Probability distribution analysis of M-QAM-modulated OFDM symbol and reconstruction of distorted data

Yoo Hyunseuk, Guilloud Frédéric, Pyndiah, Pyndiah Ramesh

Description

It is usually assumed that N samples of the time domain orthogonal frequency division multiplexing (OFDM) symbols have an identical Gaussian probability distribution (PD) in the real and imaginary parts. In this article, we analyze the exact PD of M-QAM/OFDM symbols with N subcarriers. We show the general expression of the characteristic function of the time domain samples of M-QAM/OFDM symbols. As an example, theoretical discrete PD for both QPSK and 16-QAM cases is derived. The discrete nature of these distributions is used to reconstruct the distorted OFDM symbols due to deliberate clipping or amplification close to saturation. Simulation results show that the data reconstruction process can effectively lower the error floor level.

Sujets

• Orthogonal Frequency Division Multiplexing
• Discrete probability distribution

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 4
• Langue: English

Extrait

Yoo et al. EURASIP Journal on Advances in Signal Processing 2011, 2011:135
http://asp.eurasipjournals.com/content/2011/1/135
RESEARCH Open Access
Probability distribution analysis of
M-QAMmodulated OFDM symbol and reconstruction of
distorted data
*Hyunseuk Yoo , Frédéric Guilloud and Ramesh Pyndiah
Abstract
It is usually assumed that N samples of the time domain orthogonal frequency division multiplexing (OFDM)
symbols have an identical Gaussian probability distribution (PD) in the real and imaginary parts. In this article, we
analyze the exact PD of M-QAM/OFDM symbols with N subcarriers. We show the general expression of the
characteristic function of the time domain samples of M-QAM/OFDM symbols. As an example, theoretical discrete
PD for both QPSK and 16-QAM cases is derived. The discrete nature of these distributions is used to reconstruct
the distorted OFDM symbols due to deliberate clipping or amplification close to saturation. Simulation results show
that the data reconstruction process can effectively lower the error floor level.
Keywords: OFDM, discrete probability distribution, M-QAM, nonlinear amplifier, data reconstruction.
1 Introduction complexity, such as iterative methods [6-10] and an
A significant drawback of orthogonal frequency division oversampling method [11].
multiplexing (OFDM)-based systems is their high peak- It is usually assumed that the time domain samples of
to-average power ratio (PAPR) at the transmitter, OFDM symbols are complex Gaussian distributed,
requiring the use of a highly linear amplifier which leads which is a very good approximation if the number of
to low power efficiency. For reasonable power efficiency, subcarriers is large enough. Furthermore, it is
theoretithe OFDM signal power level should be close to the cally proved in [12,13] that a bandlimited uncoded
nonlinear area of the amplifier, going through nonlinear OFDM symbol converges weakly to a Gaussian random
distortions and degrading the error performance. process as the number of subcarriers goes to infinity.
The distortion can be introduced for two main rea- In this article, we derive the discrete Probability
Dissons: nonlinear amplifier [1,2] and/or deliberate clipping tribution (PD) of the time domain samples of M-QAM/
[3]. For the first case, if an OFDM symbol is amplified OFDM symbols with a limited number of subcarriers.
in the saturation area of an amplifier, its data recovery The discrete PD can be used to reconstruct distorted
is not possible. For the second case, deliberate clipping OFDM symbols. We focus on the in-band distortion
makes an intentional noise which falls both in-band and which can be caused when OFDM symbols are
ampliout-of-band. In-band distortion results in an error per- fied in the saturation area or when deliberate clipping is
formance degradation, while out-of-band radiation used to reduce the PAPR [3]. Note that the conventional
Gaussian assumption cannot be used for the data recov-reduces spectral efficiency. Filtering methods can reduce
out-of-band radiation, but also introduces peak regrowth ery of distorted OFDM symbols. The article is organized
of OFDM signals and increases the overall system as follows: In Section 2, we derive the PD of M-QAM
impulse response [4,5]. modulated OFDM symbols. Using our derivation of PD,
Several approaches have been investigated for mitigat- we consider the data reconstruction (DRC) method in
ing the clipping noise with an amount of computational the presence of a soft limiter in Section 3. Finally, we
conclude this article in Section 4.
* Correspondence: [e-mail]
Department of Signal and Communications, Telecom Bretagne, Technopole
Brest Iroise - CS 83818, 29238 Brest cedex 3, France
© 2011 Yoo et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.Yoo et al. EURASIP Journal on Advances in Signal Processing 2011, 2011:135 Page 2 of 9
http://asp.eurasipjournals.com/content/2011/1/135

