Interference in OFDM Systems with Nonlinear
Power-Amplifiers
C. Alexandre R. Fernandes1, Jo˜ao Cesar M. Mota2, and G´erard Favier3
1
Federal University of Cear´a, Computer Engineering rua Anahid Andrade 472, 62011-000, Sobral, Brazil
alexandrefernandes@ufc.br
2
Federal University of Cear´a, Dept. of Teleinformatics Engineering Campus do Pici, 60.755-640, 6007 Fortaleza, Brazil
mota@gtel.ufc.br
3
I3S Laboratory, University of Nice-Sophia Antipolis, CNRS 2000 route des Lucioles, BP 121, 06903, Sophia-Antipolis Cedex, France
favier@i3s.unice.fr
Abstract. Due to a high peak-to-average power ratio (PAPR), orthog-onal frequency division multiplexing (OFDM) signals are often driven at the nonlinear region of power amplifiers (PAs). As a consequence, the orthogonality between the subcarriers is broken and nonlinear inter-carrier interference (ICI) is introduced. In this paper, we proposed two techniques for canceling nonlinear ICI in wireless OFDM communication systems with nonlinear PAs. The proposed techniques are based on the concept of power diversity, which consists in a transmission scheme that re-transmits the symbols several times with a different transmission power each time. The main advantage of using the power diversity is that the problem of canceling the nonlinear ICI can be viewed as a source separation problem, where the ICI terms correspond to “virtual” sources. The proposed techniques are able to provide a more robust transmission at the cost of a lower transmission rate.
Keywords: OFDM, nonlinear power amplifier, carrier inter-ference, power diversity, source separation.
1
Introduction
In this paper, two techniques for canceling nonlinear inter-carrier interference (ICI) in orthogonal frequency division multiplexing (OFDM) wireless communi-cation systems with nonlinear radio frequency power amplifiers (PAs) are pro-posed. An important drawback in OFDM systems is that the transmitted signals are characterized by a high peak-to-average power ratio (PAPR) [1, 10] and, as a consequence, in some situations, the OFDM signal is driven at the nonlinear region of the PA. Nonlinear inter-carrier interference (ICI) is then introduced, which may significantly deteriorate the recovery of the information symbols.
V. Vigneron et al. (Eds.): LVA/ICA 2010, LNCS 6365, pp. 337–345, 2010. c
The proposed techniques are based on the concept of power diversity, which consists in a transmission scheme that re-transmits the symbols several times with a different transmission power each time. The power diversity induces a multi-channel representation, allowing a perfect recovery of the information sym-bols in the noiseless case. As it will be viewed, the key point of this approach is that the problem of canceling the nonlinear ICI can be viewed as a source separation problem, where the information symbols correspond to the source of interest and the ICI can be viewed as “virtual” sources. The main drawback of this approach is the fact that the transmission rate is divided by the repetition factor, i.e. the number of times that every symbol is transmitted. However, in many cases, it is possible to use a repetition factor equal to 2. Moreover, the techniques are proposed for two different scenarios. The first one assumes that the wireless channel frequency response and the PA coefficients are known, while the second one assumes that only the PA coefficients are known. In the second scenario, it is assumed that some subcarriers are dedicated to pilot symbols that are used to estimate the wireless channel frequency response.
Input backoff and signal predistortion [1, 3–5] are popular approaches used to combat PA nonlinearities in communication systems. It should be mentioned that the backoff approach reduces the power efficiency, as the transmitter uses only a small portion of its allowed input range. Besides, our approach for com-pensating the nonlinear distortions at the receiver provides some advantages over predistortion schemes. One of these advantages is that few modifications in the portable units are necessary to accommodate the nonlinearity compensation in the uplink. Moreover, our approach may take other channel nonlinearities into account, contrarily to predistortion schemes that generally compensate the nonlinear distortions of a single nonlinear block. Other techniques for nonlinear interference rejection at the receiver side of OFDM systems have been proposed, most of them based on iterative methods as, for instance, [6, 7].
