Practical Implementation of PN Scrambler for PAPR Reduction

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After evaluating the derivative with respect to k, the joint PDF is given as f W , k ( L, N, k) N N ⎛ L ⎞ ⎛ L ⎞ ⎜ ⎛ ⎛ ⎞⎞ ⎜ ⎛ ⎛ ⎞⎞ − ln 1− ⎜1− exp⎜− ⎟⎟ ⎟ ⋅ 1− ⎜1− exp⎜− ⎟⎟ ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎝ ⎝ ⎝ 2 ⋅σ ⎠⎠ ⎠ ⎝ ⎝ ⎝ 2 ⋅σ ⎠⎠ ⎠ = 2 The probability in (31) can now be found by integrating the joint PDF over the range of k, ( L, N, k) dk = p k (33) k m k > = − ∫ f = Pr( 256) 1 (34) W , k k =0 Equation 34 can be solved for L to obtain ⎛ ⎜ ⎛ L = − ln⎜1 − ⎜ 1− p ⎜ ⎝ ⎝ 1 1 N m ⎞ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠ (35) Equation (35) can be used to find the required PAPR threshold level L for any FFT size N. The value m in (35) is the desired number of k-PN scrambling sequences not to exceed. The value p in (35) specifies the probability that a given OFDM symbol will not require more than m scrambling sequences in order to pass the objective PAPR threshold level setting L. Evaluating (35) using p=10 -4 , m=256, and N=64 yields the required PAPR threshold level setting L=2.98 or 4.74 dB. In order to find the average number of scrambling sequences, it is convenient to make use of the definition of the expected value of a random variable, ∞ ∫ −∞ X = E[ X ] = x ⋅ f ( x) dx (36) The average number of scrambling sequences k becomes = ∞ ∫ = ⎟ ⎟⎟ k ⎛ N N k ⎞ ⎜ ⎛ ⎞ ⎛ ⎞ ⎜ ⎛ ⎛ L ⎞⎞ ⎟ ⎜ ⎛ ⎛ L ⎞⎞ (37) k = k ⋅ ⎟ ⎜− ln 1− ⎜1− exp⎜− ⎟⎟ ⋅ 1− ⎜1− exp⎜− ⎟⎟ dk 2 2 ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎝ ⎝ ⎝ 2⋅σ ⎠⎠ ⎠ ⎝ ⎝ ⎝ 2⋅σ k 0 ⎠⎠ ⎠ ⎠ After simplification, −1 (38) k = N ⎛ ⎞ ⎜ ⎛ ⎛ L ⎞⎞ ln 1− ⎜1− exp⎜ − ⎟⎟ ⎟ ⎜ 2 ⎟ ⎝ ⎝ ⎝ 2⋅σ ⎠⎠ ⎠ For this example, the average number of PN scrambling sequences that will occur at this PAPR threshold level setting is ( ( ( ) ) ) 27.8 1 64 ln 1 1 exp 2. 98 = − k = (39) − − − Using 64-QAM modulation per subcarrier, the overhead v for this example is log 2 ( m) log ( 256) (40) 2 v = = = 0. 028 N ⋅ N ⋅ C 3 bps 64 ⋅ 6 ⋅ 4 where m is the number of sequences not to exceed, N is the number of subcarriers, Nbps is the number of bits per subcarrier based on the constellation size, and C is the FEC code rate. 6 of 7 In summary, in order to ensure that the OFDM transmitter does not exceed k=256 PN scrambling sequences with a probability of p=10 -4 , the peak power threshold level must be set to L=4.74 dB which will require approximately k=28 PN scrambling sequences on average. In other words, the system will have a PAPR less than or equal to 4.74 dB with a 99.9900% probability. Without the symbol scrambling (i.e. k=1), the PAPR is 11.3 dB at p=10 -4 . Using the system parameters derived in this example, this technique results in an improvement of 6.5 dB with an insignificant overhead of 2.8%. 4. PRACTICAL IMPLEMENTATION The PN-Scrambler was implemented as a PAPR reduction technique for an IEEE 802.11a OFDM modem. Without application of the PAPR reduction, the CCDF curve for the IEEE 802.11a OFDM modem closely follows the k = 1 curve of Fig. 3, indicating that the IEEE 802.11a waveform has a large PAPR. PAPR Reduction ON PAPR Reduction ON Backoff = 8 dB Backoff = 8 dB EVM = -40.34 dB EVM = -40.34 dB PAPR Reduction ON PAPR Reduction ON Backoff = 5 dB Backoff = 5 dB EVM = -33.44 dB EVM = -33.44 dB PAPR Reduction OFF PAPR Reduction OFF Backoff = 8 dB Backoff = 8 dB EVM = -34.35 dB EVM = -34.35 dB PAPR Reduction OFF PAPR Reduction OFF Backoff = 5 dB Backoff = 5 dB EVM = -25.40 dB EVM = -25.40 dB Figure 5. VSA EVM plots with and without PAPR reduction The OFDM signal from the output of the I/Q modulator was sent through a 10-Watt Stealth Microwave Class-A Power Amplifier (SM0825-40) into an Agilent Vector Signal Analyzer 89641A (VSA). The VSA demodulated the OFDM signal and the resulting constellations are shown in Fig. 5. The results show significant reduction in the Error Vector Magnitude (EVM) with PAPR reduction “on” versus with PAPR reduction “off.” Furthermore, the results show that at 8 dB backoff from the power amplifier’s one dB compression point (P1dB) with PAPR reduction turned-off produces approximately the same EVM as 5 dB backoff when the PAPR reduction is turned-on. This 3 dB reduction in backoff provides twice the RF output transmit power, or equivalently, allows a 10-Watt amplifier to be used instead of a 20-Watt amplifier.
