analog/RF IC for future
base stations at Ericsson AB, Stockholm, Sweden.
His research interests include CMOS RF power
amplifiers, transmitters, and predistortion.
Martin Enqvist (M08) was born in Lund, Sweden
in 1976. He obtained the M.Sc. degree in Applied
Physics and Electrical Engineering in 2000 and the
Ph.D. degree in Automatic Control in 2005, both
from Linköping University, Sweden. During 2006,
he worked as a postdoctoral researcher at Vrije
Universiteit Brussel in Belgium and he is currently
an associate professor at the Department of Electrical Engineering at Linköping University. His main
research interest is in the field of nonlinear system
identification.
Atila Alvandpour (M99-SM04) received the M.S.
and Ph.D. degrees from Linköping University, Sweden, in 1995 and 1999, respectively. From 1999 to
2003, he was a senior research scientist with Circuit
Research Lab, Intel Corporation. In 2003, he joined
the department of Electrical Engineering, Linköping
University, as a Professor of VLSI design. Since
2004, he is the head of Electronic Devices division.
He is also the coordinator of Linköping Center for
Electronics and Embedded Systems (LINCE). His
research interests include various issues in design
of integrated circuits and systems in advanced nanoscale technologies, with a
special focus on efficient analog-to-digital data converters, wireless transceiver
front-ends, sensor interface electronics, high-speed signaling, on-chip clock
generators and synthesizers, low-power/high-performance digital circuits and
memories, and chip design techniques. He has published about 100 papers
in international journals and conferences, and holds 24 U.S. patents. Prof.
Alvandpour is a senior member of IEEE, and has served on many technical
program committees of IEEE and other international conferences, including
the IEEE Solid-State Circuits Conference, ISSCC, and European Solid-State
Circuits Conference, ESSCIRC. He has also served as guest editor for IEEE
Journal of Solid-State Circuits.
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