2
System Model
A simplified scheme of the discrete-time equivalent baseband OFDM system used in this work is shown in Fig. 1. Let us denote byN the number of subcarriers, ¯
Fig. 1.Discrete-time equivalent baseband SISO-OFDM system
The time-domain OFDM symbolsi,n, for 1 ≤n≤N, is obtained by taking the Inverse Fast Fourier Transform (IFFT) of the frequency-domain data sym-bols and, then, a cyclic prefix (CP) is inserted in order to avoid intersymbol interference (ISI) and ICI. Indeed, when the PA is linear, the cyclic prefix en-sures the orthogonality between the subcarriers. However, as it will be shown in the sequel, this is not the case for a nonlinear PA.
The time-domain symbol with the CP is then amplified by a PA that is modeled as a polynomial of order 2K + 1. Denoting by ui,n (1 ≤ n ≤ N) the output of the PA, we have:
ui,n= K
k=0
f2k+1|si,n|2ksi,n = K
k=0
f2k+1ψ2k+1(si,n), (1)
where f2k+1, for 0 ≤ k ≤ K, are the equivalent baseband coefficients of
the polynomial that models the PA and the operator ψ2k+1(·) is defined as ψ2k+1(a) =|a|2ka.The equivalent baseband polynomial model (1) includes only
the odd-order power terms with one more non-conjugated term than conju-gated terms because the other nonlinear products of input signals correspond to spectral components lying outside the channel bandwidth, and can therefore be eliminated by bandpass filtering [8].
The signalui,n is transmitted through a frequency-selective fading wireless channel and, at the receiver, the CP is removed (RCP box in Fig. 1) from the time-domain received signal and the FFT is then calculated. Assuming that the length of the cyclic prefix is higher than or equal to the channel delay spread, it can be shown that the frequency-domain received signals can be written as [8]:
¯ x(i) =Λ
K
k=0
f2k+1ψ¯2k+1(VH¯s(i)) + ¯n(i). (2)
where ¯x(i) ∈ CN×1 is the vector of frequency-domain received signals at the
ith symbol period, Λ ∈ CN×N is a diagonal matrix containing samples of the channel frequency response λn (1 ≤ n ≤ N), V ∈ CN×N is the FFT ma-trix of dimension N, with [V]p,q = e−j2π(p−1)(q−1)/N (1 ≤ p, q ≤ N), and ¯
n(i)∈CN×1is a vector containing additive white Gaussian noise (AWGN)
com-ponents of varianceσ2. Besides, the operatorsψ
2k+1(·) and ¯ψ2k+1(·) are defined
asψ2k+1(a) = [ψ2k+1(a1)· · ·ψ2k+1(aN)]T ∈CN×1, for a= [a1· · ·aN]∈CN×1, and ¯ψ2k+1(a) =Vψ2k+1(a)∈CN×1.
Equation (2) shows that the frequency-domain received signal ¯xi,n equals a scaled version of the frequency-domain data symbol λnf1s¯i,n plus nonlin-ear ICI terms K
eliminate the nonlinear ICI and to remove the scalar factorλnf1.
The frequency-domain received signal ¯xi,n can be rewritten in a vector form:
¯
xi,n=λnfTφ¯i,n+ ¯ni,n, (3)
where f = [f1 f3· · · f2K+1]T ∈ C(K+1)×1, ¯φi,n = [¯si,n [ ¯ψ3(VH¯s(i))]n · · ·
[ ¯ψ2K+1(VH¯s(i))]n]T ∈C(K+1)×1 is a vector containing the information symbol
and the nonlinear ICI components of thenthsubcarrier andithsymbol period, and ¯ni,n is the corresponding noise component in the frequency domain.
3
Power Diversity
In this section, we introduce the concept of power diversity by presenting a transmission scheme that consists of re-transmitting the information symbols several times with different transmission powers. Then, we briefly discuss how the problem of estimating the information symbols can be viewed as a source separation problem.
3.1 Transmission Scheme
The power diversity transmission scheme can be summarized as follows. The frequency-domain information symbol ¯si,n at the nth subcarrier (1 ≤n ≤ N) and ith symbol period (1 ≤i ≤ IB) is transmitted L times with transmission powers that are multiplied by the factorsP1, ..., PL, as follows:
¯
s(((pdi−)1)L+l),n=Pl¯si,n, for 1≤l≤L, (4)
where ¯s(((pdi−)1)L+l),n is the “weighted” frequency-domain symbol associated with thenthsubcarrier and ((i−1)L+l)thsymbol period. Note that the transmission power factorsP1, ..., PL are the same for all the subcarriers.
