Clip correction in wireless LAN receivers

Abstract

OFDM signals suffer from a large peak to average power ratio, which requires large power back-offs in the transmit and receive chains. This paper presents a digital postprocessing method that mitigates clipping by the analog-to-digital converter (ADC) in the receiver. Clipped peaks cause spurious signals on empty subcarriers, which can be used to eliminate clipping artifacts and to recover the original signal. Simulations show that a significant reduction of 3 dB in the headroom of the A/D converter (ADC) is possible, when an elaborate MMSE clip correction algorithm is used. A simple algorithm still allows for 1 dB reduction of the headroom. As the ADC is consuming an ever increasing fraction of the total receiver power, the results are believed to be relevant for low-power design of OFDM receivers, for instance to prolong battery life of laptops and other portable WLAN devices.

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Clip Correction in Wireless LAN Receivers
Ronald Rietman#1, Jean-Paul Linnartz#∗2Eric Penning de Vries#3
#Philips Research Laboratories, Eindhoven, The Netherlands
∗Eindhoven University of Technology, Eindhoven, The Netherlands
1ronald.rietman@philips.com
2j.p.linnartz@philips.com
3eric.penning.de.vries@philips.com
Abstract—OFDM signals suffer from a large Peak to Aver-
age Power Ratio, which requires large power back-offs in the
transmit and receive chains. This paper presents a digital post-
processing method that mitigates clipping by the analog-to-digital
converter (ADC) in the receiver. Clipped peaks cause spurious
signals on empty subcarriers, which can be used to eliminate
clipping artifacts and to recover the original signal. Simulations
show that a significant reduction of 3 dB in the headroom of the
A/D converter (ADC) is possible, when an elaborate MMSE clip
correction algorithm is used. A simple algorithm still allows for
1 dB reduction of the headroom. As the ADC is consuming an
ever increasing fraction of the total receiver power, the results are
believed to be relevant for low–power design of OFDM receivers,
for instance to prolong battery life of laptops and other portable
WLAN devices.
I. INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM)
modulation has a number of distinct advantages for transmis-
sion over multipath channels. However, the nearly Gaussian
distribution of the OFDM transmit signal is seen as one of
its main disadvantages. Practical systems require a margin
of about 8 to 9 dB headroom between the average signal
power and the clipping level of the transmitter and the receiver.
Many papers have been written about the effect of the Peak
to Average Power Ratio (PAPR) of OFDM, and on methods
to mitigate this problem. Mostly, the high PAPR is seen as a
problem for the linearity and the power consumption of the
power amplifier at the transmitter. Several methods have been
proposed to reduce the PAPR of the transmitted and many
papers have been written on receiver algorithms for mitigating
clipping noise, see e.g., [1].
Here we take a receiver-oriented direction. We develop a
receiver algorithm that compensates modest clipping by the
ADC. This approach is relevant for instance for wireless
personal and local area network systems which transmit at
gigabits per second over a very short distance, the power
consumption of the ADC in the receiver may exceed the power
consumed by the transmit power amplifier. If a moderate
amount of clipping could be compensated digitally, the ADC’s
saturation level could be lowered, and hence its power con-
sumption reduced. This has been a motivation for us to develop
a signal processing algorithm that attempts to reconstruct
clipped peaks of the OFDM signal. In order to effectively
shape the emitter spectrum, all wireless OFDM standards use
empty, i.e. unmodulated, subcarriers at the outskirts of their
S
IFFT
sTx
h
Rx
η
rx
ADC
y
FFT
Y
Fig. 1. System diagram at baseband level
spectrum. We pursue the idea that clippig causes spurious
components which occur as non-zero subcarrier signals. Hence
these can assist the receiver to reconstruct the original (not
clipped) signal components.
In Section II we give a baseband model of an OFDM
communication system and a statistical model of the OFDM
signal at the receiver, including the receiver’s ADC. Then, in
Section III, we describe our clipping correction algorithms and
evaluate their performance for an idealized DVB-H system. In
Section IV we present the results of an 802.11a WLAN system
simulation in which we included our simple clip correction
algorithm.
II. MODEL OF THE COMMUNICATION SYSTEM
We give a baseband model for the transmitter, the channel
and the receiver of an OFDM system, including the receiver’s
ADC. The system diagram is given in fig. 1.
