Practical Implementation of PN Scrambler for PAPR Reduction

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codes need to be used and as a result, the overall efficiency of transmission becomes reduced. Finally, the third category is based on OFDM symbol scrambling and selection of the sequence that produces minimum PAPR. The pre-scrambling techniques [7-8] achieve good PAPR reduction but they require multiple FFT transforms and somewhat higher processing power. The method presented in this paper belongs to the third category of the PAPR reduction techniques. It uses conveniently chosen Pseudorandom Noise (PN) sequences applied to the input data bit stream. The method is very easy to realize in the software or hardware environment which is very important if the PAPR needs to be implemented in Application Specific Integrated Circuits (ASIC). In such a scenario, the PN-Scrambler may be implemented by the addition of external FPGA and DSP hardware to the Commercial Off-The-Shelf (COTS) ASICs. As a result, one obtains cost efficient and reliable hardware solutions. 3. OVERVIEW OF THE PN-SCRAMBLER TECHNIQUE FOR PAPR REDUCTION 3.1. System overview Block diagrams of the transmitter and receiver implementing the proposed PN-Scrambler are presented in Figs. 1 and 2, respectively. A/D Real A/D Imag Figure 1. PN-Scrambler Tx Block Diagram DC Offset DC Offset Subcarrier Demapper I / Q Imbalance I / Q Imbalance Channel Estimate Acquisition / Synchronization AGC Receive (Demodulation) Path Timing Detect Carrier Phase and Timing Drift Correction Coarse Freq Soft Bit Decisions Timing Fine Freq Deinterleaver and Convolutional decoder Sample Buffer Guard Interval Removal Descrambler FFT Data Bits Output Figure 2. PN-Scrambler Rx Block Diagram As seen in Fig. 1, two additional elements are added to a typical OFDM transmitter. The first element is the PAPR scrambler, and the second one is the PAPR threshold compare block. The PN-Scrambler utilizes a Maximal-Length Linear Feedback Shift Register (ML- LFSR) with log2(k) = l taps in order to produce k = 2 l -1 uncorrelated unique sets of data from the same input sequence. The k unique sets of data are used to generate k independent identically distributed (i.i.d.) OFDM symbols. A block of Nb bits comprising one OFDM symbol is 2 of 7 scrambled and passed along for Forward Error Correction (FEC) coding, interleaving, modulation, symbol mapping and IFFT. In any given OFDM system, Nb is a function of the number of subcarriers, the modulation scheme applied to each subcarrier, and the coding rate. By examining each individual sample coming out of the IFFT, the PAPR threshold comparator determines if the scrambler has achieved a desired PAPR on a symbol-by-symbol basis. If the PAPR of the symbol is below a desired threshold, then the data is passed along towards the RF stage of the transmitter. However, if the PAPR is still high, the data is scrambled with a different phase of the ML-LFSR’s PN sequence. Since this technique operates on the input bit stream, it is essentially independent of the OFDM modulation and may be adapted to any particular scenario. The receiver presented in Fig. 2 is a typical OFDM receiver that needs to perform the tasks of down conversion, channel estimation, and decoding. The only additional task required by the PN-Scrambler PAPR reduction technique is descrambling of the data at the receiver output. To perform descrambling, the receiver has to “know” the phase of the ML-LFSR used on the transmission side. This phase is embedded in the data stream. For example, the first l bits of the OFDM symbol may carry the information on the ML-LFSR phase. 3.2. Analytical Performance Characterization A practical implementation of the PN-Scrambler PAPR reduction technique requires selection of several parameters. These parameters are defined as follows. 1. Number of scrambling sequences (k) - defined as the number of PN sequences produced by the ML- LFSR. Each sequence is Nb bits long. 2. PAPR threshold (L) – defined as the maximum PAPR for the OFDM symbol. This value is used by the PAPR threshold comparator block in order to discard OFDM symbols with PAPR greater than L. 3. IFFT size / number of sub-carriers (N) – defined as the number of the non-zero orthogonal subcarriers per OFDM symbol. 4. Average latency ( k ) – defined as the average number of scrambling attempts per OFDM symbol in order to pass the threshold level L. 5. Probability of clipping (p) – probability that the PAPR exceeds the threshold level L after k scrambling attempts. 6. PN scrambler overhead ( v ) – defined as the ratio of the number of bits required to represent the phase of the ML-LFSR to the number of bits per OFDM symbol Nb. In any actual design, the above parameters allow different tradeoffs. The subsequent section highlights some of these design trades.
