Characterization of power spectral density for nonlinearly amplified OFDM signals based on cross-correlation coefficient
© Lee and Ochiai; licensee Springer. 2014
Received: 3 August 2014
Accepted: 4 November 2014
Published: 24 November 2014
Orthogonal frequency division multiplexing (OFDM) has been adopted in many modern communication systems due to its robustness against frequency-selective fading channels as well as its near-rectangular spectrum that can achieve high spectral efficiency. However, its major drawback is the resulting signal with high peak-to-average power ratio (PAPR), which causes severe nonlinear distortion at the power amplifier (PA) unless input backoff is chosen sufficiently large. The effect of the nonlinear distortion is two-fold: out-of-band radiation and signal quality degradation. The former causes adjacent channel interference and thus degrades the bandwidth efficiency. The latter affects the system level performance and is often measured by the error vector magnitude (EVM). It is thus important for the system designer to analyze the nonlinear distortion caused by a given PA in terms of power spectral density (PSD) and EVM, but accurate calculation of these characteristics may be generally involved. In this work, by establishing the link between the cross-correlation coefficient of the input and output signals from PA and the resulting PSD, we characterize the in-band and out-of-band distortion of nonlinearly amplified OFDM signals based exclusively on the cross-correlation coefficient. The accuracy of the proposed approach is confirmed by both simulation and measurement using a real PA.
As the mobile terminals become smaller while meeting their demand for communication with even higher data rate, the future wireless communication signals should satisfy high bandwidth efficiency without sacrificing power efficiency. Orthogonal frequency-division multiplexing (OFDM) signaling has gained significant attention due to its high bandwidth efficiency and robustness against frequency-selective fading channels. However, its well-known drawback is the high peak-to-average power ratio (PAPR) property of the resulting signals. High PAPR signal is difficult to amplify without sacrificing its power conversion efficiency at the linear power amplifier (PA). In order to maximize PA efficiency, it is essential to adjust the input signal to be amplified mostly around the saturation region. This PA operation introduces severe nonlinear distortion which degrades the signal quality and, in turn, increases bit error rate (BER). In addition, it also introduces out-of-band radiation which causes adjacent channel interference (ACI). On the other hand, when we set the operation point of the PA much lower than its saturation point, it suffers from a severe power penalty. This is a well-known trade-off between the PA efficiency and the quality of transmit signals [1, 2], a salient issue for the OFDM systems that exhibit highest PAPR among many communication systems.
The PA models are categorized into strictly memoryless, quasi-memoryless, and memory . The output signal of strictly memoryless PA models (e.g., Rapp model ) depends only on the amplitude-to-amplitude (AM-AM) characteristic, while that of the quasi-memoryless PA models (e.g., Saleh model ) depends on both the AM-AM and amplitude-to-phase (AM-PM) characteristics.
Most wireless communication standards such as IEEE 802.11 wireless LAN strictly regulate the permissible spectral sidelobe levels in order to avoid ACI and thus enhance overall spectral efficiency of the multi-user systems. The error vector magnitude (EVM) is another measure that characterizes the performance degradation caused by nonlinearity due to the system impairments and often restricted by the specification. The EVM is an alternative measure for a ratio of the power of the received signal to that of the in-band distortion and noise [2, 6].
Therefore, from the viewpoint of communication system designers, it may be helpful if performance measures such as power spectral density (PSD) and EVM of the PA output signal are easily predicted or estimated.
The PSD of the signals affected by the nonlinearity of the PA has been extensively studied, mostly in conjunction with the OFDM signals that can be characterized as a band-limited complex Gaussian process.
For example, in [7, 8], based on the autocorrelation function of memoryless PA output signal, the out-of-band spectrum has been theoretically analyzed and good agreement with the experimental result has been observed. In [9, 10], using cumulant expression as a generalization of the autocorrelation function, the closed-form polynomial expression is derived. In , the spectrum estimation is performed by autocorrelation function with curve fitting by a series of Bessel function.
One of the major issues associated with the abovementioned approaches is how to accurately model a given PA. In principle, any well-behaved nonlinear function can be approximated by Taylor series expansion or a series of special functions, but reducing the residual error in the estimated PSD requires an addition of higher order terms. Incorporating higher order terms, in turn, makes the analysis complicated or even mathematically intractable.