The number of subcarriers is denoted by Ns, which is
assumed to be even. The transmitted OFDM symbol has
duration Tsymbol=1/fs, where fsis the subcarrier spacing. The
subcarriers are indexed from −Ns/2 to Ns/2−1, and subcar-
rier m has subcarrier frequency mfs. In an OFDM symbol, Ns
complex numbers S=(S−Ns/2,...,SNs/2−1) are transmitted, by
transmitting Smon subcarrier m. Actually only a subset of the
subcarriers is used: most standards require that the leftmost
MLand the rightmost MRsubcarriers be unmodulated, so that
the corresponding Sm=0. For IEEE 802.11a, Ns=64, ML=6
and MR= 5. The set of empty subcarriers is denoted by M0,
its size |M0| = ML+MRis denoted by M.
The transmitter adds a cyclic prefix of duration Tguard, so
the transmitted signal can be written in complex baseband
representation as
s(t) =
Ns/2−1
∑
m=−Ns/2
Sme2πimfst,
t ∈ [−Tguard,Tsymbol).
(1)
The signal passes through a channel with impulse response
h(τ), where it is assumed that h(τ) is zero for τ outside of
978-2-87487-008-8 © 2008 EuMAOctober 2008, Amsterdam, The Netherlands
Proceedings of the 1st European Wireless Technology Conference
174
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[0,Tguard). The analog transmit and receive filters are included
in h(τ). Noise from the receiver’s low-noise amplifier and
interference from adjacent channel users is denoted by η. The
receiver ignores the signal received in the interval [−Tguard,0),
since it suffers from inter-symbol interference, and samples
and discretizes the in-phase and quadrature components of
the complex signal in the interval [0,Tsymbol) at the ADC.
The complex baseband signal entering the ADC is x(t) =
η(t)+r(t) with
r(t) =
Ns/2−1
∑
m=−Ns/2
Rme2πimfst,
t ∈ [0,Tsymbol)
(2)
where
Rm= Sm
ZTguard
0
h(τ)e−2πimfsτdτ =: SmHm.
(3)
Assuming sampling at the Nyquist frequency, Tsample =
(Nsfs)−1and the n-th sample is xn= x(nTsample) = rn+ηn,
n = 0,...,Ns−1. The set of all Ns samples is denoted by
x = (x0,...,xNs−1). The in-phase and quadrature components
of each sample xnare discretized. Discretization of the signal
value w gives the best approximation in a discrete set D
with L different quantisation levels, D={d1,...,dL}⊂R with
d1< d2< ... < dL, according to
q(w) = argmin
d∈D|d−w|.
After the ADC, the complex baseband representation of the
sample is
yn= q(Re(xn))+iq(Im(xn)) =: Q(xn),
(4)
or
y = Q(x),
(5)
where y = (y0,...,yNs−1).
The discretized samples are Fourier transformed, which
yields the received signal on the subcarriers:
Ym=1
Ns
Ns−1
∑
n=0
yne−2πimn/Ns,
m = −Ns/2,...,Ns/2−1.
(6)
We now return to the quantization of the signal. We take
L>2 and a uniform level spacing: d?=(2?−L+1)C/(L−2),
so that (d1+d2)/2=−C, q(w)=d1if w<−C, and q(w)=dL
if w > C. If the signal value w is less than C in absolute
value, the discretization error |q(w)−w| is at most equal to
C/(L−2), but if the input sample w>C, or w<−C the error
is unbounded and the ADC is said to clip the sample. For an
unclipped signal, the conditional expected squared error Wu=
E[|w−q(w)|2||w| <C] is well approximated by C2/3(L−2)2,
for a clipped signal the conditional expected squared error
depends on the tail of the distribution of w.
If we ignore, for the moment, the correlation of different
samples due to the empty subcarriers and, motivated by the
central limit theorem which holds in the limit Ns→∞, assume
that the n = 2Ns samples are independently and identically
Fig. 2.
(in dB) as a function of P/C2(in dB) for n=2Ns=128 and (L=256,m=0)
(thin line), (L=1024,m=0) (thick line), (L=256,m=20) (thin dashed line)
and (L = 1024,m = 20) (thick dashed line).
The relative error due to clipping and quantization according to (10)
distributed as a zero-mean Gaussian with variance P/2, then
the probability that a given sample is clipped is given by
2
√πP
C
The expected squared error, given that a sample is clipped,
equals
R∞
The expected total squared error is thus given by
?n
= n[pcWc+(1− pc)Wu].
