IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005 127 Signal Modeling Classes for Linearized OFDM SSPA Behavioral Analysis M. S. O’Droma, Senior Member, IEEE, and N. Mgebrishvili, Member, IEEE Abstract— This paper presents a comparison of four signal representation philosophies, denoted direct time domain, (DTD), mixed frequency and time domain (MFTD), statistical (Stat), and a newly proposed hybrid MFTD-statistical (MFTD-S) for the behavioral analysis of an orthogonal frequency division multiplexing (OFDM) signal through a solid state power amplifier (SSPA) with five degrees of linearization applied. The potential of the Stat and MFTD-S approaches for behavioral analysis of mul- ticarrier, multi-band and ultra wideband systems is illustrated. Also, and for the first time, the impairment amelioration obtained through the introduction of different degrees of linearization in OFDM based systems is quantified, together with an upper bound to potential linearization improvement: linearization benefit is poor unless the narrow area of benefit can be matched to the raised-sine shape of the SSPA energy efficiency curve. Index Terms— Nonlinear RF amplifier, OFDM, percentage linearization, MFTD, MFTD-S, error vector magnitude (EVM), wireless transmitters. I. I NTRODUCTION T HE impairment caused by RF SSPA’s nonlinearity is especially serious in multi-carrier systems, and there is a growing demand to develop advanced linearization schemes to ameliorate this [1]. In this it is useful to seek signal represen- tation techniques more suitable for the behavioral analysis of future simultaneous multicarrier broadband (including ultra- wideband, UWB) mobile terminals and such systems. Here a preliminary comparative investigation is presented into four signal representation approaches for behavioral modeling and analysis of the passage of OFDM signals through an RF nonlinear SSPA. These are DTD, MFTD, Stat, and MFTD- S. The latter is a hybrid of MFTD and Stat approaches newly proposed here. Five degrees of SSPA linearization are applied viz. PL of 0, 15, 34, 51 & 73%, [2]. The comparison is of the modulation fidelity (MF) degradation, measured as error vector magnitude (EVM) deterioration. The model used for the unlinearized PA, a PHEMT SSPA [3], and composite linearizer-PA, herein referred to as the ‘nonlinearity,’ is the same in all cases. It is based on an L th order complex Bessel function series approximation of the nonlinearity’s single unmodulated carrier envelope transfer Manuscript received June 11, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Gianluca Mazzini. This material is based upon work supported by TARGET-Top Amplifier Research Groups in a European Team, an EU FP6 NOE, IST- 2004-507893 (www.target-net.org), and the University of Limerick, Ireland (www.ul.ie). The authors are with the Department of Electrical and Computer En- gineering, University of Limerick, Ireland (e-mail: mairtin.odroma@ul.ie; nana.mgebrishvili@ul.ie). Digital Object Identifier 10.1109/LCOMM.2005.02030. characteristics derived from a complex Bessel-Fourier series expansion approximation of the nonlinearity’s generalized memoryless instantaneous complex nonlinear voltage transfer characteristics, [4]. This SSPA’s small 0.13rads nonlinear AM- to-PM variation over the dynamic range will have negligible effect, e.g. [5], whether linearized or not. The OFDM system examined is the 48 subcarrier, 16QAM, IEEE 802.11a system [6], which has a potential 17dB peak to average power ratio, PAPR, though most of the time it will not exceed 8dB [7]. II. SIGNAL REPRESENTATION CLASSES A. Direct Time Domain Approach (DTD) Here the envelope of an M -symbol N -subcarrier RF OFDM nonlinearity input – A(t − sT ), where s =1, 2, ··· ,M , and T is the symbol duration, represented in its single narrowband modulated carrier form – is directly applied (i.e. in the time do- main) to the nonlinearity’s single unmodulated carrier transfer characteristic, [4]; the nonlinearity’s zonal band (ZB) output, including all ZB intermodulation products (IMP), s o (t): s o (t)= M  s=1  L  k=1 b k J 1 (akA(t − sT ))e jω0(t−sT )  (1) where b k are the nonlinearity model’s complex coefficients and α is determined by its dynamic range; J (.) are Bessel functions of the first kind. After a normal OFDM demodula- tion process, EV M l degradation per subcarrier l is estimated directly from the sub-carrier’s demodulated constellation dia- gram using modified versions of the EVM algorithm in [8], and then averaged over all subcarriers. B. Mixed Frequency and Time Domain Approach (MFTD) Here, the constant envelope constituent components of the nonlinearity’s output, which collectively make up the desired output OFDM signal, plus all harmonics and all IMPs, are individually constructed on a per symbol basis, by calculating their intra-symbol, s, individual amplitudes, phases and fre- quencies (A, φ and ω) from the corresponding input symbol’s individual subcarriers’ A l,s , φ l,s and ω l . Intra-symbol, the subcarriers are constant-envelope (A l,s ). The general multi- carrier nonlinearity output may be written (by appropriately modifying the approximation in [4]): s o (t)= M  s=1 L  k=1 b k ∞  n1,n2,···,nN=−∞  N  l=1 J n l (akA l,s )  · e ∑ N l=1 jn l (ω l (t−sT )+φ l,s ) (2) 1089-7798/05$20.00 c  2005 IEEE