The Peak-to-Average Power Ratio Problem

1
The Peak-to-Average Power Ratio Problem

• Gwo-Ruey Lee

2
Outlines

• The Peak-to-Average Power Ratio Problem
• The Peak-to-Average Power Ratio 1-7
• OFDM Signal Amplitude Statistics4,13
• 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

3
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.

4
The Peak-to-Average Power Ratio Problem
2/3
PAPR number of subcarriers N
5
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)

6
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.

7
The Peak-to-Average Power Ratio PAPR 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

8
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
  .

9
The Peak-to-Average Power Ratio
4/17

• Basic waveforms of OFDM signal with 4-DFT BPSK

10
The Peak-to-Average Power Ratio
5/17

• OFDM signal with 4-DFT BPSK

11
The Peak-to-Average Power Ratio
6/17

• The histogram of peak amplitude of 4-DFT BPSK

12
The Peak-to-Average Power Ratio
7/17
4-DFT QPSK with max peak amplitude
13
The Peak-to-Average Power Ratio
8/17

• 4-DFT QPSK

14
The Peak-to-Average Power Ratio
9/17
The histogram of peak amplitude of 4-DFT QPSK
15
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
  .

16
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
  .

17
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.

18
OFDM Signal Amplitude Statistics
13/17

• The distribution of I/Q component and amplitude

(a) in-phase component/Quadrature component
histogram
(b) Amplitude histogram
19
OFDM Signal Amplitude Statistics
14/17
The distribution of Measured amplitude which the
value is large than threshold
Signal Amplitude CDF
20
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

21
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.

22
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

23
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.

24
Clipping Amplitude Methods
2/6

• Clipping a example of reducing the large peaks
  in OFDM with the use of windowing

25
Clipping Amplitude Methods
3/6

• The difference between clipping the signal and
  windowing the signal

26
Clipping Amplitude Methods
4/6

• The spectral distortion can be decreased by
  increasing the windowing

27
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.

28
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.

29
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.

30
Peak Cancellation
2/7

• The peak cancellation was done after
  parallel-to-serial conversion of signal.

31
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.

32
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.

33
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

34
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

35
Peak Cancellation
7/7

• Simulated power spectral densities of an OFDM
  system with 32 carriers by using peak
  cancellation technique
• (a) undistorted spectrum, PAPR15dB (b)
  spectrum after peak cancellation to PAPR4dB (c)
  clipping to PAPR 4dB

36
PAP Reduction Codes
1/7

• Coding techniques that using a special
  forward-error-correction code
• Golay complementary sequence
• Linear block code 17,18

37
PAP Reduction Codes Golay 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.

38
PAP Reduction Codes Golay 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.
39
PAP Reduction Codes Linear block code
4/7

• Linear block code17,18
• 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.

40
PAP Reduction Codes Linear 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

41
PAP Reduction Codes Linear block code
6/7
(a) Instantaneous power of an uncoded OFDM system
with BPSK modulation and N4 subcarriers.
(b) Instantaneous power of an uncoded OFDM system
employing the block coding scheme.
42
PAP Reduction Codes Linear block code
7/7

• Instantaneous power of an uncoded OFDM system
  with BPSK modulation and N4 subcarriers.

43
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.

44
Symbol Scrambling - ASUS
2/10

• OFDM system using ASUS (adaptive subcarrier
  selection) 20,21

45
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

46
Symbol Scrambling - SLM
4/10

• Selected Mapping (SLM)

47
Symbol Scrambling - SLM
5/10

• Selected Mapping (SLM)
• SLM requires U IDFTs 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.

48
Symbol Scrambling - SLM
6/10

• Performance of SLM
• Known side information

49
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
  .
  
  

50
Symbol Scrambling - PTS
8/10

• Partial Transmit Sequence (PTS)

51
Symbol Scrambling - PTS
9/10

• Partial Transmit Sequence (PTS)

52
Symbol Scrambling - PTS
10/10

• Performance of PTS
• Known phase rotation

53
The Peak-to-Average Power Ratio Problem

• Readings
• Ochiai, H. and Imai H. ,On the distribution of
  the peak-to-average power ratio in OFDM signals,
  Communications, IEEE Transactions on , Vol. 49,
  Issue 2, pp. 282 289, Feb. 2001.
• S. Müller and J. Huber, A Comparison of Peak
  Power Reduction Schemes for OFDM, In IEEE Global
  Telecommunications Conference (GLOBECOM '97),
  Phoenix, Arizona, USA, pp. 1-5, Nov. 1997.

54
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55
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