3.3. PAPR per OFDM Sample after Application of k- PN Scrambling Sequences The goal of the analysis presented in this section is to find the analytical PAPR probability distribution of the PN-scrambled OFDM signal on a sample-by-sample basis. This allows the PAPR to be compared with other sampled time-domain signals and techniques. Define the complex baseband signal as S = X + jY , where j = −1 (2) From the central limit theorem, the OFDM signal will have a Gaussian probability density function (PDF) for both the real X and imaginary Y time-domain signals due to the addition of multiple complex exponentials with random amplitude A and phase θ. N −1 N −1 ⎛ 2π nk ⎞ ⎛ 2πnk ⎞ ⎛ 2πnk ⎞ (3) ∑ Ak ⋅ exp⎜ j ⋅ + jθ k ⎟ = ∑ Ak ⋅ cos⎜ + θ k ⎟ + j ⋅ Ak ⋅ sin⎜ + θ k ⎟ n= 0 ⎝ N ⎠ n= 0 ⎝ N ⎠ ⎝ N ⎠ Then, the magnitude R, is R + The magnitude of the complex baseband signal can be approximated as Rayleigh distributed, since the real and imaginary components X and Y are independent Gaussian random variables with zero mean and equal variances σ 2 . Interestingly, the original derivation of this density function by Lord Rayleigh in 1880 was applied to the similar application of finding the envelope of the sum of many sine waves of different frequencies. The Rayleigh PDF is given by [9] as 2 ⎧ r ⎛ r ⎞ ⎪ exp ≥ ⎨ ⎜ ⎜− ⎟ r 0 f ( r) = 2 2 (5) R σ ⎝ 2 ⋅σ ⎠ ⎪ ⎩ 0 r < 0 The PDF of the power signal of a complex Gaussian signal is derived using a transformation of random variables, where it is desired to find the power random variable 2 W = R (6) Using a transformation of random variables dr fW ( w) = f R ( r) ⋅ (7) dw The complex 1 power PDF can be written as ⎧⎛ 1 ⎞ ⎛ w ⎞ ⎪⎜ ⎟ ⋅ exp⎜ − ⎟ w ≥ 0 f ( w) = 2 2 ⎨ (8) W ⎝ 2 ⋅σ ⎠ ⎝ 2 ⋅σ ⎠ ⎪ ⎩ 0 w < 0 The complex power CDF of a complex signal with Gaussian real and imaginary components is given by w ⎛ 1 ⎞ ⎛ u ⎞ Pr( W ≤ w) = FW ( w) = ∫ ⎜ ⎟ ⋅exp⎜ − ⎟du (9) 2 2 0 ⎝ 2⋅σ ⎠ ⎝ 2⋅σ ⎠ Since this is an indeterminate integral, it cannot be represented in closed-form solution. However, it is still 2 2 = X Y (4) 1 The term “complex” power refers to the OFDM complex baseband signal in order to avoid confusion from the power PDF of a real variable. 3 of 7 possible to find the complex power CDF 2 by taking advantage of the Rayleigh CDF. 2 W = R 2 2 = X + Y (10) The complex power CDF is given by, FW ( w) = Pr( R ≤ + w) − Pr( R ≤ − w) (11) Then, the probability that the power is less than or equal to some value w is simply the probability that the magnitude is between the values of + w and − w . The CDF for the magnitude R is the Rayleigh CDF 2 ⎧ ⎛ r ⎞ ⎪1 − exp ⎨ ⎜ ⎜− ⎟ Pr( R ≤ r) = F ( ) = 2 R r ⎝ 2 ⋅σ ⎠ ⎪ ⎩ 0 r ≥ 0 (12) r < 0 Substituting equation (12) into equation (11) and using the fact that FR ( − w) = 0 since FR ( r) = 0 when r < 0 ⎧ ⎛ w ⎞ ⎪1 − exp⎜ − Pr( W ≤ w) = F ( ) = 2 ⎟ W w ⎨ ⎝ 2 ⋅σ ⎠ ⎪ ⎩ 0 w ≥ 0 w < 0 (13) The peak power per OFDM symbol is measured as the largest power sample out of the N corresponding complex samples. It is desired to find the probability that the peak power of at least one out of N samples exceeds a predetermined peak power threshold level L. We start by defining event A as the event that the complex timedomain sample’s power w exceeds threshold L. We want to find the probability that event A occurs at least m=1 time out of the N OFDM symbol samples. Since each time-domain sample is i.i.d., the Bernoulli trials can be useful here Pr{A occurs m times} = ⎛ N ⎞ m N −m g N ( m) = ⎜ ⎟ ⋅ p q (14) ⎝ m ⎠ ⎛ N ⎞ N! ⎜ ⎟= N Cm = (15) ⎝ m ⎠ m! ( N − m)! Using the Binomial distribution, we can write Pr{A occurs at least m times in N samples}= N N ⎛ N ⎞ i N −i ∑ g N ( i) = ∑⎜ ⎟ ⋅ p q (16) i= m i= m ⎝ i ⎠ where p is the probability that a single complex timedomain sample exceeds L, defined as 2 { x + jy > L} = 1− F ( L) p = Pr (17) w q W = 1− p = F ( L) (18) 2 The probability distribution function, or cumulative distribution function (CDF), is defined by [10] to be the probability of the event that the observed random variable W is less than or equal to the allowed value w. In this case, W is the random variable and the value w will later represent a power threshold level and will be substituted with the variable L.
Page 1: PRACTICAL IMPLEMENTATION OF PN SCRA
Page 5 and 6: Probability of PAPR > x dB 10 -1 10
Page 7: 5. POWER SAVINGS The power consumpt

Practical Implementation of PN Scrambler for PAPR Reduction

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