Peak-to-average power reduction of OFDM signals ... - Semantic Scholar
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, IN PRESS, JANUARY 2010
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Peak-to-average power reduction of OFDM signals by convex optimization: experimental validation and performance optimization Charles Nader, Student Member, IEEE, Peter Händel, Senior Member, IEEE, Niclas Björsell, Member, IEEE
Abstract—We evaluated the application of convex optimization to peak-to-average power reduction on an orthogonal frequency division multiplexing (OFDM) 802.11a signal. A radio frequency power amplifier was excited with an OFDM-signal, and the peak-to-average reduced counterpart and its performance figure of merits were measured and compared. A state-of-art radio frequency test system with high accuracy was used for this purpose. Improvements due to optimization in output power and power added efficiency and the influence of the power distribution in the excitation signal on power amplifier performance were investigated. Improvements of 6dB in output power and 6.5% in power added efficiency were achieved on average near the operating region. The effect of preserving power-free guard subcarriers was introduced in the optimization algorithm and investigated regarding adjacent channel interference. An improvement of 9dB from that aspect was observed using half of the power-free subcarriers, which reveals the importance of a guard interval. Index Terms—Communication system performance, power amplifiers, power added efficiency, spectrum mask, convex optimization, dirty radio.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) is a widely used modulation scheme because of its high bandwidth efficiency and robustness against frequency fading due to multipath propagation [1], [2]. The power amplifier is a key component in a wireless communication chain as it holds the highest power level in the system. Its power added efficiency (PAE) directly influences the power consumption of the wireless system [3]. Its input-output signal nonlinearity is important for in-band error and out-of-band interference [3]. A major drawback of OFDM is the generally high peak to average ratio (PAR) of the radio frequency (RF) signal entering the power amplifier, which causes early clipping of the signal due to amplifier saturation and results in nonlinear distortions presented in the frequency domain on the shape of unwanted intermodulation products, spectral regrowth and harmonics [3]. Such nonlinear distortions cause spectral interference to adjacent channels and brake the spectral mask standard for emission [1]. Due that, the input power of the power amplifier has to be reduced; that is, a large number of dBs have to be backed-off to keep the amplifier in linear operation. However, such a back-off drastically reduces the PAE of the amplifier C. Nader and N. Björsell are with the University of Gävle, Center for RF Measurement Technology, Gävle, SE-80176, Sweden. C. Nader and P. Händel are with the Signal Processing Lab, Royal Institute of Technology, Stockholm, Sweden. Corresponding author: [email protected].
because a large amount of power (i.e., heat) must be dissipated [4]. Several methods have been proposed in the literature to reduce the PAR of OFDM signals prior to the conversion to RF, including clipping, dynamic PAR/bias adapting, or redistributing the nonlinear energy on the free subcarriers [5][8]. Recently in [9], PAR minimization was formulated as a convex optimization problem. The power spectral density of the signal to be transmitted was reshaped by minimizing the time domain peak power, subject to some constraints on the error vector magnitude (EVM) and the use of power-free guard subcarriers. By applying the fast Fourier transform (FFT) and its inverse on the OFDM signal, a customized interior point method (IPM) that finds the near-to global minimum PAR by a fast and reliable algorithm was developed [10]. This PAR optimization approach was further developed in [11], [12] by adding a spectral mask constraint and minimizing the EVM while keeping the PAR below a minimum threshold. In this work, the method in [9] is extended to reduce the adjacent channel interference that arises when the guard intervals are excited. Further, there is significant theoretical interest in applying convex optimization to obtain PAR-reduction in OFDM communication systems, although the literature, and in particular [9], [11], [12], present no in-depth experimental validation of the impact of the PAR reduction of the baseband signal on the effect of the PAE of the power amplifier. Such experimental verification is of utmost importance for improving PAE and reducing the overall energy consumption of wireless communication systems. Another important aspect to investigate is the effect of exciting the free-power guard subcarriers on the channel leakage and its induced adjacent channel interference. In this paper, we experimentally evaluate the PAR reduction method introduced in [9] with respect to how the PAR reduced signal influences the power aspects of the power amplifier, which includes the PAE and adjacent channel interference. An extension of the method [9] is developed which preserves a fraction of the power-free subcarriers as a guard interval. The impact of the extended method on the power amplifier performance and reduction of adjacent channel interference is evaluated. The paper is organized as follows. An OFDM signal was generated, and PAR was optimized, as briefly reviewed in Sec. II. The PAR-optimized signal was then used to excite a commercial power amplifier, using a state-of-art measurement setup as described in Sec. III. In Sec. IV, a study of the
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power amplifier figure of merits is presented for both methods, [9] and its extended version, and results are compared to the corresponding measures of the reference signal. Finally, conclusions are drawn in Sec. V. II. OFDM PAR
REDUCTION BY CONVEX OPTIMIZATION
In this section, a review of OFDM PAR reduction using convex optimization, as formulated in [9], is briefly reviewed. To combat the demonstrated effects of exciting guard intervals, such as adjacent channel distortion, an extended method is proposed. The Section ends with some remarks on algorithmic details. A. PAR reduction by convex optimization According to the standards, WLAN 802.11a used an OFDM signal comprising 64 subcarriers, distributed into 48 data, 4 pilot, and 12 free subcarriers. The 52 modulated subcarriers used binary or quadrature phase shift keying (BPSK/QPSK), 16-quadrature amplitude modulation (16-QAM), or 64-QAM. A baseband OFDM signal was generated by dividing the information data into multiple data streams. Each data stream was passed to a subcarrier for modulation. The modulated data streams (symbol streams) were sent in parallel on the orthogonal subcarriers. The frequency constellation was then time domain transformed through IFFT. Cyclic prefix (guard interval) as well as windowing (Hamming) was applied for time-spreading handling and intersymbol interference elimination (side lobes suppression). The time domain symbols were then serially packed and sent for in-phase and quadrature (IQ) modulation [1]. Consider c = (c1 , . . . , cn )T as a transmitted OFDM frequency constellation, which is a complex-valued vector of length n. Further, let x be the corresponding time domain signal obtained by a ℓ-times oversampling, that is x = IFFTℓ [c]
(1)
where IFFTℓ [·] denotes the inverse discrete Fourier transform of the zero-padded vector c, resulting in the length-nℓ vector x. To learn more about the role of oversampling in this context, see the references in [9]. Now, the PAR can be defined as PAR =
n2 maxi (|xi |2 ) ¯H c ¯ βc
(2)
where xi denotes the entries in x = (x1 , . . . , xnℓ )T , β is a real-valued FFT scaling used to lower bound PAR to 1 and ¯ only contains the contribution by the data carriers, that is c ¯ = S c, where S is a diagonal carrier selection matrix with c diagonal elements Si,i = 1 for those d carriers i1 , . . . , id that contain data, and zero otherwise – in our context d = 52 and n = 64. For a given use of the subcarriers, minimizing PAR is equivalent to minimizing the peak-power p = maxi (|xi |2 ), where for all i = 1, . . . , nℓ it holds that |xi |2 ≤ p. The minimization of PAR is obtained both by i) adding power to the free carriers, which are given by (I − S)c, and ii) distorting the data/pilot carriers S c. The introduced distortion of the transmitted constellation has to be bounded, given as a
