Directed Selected Mapping for Peak-to-Average Power Ratio
Reduction in MIMO OFDM
Robert F.H. Fischer, Martin Hoch
Lehrstuhl für Informationsübertragung,
Friedrich–Alexander–Universität Erlangen–Nürnberg,
Cauerstraße 7/LIT, 91058 Erlangen, Germany, email: {fischer,hoch}@LNT.de
Abstract
Peak-to-average power ratio reduction in OFDM using antenna arrays (MIMO OFDM) is considered. In
particular, a generalisation of selected mapping (SLM), we call it directed SLM (dSLM), is introduced. It
is shown that, in contrast to schemes proposed in literature, dSLM utilises the potential offered by MIMO
transmission. As a result, the complementary distribution function of the PAR exhibits a steeper decay,
increased by a factor equal to the number of transmit antennas. This effect is similar to the diversity gain in
error performance when using MIMO transmission.
This work was supported in parts by Deutsche Forschungsgemeinschaft (DFG) within the framework TakeOFDM under grant FI 982/1-1.
2
Introduction — Multicarrier modulation, in particular its implementation as orthogonal frequencydivision multiplexing (OFDM), is very popular for transmission over frequency-selective channels [3]. Since
for future applications the use of antenna arrays is envisaged, parallel OFDM transmission, often denoted as
MIMO OFDM (multiple-input/multiple-output), is the most important candidate for next-generation wireless
communication, e.g., [7].
We assume NT transmit antennas, over which independent data streams should be communicated. In
each of the NT parallel OFDM transmitters a block of D distinct complex-valued carriers Aµ,ν , ν =
0, . . . , D − 1, µ = 1, . . . , NT , (OFDM frame) is transformed into time-domain (equivalent complex-valued
baseband signals) using the inverse discrete Fourier transform (IDFT), i.e., aµ,k =
√1
D
D−1
ν=0
Aµ,ν ·ej2πkν/D ,
k = 0, . . . , D − 1, µ = 1, . . . , NT . As usual in wireless applications, all frequency-domain samples Aµ,ν
are expected to be drawn from the same constellation with variance σa2 .
Unfortunately, because of the statistical independence of the carriers, the time-domain samples aµ,k are
approximately Gaussian distributed. This results in a high peak-to-average power ratio (PAR)1 PARµ =
def
maxk |aµ,k |2
E{|aµ,k |2 }
=
maxk |aµ,k |2
.
σa2
To avoid non-linear distortion in the power amplifiers and in turn the generation
of out-of-band radiation, the PAR of all NT transmit signals should be simultaneously as small as possible.
Since performance is governed by the worst-case PAR, we consider
PAR =
def
max
µ=1,...,NT
PARµ =
maxµ=1,...,NT maxk |aµ,k |2
.
σa2
(1)
In literature, cf. [5], a number of PAR reduction techniques for (single-antenna) OFDM are known. A
first extension to MIMO OFDM was given in [1]. In the present letter we introduce a generalisation of
selected mapping (SLM) [2], called directed SLM which, in contrast to [1], uses the inherent potential of
MIMO transmission.
Directed Selected Mapping — In (single-antenna) selected mapping each OFDM frame is mapped
to a number of U (independent) candidates representing the same information. From these that one with
the lowest PAR (or any other criteria) is selected. In [2], [6] the candidates are generated by multiplying
(u)
(u)
carrier-wise the initial OFDM frame A = [A0 , . . . , AD−1 ] with U phase vectors P (u) = [P0 , . . . , PD−1 ],
(u)
u = 1, . . . , U , Pν
(u)
(u)
= ejϕν , and ϕν ∈ [0, 2π) randomly selected and known to transmitter and receiver.
The candidates are transformed to time-domain (U IDFTs are required) and the “best” OFDM frame is
∗)
actually transmitted. In order to recover data log2 (U ) bits of side information to indicate the vector P (u
have to be communicated to the receiver (x: smallest integer ≥ x). Using SLM and assuming Gaussian
1
In the present letter, we concentrate on the PAR of the discrete-time samples aµ,k , which usually gives a good estimate for the PAR of
the continuous-time transmit signal, obtained after inserting the “guard interval”, pulse shaping (including digital-to-analog conversion), and
modulation to radio frequency.