In this work, we propose an approach to develop a simple model that can accurately characterize the PSD and EVM of the resulting OFDM signals. Our approach is based on the use of the cross-correlation coefficient between the input and output signals from the PA, which can be easily calculated from its AM-AM and AM-PM characteristics and input backoff (IBO) operation. This cross-correlation coefficient is directly related to the concept of the total signal-to-distortion power ratio (SDR) discussed in . Here, we emphasize that unlike conventional curve fitting approaches, our approach does not necessarily require the precise expressions for the AM-AM and AM-PM curves of a given PA in order to characterize the resulting PSD and EVM. The accuracy of the proposed approach is confirmed by simulation as well as an experimental measurement using a real PA.
The major contributions of this work are summarized as follows: 1) We theoretically establish the link among the cross-correlation coefficient, total SDR as well as in-band SDR (or EVM), and the resulting PSD. In particular, the effect of the higher-order distortion terms is theoretically analyzed for two representative and analytically tractable nonlinear models. 2) We propose a simple PSD estimation approach that only makes use of the cross-correlation coefficient and the spectral shape of the third-order distortion. 3) The effectiveness of our approach is verified by both simulation and measurement using real PA with OFDM signal input. As is common in the statistical analysis of OFDM signals [8, 9, 11, 16, 17], the PA input signal is assumed to be a zero-mean circular symmetric stationary complex Gaussian process  throughout this work.
This paper is organized as follows. A general mathematical expression of PSD for nonlinearly amplified Gaussian signals in terms of the correlation coefficients of input signals is described in Section 2, followed by its examination through two specific nonlinearity examples in Section 3. In Section 4, the proposed estimation of PSD based on the cross-correlation coefficient of input and output signals is developed and its application to OFDM signaling is discussed. The simulation and experimental results are compared with those based on the proposed theoretical approach in Section 5. Finally, the concluding remarks are given in Section 6.
2 PSD expression of nonlinearly amplified Gaussian signals
2.1 Input signal model
where is a real-valued correlation coefficient, and thus, is a real-valued function as well, satisfying .
2.2 Correlation coefficient of output signals
2.3 PSD of output signals
for any positive integer m.
In principle, once the input signal PSD and C n are known, the output PSD can be determined through the above equations. In the case of OFDM, it is reasonable to assume that the PSD is rectangular, and in this case, the closed-form expression can be obtained for their self-convolution terms as will be discussed in Section 4.4. On the other hand, the coefficient C n involves the nonlinear function G(r) and whether it can be given in a tractable form or not depends on the mathematical structure of G(r).
3 Examples of distortion coefficients for nonlinearly amplified Gaussian signals
In this section, we examine the effect of sharpness and smoothness of the nonlinearity on the resulting PSD by analyzing the coefficients C n /ξ in (19). The two specific example AM-AM models, i.e., soft envelope limiter model and erf model, are considered as our representative examples.
3.1 Analysis of coefficients
3.1.1 Soft envelope limiter model
3.1.2 erf model
where Amax in this case corresponds to the maximum output envelope level if the power gain is normalized to unity.
In the case of the soft envelope limiter, when γdB is low, the third-order distortion term (C1) becomes dominant, but as γdB increases, higher-order terms eventually dominate. This particular behavior is mostly specific to the case of the soft envelope limiter which has a piecewise linear characteristic such that the envelope undergoes severe nonlinearity at the saturation point. On the other hand, in the case of the erf model, the third-order distortion term (C1) is always dominant and higher-order terms become less dominant, and this holds almost regardless of the IBO value. Therefore, for many practical amplifiers that have smooth nonlinearity, the third-order term may be considered as the most effective factor in the PSD analysis.
4 Simple expressions for approximate PSD and EVM upper bound
In this section, we first establish the relationship between the nonlinearity behavior and signal-to-distortion power ratio under the assumption of Gaussian input signals. We will then develop a simple approach that does not require any curve fitting but still can estimate the PSD as well as the resulting in-band distortion even if the function G(r) is only partially measured or the signal-to-distortion power ratio is known only at the detector.