2
constraints imposed on the EVM. Let c0 = (c0,1 , . . . , c0,n )T be a reference constellation, then EVM is defined as [1] v u id u1 ∑ u √ |ci − c0,i |2 ud t i=i1 1 ||S(c − c0 )||2 EVM = = (3) P0 d P0 where i1 , . . . , id denotes the location of data/pilot subcarriers, that is determined by the non-zero diagonal elements of S. In the second equality || · || denotes the Euclidian vector norm. Note that c and c0 are scaled to the same average power for evaluation, that is ||c||2 = ||c0 ||2 [9]. The scalar P0 is the average power of the modulation scheme used. A convex formulation of the problem of minimizing the PAR in (2) of an OFDM baseband signal x in (1) by adding power at the free carriers and distorting the transmitted constellation c away from the ideal constellation c0 , subject to a constraint imposed on the EVM (3) is [9] minimize peak power subject to
p = max(|xi |2 ) i
||S(c − c0 )|| ≤ ϵ
(4)
2 ℜ[cH 0 S (c − c0 )] ≥ −ϵ /2
The first constraint in (4) is a bound on the maximum allowed EVM in (3), where ϵ is a real-valued positive parameter proportional to the allowed EVM and given by √ ϵ = EVMmax d P0 , where EVMmax is the maximum allowed EVM for a given bit error rate. The second constraint in (4) is a relaxed constraint on the average transmitted data power ||S c||2 , that is a relaxed constraint corresponding to ||S c||2 ≥ ||S c0 ||2 ; see [9] for details.
B. Channel leakage and an extended method Radio frequency receivers-transmitters (Rx-Tx) are an enhanced type of equipments with performance improving drastically with time and technology. Digital processing has been introduced as a tool to achieve perfection and reduce equipment errors caused by the non-ideality of the analog counterparts [13]. Such errors include IQ-imbalance, analog to digital converter impairments, non-ideal filters and frequency offset, which affect the spectral occupancy characteristics and potentially generate interference [14], [15]. Exciting the free subcarriers of the guard intervals in the minimization process to redistribute the channel power spectrum and reduce PAR has raised questions regarding its applicability because adjacent channel interference can popup as a problematic drawback. Using RF channels with equal bandwidth and spacing put strong requirements on the Rx-Tx RF equipments if a 100% bandwidth is excited, without any spectrum guard margin. To combat the aspect of leakage, the method introduced in [9] is extended by preserving a fraction of the power-free subcarriers as guards. The modification is achieved by adjusting the matrix I to have zeros on the diagonal elements relative to the preserved subcarriers.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, IN PRESS, JANUARY 2010
SMU 200A (R&S) DAC
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the power amplifier is replaced by a connector. EVM of the transmitted baseband frequency constellation and the received counterpart was calculated with respect to input power Pin and carrier frequency. An average system error of -45dB was found, showing good performance regarding in-band error. B. Device under test
PC Matlab
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The measurement setup.
A class AB LDMOS high power (47dBm) amplifier was used for the validation process. It has the capability to handle high PAR up to 15dB. WLAN 802.11a uses a bandwidth of 20MHz. In order to ensure inband flatness, the power amplifier was operated at 2GHz with a gain variation of 0.4dB over an 80MHz bandwidth. Such characteristics simulate the behavior of a typical WLAN power amplifier.
C. Algorithmic details The convex optimization problem (4), and its extension, can be solved by standard methods, yielding a global optimum p∗ , c∗ , x∗ . The reader is referred to [9] for details on general solvers, as well as specific solvers for the problem at hand. In the experimental verifications presented here, the logarithmbarrier-IPM algorithm presented in [9] is employed. The algorithm starts with a strictly feasible point (c, p) and finds a search direction (v, vp ) and a step size α that update the point with a factor αv and αvp , respectively. The updating procedure is designed to respect the feasibility condition of the point and reduce the barrier function value [10]. The procedure is iterated until a global optimum point is reached, or almost reached, which solves the PAR minimization problem [9]. III. M EASUREMENT SETUP AND DEVICE UNDER TEST Two major characteristics of RF measurement systems for power amplifier testing are accuracy and the ability to track fast variations in the signal envelope. Because the aim of this study is to validate the improvements in power amplifier performance, a state-of-art measurement system was needed. A. Measurement set-up The test-setup presented in Fig. 1 was mainly based on an R&S SMU200A vector signal generator, an Anritsu MS2692A signal analyzer, an Agilent 54610B oscilloscope, an Agilent N2783A high bandwidth hall sensor current probe, an Ericsson LDMOS highly linear driver amplifier, an Agilent E3631A controllable power supply, and a personal computer (PC). The instruments were connected to the PC via LAN or GPIB interface. The output power from the amplifier is measured by the MS2692A, which is accurately calibrated in amplitude and phase over the measured bandwidth of interest to obtain ±0.3dB power accuracy, even for modulated time variant signals. To accurately monitor the drain current vector and obtain accurate PAE readings, a high bandwidth hall sensor current probe was used. Measuring the current envelope through an oscilloscope allowed an envelope-tracking dynamic power consumption up to 100MHz. Additional details on the testbed can be found in [16], [17]. To study the performance of the test-setup, an evaluation of the EVM as a function of the input power is realized where
IV. R ESULTS AND E VALUATION The reference WLAN OFDM 20MHz signal was generated based on 802.11a standards. It had 64 subcarriers with 128 OFDM symbols, a cyclic prefix of 1/4, an oversampling rate of 4 and 14dB PAR after being Hamming windowed. The PARoptimized counterpart, based on full use of the guard subcarriers, reached a PAR of 9.5dB after three Newton iterations in the employed algorithm. Optimizing OFDM signals allocates power in the free subcarriers that reside at the channel sides. Such allocation extends the signal bandwidth from 16.6MHz for the reference signal to 20.0MHz for the optimized one. To combat this effect, the reference OFDM signal is optimized based on the method in Sec. II-C, where half of the power-free subcarriers are preserved as a guard interval, which results in an effective bandwidth of 18.0MHz and a PAR of 9.75dB. This guard margin is sufficient for reducing channel leakage, but requires increasing the number of Newton iterations because two additional iterations were needed to reach the optimal solution in the algorithm. A. Power added efficiency The main goals of reducing PAR are extending the input power level at the 1 dB compression point of the amplifier, reducing the back-off margin, and allowing an efficient use of the available power. Fig. 2 shows the PAE of the amplifier as function of Pin for both signals, reference and PARoptimized with six reserved guard subcarriers, where the 1dB compression points have been specified. As shown in Fig. 2, the 1dB compression point of the amplifier was extended by 1.5dB which improved the PAE by 4.2% near the compression region. The power amplifier gain was found to have similar shaping with a 1.5dB extended compression region. Considering the operating region of the amplifier in a real application, which is the compression region backed-off by the respective PAR, an improvement of 6.5% in PAE can be seen between the reference and optimized signal. In fact, reducing the PAR by 4.5dB and extending the compression region by 1.5dB leads to a total output power improvement of 6dB. Such improvement (that is, 6.5%) varies with the amplifier technology and design. Measurements based on PAR-optimized signal without guard subcarriers resulted in similar performance regarding
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PAE and output power compared to that of a PAR-optimized signal with guard subcarriers.