ˆ2 IDFT forM-QAM symbols The characteristic function of and , l Î {0, 1, ...,X Xl l
AnOFDMsignalinthetimedomainisthesumof N N - 1}, is given by [14]
independent signals over sub-channels of equal
bandϕ (ω)= ϕ (ω)ˆ width 1/(T + T ) and regularly spaced with frequency Xcp l
Xl 1/(T + T ), where T is the orthogonality period and Tcp cp
ˆ E exp jX ωlis the duration of cyclic prefix.
(3)
√At the transmitter, a frequency domain OFDM symbol M−1 √1X with N samples X={X , X , ..., X } is transformed0 1 N-1 = √ exp j( M −2k −1)τω ,
via an N-point inverse discrete Fourier transform M
k=0
(IDFT) to a time domain OFDM symbol x with N
samwhere E [·] is the expectation operator. We will useples x={x , x , ..., x }:0 1 N-1
this characteristic function in order to obtain the PD of
N−1 time domain OFDM samples.1 2πlm
x = X ·exp j , (1)m l We first consider the real part given byxˆ { x }N N m m
l=0
N−1 where m, l Î{0,1,..., N - 1}. Note that the trans- 1
ˆxˆ = X · c(l,m)+ X · s(l, m) , (4)m l lmitted signal is made of the time domain OFDM sym- N
l=0
bol together with the cyclic prefix. Since the cyclic

prefixisthecopyofapartof x, the derivation of the −2πlmwhere c(l, m) cos andNdistribution of the samples in x completely determines
−2πlmthe distribution of the transmitted signal. s(l, m) sin .N
We assume hereafter that all the frequency domain
Given l and m, since both c (l, m)ands(l, m)are
samples X are uniformly distributed in the set of al
ˆconstants, the characteristic functions of X · c(l, m)lsquare M-QAM constellation S; for example:

+1+j +1−j −1+j −1−j and are obtained as√ √ √ √ X · s(l, m)S = { , , , } in the QPSK case. In addi- l
2 2 2 2
√tion, the real and imaginary parts of X, denoted, respec- M−1 l √1
ϕ (ω)= ϕ (c(l, m) · ω)= √ exp j( M −2k −1)τ · c(l, m) · ω , ˆ ˆXl·c(l,m) Xl Mˆtively, , , are uniformly distributed k=0X { X } X { X }l l l l √ (5)
M−1 √1as depicted in Figure 1. The minimum Euclidean dis- ϕ (ω)= ϕ (s(l, m) · ω)= √ exp j( M −2k −1)τ · s(l, m) · ω .
X ·s(l, m) Xl l M
k=0tance of the constellation is given by 2τ. Then, a general

ˆ Then, the characteristic function ofexpression for the PD of , l Î {0, 1, ..., N-1}is{X,X }l l

given by ˆ is given byX · c(l, m)+X · s(l, m)l l

√ √ 1 ϕ (ω)ˆ Xˆ ·c(l, m)+X ·s(l, m)l lPr X = M −2k −1 τ =Pr X = M −2k −1 τ = √ , (2)l l ⎡ √ √ ⎤ ⎡ √ √ ⎤
M M M M M
⎢sin τ · c(l, m)ω cos τ · c(l, m)ω ⎥ ⎢sin τ · s(l, m)ω cos τ · s(l, m)ω ⎥2 2 2 2⎢ ⎥ ⎢ ⎥ (6)4 ⎢ ⎥ ⎢ ⎥= ·⎢ ⎥ ⎢ ⎥M sin(τ · c(l, m)ω) sin(τ · s(l, m)ω)√ ⎣ ⎦ ⎣ ⎦
where .k∈{0, 1,..., M −1}
1√
M
... ...
+3τ-3τ -1τ +1τ +5τ-5τ
ˆ ˘X or X

Figure 1 PD of the M-QAM symbol. PD of the M-QAM modulated symbol in each real or imaginary part, ˆ or .X X
Probability
Yoo et al. EURASIP Journal on Advances in Signal Processing 2011, 2011:135 Page 3 of 9
http://asp.eurasipjournals.com/content/2011/1/135
which is proved in Appendix. Referring to Equations (2) and (3), the discrete PD of
ˆ ˆSince and , l Î {0, 1, ..., N - 1}, are mutually Pr{xˆ }, Pr{xˆ }, is given byX X m ml l
independent, ϕ (ω) is given by Equation (7). Nxˆm 1 N
Pr{xˆ =0} = ,m
N N/22ϕ ˆ (ω)=ϕ (ω)=Nxm N−1 ˆXl·c(l,m)+Xl·s(l, m)

(12)l=0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2k 2k 1⎛ ⎡ √ √ ⎤ ⎡ √ √ ⎤⎞ N
M M M M Pr xˆ = τ 1 − =Pr xˆ = τ −1 = ,⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ m m⎜ ⎢sin⎝ τ·c(l, m)ω⎠ cos⎝ τ·c(l, m)ω⎠⎥ ⎢sin⎝ τ·s(l, m)ω⎠ cos⎝ τ·s(l, m)ω⎠⎥⎟ NN−1 (7) k⎜ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥⎟ N N 2
4⎜ ⎢ ⎥ ⎢ ⎥⎟· .⎜M⎢ sin (τ·c(l, m)ω) ⎥ ⎢ sin (τ·s(l, m)ω) ⎥⎟
⎝ ⎣ ⎦ ⎣ ⎦⎠l=0
Nwhere k∈{0,1,..., −1}.
2
Therefore, Similarly, the PD of can be derived asx { x }m m
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎡ √ √ ⎤ ⎡ √ √ ⎤⎞
M M M M Pr {x }=Pr {xˆ }.⎜ &#