4.1 Total signal-to-distortion power ratio
where is defined in (17). This is referred to as a design SDR in  and can be easily calculated through the measurement of only the cross-correlation coefficient (18).
4.2 Effective signal-to-distortion power ratio
where and represent the frequency regions corresponding to in-band and out-of-band, respectively.
4.3 Simple approximate expression of power spectral density using cross-correlation coefficient
In Section 3.2, we have seen that in both the cases of smooth and sharp (including piecewise linear) nonlinearity, the third-order component (C1/ξ) is a dominant factor of distortion. Using this fact, we establish the simple approximate PSD expression based only on the input/output cross-correlation coefficient and the PSD of input signals.
4.4 Power spectrum expression for band-limited OFDM signals
where N is the number of subcarriers, Xl,k denotes a QAM (or PSK) symbol on the k th subcarrier of the l th OFDM symbol, and T is a symbol period.
where w(t) is a windowing function of length T s >T that controls the smoothness of the transition between the consecutive OFDM symbols. Strict characterization of the PSD requires the knowledge of w(t) as it also causes the spectral leakage. In this paper, however, we focus only on the out-of-band radiation caused by nonlinear distortion through examination of only one OFDM symbol for simplicity, and the effect of the spectral leakage caused by this windowing will not be considered.
That is, the effective SDR is at least 1.76 dB higher than the total (or design) SDR in the case of OFDM signals with near rectangular spectrum, which agrees with the observation given in .
5 Numerical and experimental results
5.1 Calculation of PSD and EVM for simulation and measurement
In this work, we calculate the periodogram by taking an ensemble average of the square of discrete Fourier transform of the generated and power amplified complex baseband OFDM signals. Upon evaluating the periodogram through simulation and measurement, the N-subcarrier OFDM signal is sampled only for one OFDM symbol period with J times oversampling, i.e., with J N-point FFT. In this manner, the effect of the spectral leakage associated with OFDM symbol transition is eliminated from our PSD calculation, and the results become consistent with our theoretical analysis.
This will be compared by the upper bound based on the total SDR, i.e., (45) with .
5.2 Calculation of cross-correlation coefficient for measurement
where corresponds to the IBO with the reference maximum envelope level denoted by A0.
5.3 Simulation setup
For the Monte-Carlo simulation, we generate 1,024,000 OFDM symbols with the number of subcarriers N=256, where each subcarrier is modulated by QPSK except for the center subcarrier that is set to be null. Since arbitrary PA characteristics can be generated for simulation, in addition to the soft envelope limiter and erf models described in Section 3.1, we consider the well-adopted Rapp and Saleh models described below as our reference of more practical PA models. Their complex gains are plotted in Figure 1. For both cases, the coefficients C n of (23) may not be expressed in a convenient analytical form, and thus they should be calculated numerically.
5.3.1 Rapp model
where the definition of Amax is the same as that of the erf model, and p is a smoothness factor that controls transition from linear to nonlinear region of PA. Note that there is no AM-PM effect in the Rapp model, that is, ϕ(r)=0. Throughout this work, p=3.0 will be adopted as a relatively good AM-AM example.
5.3.2 Saleh model
5.4 Experimental setup
5.5 Numerical and experimental results
For all the theoretical results on PSD, we refer to the results as ‘exact’ when it is calculated using (19) with the coefficients C n given by (23), where the summation is taken up to n=10 terms. On the other hand, the proposed simple approximate form (which will be referred to as ‘approximation’ in the results) is based on (47) where the cross-correlation coefficient is calculated from (17).
For the exact case, the required self-convolution of the spectrum is calculated using (51), whereas only (53) is used for the approximation.
5.5.1 Power spectral density
5.5.2 Error vector magnitude
We observe that both the simulation results and upper bounds well agree, which may justify the accuracy of our proposed analytical approach. It is interesting to observe that when the two curves are compared with the same IBO value, soft envelope limiter has higher EVM value when IBO is lower than 3 dB, even though it rapidly decreases as IBO increases compared to that of the erf model. We note that similar agreement behavior has been also observed for the Rapp and Saleh models.