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Fig. 2. Power added efficiency of the device under test versus input power for the reference signal (dashed line) and PAR-optimized signal with 6 guard subcarriers (solid line).
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Adjusting the frequency constellation power might raise questions regarding in-band errors in the system as well as out-of-band errors due to spectrum regrowth. An evaluation of the output signal EVM versus input power, before and after optimization, is presented in Fig. 3. Despite the 12.6dB EVM difference between the signals in the backed-off region, the EVM of the optimized signal (with guard subcarriers) follows the standard limit value (-19dB for the used signal [1]) with a margin error of -0.3dB. Similar results were obtained when exciting with the PAR-optimized signal without guard intervals. Such behavior is expected as the EVM constraint in the optimization algorithm was set to the standard limit in order to study the maximum improvements in PAE and ACPR. One should mention that the state of the art coding/decoding techniques are successful in correcting for in-band errors as long as the standards limits for EVM are fulfilled. By that, the out-of-band emissions are the most problems tackled nowadays as their effects (intermodulation products and spectrum regrowth) interfere with the adjacent channels and violate the regulations for spectral emission. In summary, despite the changes made to the constellation diagram after optimization, the EVM achieved at the output of the power amplifier in the operation region (backed-off) is still sufficient for allowing decoding algorithms to correct caused errors and restore original information. It complies with the IEEE 802.11a error standard [1] in that region of interest.
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C. Spectral mask and out-of-band errors Adding power to the sideband free subcarriers increases the bandwidth of the optimized signal. Fig. 4 presents the power spectrums of both reference and optimized signals without guard interval at back-off region, compared to the 802.11a spectral mask standard for emission [1]. As shown, a 16.6MHz bandwidth is found for the reference signal, while exciting the free subcarriers leads to a 20.0MHz bandwidth. Such an increase in the spectrum breaks the standard constraint mask for emission on the lower side of the channel by 30KHz and raises questions about the applicability of such a method. A worse behavior is found at compression region where the spectral mask is violated at both the upper and lower channels. Violating the spectral mask for emission by 30KHz will generate strong interference to neighboring channels which marks the importance of preserving a fraction of the powerfree subcarriers as guard interval. Fig. 5 presents the power spectrum of both signals, reference and optimized with 6 guard subcarriers, at back-off region. It shows complete agreement with the IEEE 802.11a standards spectral mask emission. Regarding the out-of-band errors, measurements of the adjacent channel power ratios (ACPRs) of the three signals at 20.0MHz channel spacing and bandwidth, at different input power levels, as presented in Fig. 6, show a large variation between the reference signal and the guard-free optimized one, which is clearly revealed in the lower channel side. Such an
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Fig. 3. Error vector magnitude of the device under test versus input power for the reference signal (dashed line) and PAR-optimized signal with 6 guard subcarriers (pointed line).
increase in the ACPR is due to the 30kHz leakage/interference from the main channel, which biases the value of ACPR. Comparing the above result with the ACPR of the optimized signal with a guard interval shows an improvement of 9dB in the lower channel near the back-off region with respect to not using guard margin, while a 1dB improvement was found in the transition region with respect to the reference signal. Considering the upper adjacent channel, an average ACPR improvement of 1.5dB in the transition region between the backed-off and compression regions was found between the guard-free optimized signal and the reference one; while preserving a guard interval in the optimization process showed an ACPR improvement of 2dB in the back-off region compared to not using a guard margin.
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Fig. 6. Comparison of adjacent channel power ratio for both optimized signals, with and without power-free guard subcarriers, and the reference signal.
reduce the adjacent channel interference and the associated errors, but more importantly, this improvement is advantageous compared to the clipping-based algorithms that usually generate undesired regrowth in the output spectrum of the power amplifier and require sophisticated methods to filter the regrowth without regenerating the peaks.
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Fig. 5. Power spectrum of both the reference signal (pointed line) and the PAR-optimized signal with 6 guard subcarriers (dashed line) at back-off region, compared to 802.11a spectral mask standard for emission (solid line).