3
time-domain samples, the probability that the PAR of an OFDM frame exceeds a certain threshold PAR0 is
given by Pr{PAR > PAR0 } = (1 − (1 − e−PAR0 )D )U [2].
It is natural to individually apply SLM to each of the NT parallel schemes in MIMO OFDM, a procedure
called ordinary SLM (oSLM) in [1]. For each of the parallel OFDM frames the best phase modification out
of the U possible is individually selected. Now, NT U IDFTs and NT log2 (U ) bits of side information are
required. Straightforwardly it can be shown that in this case the worst-case PAR defined in (1) is distributed
according to Pr{PAR > PAR0 } = 1 − (1 − (1 − (1 − e−PAR0 )D )U )NT .
In order to reduce signalling overhead, simplified SLM (sSLM) was proposed in [1]. Here, all NT OFDM
(u)
, ∀µ. Consequently, only log2 (U ) bits
frames are modified with the same phase vector, i.e., P (u)
µ = P
of side information are required, however no complexity reduction is achieved as still NT U IDFTs have
to be calculated. In this case, the worst-case PAR is distributed according to [1] Pr{PAR > PAR0 } =
(1 − (1 − e−PAR0 )DNT )U , i.e., the same performance as (single-antenna) SLM with the number of carrier
increased by the factor NT of antennas is achieved.
Neither ordinary nor simplified SLM indeed use the potential of MIMO transmission for PAR reduction.
In MIMO communication, data rate or diversity order can be improved by exploiting the spatial dimension.
In the same spirit, treating the parallel transmit signals jointly, PAR reduction can be improved by “reallocate
the peak power over the antennas”.
We now present a scheme which we call directed SLM (dSLM), capable to use this advantage. Main idea
is to invest complexity only where PAR reduction is really needed. Instead of performing U trials for each
of the NT transmitters, the budget of NT U IDFTs is used to successively improve the highest PAR over the
antennas. For that, in the first step the PAR of the NT initial (original) OFDM frames is calculated. Then,
in each successive step, the OFDM frame with currently highest PAR is considered and using a next phase
vector P (u) , a reduction of PAR is tried. This procedure is continued NT (U − 1) times, leading to the same
complexity as ordinary or simplified SLM. The pseudo code of the algorithm is depicted in Fig. 1.
Since, including the initial step, at maximum NT (U − 1) + 1 trials may be performed for one antenna,
NT log2 (NT (U − 1) + 1) bits side information are required. However, it should be remarked that replacing
SLM as described above with the version presented in [4] (using scramblers instead of phase vectors), no explicit side information at all has to be communicated. Yet, both version of SLM show the same performance.
Finally, the same procedure can be applied to partial transmit sequences (PTS) [6], leading to directed PTS
for MIMO OFDM.
Numerical Results and Conclusion — The performance of the proposed dSLM scheme has been assessed by means of numerical simulations. The number of carriers (all used) is D = 512, the modulation
4
in each carrier is 4PSK, and transmission of NT = 4 parallel data streams is assumed. U = 4 or 16 phase
vectors with phases randomly selected from the set {0, π/2, π, 3π/2} (pure inversion or interchange of the
quadrature components [2]) are used; in each PAR reduction scheme 4 · 4 = 16, and 4 · 16 = 64 IDFTs,
respectively, have to be calculated per OFDM frame duration.
In Fig. 2, the complementary cumulative distribution function (ccdf) Pr{PAR > PAR0 } is plotted for
the variants of SLM. Additionally (rightmost, dotted curve), the ccdf of OFDM without PAR reduction is
included. It is clearly visible that dSLM shows better performance than oSLM or sSLM (same number of
IDFTs). For clipping levels lower than 10−5 gains over 1 dB over oSLM and 1.5 dB over sSLM are possible.