Note that the relative errors Wu/P and Wc/P depend on the
power P and clip level C only through the ratio P/C2.
Now suppose that we have an algorthm that is able to fully
mitigate the effect of up to m clippings, e.g., as in (14) in the
next section. Then that algorithm reduces the expected error
W to
n
∑
k=m+1
k
As an example, we have plotted in fig. 2 the relative squared
error per complex sample 2W/nP (in dB) as function of P/C2
(in dB) for n = 128, L ∈ {28,210} and m ∈ {0,20}. We want
to operate the ADC in a regime where this relative error is
well below the relative error due to thermal noise, for instance
2W/nP ? −30dB. The graphs show that the lower limit of
this regime depends on the number of quantization levels, L,
whereas the upper limit appears to be independent of L. It
also appears that a clip mitigation algorithm can increase the
upper limit of the regime by an amount that does not depend
on L, whereas the lower limit is unaffected. Futhermore the
results suggest that a good clip correction algorithm may give
a gain (at the upper limit) that is similar to the gain (at the
lower limit) that would be obtained if the number of bits of the
pc=
Z∞
exp[−w2/P]dw = erfc(C/√P).
(7)
Wc=
C(w−dL)2exp[−w2/P]dw
R∞
?
Cexp[−w2/P]dw
.
(8)
W =
n
∑
k=0
k
(1− pc)n−kpk
c[kWc+(n−k)Wu]
(9)
W = nWu+
?n
?
(1− pc)n−kpk
c(k−m)(Wc−Wu). (10)
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ADC were increased by two. In other words: a good clipping
algorithm may allow for a reduction in the power consumption
of the ADC by 6 dB (by reducing the number of bits) without
significantly lowering its dynamic range.
We assume from now on that L is large enough to consider
only the error introduced by clipping, i.e.:


III. CLIPPING COMPENSATION ALGORITHMS
The most promising approach to clip correction is to find an
estimate of the received signal, given that it is observed after
being disturbed by additive thermal noise and the clipping
ADC. This MMSE algorithm is worked out in [2]. In that
letter we show that the MMSE algorithm has close to optimal
performance, but is hard to implement in practice. Here
we focus on a simple clip correction algorithm that can be
implemented easily in the digital receiver. It estimates the
degree of clipping independently for all subcarriers involved
and weighs the measured out-of-band artifacts equally.
We assume that there is no noise, i.e., η = 0 so that any
sample that is not clipped by the ADC is not distorted at all.
Then we can write rn= yn+cnwith
q(w) =



−C
w
C
if w ≤ −C
if −C < w <C
if w ≥C.
(11)
Re(cn) = 0 if |Re(yn)| <C,
Im(cn) = 0 if |Im(yn)| <C.
(12)
Denoting the number of clippings by Nc, there are Ncnon-zero
parameters, one for each clipped real or imaginary part of a
sample.
After the Fourier transform, we have that Rm= Ym+Cm.
Since the real and imaginary parts of Rm are zero if m
corresponds to an empty subcarrier, we have that Cm= −Ym
for m ∈ M0. These equations are equivalent to 2M equations
for Nc unknown parameters, which can be written in matrix
form as
v = Au,
(13)
where v is a known 2M-vector with the real and imaginary
parts of −Ym for the subcarriers where Rm= 0, u is an Nc-
vector with the unknown values of Re(cn) or Im(cn) at a
clipped sample and A is an 2M × Nc-matrix of which the
elements are the appropriate real and imaginary parts of the
Fourier-transform matrix. If Nc≤ 2M, u can be uniquely
determined by solving any subset of Ncequations, or by
u = (ATA)−1ATv.
(14)
If Nc> 2M, ATA has Nc−2M eigenvalues equal to zero, so u
cannot be determined uniquely and this equation-solving clip
correction algorithm doesn’t work. The above results suggest
that each empty subcarrier allows for the correction of two
clippings, so that at most 2M clippings can be corrected for.
We now make an approximation that simplifies the clip
correction significantly. Each of the diagonal elements of
ATA is equal to M/N2
s. The off-diagonal elements are partial
correlations of an integer number of rotations of complex
exponentials, so these are usually non-zero. However, off-
diagonal components are smaller than the diagonal elements.