Denoting by ¯x(((pdi−)1)L+l),n, 1≤l≤L, theLfrequency-domain received signal samples associated with the frequency-domain information symbol ¯si,n, we can define the following vector ¯x(pd)(i) = [¯x(pd)
((i−1)L+1),n x¯
(pd)
((i−1)L+2),n · · · x¯
(pd)
iL,n]T ∈ CL×1. Assuming that the samples of the channel frequency response channelλ
n (1≤n≤N) and PA coefficients f2k+1 (0≤k≤K) are time-invariant overL
symbol periods, we can write from (3) and (4):
¯
where
H= ⎡
⎢ ⎢ ⎣
P12
1 · · ·P
2K+1 2
1
..
. . .. ...
P12
L · · ·P
2K+1 2
L ⎤
⎥ ⎥ ⎦ ⎡
⎢ ⎣
f1· · · 0
..
. . .. ... 0 · · ·f2K+1
⎤
⎥ ⎦∈
CL×(K+1), (6)
and ¯n(i,npd)∈CL×1is a noise vector.
3.2 Source Separation Interpretation
The main motivation for using power diversity is that the problem of estimating the information symbols can be viewed as a source separation problem, where the nonlinear ICI terms are “virtual” sources and the information symbols cor-respond to the source of interest. The proposed transmission scheme inducesL
subchannels, where each re-transmission period corresponds to a subchannel. It is interesting to remark that a classical antenna/sensor array would not succeed to induce a useful multi-channel representation, as all sources (real and virtual) are located in the same point in the space domain. In this case, the array response matrix would have unit rank, exhibiting degenerate discrimination. On the other hand, when diversity is used in power domain, it can be demonstrated that the matrixλnH, which plays the role of “array response” of thenth sub-carrier, is full-rank when Pi = Pj, for 1 ≤ i = j ≤ L. Under that condition, ifL≥(K+ 1), this multi-channel representation allows the perfect recovery of ¯
si,n in the noiseless case.
The main drawback of this approach is the fact that the transmission rate is divided by L. However, based on a realistic assumption that the PA can be modeled using a third-order polynomial (K+ 1 = 2) [3–5, 10], we can useL= 2, which minimizes the transmission rate loss.
4
Power Diversity-Based Receivers (PDRs)
ˆ¯
si,n=wn¯xi,n , (7)
for 1 ≤ n ≤ N, where wn ∈ C1×L contains the coefficients of the separator, calculated by minimizing the following mean square error (MSE) cost function:
Jn=E[|s¯i,n−wnx¯(i,npd)|2], whose solution is given by:
wn =λ∗
nrφ¯HH
HRφ¯HH|λn|2+ILσ2 −1
∈C1×L, (8)
where Rφ¯ =E[ ¯φi,nφ¯ H
i,n]∈ C(K+1)×(K+1), rφ¯ =E[¯si,nφ¯ H
i,n]∈C1×(K+1) and IL is the identity matrix of orderL. The expressions for the values ofRφ¯ andrφ¯
are omitted due to a lack of space.
4.2 PDR with Channel Estimation (PDR-CE)
The PDR-CE assumes that pilot symbols are allocated in subcarriers regularly spaced in the channel bandpass. These subcarriers are denoted by the setN =
{n1,· · ·, nD}, where Dis the number of subcarriers dedicated to pilot symbols. This technique provides an initial estimate of the channel coefficients assuming that the PA is linear, then, it iteratively re-estimates the channel coefficients and information symbols. The initial estimate of channel frequency response on the pilot subcarriers is obtained assuming that the PA is linear. In that case, the optimal estimate ofλn is found by using the maximum ratio combining (MRC) method:
ˆ
λ(0)n =
[P 1 2
1 · · · P
1 2
L] L
l=1Pl2 ¯ x(i,npd)
1 ¯
si,n
(9)
forn∈ N. The initial channel frequency response on the other subcarriers are estimated from the coefficients obtained in Step 1 by interpolation, as in Step 4 of the algorithm.
At each symbol periodi, the proposed receiver carries out the following steps:
Forit=it+ 1:
1. Estimate ˆ¯s(i,nit)(1≤n≤N) from (7) and (8), using ˆλ
(it−1)
n .