Second, since the OFDM signals with high peak power occur less frequently than those with low peak power , the event that the OFDM signal is affected by nonlinear distortion becomes rare , especially in the high-SDR region where the IBO is set high.
Finally, the measured results are affected not only by the nonlinear distortion but also by the additive white Gaussian noise (AWGN) (due to the thermal noise) as well as the quantization noise, where the latter two factors are not taken into consideration in our theoretical calculation.
In this work, based on the exact mathematical model, we have first established the relationship between the cross-correlation coefficient of the input and output envelope of the nonlinearly amplified Gaussian signals (or total SDR) and its power spectral density. Based on this result, we have proposed a simple approximate expression for the PSD and EVM of the nonlinearly amplified OFDM signals that can be derived using the cross-correlation coefficient calculated only from the AM-AM and AM-PM characteristics of a PA.
Through computer simulation and actual measurement, the effectiveness and accuracy of our approach have been demonstrated. Even though this approach is theoretically valid only for the Gaussian signals with rectangular spectra, which can be approached by the OFDM with a large number of subcarriers, it is expected to offer an approximate solution for other linearly modulated signaling cases. Further investigation may be necessary to investigate the applicability of the proposed approach to other modulation formats.
- Liang CP, Jong JH, Stark WE, East JR: Nonlinear amplifier effects in communications systems. IEEE Trans. Microw. Theory Tech 1999, 47(8):1461-1466. 10.1109/22.780395View ArticleGoogle Scholar
- Ochiai H: An analysis of band-limited communication systems from amplifier efficiency and distortion perspective. IEEE Trans. Commun 2013, 61(4):1460-1472.MathSciNetView ArticleGoogle Scholar
- Bosch W, Gatti G: Measurement and simulation of memory effects in predistortion linearizers. IEEE Trans. Microw. Theory Tech 1989, 37(12):1885-1890. 10.1109/22.44098View ArticleGoogle Scholar
- Rapp C: Effects of HPA-nonlinearity on a 4-DPSK/OFDM-signal for a digital sound broadcasting system. In Proceedings of the Second European Conference on Satellite Communications. Liége, Belgium; 22–24 Oct 1991:1-5.Google Scholar
- Saleh AAM: Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers. IEEE Trans. Commun 1981, 29(11):1715-1720.View ArticleGoogle Scholar
- Yu Z, Baxley R, Zhou GT: EVM and achievable data rate analysis of clipped OFDM signals in visible light communication. EURASIP J. Wireless Commun. Netw 2012, 2012(1):321. 10.1186/1687-1499-2012-321View ArticleGoogle Scholar
- Gard KG, Gutierrez HM, Steer MB: Characterization of spectral regrowth in microwave amplifiers based on the nonlinear transformation of a complex Gaussian process. IEEE Trans. Microw. Theory Tech 1999, 47(7):1059-1069. 10.1109/22.775437View ArticleGoogle Scholar
- Ermolova NY: Spectral analysis of nonlinear amplifier based on the complex gain Taylor series expansion. IEEE Commun. Lett 2001, 5(12):465-467.View ArticleGoogle Scholar
- Zhou GT, Raich R: Spectral analysis of polynomial nonlinearity with applications to RF power amplifiers. EURASIP J. Appl. Signal Process 2004, 2004: 1831-1840. 10.1155/S1110865704312114View ArticleGoogle Scholar
- Zhou GT: Analysis of spectral regrowth of weakly nonlinear power amplifiers. IEEE Commun. Lett 2000, 4(11):357-359.View ArticleGoogle Scholar