5. POWER SAVINGS The power consumption of communication systems is typically determined by the power requirements of the final power amplifier. From the CCDF curves given in Fig. 3, it can be observed that the PN-Scrambler technique provides considerable reduction of the PAPR. As a result, the average power of the PA may be set closer to its maximum power level. For a Class-A Linear PA, a 3 dB lower P1dB-point typically corresponds to a reduction of DC power consumption by half (see Table 1 for examples). Table 1- Class-A Power Amplifiers (Courtesy of Stealth Microwave) Class-A Linear Frequency RF Output Power RF Output DC Voltage Current Power Power Amplifier Range (MHz) P1dB (dBm) Power (Watts) (V) (Amps) (Watts) SM1720-37H 1700-2000 37 5.0 12 1.9 22.8 SM1720-41L 1700-2000 41 12.6 12 5.5 66 SM1720-44L 1700-2000 44 25.1 12 10 120 SM1720-48L 1700-2000 48 63.1 12 16 192 SM1720-50L 1700-2000 50 100.0 12 19 228 As a further example of the benefits, consider a design that utilizes a PA with an RF output power of 25.1 Watts (P1dB = 44 dBm). According to Table 1, this corresponds to DC power consumption of 120 W. With 3 dB of PAPR reduction, a 12.6 W (41 dBm) PA which consumes 66 W of DC power can be utilized to transmit the communications waveform with the same average output power. This directly results in a 54 W power savings. Additionally, the DC-DC power converter used to supply the 12 V to the PA is typically about 80% efficient. Therefore, to power the 25.1 W PA requires the system to supply about 150 W of DC power. The same system with the PN-Scrambler PAPR reduction technique applied requires 82.5 W to supply the 12.6 W PA. In summary, the total system DC power saved from the PAPR reduction is 67.5 W, which results in 45% power savings. 6. CONCLUSION The PN-Scrambler technique was presented in this paper and its performance was accurately characterized. All of the analytical equations derived in this paper were verified against simulated data. The PN-Scrambler was shown to provide exceptional PAPR reduction for a low number of subcarriers. For example, given a PAPR requirement that the peak power must not exceed L more than once in 1000 samples (i.e. p=10 -3 or 0.1%) and by using k=256 PN scrambling sequences, it was shown that the achievable PAPR level is L=4.3 dB for N=64 subcarriers, L=5.2 dB for N=128, and L=5.9 dB for N=256. Compared to conventional OFDM with N=64 subcarriers and PAPR level L=8.4 dB at p=10 -3 , this technique provides an improvement of ΔL=4.1 dB. The PN-Scrambler technique does not require significant modification of the transmitter chain and virtually no modification of the receiver chain. On the transmitter side, all that is needed is some external 7 of 7 hardware and on the receiver side, the same ML-LFSR. This makes the technique very attractive for many practical applications where low-cost, robust solutions are desired. 7. ACKNOWLEDGEMENTS The authors would like to express a sincere appreciation to Harris Corporation and Wireless Center of Excellence at Florida Institute of Technology for support of the research presented in this paper. 8. REFERENCES [1] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. 1st ed., Norwood, MA: Artech House Publishers, 2000. [2] D. Wulich, and L. Goldfeld, “Reduction of Peak Factor in Orthogonal Multicarrier Modulation by Amplitude Limiting and Coding,” IEEE Transactions on Communications, Vol. 47, No. 1, January 1999. [3] A. D. S. Jayalath and C. Tellambura, “Peak-to-Average Power ratio of IEEE 802.11a PHY layer Signals,” 6th International Symposium on Digital Signal Processing for Communication System DSPCS'02, (Sydney-Manly), pp.31- 36, January 2002. [4] W. Xianbin, T. Tjhung, and C. S. Ng, “Reduction of Peak-to- Average Power Ratio of OFDM System Using a Companding Technique,” IEEE Transactions and Broadcasting, Vol. 45, No. 3, September 1999. [5] A. N. D’Andrea., V. Lottici, and R. Reggiannini, “Nonlinear Predistortion of OFDM Signals over Frequency-Selective Fading Channels,” IEEE Transactions on Communications, Vol. 49, No. 5, May 2001. [6] K. G. Paterson, and V. Tarokh, “On the Existence and Construction of Good Codes with Low Peak-to-Average Power Ratios,” IEEE Transactions and Information Theory, Vol. 46, No. 6, September 2000. [7] S. H. Muller, R. W. Bauml, R. F. H. Fischer, and J. B. Huber, “OFDM with Reduced PAPR by Multiple Signal Representation,” Annals of Telecommunications, Vol. 52, No. 1-2, pp 58-67, Fen 1997. [8] K. D. Choe, S. C. Kim, and S. K. Park, “Pre-scrambling methods for PAPR Reduction in OFDM Communication Systems,” IEEE Transactions on Consumer Electronics, Vol.50, No. 4, November 2004. [9] G. R. Cooper and C. D. McGillem, Probabilistic Methods of Signal and System Analysis. 2nd Ed., New York: Oxford University Press, 1986. [10] M. Sharif, M. Gharavi-Alkhansari, and B. H. Khalaj, “On the Peak-to-Average Power of OFDM,” IEEE Transactions on Communications, Vol. 49, No. 2, February 2001.
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