The achieved improvements between both optimized signals points the necessity of having a frequency guard interval in the signal for adjacent channel interference reduction. From an application point-of-view, the method introduced in [9], and its derivations, need to consider this aspect. Exciting part of the power-free guard subcarriers is sufficient to achieve similar-tobetter power performance compared to that with the full use of the guard interval, with just two extra Newton iterations in the optimization algorithm. In summary, optimizing the signal while preserving a fraction of the power-free subcarriers as a guard interval improves the ACPR by an average of 2dB. This is caused by the absence of the clipped high peaks that usually cause early bird spectrum regrowth. Such small improvements can still
D. Amplifier saturation Reducing the PAR should generally increase the average power at the output of the amplifier by a couple of dBs, which will lead to a compression point at a higher input power level, because fewer peaks excite the amplifier’s nonlinearities. However, contrary to what was expected, an average increase of 1.5dB was found near compression, and requires further investigation to explain this behavior. A study of the complementary cumulative density function (CCDF) of the peaks distribution in the measured signals, reference and optimized with guard interval, would justify such a result. Fig. 7 presents the CCDF of all signal peaks normalized to the signal average, for both measured signals while Fig. 8 shows the CCDF of the normalized envelope signals with respect to their highest peak value, respectively. As shown in Fig. 7, a 4.5dB PAR reduction was observed after optimization. However, by analyzing Fig. 8, one realizes that such reduction was achieved at the cost of increasing the lower peaks density distribution, which in turn limited the improvement near saturation. In fact, despite the reduction of the high peaks, the optimization algorithm allowed the “smaller” peaks to increase. Ultimately the total energy, due to nonlinearity, retained a comparable value to the non-optimized case, sufficient to push the power amplifier into compression. Such aspect in power improvement near saturation represents the tradeoff that must be considered when choosing between advanced optimization method like convex optimization and the low-complexity clipping based methods. In fact, clipping the signal only reduce the dominant peaks, resulting in
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operates, it had a 6.5% PAE improvement with a gain of 6dB in the output power. One reason for such limited gain improvement near compression is the increase in lower peak density distribution in the excitation signal. Spectral emission was considered to be a drawback in the method because power leakage was seen by the high ACPR. The main cause of the leakage is the absence of guard subcarriers. An extended version of the method in [9] was implemented in which half of the guard subcarriers were used to prevent such interference. The extended method showed the necessity of preserving part of the guard interval with a low iterative cost in optimization. An improvement up to 9dB was found in ACPR for the lower channel side, while the power performance maintained its merits values. Even though PAR reduction by advanced methods, such as convex optimization, costs more in terms of additional digital signal processing, it is commonly considered to be a worthwhile technology because the digital processing is “free”; according to Moore’s law, it will be cheaper and cheaper with time. We have experimentally shown an extra 6dB in output power and 6.5% in power added efficiency due to digital processing of the transmitted signal, without any additional requirements from the hardware. This result is believed to be of significance in a world “where every dB is worth a Billion” [18].
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This work was supported by Ericsson AB, Freescale Semiconductor Nordic AB, Infineon Technologies Nordic AB, Knowledge Foundation, NOTE AB, Rohde&Schwarz AB and Syntronic AB.
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an extra power margin for saturation to be reached. However, due to the distortion created in the excitation signal, higher side band levels are expected to arise which cause higher outof-band errors. V. C ONCLUSION Optimizing the PAR of OFDM signals based on convex optimization algorithms was performed, and before/after figure of merits of a contemporary RF power amplifier were evaluated. The employed PAR optimization algorithm based on [9] resulted in limited improvement in power performance near the saturation region of the power amplifier: an extra 1.5 dB in output power and 4.2% in power added efficiency. However, in the backed-off region where the amplifier normally will
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[12] Q. Liu, R.J. Baxley, X. Ma and G.T. Zhou, “Error vector magnitude optimization for OFDM systems with a deterministic peak-to-average power ratio constraint,” Information Sciences and Systems, 42nd Annual Conference, CISS 2008, pp. 101-104, Mar 2008. [13] G. Fettweis, M. Lohning, D. Petrovic, M. Windisch, P. Zillmann, and W. Rave, “Dirty RF: a new paradigm,” International Journal of Wireless Information Networks, vol. 14, no. 2, pp. 133-148, June 2007. [14] A. Behzad, Wireless LAN Radios, IEEE Press on Digital and Mobile Communication, 2007. [15] T.C.W. Schenk, and E.R. Fledderus, “RF impairments in high-rate wireless systems - understanding the impact of TX/RX-asymmetry,” in 3rd International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, pp. 117-122, Mar 2008. [16] C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keskitalo, and H. de la Rosa, “Automated multidimensional characterization of power amplifier for design and production,” in International Instrumentation and Measurement Technology Conference Proceedings, I 2 M T C 2009, pp. 144-148, May 2009. [17] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidth of an RF power amplifier test bed,” IEEE Transactions on Instrumentation and Measurement, vol. 56, pp. 1488-1494, 2007. [18] C. Beckman, L. Eklund, B. Karlsson, B. Lindmark, D. Ribbenfjärd, and P. Wirdemark, “Verifying 3G license requirements when every dB is worth a billion,” in First European Conference on Antennas and Propagation, EuCAP 2006, France, Nov 2006.
Charles Nader (S’08) received an M.E. in electrical engineering from the Lebanese University ULFG2, Lebanon, in 2005 and an M.Sc in EE/Telecommunication from the University of Gävle, Gävle, Sweden in 2006. In 2007, he was a consultant for Multilane inc. in Lebanon, working with microwave devices and signal integrity. In 2008, he joined the University of Gävle, center for RF measurement technology, and The Royal Institute of Technology, signal processing lab, where he is currently working on his Ph.D. with a research focus on signal processing applied to radio frequency measurement systems for microwave device characterization and testing.
Peter Händel (S’88-M’94-SM’98) received his Ph.D. at Uppsala University in 1993. From 1987 to 1993, he was at Uppsala University. During 19931997, he was with Ericsson AB, Kista, Sweden. During 1996-1997, he was also with Tampere University of Technology, Finland. Since 1997, he has been with the Royal Institute of Technology, Stockholm, Sweden, where he is currently a Professor of signal processing. From 2000 to 2006, he was with the Swedish Defense Research Agency. He is currently guest professor at the University of Gävle, Sweden. He has served as an Editorial Board Member of the EURASIP Journal of Advances in Signal Processing, and an Editorial Advisory Board Member of Recent Patents on Electrical Engineering. He is a member of the Editorial Board of Hindawi’s Research Letters in Signal Processing, and Journal of Electrical and Computer Engineering. He has served as Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING.
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Niclas Björsell (S’02-M’08) was born in Falun, Sweden, in 1964. He received his B.Sc. in Electrical Engineering and his Lic. Ph. in Automatic control from Uppsala University, Sweden in 1994 and 1998, respectively. His Ph. D. in Telecommunication was received at the Royal Institute of Technology, Stockholm, Sweden, in 2007. He has several years of experience from research and development projects that fostered collaborations between industry and the academy. He currently holds positions in the academy as well as in industry, and has worked as project manager for some of the R&D projects. Since 2006 he has served as the head of Division of Electronics at the Department of Technology and Built Environment, University of Gävle, Sweden. He has published more than 20 papers in journals and conferences, and his research interests include radio frequency measurement technology and analog-to-digital conversion. Dr. Björsell is a voting member of the IEEE, Instrumentation and Measurement, TC-10.
