chapter 6 the peak-to-average power ratio problem

NCCU Wireless Comm. Lab.

Chapter 6The Peak-to-Average Power Ratio Problem

NCCU Wireless Comm. Lab.2

Chapter 66 The Peak-to-Average Power Ratio Problem

6.1 The Peak-to-Average Power Ratio [1-7]6.1.1 OFDM Signal Amplitude Statistics[4,13]6.1.2 The distribution of The Peak-to-Average Power Ratio [ 1,4,16]

6.2 Clipping and Peak Window [1,4,10,11]6.2.1 Clipping Amplifier Methods6.2.2 Clipping Amplifier Simulations

6.3 Peak Cancellation [1,4,8,9,14,15]6.4 PAP Reduction Codes [14,17,18.19]6.5 Symbol Scrambling [12,14,20,21]


The Peak-to-Average Power Ratio ProblemIt 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-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

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PAPR ~ number of subcarriers =N


The Peak-to-Average Power Ratio ProblemThe 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.

ClippingPeak windowPeak Cancellation

2.Coding techniques that using a special forward-error-correction code

PAP Reduction Code3. 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)


6.1 The Peak-to-Average Power RatioSignal Expression

Let x(t) and y(t) 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.

[ ]0, st T∈

( ) ( ) ( ) ( )( )( )12 1 2

0

1 sN

j k N T tk

k

s t x t jy t A eN

π−

− −

=

= + = ∑kA sT


6.1 The Peak-to-Average Power RatioPAPR Definition

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

( ) ( ){ } ( ) ( )2 cos2 sin2cj f tc cs t s t e x t f t y t f tπ π π=ℜ = −

cf

( )( ){ }

20

2

m a x t T s s t

E s t≤ ≤Ρ


6.1 The Peak-to-Average Power RatioIf 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 exampleIt 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 .

2M2

2 NN

M MM

−=


6.1 The Peak-to-Average Power RatioBasic waveforms of OFDM signal with 4-DFT BPSK

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Maximum PAPR case[1,1,1,1] [4,0,0,0][ 1, 1, 1, 1] [ 4,0,0,0][1, 1,1, 1] [0,0,4,0][ 1,1, 1,1] [0,0, 4,0]

x Xx Xx Xx X

= ⇒ == − − − − ⇒ = −= − − ⇒ == − − ⇒ = −

Re

Im

1-1


6.1 The Peak-to-Average Power RatioOFDM signal with 4-DFT BPSK

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6.1 The Peak-to-Average Power RatioThe histogram of PAPR value

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4 4


6.1.1 OFDM Signal Amplitude StatisticsThe 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 . 1

2σ =


6.1.1 OFDM Signal Amplitude StatisticsThe 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.


6.1.1 OFDM Signal Amplitude Statistics

-4 -3 -2 -1 0 1 2 3 40

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abili

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9dB 6dB 3dB 12dB 0dB

(a) in-phase component/Quadrature component histogram (b) Amplitude histogram

Measured amplitude distribution


6.1.1 OFDM Signal Amplitude Statistics

Signal Amplitude CDF0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-5

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100

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0dB 3dB 6dB 9dB 12dB

Measured signal CDF distribution


6.1.2 The distribution of The Peak-to-Average Power Ratio

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

( )1

1n

Nj t

nn

x t a eN

ω

=

= ∑( )x t

( ) 1 zF z e −= −



Cumulative distribution functionAssuming 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.

( ) ( ) (1 )N z Np PAPR z F z e−≤ = = −

( ) ( )1Nzp PAPR z e

α⋅−≤ = −

αNα



PAPR distribution without sampling for a number of subcarriers of (a) 16 (b)32 (c) 64 (d) 128 (e) 256 and (f) 1024

0 2 4 6 8 10 12 14 1610

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CD

F)

(a )N=16 (b) N=32 (c) N=64 (d) N=128 (e ) N=256 (f) N=1024

(f)

(e )

(d) (c)

(b)

(a )


6.2 Clipping and Peak WindowClipping the signal

The simplest way to reduce the PAPRThe peak amplitude becomes limited to some desired levelBy distorting the OFDM signal amplitude, a kind of self-interference is introduced that degrades the BER. Nonlinear distortion increase out-of-band radiation

Peak WindowingTo remedy the out-of-band problem of clippingMultiply large signal peaks with 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.


6.2.1 Clipping Amplifier MethodsClipping – a example of reducing the large peaks in OFDM with the use of windowing


6.2.1 Clipping Amplifier MethodsThe difference between clipping the signal and windowing the signal


6.2.1 Clipping Amplifier MethodsThe spectral distortion can be decreased by increasing the windowing


6.2.2 Clipping Amplifier Simulations

Packet error rate versus Eb/N0 in AWGN. OFDM signal is clipped to PAPR of (a) 16 (=no distortion), (b) 6, (c) 5, and (d) 4 dB.


6.2.2 Clipping Amplifier Simulations

Packet error rate versus Eb/N0 in AWGN. Peak windowing is applied with a window width of 1/16 of the FFT duration.OFDM signal is clipped to PAPR of (a) 16 (=no distortion), (b) 6, (c) 5, and (d) 4 dB.


6.3 Peak CancellationThe 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.


6.3 Peak CancellationThe peak cancellation was done after parallel-to-serial conversion of signal.


6.3 Peak CancellationThe 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.

, , ,i i ia andϕ τ


6.3 Peak CancellationIt is also possible to do the cancellation immediately after theIFFT 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.


6.3 Peak CancellationIt shows an example of the signal envelopes of one arbitrary OFDM symbol and corresponding cancellation signal.

(a) OFDM symbol envelope (b) cancellation signal envelope


6.3 Peak CancellationAfter 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


6.3 Peak CancellationSimulated 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


6.4 PAP Reduction CodesCoding techniques that using a special forward-error-correction code

Golay complementary sequence Linear block code [17,18]


6.4 PAP Reduction CodesGolay complementary sequence

Golay complementary sequenceGolay 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 translates into a relatively small PAPE of 3 dB when the codes are used to modulate an OFDM signal.


6.4 PAP Reduction CodesGolay complementary sequence

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.


6.4 PAP Reduction CodesLinear block code

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 diagram of the OFDM signal with the proposed block coding scheme



(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.

101610log ( ) 6.0205 ( )4

PAPR dB= = 107.0710log ( ) 2.4735 ( )

4PAPR dB= =


6.5 Symbol ScramblingThe 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 selectionWith 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.


6.5 Symbol ScramblingOFDM system using ASUS (adaptive subcarrier selection) [20,21]


6.5 Symbol ScramblingSelected Mapping (SLM)

Generate U transmit sequences representing the same information for each OFDM symbols. Select the lowest PAPR in time-domain of U sequence to transmitDefine U distinct vector , ,

, (number of subcarrier), .Each OFDM frame is multiplied carrierwise with U vectors:

( )uaµ

( ) ( ) ( )1 ,...,u u u

DP P P = ( ) ( )u

vu jVP e φ=

( ) [ )0,2uvφ π∈

( ) [ ] [ ] ( ), 1: , 1:

uvu jA v A v e v D u Uφ

µ µ= ⋅ = =

1:v D= 1:u U=



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 numbers of carriers and any signal constellation.Distortionless.

[ ]2log U


6.5 Symbol ScramblingPartial 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 rotation factor that minimize PAPR. Optimum transmit sequence . ( ) ( )

1

optimum rotation factor

vV

v

v

ba aµµ µ=

= ⋅

↑

∑

( )

1

Vv

v

A Aµ µ=

= ∑

Aµ( ) , 1, , .vA v Vµ = …

( ) ( ) ( ) [ ), 0,2vjv vb e µφ

µ µφ π= ∈( ) ( )

1

Vv v

v

A b Aµ µ µ=

= ⋅∑Aµ


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 ISCOM’99, 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 Multicarrier Modulation, Wiley-Interscience.[8] S. Müller and J. Huber, “A Novel Peak Power Reduction Scheme for OFDM,” In IEEE Int. Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '97), Helsinki, