Practical Implementation of PN Scrambler for PAPR Reduction

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The probability that the power W > L occurs at least m=1 time out of N samples is Pr{A occurs at least 1 time in N samples} = N ⎛ N ⎞ i N −i ∑= ⎜ ⎟ ⋅ p q i 1 ⎝ i ⎠ (19) In order to simplify the equation, note that the probability W > L occurs at least m=1 time out of N samples is the same as the complementary probability that exactly no samples (m=0) have W > L, which can be written as 1-Pr{A occurs exactly 0 times out of N samples} = ⎛ N ⎞ N N N 1− g N ( 0) = 1− ⎜ ⎟ ⋅ p q = 1− q = 1− ( 1− p) = 1− F ⎝ 0 ⎠ 0 N W ( L) (20) Thus, we now define the CCDF of the peak power per OFDM symbol as the probability that the peak power of one sample out of N samples exceeds threshold L N ⎛ ⎛ L ⎞⎞ Fsym = Pr{ PeakPowerOFDM _ Symbol > L} = 1− ⎜1 − exp⎜ − ⎟⎟ 2 ⎝ ⎝ 2 ⋅σ ⎠⎠ (21) Using the PN scrambler to generate k OFDM symbols and then selecting the OFDM symbol with the lowest PAPR for transmission reduces the peak power probability to p k . For example, if an OFDM symbol has the probability of p=0.1 of exceeding threshold L, then the probability of k=2 OFDM symbols exceeding threshold L is (0.1) 2 =0.01. This is true because of the statistical independence of the peak power between OFDM symbols generated by the PN- Scrambler (i.e. uncorrelated data sets). That is, if the probability of event S1 does not depend upon the outcome of S2 and visa versa, such events are said to be statistically independent and Pr(S , S ) = 1 2 2 = Pr(S1) ⋅ Pr(S2) = p ⋅ p p (22) For k statistically independent events, this becomes Pr(S , S ,..., S ) = 1 2 k k = Pr(S1) ⋅Pr(S2 ) ⋅⋅⋅ Pr(Sk ) = p⋅ p⋅ ⋅⋅ p p (23) For k scrambling sequences, the new PAPR CCDF per OFDM symbol can now be formally written as Pr { PAPR k > L} OFDM _ Symbol sequences N ⎛ ⎞ ⎜ ⎛ ⎛ L ⎞⎞ = 1− ⎜1− exp⎜ − ⎟⎟ ⎟ ⎜ 2 2 ⎟ ⎝ ⎝ ⎝ ⋅σ ⎠⎠ ⎠ k 2 1 σ = 2 (24) Substituting equation (13) into (21) yields F = 1 − F N (25) sym W 4 of 7 Rearranging, the sample-based power distribution is related to the OFDM symbol-based peak power distribution by ( ) N 1 FW = 1− F (26) sym Next, we define the new PAPR distribution Fsym ˆ as the OFDM symbol-based peak-power distribution after k scrambling sequences k N k F ˆ sym = Fsym = ( 1 − FW ) (27) Finally, in order to determine FW ˆ , which is the new PAPR reduced sample-based power distribution after application of k scrambling sequences, we simply substitute Fsym ˆ into equation (26) relating the sample-based power distribution to the OFDM symbol-based peak power distribution, Fˆ W = 1 N N k ( 1 − Fˆ ) = ( 1 − ( 1 − F ) ) sym W 1 N 1 N k N ⎛ ⎞ ⎜ ⎛ L ⎞ 1 ⎜ ⎛ ⎛ ⎞⎞ ⎟ = ⎜ − 1 − ⎜1 − exp⎜ − ⎟ ⎟ 2 ⎟ 2 ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎝ ⎝ ⋅σ ⎠⎠ ⎠ ⎠ (28) In the above equation, FW ˆ is the CDF of the PAPR reduced sample-based distribution using k-scrambling sequences, which is the probability that each individual sample’s peak power is less than or equal to threshold level L using an IFFT size of N. It is more useful to evaluate the probability that each individual sample’s peak power is greater than threshold level L, which is the complementary CDF, CCDF=1-CDF. 1 k N N ⎛ ⎞ ⎜ ⎛ L ⎞ CCDF( L, N, k) 1 1 ⎜ ⎛ ⎛ ⎞⎞ ⎟ = − ⎜ − 1− ⎜1− exp⎜ − ⎟ ⎟ 2 ⎟ 2 ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎝ ⎝ ⋅σ ⎠⎠ ⎠ ⎠ (29) After normalization of the complex time-domain signal’s average power PAPR CCDF 1 N k N = 1− ⎜ ⎛1− ( 1− ( 1− exp( − L) ) ) ⎞ (30) ( L, N, k) ⎟ ⎝ ⎠ It is important to note that due to the equivalence theorem, the analysis and linear signal processing of baseband waveform presented herein will be identical to the analysis of the bandpass waveform with the exception of oversampling error. A rigorous discussion of the PAPR measurement error due to Nyquist sampling versus continuous sampling can be found in [10]. Because most applications are typically concerned with the probability that a signal’s power exceeds a certain level L, instead of using the CDF, the CCDF is typically plotted and is shown in Fig. 3 for FFT sizes N=64, N=128, N=256 and number of scrambling sequences k=1 (i.e. no PAPR reduction), k=8, and k=256. The plot shows both the theoretical curves obtained using equation (30) and the empirical curves obtained through computer simulations.