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Peak-to-average power reduction of OFDM signals by convex optimization: experimental validation and performance optimization Charles Nader, Student Member, IEEE, Peter Händel, Senior Member, IEEE, Niclas Björsell, Member, IEEE
Abstract—We evaluated the application of convex optimization to peak-to-average power reduction on an orthogonal frequency division multiplexing (OFDM) 802.11a signal. A radio frequency power amplifier was excited with an OFDM-signal, and the peak-to-average reduced counterpart and its performance figure of merits were measured and compared. A state-of-art radio frequency test system with high accuracy was used for this purpose. Improvements due to optimization in output power and power added efficiency and the influence of the power distribution in the excitation signal on power amplifier performance were investigated. Improvements of 6dB in output power and 6.5% in power added efficiency were achieved on average near the operating region. The effect of preserving power-free guard subcarriers was introduced in the optimization algorithm and investigated regarding adjacent channel interference. An improvement of 9dB from that aspect was observed using half of the power-free subcarriers, which reveals the importance of a guard interval. Index Terms—Communication system performance, power amplifiers, power added efficiency, spectrum mask, convex optimization, dirty radio.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) is a widely used modulation scheme because of its high bandwidth efficiency and robustness against frequency fading due to multipath propagation [1], [2]. The power amplifier is a key component in a wireless communication chain as it holds the highest power level in the system. Its power added efficiency (PAE) directly influences the power consumption of the wireless system [3]. Its input-output signal nonlinearity is important for in-band error and out-of-band interference [3]. A major drawback of OFDM is the generally high peak to average ratio (PAR) of the radio frequency (RF) signal entering the power amplifier, which causes early clipping of the signal due to amplifier saturation and results in nonlinear distortions presented in the frequency domain on the shape of unwanted intermodulation products, spectral regrowth and harmonics [3]. Such nonlinear distortions cause spectral interference to adjacent channels and brake the spectral mask standard for emission [1]. Due that, the input power of the power amplifier has to be reduced; that is, a large number of dBs have to be backed-off to keep the amplifier in linear operation. However, such a back-off drastically reduces the PAE of the amplifier C. Nader and N. Björsell are with the University of Gävle, Center for RF Measurement Technology, Gävle, SE-80176, Sweden. C. Nader and P. Händel are with the Signal Processing Lab, Royal Institute of Technology, Stockholm, Sweden. Corresponding author: [email protected].
because a large amount of power (i.e., heat) must be dissipated [4]. Several methods have been proposed in the literature to reduce the PAR of OFDM signals prior to the conversion to RF, including clipping, dynamic PAR/bias adapting, or redistributing the nonlinear energy on the free subcarriers [5][8]. Recently in [9], PAR minimization was formulated as a convex optimization problem. The power spectral density of the signal to be transmitted was reshaped by minimizing the time domain peak power, subject to some constraints on the error vector magnitude (EVM) and the use of power-free guard subcarriers. By applying the fast Fourier transform (FFT) and its inverse on the OFDM signal, a customized interior point method (IPM) that finds the near-to global minimum PAR by a fast and reliable algorithm was developed [10]. This PAR optimization approach was further developed in [11], [12] by adding a spectral mask constraint and minimizing the EVM while keeping the PAR below a minimum threshold. In this work, the method in [9] is extended to reduce the adjacent channel interference that arises when the guard intervals are excited. Further, there is significant theoretical interest in applying convex optimization to obtain PAR-reduction in OFDM communication systems, although the literature, and in particular [9], [11], [12], present no in-depth experimental validation of the impact of the PAR reduction of the baseband signal on the effect of the PAE of the power amplifier. Such experimental verification is of utmost importance for improving PAE and reducing the overall energy consumption of wireless communication systems. Another important aspect to investigate is the effect of exciting the free-power guard subcarriers on the channel leakage and its induced adjacent channel interference. In this paper, we experimentally evaluate the PAR reduction method introduced in [9] with respect to how the PAR reduced signal influences the power aspects of the power amplifier, which includes the PAE and adjacent channel interference. An extension of the method [9] is developed which preserves a fraction of the power-free subcarriers as a guard interval. The impact of the extended method on the power amplifier performance and reduction of adjacent channel interference is evaluated. The paper is organized as follows. An OFDM signal was generated, and PAR was optimized, as briefly reviewed in Sec. II. The PAR-optimized signal was then used to excite a commercial power amplifier, using a state-of-art measurement setup as described in Sec. III. In Sec. IV, a study of the
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power amplifier figure of merits is presented for both methods, [9] and its extended version, and results are compared to the corresponding measures of the reference signal. Finally, conclusions are drawn in Sec. V. II. OFDM PAR
REDUCTION BY CONVEX OPTIMIZATION
In this section, a review of OFDM PAR reduction using convex optimization, as formulated in [9], is briefly reviewed. To combat the demonstrated effects of exciting guard intervals, such as adjacent channel distortion, an extended method is proposed. The Section ends with some remarks on algorithmic details. A. PAR reduction by convex optimization According to the standards, WLAN 802.11a used an OFDM signal comprising 64 subcarriers, distributed into 48 data, 4 pilot, and 12 free subcarriers. The 52 modulated subcarriers used binary or quadrature phase shift keying (BPSK/QPSK), 16-quadrature amplitude modulation (16-QAM), or 64-QAM. A baseband OFDM signal was generated by dividing the information data into multiple data streams. Each data stream was passed to a subcarrier for modulation. The modulated data streams (symbol streams) were sent in parallel on the orthogonal subcarriers. The frequency constellation was then time domain transformed through IFFT. Cyclic prefix (guard interval) as well as windowing (Hamming) was applied for time-spreading handling and intersymbol interference elimination (side lobes suppression). The time domain symbols were then serially packed and sent for in-phase and quadrature (IQ) modulation [1]. Consider c = (c1 , . . . , cn )T as a transmitted OFDM frequency constellation, which is a complex-valued vector of length n. Further, let x be the corresponding time domain signal obtained by a ℓ-times oversampling, that is x = IFFTℓ [c]
(1)
where IFFTℓ [·] denotes the inverse discrete Fourier transform of the zero-padded vector c, resulting in the length-nℓ vector x. To learn more about the role of oversampling in this context, see the references in [9]. Now, the PAR can be defined as PAR =
n2 maxi (|xi |2 ) ¯H c ¯ βc
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where xi denotes the entries in x = (x1 , . . . , xnℓ )T , β is a real-valued FFT scaling used to lower bound PAR to 1 and ¯ only contains the contribution by the data carriers, that is c ¯ = S c, where S is a diagonal carrier selection matrix with c diagonal elements Si,i = 1 for those d carriers i1 , . . . , id that contain data, and zero otherwise – in our context d = 52 and n = 64. For a given use of the subcarriers, minimizing PAR is equivalent to minimizing the peak-power p = maxi (|xi |2 ), where for all i = 1, . . . , nℓ it holds that |xi |2 ≤ p. The minimization of PAR is obtained both by i) adding power to the free carriers, which are given by (I − S)c, and ii) distorting the data/pilot carriers S c. The introduced distortion of the transmitted constellation has to be bounded, given as a
2
constraints imposed on the EVM. Let c0 = (c0,1 , . . . , c0,n )T be a reference constellation, then EVM is defined as [1] v u id u1 ∑ u √ |ci − c0,i |2 ud t i=i1 1 ||S(c − c0 )||2 EVM = = (3) P0 d P0 where i1 , . . . , id denotes the location of data/pilot subcarriers, that is determined by the non-zero diagonal elements of S. In the second equality || · || denotes the Euclidian vector norm. Note that c and c0 are scaled to the same average power for evaluation, that is ||c||2 = ||c0 ||2 [9]. The scalar P0 is the average power of the modulation scheme used. A convex formulation of the problem of minimizing the PAR in (2) of an OFDM baseband signal x in (1) by adding power at the free carriers and distorting the transmitted constellation c away from the ideal constellation c0 , subject to a constraint imposed on the EVM (3) is [9] minimize peak power subject to
p = max(|xi |2 ) i
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2 ℜ[cH 0 S (c − c0 )] ≥ −ϵ /2
The first constraint in (4) is a bound on the maximum allowed EVM in (3), where ϵ is a real-valued positive parameter proportional to the allowed EVM and given by √ ϵ = EVMmax d P0 , where EVMmax is the maximum allowed EVM for a given bit error rate. The second constraint in (4) is a relaxed constraint on the average transmitted data power ||S c||2 , that is a relaxed constraint corresponding to ||S c||2 ≥ ||S c0 ||2 ; see [9] for details.
