IMPACT OF LINEAR POWER AMPLIFIER EFFICIENCY ON CAPACITY OF OFDM SYSTEMS WITH CLIPPING Gill R. Tsouri Rochester Institute of Technology Department of Electrical Engineering
[email protected]
Dov Wulich Ben-Gurion University of the Negev Department of Electrical and Computer Engineering
[email protected]
ABSTRACT
the clipping ratio increases monotonically for any SNR and for any over-sampling ratio. In this work we analyze the capacity of OFDM systems with clipping, while taking into account the power loss due to the limited efficiency of practical and commonly used LPAs [6]. The motivation for this analysis stems from recent work on the effects of PAPR reduction coupled with LPA power efficiency considerations on overall system performance [7,8]. In [7] the optimal PAPR for achieving minimal Bit Error Rate (BER) was found while taking LPA power efficiency into account. It was shown that the optimal PAPR is not necessarily the lowest one and that decreasing the PAPR too much results in performance loss. LPA power efficiency was also considered in [8] for capacity analysis of distortion-less PAPR reduction methods, such as selected mapping and tone reservation. We are unaware of past work regarding the impact of practical LPA power efficiency on the capacity of clipped OFDM. An intrinsic property of any LPA is that its power efficiency decreases as its required linear dynamic range increases [6,9]. This translates to a decrease in channel SNR, and hence a decrease in capacity. In a clipped OFDM system, the required dynamic range is determined by the clipping ratio. The following tradeoff with regard to the clipping ratio emerges: a low clipping ratio causes signal distortion and decreases capacity [4]. However, it also increases the power efficiency of the power amplifier [6], which increases capacity. The main contribution of our work1 is in evaluating the clipping ratio which resolves this tradeoff and the capacity lost incurred when using a sub-optimal clipping ratio. We demonstrate that there is an optimal clipping ratio which achieves maximal system capacity, and that increasing the clipping ratio beyond the optimal ratio reduces capacity. This result is different from the one obtained for an ideal lossless LPA [4], for which increasing the clipping ratio never decreases capacity. We assume a clipped OFDM system with out of band radiation filtering as is defined in the block scheme of [4] with an additional LPA block after the low pass filter.
Capacity of clipped OFDM systems is analyzed. In contradistinction to previous work, we take into account intrinsic properties of the linear power amplifier efficiency the dependence of power efficiency on clipping ratio. The following tradeoff emerges: a low clipping ratio causes signal distortion and decreases capacity. However, it also increases the efficiency of the power amplifier, which increases capacity. We resolve this tradeoff by evaluating the optimal clipping ratio for achieving maximal capacity. We show that increasing the clipping ratio doesn’t necessarily result in higher capacity. Instead there exists an optimal clipping ratio for given channel SNR. Increasing the clipping ratio beyond optimal significantly reduces capacity. Index Terms— OFDM, PAPR, Clipping, Capacity 1. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is widely recognized as an attractive modulation method for high-speed communications. A main drawback of OFDM is its high Peak to Average Power Ratio (PAPR). High PAPR requires the use of a Linear Power Amplifier (LPA) with large linear dynamic range in order to avoid signal distortion and spectral spreading. Many methods for reducing the PAPR were suggested in the past [1,2,3], amongst them is clipping using a soft limiter coupled with out of band filtering [4]. Clipping was shown to be a simple and effective approach to PAPR reduction – see [3], [4] and references therein. Rather recently, the capacity of OFDM systems with clipping was analyzed [4]. Such analysis has practical meaning for current OFDM systems, due to advances in coding theory [5] which allow for marginally reaching capacity. In [4] a completely linear LPA with ideal power efficiency (lossless LPA) was considered. Capacity analysis of OFDM with clipping was carried out, accounting for inband distortion, out of band filtering and peak re-growth. The capacity as function of channel Signal to Noise Ratio (SNR) was depicted for various clipping ratios and oversampling ratios. It was shown that capacity as a function of
1 The basis for the work presented in this paper was developed while being at Ben-Gurion University.
η UB ( γ ) = Gγ −20 g /ln10 .
2. THE POWER AMPLIFIER We use a soft limiter with negligible non-linear effects within [− Asat , Asat ] to model the LPA as was also done in [4,10]. In a clipped OFDM system max s ( t ) = Asat , (1) 0≤t ≤T
where s ( t ) is the signal at the LPA input. As will be explained later, this guarantees maximal usage of the LPA power. max s ( t ) is tightly related to PAPR at the LPA 0≤t ≤T
max s ( t )
def
Γ =
E
0≤t ≤T 1T
T
0
max s ( t )
0≤t ≤T
=
2
s ( t ) dt
Pav
2
,
(2)
where T is the OFDM symbol time. As a result of (1) the average power at the LPA output equals to
Pout = Pav =
max s ( t )
0≤t ≤T
Γ
2
=
2 Asat
Γ
.
(3)
From (3) it follows that for any Asat a lower Γ means higher average power from the LPA. Let us consider class A LPAs which are the most linear. They consume a constant and limited amount of power, PDC , regardless of the level of the input signal to keep the amplifier at the operating point. We use the power efficiency as was defined in [10]: η = Pout ( Pout + PDC ) . (4) For class A LPA (4) is given by
η=
0.5
Γ
.
