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The Peak-to-Average Power Ratio Problem

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## The Peak-to-Average Power Ratio Problem

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**The Peak-to-Average Power Ratio Problem**Gwo-Ruey Lee**Outlines**• The Peak-to-Average Power Ratio Problem • The Peak-to-Average Power Ratio [1-7] • OFDM Signal Amplitude Statistics[4,13] • The Distribution of The Peak-to-Average Power Ratio [ 1,4,16] • Clipping and Peak Window [1,4,10,11] • Clipping Amplifier Methods • Clipping Amplifier Simulations • Peak Cancellation [1,4,8,9,14,15] • PAP Reduction Codes [14,17,18.19] • Symbol Scrambling [12,14,20,21]**The Peak-to-Average Power Ratio Problem**1/3 • It is plausible that the OFDM signal - which is the superposition of a high number of modulated subchannel signals – may exhibit a high instantaneous signal peak with respect to the average signal level. • An OFDM signal consists of a number of independently modulated subcarriers, which can give a large peak-to-average power (PAP) ratio. • High peak-to-average power ratio • Problem 1. It increased complexity of the analog-to-digital and digital-to-analog converters • Problem 2. It reduced efficiency of the RF power amplifier • The PAPR puts a stringent requirement on the power amplifier and reduces the efficiency in the sense that a higher input backoff factor is needed before the peaks in the signal experience significant distortion due to power amplifier nonlinearity.**The Peak-to-Average Power Ratio Problem**2/3 PAPR ~ number of subcarriers =N**The Peak-to-Average Power Ratio Problem**3/3 • The existing solutions of PAPR • 1. Signal distortion techniques,which reduce the peak amplitudes simply by nonlinearly distorting the OFDM signal at or around the peaks. • Clipping • Peak window • Peak cancellation • 2. Coding techniques that using a special forward-error correct code • PAP reduction code • 3. It is based on scrambling each OFDM symbol with different scrambling sequences and selecting that sequence that gives the smallest PAP ratio. • Adaptive subcarrier selection (ASUS) • Selected mapping (SLM) • Partial transmit sequence (PTS)**The Peak-to-Average Power Ratio**1/17 • Signal expression • Let and denote the real and imaginary parts of the output signal. • A complex baseband signal, defined over the time interval , can be expressed as • where is the complex data of the kth subcarrier and is the OFDM symbol period.**The Peak-to-Average Power RatioPAPR Definition**2/17 • OFDM bandpass signal • is the carrier frequency of RF signals. • The peak power is defined as the power of a sine wave with an amplitude equal to the maximum envelope value. • The PAPR of the baseband OFDM signals can be defined as**The Peak-to-Average Power Ratio**3/17 • If all the subcarrier are modulated by phase-shift keying (PSK), the theoretical upper bound of the PAPR in OFDM signals with N subcarriers is N. • For example • It can be shown that for an M-ary PSK OFDM system, there are at most patterns that yield the highest PAPR, namely, N. • The probability of observing such a PAPR is .**The Peak-to-Average Power Ratio**4/17 • Basic waveforms of OFDM signal with 4-DFT BPSK**The Peak-to-Average Power Ratio**5/17 • OFDM signal with 4-DFT BPSK**The Peak-to-Average Power Ratio**6/17 The histogram of peak amplitude of 4-DFT BPSK**The Peak-to-Average Power Ratio**7/17 4-DFT QPSK with max peak amplitude**The Peak-to-Average Power Ratio**8/17 • 4-DFT QPSK**The Peak-to-Average Power Ratio**9/17 The histogram of peak amplitude of 4-DFT QPSK**The Peak-to-Average Power Ratio**10/17 • N-point DFT M-ary PSK • It can be shown that for an M-ary PSK OFDM system, there are at most patterns that yield the highest PAPR, namely, N. • The probability of observing such a PAPR is .**OFDM Signal Amplitude Statistics**11/17 • The time domain OFDM signal is constituted by the sum of complex exponential functions, whose amplitudes and phases are determined by the data symbols transmitted over the different carriers. • Assuming random data symbols, the resulting time domain signal exhibits an amplitude probability density function (PDF) approaching the two-dimensional or complex Gaussian distribution for a high number of subcarriers. • Figure listed below explicitly shows that the measured amplitude histogram of the (a) in-phase component/Quadrature component and (b) amplitude of the a 256-subcarrier OFDM signal obeys a (a) Gaussian distribution and (b) Rayleigh distribution with a standard deviation of .