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The Peak-to-Average Power Ratio Problem

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Title: 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
References
  • 1 Richard van Nee, Ramjee Prasad, OFDM wireless
    multimedia communication, Artech House Boston
    London, 2000.
  • 2 Ahmad R. S. Bahai and Burton R. Saltzberg,
    Multi-carrier digital communications - Theory and
    applications of OFDM, Kluwer Academic / Plenum
    Publishers New York, Boston, Dordrecht, London,
    Moscow 1999.
  • 3 Ramjee Prasad, OFDM based wireless broadband
    multimedia communication, Letter Notes on
    ISCOM99, Kaohsiung, Taiwan, Nov. 7-10, 1999.
  • 4 L. Hanzo, W. Webb and T. Keller, Single- and
    multi-carrier quadrature amplitude modulation
    Principles and applications for personal
    communications, WLANs and broadcasting, John
    Wiley Sons, Ltd, 2000.
  • 5 Mark Engels, Wireless Ofdm Systems How to
    Make Them Work? Kluwer Academic Publishers.
  • 6 Lajos Hanzo, William Webb, Thomas Keller,
    Single and Multicarrier Modulation Principles
    and Applications, 2nd edition, IEEE Computer
    Society.
  • 7 John A. C. Bingham, ADSL, VDSL, and
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    pp. 1090-1094, Sep. 1997.
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    Power Reduction Schemes for OFDM, In IEEE Global
    Telecommunications Conference (GLOBECOM '97),
    Phoenix, Arizona, USA, pp. 1-5, Nov.1997.
  • 10 Ochiai, H. Imai, H, Performance of the
    deliberate clipping with adaptive symbol
    selection for strictly band-limited OFDM systems,
    Selected Areas in Communications, IEEE Journal
    on , Vol. 18 Issue 11, pp. 2270 2277, Nov.
    2000.

55
References
  • 11 Wulich, D. Dinur, N. Glinowiecki, A,Level
    clipped high-order OFDM, Communications, IEEE
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    June 2000.
  • 12 S. Müller and J. Huber, OFDM with Reduced
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  • 17 Hyo-Joo Ahn, Yoan Shin and Sungbin Im, A
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56
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