B. Channel leakage and an extended method Radio frequency receivers-transmitters (Rx-Tx) are an enhanced type of equipments with performance improving drastically with time and technology. Digital processing has been introduced as a tool to achieve perfection and reduce equipment errors caused by the non-ideality of the analog counterparts [13]. Such errors include IQ-imbalance, analog to digital converter impairments, non-ideal filters and frequency offset, which affect the spectral occupancy characteristics and potentially generate interference [14], [15]. Exciting the free subcarriers of the guard intervals in the minimization process to redistribute the channel power spectrum and reduce PAR has raised questions regarding its applicability because adjacent channel interference can popup as a problematic drawback. Using RF channels with equal bandwidth and spacing put strong requirements on the Rx-Tx RF equipments if a 100% bandwidth is excited, without any spectrum guard margin. To combat the aspect of leakage, the method introduced in [9] is extended by preserving a fraction of the power-free subcarriers as guards. The modification is achieved by adjusting the matrix I to have zeros on the diagonal elements relative to the preserved subcarriers.
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SMU 200A (R&S) DAC
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the power amplifier is replaced by a connector. EVM of the transmitted baseband frequency constellation and the received counterpart was calculated with respect to input power Pin and carrier frequency. An average system error of -45dB was found, showing good performance regarding in-band error. B. Device under test
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A class AB LDMOS high power (47dBm) amplifier was used for the validation process. It has the capability to handle high PAR up to 15dB. WLAN 802.11a uses a bandwidth of 20MHz. In order to ensure inband flatness, the power amplifier was operated at 2GHz with a gain variation of 0.4dB over an 80MHz bandwidth. Such characteristics simulate the behavior of a typical WLAN power amplifier.
C. Algorithmic details The convex optimization problem (4), and its extension, can be solved by standard methods, yielding a global optimum p∗ , c∗ , x∗ . The reader is referred to [9] for details on general solvers, as well as specific solvers for the problem at hand. In the experimental verifications presented here, the logarithmbarrier-IPM algorithm presented in [9] is employed. The algorithm starts with a strictly feasible point (c, p) and finds a search direction (v, vp ) and a step size α that update the point with a factor αv and αvp , respectively. The updating procedure is designed to respect the feasibility condition of the point and reduce the barrier function value [10]. The procedure is iterated until a global optimum point is reached, or almost reached, which solves the PAR minimization problem [9]. III. M EASUREMENT SETUP AND DEVICE UNDER TEST Two major characteristics of RF measurement systems for power amplifier testing are accuracy and the ability to track fast variations in the signal envelope. Because the aim of this study is to validate the improvements in power amplifier performance, a state-of-art measurement system was needed. A. Measurement set-up The test-setup presented in Fig. 1 was mainly based on an R&S SMU200A vector signal generator, an Anritsu MS2692A signal analyzer, an Agilent 54610B oscilloscope, an Agilent N2783A high bandwidth hall sensor current probe, an Ericsson LDMOS highly linear driver amplifier, an Agilent E3631A controllable power supply, and a personal computer (PC). The instruments were connected to the PC via LAN or GPIB interface. The output power from the amplifier is measured by the MS2692A, which is accurately calibrated in amplitude and phase over the measured bandwidth of interest to obtain ±0.3dB power accuracy, even for modulated time variant signals. To accurately monitor the drain current vector and obtain accurate PAE readings, a high bandwidth hall sensor current probe was used. Measuring the current envelope through an oscilloscope allowed an envelope-tracking dynamic power consumption up to 100MHz. Additional details on the testbed can be found in [16], [17]. To study the performance of the test-setup, an evaluation of the EVM as a function of the input power is realized where
IV. R ESULTS AND E VALUATION The reference WLAN OFDM 20MHz signal was generated based on 802.11a standards. It had 64 subcarriers with 128 OFDM symbols, a cyclic prefix of 1/4, an oversampling rate of 4 and 14dB PAR after being Hamming windowed. The PARoptimized counterpart, based on full use of the guard subcarriers, reached a PAR of 9.5dB after three Newton iterations in the employed algorithm. Optimizing OFDM signals allocates power in the free subcarriers that reside at the channel sides. Such allocation extends the signal bandwidth from 16.6MHz for the reference signal to 20.0MHz for the optimized one. To combat this effect, the reference OFDM signal is optimized based on the method in Sec. II-C, where half of the power-free subcarriers are preserved as a guard interval, which results in an effective bandwidth of 18.0MHz and a PAR of 9.75dB. This guard margin is sufficient for reducing channel leakage, but requires increasing the number of Newton iterations because two additional iterations were needed to reach the optimal solution in the algorithm. A. Power added efficiency The main goals of reducing PAR are extending the input power level at the 1 dB compression point of the amplifier, reducing the back-off margin, and allowing an efficient use of the available power. Fig. 2 shows the PAE of the amplifier as function of Pin for both signals, reference and PARoptimized with six reserved guard subcarriers, where the 1dB compression points have been specified. As shown in Fig. 2, the 1dB compression point of the amplifier was extended by 1.5dB which improved the PAE by 4.2% near the compression region. The power amplifier gain was found to have similar shaping with a 1.5dB extended compression region. Considering the operating region of the amplifier in a real application, which is the compression region backed-off by the respective PAR, an improvement of 6.5% in PAE can be seen between the reference and optimized signal. In fact, reducing the PAR by 4.5dB and extending the compression region by 1.5dB leads to a total output power improvement of 6dB. Such improvement (that is, 6.5%) varies with the amplifier technology and design. Measurements based on PAR-optimized signal without guard subcarriers resulted in similar performance regarding
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PAE and output power compared to that of a PAR-optimized signal with guard subcarriers.