**OFDM Signal Amplitude Statistics**12/17 • The observed amplitude histogram of the 256-subcarrier OFDM signal is correspond to Rayleigh distribution. • Note that the standard deviation of the probability density function is independent of the number of subcarriers employed, since the mean power of the signal is normalized to 1.**OFDM Signal Amplitude Statistics**13/17 • The distribution of I/Q component and amplitude (a) in-phase component/Quadrature component histogram (b) Amplitude histogram**OFDM Signal Amplitude Statistics**14/17 The distribution of Measured amplitude which the value is large than threshold Signal Amplitude CDF**The Distribution of The Peak-to-Average Power Ratio**15/17 • For one OFDM symbol with N subcarrier, the complex baseband signal can be written as • For large N, the real and imaginary values of become Gaussian distributed, each with a mean of zero and a variance ½. • The amplitude of the OFDM signal therefore has a Rayleigh distribution, while the power distribution becomes a central chi-square distribution given by**The Distribution of The Peak-to-Average Power Ratio**16/17 • Cumulative distribution function • Assuming the samples are mutually uncorrelated – which is true for non-oversampling – the probability that the PAPR is below some threshold level can be written as • Assuming the distribution of N subcarriers and oversampling can be approximated by the distribution for subcarriers without oversampling with larger than one.**The Distribution of The Peak-to-Average Power Ratio**17/17 • PAPR distribution without oversampling for a number of subcarriers of (a) 16 (b)32 (c) 64 (d) 128 (e) 256 and (f) 1024**Clipping and Peak Window**1/6 • Clipping the signal • The simplest way to reduce the PAPR • The peak amplitude becomes limited to some desired level • By distorting the OFDM signal amplitude, a kind of self-interference is introduced that degrades the BER. • Nonlinear distortion increases out-of-band radiation • Peak windowing • To remedy the out-of-band problem of clipping • To multiply large signal peaks by nonrectangular window • To minimize the out-of-band interference, ideally the window should be as narrowband as possible. • The windows should not be too long in the time domain, because that implies that many signal samples as affected, which increases the BER.**Clipping Amplitude Methods**2/6 • Clipping – a example of reducing the large peaks in OFDM with the use of windowing**Clipping Amplitude Methods**3/6 • The difference between clipping the signal and windowing the signal**Clipping Amplitude Methods**4/6 • The spectral distortion can be decreased by increasing the windowing**Clipping Amplitude Simulations**5/6 Symbol error rate versus Eb/N0 in AWGN. OFDM signal is clipped to PAPR of (a) no distortion (b) 5 (c) 3 and (d) 1 dB.**Clipping Amplitude Simulations**6/6 Symbol error rate versus Eb/N0 in AWGN. Peak windowing is applied with a window width of 1/16 of the FFT duration.**Peak Cancellation**1/7 • The undesired effect of nonlinear distortion can be avoided by doing a linear peak cancellation technique, whereby a time-shifted and scaled reference function is subtracted from the signal, such that each subtracted reference function reduced the peak power of the least one signal sample. • By selecting an appropriate reference function with approximately the same bandwidth as the transmitted signal, it can be assured that the peak power reduction does not cause any out-of-band interference. • Peak cancellation can be done digitally after generation of the digital OFDM symbols.**Peak Cancellation**2/7 • The peak cancellation was done after parallel-to-serial conversion of signal.**Peak Cancellation**3/7 • The peak cancellation is identical to clipping followed by filtering • Supposed the clipped signal is filtered by an ideal LPF with impulse response of . • are the amplitude, phase, and delay of the correction that is applied to the ith sample in order to reach the desired clipping level.**Peak Cancellation**4/7 • It is also possible to do the cancellation immediately after the IFFT that is done on a symbol-by-symbol basis. • An efficient way to generate the cancellation signal without using a stored reference function is to use a lowpass filter in the frequency domain.