- Banelli P, Cacopardi S: Theoretical analysis and performance of OFDM signals in nonlinear AWGN channels. IEEE Trans. Commun 2000, 48(3):430-441. 10.1109/26.837046View ArticleGoogle Scholar
- Kim HS, Daneshrad B: Power optimized PA clipping for MIMO-OFDM systems. IEEE Trans. Wireless Commun 2011, 10(9):2823-2828.View ArticleGoogle Scholar
- Ahmad I, Sulyman AI, Alsanie A, Alasmari AK, Alshebeili S: Spectral broadening effects of high-power amplifiers in MIMO-OFDM relaying channels. EURASIP J. Wireless Commun. Netw 2013, 2013(1):32. 10.1186/1687-1499-2013-32View ArticleGoogle Scholar
- Le NP, Safaei F, Tran LC: Transmit antenna subset selection for high-rate MIMO-OFDM systems in the presence of nonlinear power amplifiers. EURASIP J. Wireless Commun. Netw 2014, 2014(1):27. 10.1186/1687-1499-2014-27View ArticleGoogle Scholar
- Sabbaghian M, Sulyman A, Tarokh V: Capacity analysis for Gaussian channels with memoryless nonlinear hardware. In IEEE International Conference on Communications (ICC). Budapest, Hungary; 9–13 June 2013:3403-3407.Google Scholar
- Ochiai H, Imai H: On the distribution of the peak-to-average power ratio in OFDM signals. IEEE Trans. Commun 2001, 49(2):282-289. 10.1109/26.905885View ArticleMATHGoogle Scholar
- Wei S, Goeckel D, Kelly PA: Convergence of the complex envelope of bandlimited OFDM signals. IEEE Trans. Inform. Theory 2010, 56(10):4893-4904.MathSciNetView ArticleGoogle Scholar
- Gardner WA: Introduction to Random Processes: With Applications to Signals and Systems. Macmillan, New York; 1986.MATHGoogle Scholar
- Blachman NM: The output signals and noise from a nonlinearity with amplitude-dependent phase shift. IEEE Trans. Inform. Theory 1979, 25(1):77-79. 10.1109/TIT.1979.1055981View ArticleGoogle Scholar
- Yoo H, Guilloud F, Pyndiah R: Amplitude PDF analysis of OFDM signal using probabilistic PAPR reduction method. EURASIP J. Wireless Commun. Netw. 2011, 2011(1):983915.View ArticleGoogle Scholar
- Olver FW, Lozier DW, Boisvert RF, Clark CW: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge; 2010.MATHGoogle Scholar
- Rowe HE: Memoryless nonlinearities with Gaussian inputs: elementary results. Bell Syst. Tech. J 1982, 61(7):1519-1525. 10.1002/j.1538-7305.1982.tb04356.xView ArticleGoogle Scholar
- Raich R, Qian H, Zhou GT: Optimization of SNDR for amplitude-limited nonlinearities. IEEE Trans. Commun 2005, 53(11):1964-1972. 10.1109/TCOMM.2005.857141MathSciNetView ArticleGoogle Scholar
- Galejs J: Signal-to-noise ratios in smooth limiters. IRE Trans. Inform. Theory 1959, 5(2):79-85. 10.1109/TIT.1959.1057490MathSciNetView ArticleGoogle Scholar
- Ochiai H, Imai H: Performance analysis of deliberately clipped OFDM signals. IEEE Trans. Commun 2002, 50(1):89-101. 10.1109/26.975762View ArticleGoogle Scholar
- Dardari D: Joint clip and quantization effects characterization in OFDM receivers. IEEE Trans. Circuits Syst. I, Reg. Papers 2006, 53(8):1741-1748.View ArticleGoogle Scholar
- Morgan DR: Finite limiting effects for a band-limited Gaussian random process with applications to A/D conversion. IEEE Trans. Acoust. Speech Signal Process 1988, 36(7):1011-1016. 10.1109/29.1624View ArticleMATHGoogle Scholar
- Cann AJ: Nonlinearity model with variable knee sharpness. IEEE Trans. Aerosp. Electron. Syst 1980, AES-16(6):874-877.View ArticleGoogle Scholar
- Cann AJ: Improved nonlinearity model with variable knee sharpness. IEEE Trans. Aerosp. Electron. Syst 2012, 48(4):3637-3646.View ArticleGoogle Scholar
- Bahai ARS, Singh M, Goldsmith AJ, Saltzberg BR: A new approach for evaluating clipping distortion in multicarrier systems. IEEE J. Select. Areas Commun 2002, 20(5):1037-1046. 10.1109/JSAC.2002.1007384View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.