Probability of PAPR > x dB 10 -1 10 -2 10 -3 10 -4 10 -5 CCDF Curve Analytical PAPR CCDF Empirical Prob k: 1 FFT SIZE: 64 Derived Prob k: 1 FFT SIZE: 64 Empirical Prob k: 16 FFT SIZE: 64 Derived Prob k: 16 FFT SIZE: 64 Empirical Prob k: 256 FFT SIZE: 64 Derived Prob k: 256 FFT SIZE: 64 Empirical Prob k: 1 FFT SIZE: 128 Derived Prob k: 1 FFT SIZE: 128 Empirical Prob k: 16 FFT SIZE: 128 Derived Prob k: 16 FFT SIZE: 128 Empirical Prob k: 256 FFT SIZE: 128 Derived Prob k: 256 FFT SIZE: 128 Empirical Prob k: 1 FFT SIZE: 256 Derived Prob k: 1 FFT SIZE: 256 Empirical Prob k: 16 FFT SIZE: 256 Derived Prob k: 16 FFT SIZE: 256 Empirical Prob k: 256 FFT SIZE: 256 Derived Prob k: 256 FFT SIZE: 256 10 0 2 4 6 8 10 12 -6 x dB Figure 3. PAPR probability distribution per sample using k-PN scrambling sequences Figure 3 shows that the PAPR probability given in equation (30) matches the simulated data very closely. The figure shows a significant reduction in PAPR due to application of the k-PN scrambling sequences. For example, for N=64 subcarriers, the figure indicates the probability that the PAPR exceeds L=4.5 dB reduces from p=6.0% at k=1 down to p=0.01% at k=256. In order to ensure that the PAPR is less than L at p=0.01% of the time for conventional OFDM (i.e. k=1), the equivalent PAPR threshold is L=9.6dB. If the OFDM signal is input into an ideal amplifier that clips the signal when it is larger than the saturation point and it is desired to not clip the signal more than p=0.01% of the time, then application of the k=256 scrambling sequences requires 9.6-4.5=5.1 dB less backoff. Often a larger amount of in-band noise and outof-band noise can be tolerated and therefore a higher clipping probability is acceptable. Therefore, in a typical system, less than 5.1 dB performance is gained due to application of the k=256 scrambling sequences. Probability of PAPR > L dB 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 CCDF Curve Vs. Threshold L and k-PN Scrambling Sequences k=256 Increasing k Scrambling Sequences 10 0 2 4 6 8 10 12 -6 L dB Figure 4. CCDF Curve Probability PAPR>L, multiple k- PN scrambling sequences overlaid (k=1:1:256) k=1 5 of 7 The CCDF curves in Fig. 4 show rapid PAPR reduction during the first few k scrambling sequences. After about k=8, the PAPR reduction begins to approach diminishing returns. For example, at k=128 and k=129 the probability p=Pr{PAPR>L}=0.01% occurs at threshold L=4.7786 dB and L=4.7753 dB, respectively. This results in a small improvement of ΔL=0.0033 dB, which is insignificant compared to ΔL=1.4 dB from k=1 to k=2. 3.4. Practical Example As a practical example, consider the following design problem. An OFDM transmitter with N=64 subcarriers must be designed. As part of the design problem, the transmitter must achieve a low latency and therefore use no more than k=256 scrambling sequences per OFDM symbol. Each OFDM symbol is compared to a threshold level L. If the OFDM symbol’s peak power exceeds the threshold level L, it is re-generated using the PN- Scrambler until the symbol’s peak power is smaller than L. As soon as the OFDM symbol has a peak power less than L, it is stored in a packet buffer where it waits to be transmitted. Once the desired number of “symbols per packet” is stored in the packet buffer, the packet is immediately transmitted out the digital-to-analog converter (DAC). Since the PN-Scrambler in this example is only l=8-bits, the ML-LFSR can only generate k=2 8 -1=255 possible unique OFDM symbols that represent the same data. Since the transmitter already has a data whitener, an additional bypass mode is added to provide a total of k=255+1=256 possible unique OFDM symbols that represent the same data. If any of the symbols exceed the k=256 scrambling sequences, the symbol with the lowest PAPR out of the 256 will be selected for transmission. However, this means that one or more of the symbols in the packet buffer did not pass the threshold level L and may be clipped by the DAC or PA. Therefore, in order to ensure that this does not happen often, we desire a very low probability of 1 in 10,000 OFDM symbols requiring more than k=256 PN-Sequences, or equivalently, p=10 -4 . The problem is to determine the value of the threshold level L and the average number of k-PN scrambling sequences that will be generated as a result of the corresponding threshold level setting. We start by noting the latency requirement and let m=256, the number of sequences not to exceed. For this example, −4 p = Pr( k > m) = Pr( k > 256) = 10 (31) Both L and N are predetermined constants in the transmitter and receiver and k is the random variable, f W , k ( L, N, k) dF = W , k ( L, N, k) dk k ⎛ N ⎞ ⎜ ⎛ L ⎞ ⎟ d ⎜ ⎛ ⎛ ⎞⎞ ⎟ ⎜1− 1− ⎜1− exp⎜− ⎟ 2 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎝ ⎝ 2⋅σ ⎠⎠ ⎠ = ⎠ dk (32)
Page 1 and 2: PRACTICAL IMPLEMENTATION OF PN SCRA
Page 3: 3.3. PAPR per OFDM Sample after App
Page 7: 5. POWER SAVINGS The power consumpt

Practical Implementation of PN Scrambler for PAPR Reduction

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