**Peak Cancellation**5/7 • It shows an example of the signal envelopes of one arbitrary OFDM symbol and corresponding reference signal. (a) OFDM symbol envelope (b) corresponding reference signal envelope**Peak Cancellation**6/7 • After subtraction, the peak amplitude is reduced to a maximum of 3dB above the RMS value. (a) OFDM symbol envelope (b) signal envelope after peak cancellation**Peak Cancellation**7/7 • Simulated power spectral densities of an OFDM system with 32 carriers by using peak cancellation technique (a) undistorted spectrum, PAPR=15dB (b) spectrum after peak cancellation to PAPR=4dB (c) clipping to PAPR =4dB**PAP Reduction Codes**1/7 • Coding techniques that using a special forward-error-correction code • Golay complementary sequence • Linear block code [17,18]**PAP Reduction CodesGolay complementary sequence**2/7 • Golay complementary sequence • Golay complementary sequences are sequence pairs for which the sum of auto-correlation function is zero for all delay shifts unequal to zero. • The correlation properties of complementary sequences translate into a relatively small PAPR of 3 dB when the codes are used to modulate an OFDM signal.**PAP Reduction CodesGolay complementary sequence**3/7 • For this case of 16 channels, the PAPR is reduced by approximately 9 dB in comparison with the uncoded case. (a) Square root of PAPR for a 16 channel OFDM signal, modulated with the same initial phase for all subcarrier ((b) Square root of PAPR for a 16 channel OFDM signal, modulated with a complementary code.**PAP Reduction CodesLinear block code**4/7 • Linear block code[17,18] • A block coding scheme provides error correction capability, and also achieves the minimum PAPR for the OFDM system utilizing QPSK modulation and 4 subcarriers. • Block coding approach : by selecting only those codewords with small PAPR. Well-designed block codes provide error correction capability.**PAP Reduction CodesLinear block code**5/7 • Block diagram of the OFDM signal with the proposed block coding scheme • The 8 bit vector x becomes 4 complex anti-podal symbols**PAP Reduction CodesLinear block code**6/7 (a) Instantaneous power of an uncoded OFDM system with BPSK modulation and N=4 subcarriers. (b) Instantaneous power of an uncoded OFDM system employing the block coding scheme.**PAP Reduction CodesLinear block code**7/7 • Instantaneous power of an uncoded OFDM system with BPSK modulation and N=4 subcarriers.**Symbol Scrambling**1/10 • The basic idea of symbol scrambling is that for each OFDM symbol, the input sequence is scrambled by a certain number of scrambling sequence, and the output signal is transmitted with the smallest PAPR. • Symbol scrambling techniques • Adaptive subcarrier selection • With the subcarrier allocation scheme • Selected Mapping (SLM) • The transmitter selects one favorable transmit signal from a set of sufficiently different signals which all represent the same information. • Partial Transmit Sequence (PTS) • The transmitter constructs its transmit signal with low PAR by coordinated addition of appropriately phase rotated signal parts. • The difference between SLM and PTS is that the first applies independent scrambling rotations to all subcarriers, while the latter only applies scrambling rotations to group of subcarriers.**Symbol Scrambling - ASUS**2/10 • OFDM system using ASUS (adaptive subcarrier selection) [20,21]**Symbol Scrambling - SLM**3/10 • Selected Mapping (SLM) • Generate U transmit sequences , representing the same information for each OFDM symbols. • Select the lowest PAPR in time-domain of U sequences to transmit • Define U distinct vectors , , (number of subcarriers) , . • Each OFDM frame is multiplied carrierwise with U vectors:**Symbol Scrambling - SLM**4/10 • Selected Mapping (SLM)**Symbol Scrambling - SLM**5/10 • Selected Mapping (SLM) • SLM requires U IDFT’s in the transmitter, while the receiver still needs only one DFT. • bits are required to explicitly represent the side information. • Moderate complexity. • For arbitrary number of carriers and any signal constellation. • Distortionless.**Symbol Scrambling - SLM**6/10 • Performance of SLM • Known side information**Symbol Scrambling - PTS**7/10 • Partial Transmit Sequence (PTS) • The information bearing subcarrier block is subdivide into V pairwise disjoint carrier subblocks . • All subcarrier positions in which are already represented in another subblock are set to zero . • Rotation factor for each subblock v and the modified subcarrier vector represents the same information as . • The subblocks are transformed by V separate IDFTs. • Choose the rotation factor that minimize PAPR. • Optimum transmitted sequence .**Symbol Scrambling - PTS**8/10 • Partial Transmit Sequence (PTS)