Approximating ATA by its diagonal leads to the simpler
approximation
˜ u =(Ns)2
M
The corresponding clip-corrected complex-valued signal in the
frequency domain is given by
˜R =Y −Ns
where F is the Fourier transform matrix, with elements Fmn=
exp(−2πimn/Ns)/Ns, F−1its inverse, Pemptyprojects onto the
M empty subcarriers, i.e., (PemptyY)m=Ymif m∈M and zero
otherwise, and Pclip,yprojects the real and imaginary parts of
its argument onto the positions where the corresponding real
or imaginary parts of y are clipped, i.e.,
?
0
?
0
Intuitively, this algorithm is equivalent to making the approx-
imation that the out-of-band clip artifacts u for every clip
location are orthogonal, so that clipping can be corrected
by inverting the clip artifacts one by one. Note that this
algorithm only uses simple digital processing and is of order
O(Nslog(Ns)),
IV. CLIP CORRECTION IN AN IEEE 802.11A SIMULATION
In order to evaluate the effectiveness of the simple clip
correction algorithm in a somewhat more realistic scenario,
we introduced the correction (16) our WLAN simulation code.
This code simulates a complete 802.11a WLAN system, with
optional MIMO extensions. The digital processing chain of
the receiver is shown in fig. 3.
We have simulated a scenatio where clipping is the domi-
nant source of errors, therefore we assume perfect synchroni-
sation of transmitter and receiver, perfect channel estimation at
the receiver and an ADC with enough bits so that only clipping
needs to be considered. The transmitter sends 8000 bit packets
at the highest data rate of 54Mb/s, the channel is modeled
as having an exponential delay profile with RMS delay of
50ns. The automatic gain controller in the receiver scales the
received signal power to 10−BackOffdB/10, where BackOffdB
is the back-off in decibel. Fig. 4 show the resulting packet
error rate (PER) versus the SNR for various values of the
back-off, both with and without clip correction. The PER was
obtained by simulating random channel and noise realizations
until 100 packet errors were observed, hence the estimates of
PER are accurate to about 10% at low PER, which explains
the non-smoothness of the graphs al high SNR. Nonetheless,
these graphs suggest that the simple clip correction algorithm
allows for a reduction of back-off by about 1 dB at a SNR of
30 dB while delivering the same PER.
ATv.
(15)
MFPclip,yF−1PemptyY,
(16)
Re(Pclip,yx)n=
Re(xn)
if |Re(yn)| =C
otherwise
Im(Pclip,yx)n=
Im(xn)
if |Im(yn)| =C
otherwise
(17)
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Fig. 3.Block diagram of the digital part of the WLAN receiver simulation chain.
0510 152025303540
10
−2
10
−1
10
0
SNR (dB)
PER
3 dB
5 dB
7 dB
9 dB
11 dB
Fig. 4.
simulation for varioous values of the AGC back-off from 3 to 11 dB, without
using any clip correction (lines) and using the simple clip correction algorithm
(lines with circles).
Packet error rate (PER) versus SNR for an 802.11a WLAN system
V. CONCLUDING REMARKS
The advantage of a clip compensation algorithm is that the
design specifications of the ADC can be relaxed.
We found that for systems operating at high SNR the
clipping threshold can be reduced by up to 3 dB, when an
advanced clip correction algorithm is used. So for a specified
signal-to-quantization noise, one may use only 70% of the
number of quantization steps needed hitherto. As the power
consumption of an ADC is proportional to the number of
quantization steps, this corresponds to a reduction of power
consumption by the ADC by 30%. Nonetheless our solution
demands more digital signal processing operations. We have
developed a much simpler algorithm that gives less gain,
about 1 dB, but does so for very little signal processing cost.
Our proposed algorithms do not require any modification to
existing OFDM standards. Its effectiveness has been tested in
our simulation chain for 802.11a WLAN.
REFERENCES
[1] D. Kim and G. L. St¨ uber, “Clipping noise mitigation for OFDM by
decision-aided reconstruction,” IEEE Communications Letters, vol. 3,
no. 1, pp. 4–6, Jan. 1999.
[2] R. Rietman and J. P. Linnartz, “Peak restoration in OFDM receiver with
clipping A/D converter,” to appear in IEEE Trans. Wireless Comm.
177
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