The simulation fit well with the above given analytic results (gray curves).
It is worth noting that the PAR curves for dSLM exhibit a larger slope than o/sSLM. Their slope is the
same as (single-antenna) SLM with U candidates. In dSLM, a slope corresponding to NT U candidates is
achieved. Hence an effect similar to the diversity order (slope of error rate curves) in MIMO communication
is present in MIMO PAR reduction as well. It can thus be argued that the proposed dSLM scheme exploits
the potential of MIMO OFDM for PAR reduction.
R EFERENCES
[1] M.-S. Baek, M.-J. Kim, Y.-H. You, H.-K. Song. Semi-Blind Channel Estimation and PAR Reduction for MIMO-OFDM
System With Multiple Antennas. IEEE Transactions On Broadcasting, pp. 414–424, Dec. 2004.
[2] R. Bäuml, R.F.H. Fischer, J.B. Huber. Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by Selected
Mapping. Electronics Letters, pp. 2056–2057, Nov. 1996.
[3] J.A.C. Bingham. Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come. IEEE Communications
Magazine, pp. 5–14, May 1990.
[4] M. Breiling, S.H. Müller–Weinfurtner, J.B. Huber: SLM Peak-Power Reduction Without Explicit Side Information. IEEE
Communication Letters, pp. 239–241, June 2001.
[5] S.H. Han, J.H. Lee. An Overview of Peak-to-Average Power Ratio Reduction Techniques for Multicarrier Transmission. IEEE
Wireless Communications, pp. 56–65, April 2005.
[6] S.H. Müller, R.W. Bäuml, R.F.H. Fischer, J.B. Huber. OFDM with Reduced Peak-to-Average Power Ratio by Multiple Signal
Representation. Annal of Telecommunications, pp. 58–67, Feb. 1997.
[7] H. Yang. A Road to Future Broadband Wireless Access: MIMO-OFDM-Based Air Interface. IEEE Communications Magazine, pp. 53–60, Jan. 2005.
5
L IST OF F IGURES
1
Pseudocode of directed selected mapping for MIMO OFDM. The function max returns the maximum
and the corresponding index and denotes element-wise multiplication. . . . . . . . . . . . . . . . . .
2
6
Complementary cumulative distribution function (ccdf) Pr{PAR > PAR0 } for MIMO OFDM with
dSLM, oSLM, and sSLM. D = 512 carriers, NT = 4 antennas, U = 4, 16 candidate vectors. Rightmost
curve: MIMO OFDM without PAR reduction; gray: theoretical curves assuming Gaussian time-domain
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Figures
6
given: U , [P (1) , . . . , P (NT (U −1)) ]
function [a1 , . . . , aNT ] = dSLM([A1 , . . . , ANT ])
1 aµ = IDFT{Aµ }, calc. PARµ , µ = 1, . . . , NT
2 for u = 1, . . . , NT (U − 1)
3
[PARmax , µmax ] = max{PAR1 , . . . , PARNT }
4
anew = IDFT{Aµmax P (u) }, calc. PARnew
5
if (PARnew < PARµmax )
6
aµmax = anew , PARµmax = PARnew
7
endif
8 endfor
Fig. 1. Pseudocode of directed selected mapping for MIMO OFDM. The function max returns the maximum and the corresponding index and
denotes element-wise multiplication.
Figures
7
0
Pr{PAR > PAR0 } −→
10
dSLM
oSLM
sSLM
U=4
U = 16
theory
−1
10
−2
10
−3
10
−4
10
−5
10
7
8
9
10
11
10 log10 (PAR0 ) [dB] −→
Fig. 2. Complementary cumulative distribution function (ccdf) Pr{PAR > PAR0 } for MIMO OFDM with dSLM, oSLM, and sSLM.
D = 512 carriers, NT = 4 antennas, U = 4, 16 candidate vectors. Rightmost curve (dotted): MIMO OFDM without PAR reduction; gray:
theoretical curves assuming Gaussian time-domain samples.