2. Project ˆ¯s(i,nit) (1 ≤n ≤ N) onto the QAM alphabet and construct ˆ¯φ(i,nit) =
[ˆ¯s(i,nit) [ ¯ψ3(VHˆ¯s (it)
(i))]n · · · [ ¯ψ2K+1(VHˆ¯s(it)(i))]n]T from ˆ¯s(i,nit), for 1≤n≤
N.
3. Estimate the channel frequency response on the pilot subcarriers as: ˆ
λ(nit)=
[u(i,nit)]Hx¯
(pd)
i,n
1
[u(i,nit)]Hu
(it)
i,n
, (10)
forn∈ N, whereu(i,nit)=Hφˆ¯
(it)
i,n. That corresponds to the optimal estimate
ofλn givenHand ˆ¯φ
(it)
4. The channel frequency response on the other subcarriers (ˆλ(nit), for 1≤n≤
N) are calculated by interpolation from the channel frequency response on the pilot subcarriers (ˆλ(nit), for n ∈ N) obtained in Step 3. See [7, 11], for more details about the interpolation procedure. In the simulation results section, the interpolation is done by using truncated FFT matrices [7, 11]. 5. IfN
n=1|λˆ (it)
n −ˆλ(nit−1)|2/nN=1|ˆλ(nit−1)|2< ǫ, stop. Otherwise, go to Step 1.
5
Simulation Results
In this section, the proposed techniques are evaluated by means of simulations. A OFDM system with a third-order polynomial PA with coefficients equal to
f1= 0.9798−0.2887jandf3 =−0.2901 + 0.4350j[9], and a wireless link with
frequency selective fading due to multipath propagation has been considered for the simulations. The results were obtained withN = 64 subcarriers and BPSK (binary phase shift keying) transmitted signals, via Monte Carlo simulations using 500 independent data realizations. In all the simulations, the PDRs use a repetition factorL= 2, with P1= 1.2 andP2= 0.6.
Fig. 2 shows the bit error rate (BER) versus the noise variance for 6 different techniques. The PDR-CK and PDR-CE correspond to the proposed techniques, while 1-tap eqz-CK and 1-tap eqz-CE correspond to 1-tap equalizers (complex automatic gain controls) that simply divide the received signal ¯xi,n by the cor-responding channel coefficient, with channel knowledge and channel estimation, respectively. For the 1-tap equalizers, we have used the transmission power that maximizes signal to noise plus interference (SNIR) ratio. The derivation of this optimal power is omitted due to a lack of space. Moreover, we also show the BER provided by 1-tap equalizers in the case of a linear PA, with channel knowledge and channel estimation. All the techniques that use channel estimation assume that 8 subcarries regularly spaced in the channel bandpass are dedicated to pi-lot symbols. The channel output signal to noise ratio (SNR) is not used as the
−20 −15 −10 −5 0
10−4 10−3 10−2 10−1 100
Noise Variance (dB)
BER
1−tap eqz−CE 1−tap eqz−CK PDR−CE PDR−CK Linear PA−CE Linear PA−CK
standard OFDM receiver. Besides, PDR-CK provides a gain of 2.5 to 5dB in noise variance with respect to the PDR-CE. Note also that, as expected, when the PA is linear, the BERs are much lowers than the ones obtained with the nonlinear PA.
6
Conclusion
Two techniques for canceling nonlinear ICI in OFDM systems with nonlinear PAs have been proposed in this paper. These techniques are based on the power diversity transmission scheme that re-transmits all the symbols several times with a different transmission power each time. We have tested the proposed re-ceivers by means of simulations and they have significantly outperformed the 1-tap equalizers, which correspond to a standard approach to recover OFDM symbols. The main drawback of the PDRs is the fact that the transmission rate is divided by the repetition factor. However, based on a realistic assumption that the PA can be modeled using a third-order polynomial, we can use a repe-tition factor equal to 2. That means that the PDRs may provide a more robust transmission at the cost of a lower transmission rate. In a future work, we will compare the performance of the proposed techniques with other methods for in-terference rejection in nonlinear OFDM systems and the PDRs will be extended to the case of a nonlinear PA with memory.
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