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Adjusting the frequency constellation power might raise questions regarding in-band errors in the system as well as out-of-band errors due to spectrum regrowth. An evaluation of the output signal EVM versus input power, before and after optimization, is presented in Fig. 3. Despite the 12.6dB EVM difference between the signals in the backed-off region, the EVM of the optimized signal (with guard subcarriers) follows the standard limit value (-19dB for the used signal [1]) with a margin error of -0.3dB. Similar results were obtained when exciting with the PAR-optimized signal without guard intervals. Such behavior is expected as the EVM constraint in the optimization algorithm was set to the standard limit in order to study the maximum improvements in PAE and ACPR. One should mention that the state of the art coding/decoding techniques are successful in correcting for in-band errors as long as the standards limits for EVM are fulfilled. By that, the out-of-band emissions are the most problems tackled nowadays as their effects (intermodulation products and spectrum regrowth) interfere with the adjacent channels and violate the regulations for spectral emission. In summary, despite the changes made to the constellation diagram after optimization, the EVM achieved at the output of the power amplifier in the operation region (backed-off) is still sufficient for allowing decoding algorithms to correct caused errors and restore original information. It complies with the IEEE 802.11a error standard [1] in that region of interest.
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C. Spectral mask and out-of-band errors Adding power to the sideband free subcarriers increases the bandwidth of the optimized signal. Fig. 4 presents the power spectrums of both reference and optimized signals without guard interval at back-off region, compared to the 802.11a spectral mask standard for emission [1]. As shown, a 16.6MHz bandwidth is found for the reference signal, while exciting the free subcarriers leads to a 20.0MHz bandwidth. Such an increase in the spectrum breaks the standard constraint mask for emission on the lower side of the channel by 30KHz and raises questions about the applicability of such a method. A worse behavior is found at compression region where the spectral mask is violated at both the upper and lower channels. Violating the spectral mask for emission by 30KHz will generate strong interference to neighboring channels which marks the importance of preserving a fraction of the powerfree subcarriers as guard interval. Fig. 5 presents the power spectrum of both signals, reference and optimized with 6 guard subcarriers, at back-off region. It shows complete agreement with the IEEE 802.11a standards spectral mask emission. Regarding the out-of-band errors, measurements of the adjacent channel power ratios (ACPRs) of the three signals at 20.0MHz channel spacing and bandwidth, at different input power levels, as presented in Fig. 6, show a large variation between the reference signal and the guard-free optimized one, which is clearly revealed in the lower channel side. Such an
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Fig. 3. Error vector magnitude of the device under test versus input power for the reference signal (dashed line) and PAR-optimized signal with 6 guard subcarriers (pointed line).
increase in the ACPR is due to the 30kHz leakage/interference from the main channel, which biases the value of ACPR. Comparing the above result with the ACPR of the optimized signal with a guard interval shows an improvement of 9dB in the lower channel near the back-off region with respect to not using guard margin, while a 1dB improvement was found in the transition region with respect to the reference signal. Considering the upper adjacent channel, an average ACPR improvement of 1.5dB in the transition region between the backed-off and compression regions was found between the guard-free optimized signal and the reference one; while preserving a guard interval in the optimization process showed an ACPR improvement of 2dB in the back-off region compared to not using a guard margin.
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reduce the adjacent channel interference and the associated errors, but more importantly, this improvement is advantageous compared to the clipping-based algorithms that usually generate undesired regrowth in the output spectrum of the power amplifier and require sophisticated methods to filter the regrowth without regenerating the peaks.
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The achieved improvements between both optimized signals points the necessity of having a frequency guard interval in the signal for adjacent channel interference reduction. From an application point-of-view, the method introduced in [9], and its derivations, need to consider this aspect. Exciting part of the power-free guard subcarriers is sufficient to achieve similar-tobetter power performance compared to that with the full use of the guard interval, with just two extra Newton iterations in the optimization algorithm. In summary, optimizing the signal while preserving a fraction of the power-free subcarriers as a guard interval improves the ACPR by an average of 2dB. This is caused by the absence of the clipped high peaks that usually cause early bird spectrum regrowth. Such small improvements can still
D. Amplifier saturation Reducing the PAR should generally increase the average power at the output of the amplifier by a couple of dBs, which will lead to a compression point at a higher input power level, because fewer peaks excite the amplifier’s nonlinearities. However, contrary to what was expected, an average increase of 1.5dB was found near compression, and requires further investigation to explain this behavior. A study of the complementary cumulative density function (CCDF) of the peaks distribution in the measured signals, reference and optimized with guard interval, would justify such a result. Fig. 7 presents the CCDF of all signal peaks normalized to the signal average, for both measured signals while Fig. 8 shows the CCDF of the normalized envelope signals with respect to their highest peak value, respectively. As shown in Fig. 7, a 4.5dB PAR reduction was observed after optimization. However, by analyzing Fig. 8, one realizes that such reduction was achieved at the cost of increasing the lower peaks density distribution, which in turn limited the improvement near saturation. In fact, despite the reduction of the high peaks, the optimization algorithm allowed the “smaller” peaks to increase. Ultimately the total energy, due to nonlinearity, retained a comparable value to the non-optimized case, sufficient to push the power amplifier into compression. Such aspect in power improvement near saturation represents the tradeoff that must be considered when choosing between advanced optimization method like convex optimization and the low-complexity clipping based methods. In fact, clipping the signal only reduce the dominant peaks, resulting in
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operates, it had a 6.5% PAE improvement with a gain of 6dB in the output power. One reason for such limited gain improvement near compression is the increase in lower peak density distribution in the excitation signal. Spectral emission was considered to be a drawback in the method because power leakage was seen by the high ACPR. The main cause of the leakage is the absence of guard subcarriers. An extended version of the method in [9] was implemented in which half of the guard subcarriers were used to prevent such interference. The extended method showed the necessity of preserving part of the guard interval with a low iterative cost in optimization. An improvement up to 9dB was found in ACPR for the lower channel side, while the power performance maintained its merits values. Even though PAR reduction by advanced methods, such as convex optimization, costs more in terms of additional digital signal processing, it is commonly considered to be a worthwhile technology because the digital processing is “free”; according to Moore’s law, it will be cheaper and cheaper with time. We have experimentally shown an extra 6dB in output power and 6.5% in power added efficiency due to digital processing of the transmitted signal, without any additional requirements from the hardware. This result is believed to be of significance in a world “where every dB is worth a Billion” [18].
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Fig. 8. Complementary cumulative density function of the normalized envelope for both the reference signal (dashed line) and the PAR-optimized signal with guard margin (solid line).
an extra power margin for saturation to be reached. However, due to the distortion created in the excitation signal, higher side band levels are expected to arise which cause higher outof-band errors. V. C ONCLUSION Optimizing the PAR of OFDM signals based on convex optimization algorithms was performed, and before/after figure of merits of a contemporary RF power amplifier were evaluated. The employed PAR optimization algorithm based on [9] resulted in limited improvement in power performance near the saturation region of the power amplifier: an extra 1.5 dB in output power and 4.2% in power added efficiency. However, in the backed-off region where the amplifier normally will
[1] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems, IEEE Standard 802.11a, Sept 1999. [2] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Standard 802.16-2004 (Revision of IEEE Std. 802.16-2001), 2004. [3] S. Cripps, RF Power Amplifiers for Wireless Communications, Second ed. Norwood, MA: Artech House, 2006. [4] J.L. Dawson, “Power Amplifier Linearization Techniques: An Overview,” in Workshop on RF Circuits for 2.5G and 3G Wireless Systems, Feb 2001. [5] H. Ochiai, and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Transactions on Communications, vol. 50, no. 1, pp. 89-101, Jan 2002. [6] W. Jiang, B. Xing, J. Wang, Y. Ni, C. Peng, X. Zhu, and W. Hang, “Performance improvement of power amplifiers with digital linearization technology,” in Asia-Pacific Microwave Conference Proceedings, APMC 2007, pp. 1-4, Dec 2007. [7] S. Sen, R. Senguttuvan, and A. Chatterjee, “Concurrent PAR and power amplifier adaptation for power efficient operation of WiMAX OFDM transmitters,” IEEE, Radio and Wireless Symposium, pp. 21-24, Jan 2008. [8] J. Tellado, “Multicarrier Modulation with Low PAR: Applications to DSL and Wireless, Norwell, MA: Kluwer, Sept 2000. [9] A. Aggarwal, and T.H. Meng, “Minimizing the peak-to-average power ratio of OFDM signals using convex optimization,” IEEE Transactions on Signal Processing, vol. 54, pp. 3099-3110, Aug 2006. [10] S. Boyd, and L. Vandenberghe, Convex Optimization, Cambridge, UK: Cambridge Univ. Press, Mar 2004. [11] Q. Liu, R.J. Baxley, and G.T. Zhou, “Free subcarrier optimization for peak-to-average power ratio minimization in OFDM systems,” in Proceedings IEEE ICASSP, Las Vegas, NV, Mar 2008.
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[12] Q. Liu, R.J. Baxley, X. Ma and G.T. Zhou, “Error vector magnitude optimization for OFDM systems with a deterministic peak-to-average power ratio constraint,” Information Sciences and Systems, 42nd Annual Conference, CISS 2008, pp. 101-104, Mar 2008. [13] G. Fettweis, M. Lohning, D. Petrovic, M. Windisch, P. Zillmann, and W. Rave, “Dirty RF: a new paradigm,” International Journal of Wireless Information Networks, vol. 14, no. 2, pp. 133-148, June 2007. [14] A. Behzad, Wireless LAN Radios, IEEE Press on Digital and Mobile Communication, 2007. [15] T.C.W. Schenk, and E.R. Fledderus, “RF impairments in high-rate wireless systems - understanding the impact of TX/RX-asymmetry,” in 3rd International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, pp. 117-122, Mar 2008. [16] C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keskitalo, and H. de la Rosa, “Automated multidimensional characterization of power amplifier for design and production,” in International Instrumentation and Measurement Technology Conference Proceedings, I 2 M T C 2009, pp. 144-148, May 2009. [17] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidth of an RF power amplifier test bed,” IEEE Transactions on Instrumentation and Measurement, vol. 56, pp. 1488-1494, 2007. [18] C. Beckman, L. Eklund, B. Karlsson, B. Lindmark, D. Ribbenfjärd, and P. Wirdemark, “Verifying 3G license requirements when every dB is worth a billion,” in First European Conference on Antennas and Propagation, EuCAP 2006, France, Nov 2006.
Charles Nader (S’08) received an M.E. in electrical engineering from the Lebanese University ULFG2, Lebanon, in 2005 and an M.Sc in EE/Telecommunication from the University of Gävle, Gävle, Sweden in 2006. In 2007, he was a consultant for Multilane inc. in Lebanon, working with microwave devices and signal integrity. In 2008, he joined the University of Gävle, center for RF measurement technology, and The Royal Institute of Technology, signal processing lab, where he is currently working on his Ph.D. with a research focus on signal processing applied to radio frequency measurement systems for microwave device characterization and testing.
Peter Händel (S’88-M’94-SM’98) received his Ph.D. at Uppsala University in 1993. From 1987 to 1993, he was at Uppsala University. During 19931997, he was with Ericsson AB, Kista, Sweden. During 1996-1997, he was also with Tampere University of Technology, Finland. Since 1997, he has been with the Royal Institute of Technology, Stockholm, Sweden, where he is currently a Professor of signal processing. From 2000 to 2006, he was with the Swedish Defense Research Agency. He is currently guest professor at the University of Gävle, Sweden. He has served as an Editorial Board Member of the EURASIP Journal of Advances in Signal Processing, and an Editorial Advisory Board Member of Recent Patents on Electrical Engineering. He is a member of the Editorial Board of Hindawi’s Research Letters in Signal Processing, and Journal of Electrical and Computer Engineering. He has served as Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING.
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Niclas Björsell (S’02-M’08) was born in Falun, Sweden, in 1964. He received his B.Sc. in Electrical Engineering and his Lic. Ph. in Automatic control from Uppsala University, Sweden in 1994 and 1998, respectively. His Ph. D. in Telecommunication was received at the Royal Institute of Technology, Stockholm, Sweden, in 2007. He has several years of experience from research and development projects that fostered collaborations between industry and the academy. He currently holds positions in the academy as well as in industry, and has worked as project manager for some of the R&D projects. Since 2006 he has served as the head of Division of Electronics at the Department of Technology and Built Environment, University of Gävle, Sweden. He has published more than 20 papers in journals and conferences, and his research interests include radio frequency measurement technology and analog-to-digital conversion. Dr. Björsell is a voting member of the IEEE, Instrumentation and Measurement, TC-10.
