Soft Decision Aided Suboptimal ML Detection Receiver for Clipped COFDM Transmissions Romain Déjardin, Maxime Colas and Guillaume Gellé CReSTIC - SysCom - University of Reims Champagne-Ardenne Moulin de la Housse, BP 1039 51687 Reims Cedex 2, France
[email protected],
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Abstract—A soft decision aided suboptimal maximum likelihood detection algorithm for clipped COFDM transmissions is presented. Then clipping mitigation is performed by means of an iterative fashion receiver. Here the goal is to provide a posteriori code word bits soft decision instead of observed channel signal correction. From the channel decoder output information, a reduced set of channel neighbour signals is built aiding the maximum likelihood-based detection. Furthermore, we investigate the receiver performance for both non-coded and coded transmission schemes as well as over a frequency-selective channel scenario. Numerical results show that the proposed receiver is able to perform very close to the linear case even at severe clipping ratio.
I. I NTRODUCTION Despite that multi-carrier signals amplification is pratically difficult due to their Gaussian-like distribution and the nonlinear behaviour of power amplifiers, the orthogonal frequency division multiplexing (OFDM) modulation has been successfully brought in many communication standards for wired and wireless applications as ADSL [1], WiFi standards IEEE 802.11a/g [2], [3] and E-UTRA 3G [4] communication systems. Due to the tones orthogonality property allowing them to partially overlap in frequency-domain, OFDM maximizes its spectral efficiency. Furthermore, OFDM systems use cyclic prefix (CP) as specific time guard interval and need only easy frequency-domain channel equalization [5]. However, high peak-to-average power ratio (PAPR) of OFDM signals stands for the modulation major drawback and has been widely discussed in the literature. Then many different stratigies can be cited to manage PAPR at transmitter side [6] but most of these techniques may be computationally expensive or need a dedicated process at receiver, relying on side information transmission to perform. The deliberate amplitude clipping and filtering before amplification [7] is an easy way to restrain PAPR at a desired level but the resulting non-linear signal distortions lead to heavy performance degradation. If the out-of-band noise caused by clipping can be easilly discarded by filtering, the in-band distortion noise, also refered as clipping noise, must be mitigated at receiver side in order to restore the transmission performance. In case of clipping the complex OFDM signal envelope, the distortion noise can be reasonably modeled as Gaussian and receivers based
on the turbo-principle are known to give potentially good performance in such condition. Decision aided reconstruction [8], [9], clipping noise cancellation [10], [11], [12] and clipping function inversion [13], [14] are noteworthy iterative techniques mitigating the clipping non-linearity effects by observed channel signal direct correction. Unlike these techniques, this work proposes a new scheme dealing with code word bits a posteriori probability enhancement. This soft decision is based on a suboptimal maximum likelihood (SML) detection. The SML uses a limited subset of channel observation neighbour signals to perform the transmitted code word detection. Actually, the subset is given by a low complexity designed bit flipping process after SISO channel decoding. Then the receiver performance study shows that the algorithm can operate under severe clipping ratio constraint and is able to perform close to the non-clipped transmission case over both AWGN and frequency-selective channels. In the next section, we show the system model used as transmitter and recall some common definitions and properties about clipped OFDM signals. In Section III, the SML receiver is presented and the detailled algorithm is proposed. Finally, Section IV shows numerical results and discussion following conclusion are given. II. S YSTEM M ODEL A. Clipping and filtering U and C are respectively non-coded and coded binary streams. C is a code word of size V encoded with ratio R. The considered coded OFDM (COFDM) system uses N frequency tones Xn . Each tone is modulated by a complex signal issued from a 2M -size constellation S and we note M the number of bits labelling one symbol Sm , m ∈ [0, 2M −1]. The transmitter block diagram is depicted in Figure 1. U
C
π
Cπ
xc
Xos,c
xos,c
Figure 1.
xos
Clipped and filtered coded OFDM transmitter block diagram.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
OFDM signals are usually obtained by N -points inverse discrete Fourier transform (IDFT) operation over the frequencydomain tones. The clipped and filtered OFDM signal is processed by oversampling with rate J, i.e. zero-padding X in frequency-domain, as [15] X os = {X0 , . . . , XN −1 , 0, . . . , 0}.
to be investigated. The complementary cumulative density function (CCDF) of PAPR gives a practical information on OFDM signal envelope high amplitude occurrences. Actually, the CCDF is used to check the gain of PAPR reduction schemes at transmitter.
(1) 0
10
N (J−1)
The oversampled OFDM time-domain signal xos is given by a JN -points IDFT nk Xnos ej2π JN
,
0 ≤ k < JN.
CR=1dB
0
1 =√ JN
−1
10 Pr(PAPR(x) > PAPR )
xos k
JN −1
(2)
n=0
Then the clipping function acts on the xos envelope while the phase remains unchanged. Refered as a soft limiter in the literature, this non-linear function is written os |xos xk k |≤A = , 0 ≤ k < JN, (3) xos,c jφ(xos ) k k |xos Ae k |>A
CR=3dB −2
10
CR → ∞ J=1 J=2 J=4 J=16
−3
10
os
jφ(xk ) the polar coordinate notation of sample xos with |xos k |e k . The clipping threshold A depends on the clipping ratio (CR)
CR = 20 log10 (
A ) dB, σx
(4)
with σx2 = E{|x2 |} the mean energy per complex OFDM signal x before clipping and E{.} being the mathematical expectation function. The out-of-band radiations are filtered by considering only the N first discrete Fourier transform (DFT) terms of Xnos,c = √
JN −1 nk 1 −j2π JN xos,c , k e JN k=0
0 ≤ n < JN,
(5)
os,c os,c os,c X os,c = {X0os,c , . . . , XN −1 , XN , . . . , XJN −1 }.
(6)
Xc
Out−of−band noise
In the sequel, we note the oversampling, clipping and filtering operations in frequency-domain, including Equations (1-6), as (7) X c = CF JCR (X). Finally, a N -points IDFT operation over X c gives the clipped and filtered complex baseband OFDM signal N −1 1 c j2π nk Xn e N , xck = √ N n=0
0 ≤ k < N.
(8)
B. Peak-to-average power ratio of OFDM signals
0
1
2
P AP R(x) =
(9)
Due to the Fourier transform, assumption of Gaussian distributed OFDM signal samples is commonly accepted and their complex envelope is Rayleigh distributed. Then OFDM signals may have high PAPR and reduction techniques have
4
5 6 PAPR0 (dB)
7
8
9
10
Figure 2. PAPR CCDF versus CR of clipped and filtered 16-QAM OFDM signal, N = 64.
Figure 2 shows the occurrence of 64 tones 16-QAM OFDM signal exceeding a reference threshold set as P AP R0 . The reference curve labelled CR → ∞ stands for the unclipped transmission case. At Nyquist rate, i.e. J = 1, the probability that P AP R(xc ) > P AP R0 is far from the theorical null value for P AP R0 = CR, due to peak regrowth effect [19] increasing PAPR back. Furthermore, filtering by oversampling is no more possible and out-of-band noise radiations fall in the signal band. For J > 1, oversampling allows filtering but peak regrowth remains. However, it decays as J increases. For J > 4, PAPR can not be further diminished by clipping. Moreover, out-ofband noise filtering is still ensured, limiting BER performance degradation of the transmission. This particular setting value is used in the following discussion. C. Clipped OFDM signals properties As viewed in (3), the clipping is a memoryless non-linear process and it seems difficult to establish a formal relation with the original non-clipped signal. However, under the assumption that the OFDM signal samples are described by a stationary Gaussian process, we can write from [16]
Peak-to-average power ratio is a random variable measuring signal variations as max |x2k | 0≤k