(5)
Alternatively, an upper bound on η ( Γ ) for commonly used LPAs is given in [6], and may be expressed as [7]:
η UB ( Γ ) = GΓ −10 g /ln10 ,
(6)
where G = 0.587 , g = 0.1247 for class A LPA, and G = 0.907 g = 0.1202 for class B LPA. The clipping ratio ( γ ) was defined in [4] as:
γ =
Amax Pin
,
It follows that η is upper bounded by a monotonic decreasing convex function of γ . We use (8) from here on for generality, but note that the following derivations hold for any case where η ( γ ) is a monotonic decreasing and convex function. We define the channel SNR after power-loss at the LPA as follows: Pavη ( γ ) SNRc = , (9) Pnoise where Pnoise is the average power of noise at the receiver after a rectangular filter. For η = 1 (9) is exactly the
input according to: 2
(8)
(7)
where Amax is the maximum permissible amplitude over which the signal is clipped, and Pin is the signal average power prior to clipping. When there is no peak re-growth
Γ = γ 2 . Since there is some peak re-growth due to filtering, Γ > γ 2 , and we may write a new upper bound based on (6):
definition of SNRc given in [4]. We assume that Pav is limited and cannot be made larger to compensate for the limited efficiency of the LPA. It follows that SNRc is upper bounded by:
SNRcUB =
PavηUB ( γ ) Pnoise
.
(10)
3. SYSTEM CAPACITY In [4, eqs (40)-(45)] explicit expressions are given for the capacity per sub-carrier of the considered clipped OFDM system as function of SNRc , while assuming a lossless LPA ( η = 1 ) for an AWGN channel. Using these expressions with (10) results in upper bounds on the capacity of such channels, with the LPA power efficiency (8) taken into account for the commonly used class A and class B LPAs. Fig. 1 displays the upper bound for the AWGN channel as function of the clipping ratio for various SNRc with clipping using an oversampling ratio of 4. The lossless LPA ( η = 1 ) is displayed for reference. For the lossless LPA the capacity increases when γ is increased. However, when the LPA power efficiency is taken into account there exists some optimal γ opt for which the capacity is maximal. For
γ > γ opt or γ < γ opt the capacity decreases. For example, when SNRc = 30dB γ opt ≈ 2.2 and the corresponding maximal capacity for class A and class B LPAs is about 8 bits per sub-carrier. This is opposed to the lossless LPA which indicates a maximal capacity of 10 bits per sub-carrier. If we choose γ = 8 the lossless LPA indicates a capacity of 8 bits per sub-carrier, while the capacity of class A LPA degraded to 6 bits per sub-carrier and the capacity of class B LPA degrades to 7 bits per subcarrier. The same trends were observed for the Rayleigh fading channel. The same trends were also observed when using (6) to accurately describe η ( Γ ) for class A LPA
18 lossless PA class A PA class B PA
16
[5] C. Berrou, A. Glavieux, “Near optimum error correcting coding and decoding: Turbo-codes,” IEEE Transactions on Communication, vol. 44, pp. 1261-1271, Oct. 1996.
50 dB
capacity, bits/subcarrier
14
[6] P. Antognetti, Power Integrated Circuits: Physics, Design and Applications, New York, McGraw-Hill, 1986.
12 10
[7] D. Wulich, “Definition of Efficient PAPR in OFDM,” IEEE Communication Letters, vol. 9, No. 9, Sept. 2005, pp 832-834.
30 dB
8 6
[8] G. R. Tsouri and D. Wulich, “Capacity Analysis and Optimization of OFDM with Distortionless PAPR Reduction”, European Transactions on Telecommunications (ETT), Aug. 2008.
4 10 dB
2 0
0
1
2
3
4 clipping ratio
5
6
7
8
Fig.1 - Capacity vs. clipping ratio for various SNRs instead of the upper bound in (8). Results are omitted for brevity.
4. CONCLUSION The impact of power efficiency of practical linear power amplifier on the capacity of OFDM systems with clipping was evaluated. The intrinsic property of decreased efficiency for increased dynamic range was taken into account, along with the degrading effects of clipping. It was shown that there exists an optimal clipping ratio for given channel SNR, which achieves maximal capacity. This maximal capacity was shown to be lower than the capacity achieved when considering a lossless power amplifier. It was also shown that setting the clipping ratio too high may result with considerable degradation of capacity. It follows that careful consideration should be taken when setting the clipping ratio in practical OFDM systems. While for the ideal lossless LPA the clipping ratio has to be high enough, for practical LPAs the clipping ratio should be high enough to avoid degradation associated with clipping, but not too high to avoid excessive decrease in power amplifier efficiency.
5. REFERENCES [1] S. H. Han, J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Communication, Apr. 2005, pp. 56-65 [2] J. Tellado, Multicarrier Modulation with Low PAR, Applications to DSL and Wireless, Kluwer 2000. [3] X. Li, L. J. Cimini, “Effects of clipping and filtering on the performance of. OFDM,” IEEE Communication Letters, vol. 2, pp. 131-133, 1998. [4] H. Ochiai, H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Transactions on Communication, vol. 50, pp. 89–101, Jan. 2002
[9] H. Krauss, C. Bostian, and F. Raab, Solid State Radio Engineering, New York, Wiley, 1980. [10] R. Baxley and G. T. Zhou, “Power Savings Analysis of Peakto-Average Power Ratio in OFDM,” IEEE Transactions on Consumer Electronics, vol. 50, no. 3, pp. 792 - 798, Aug. 2004.
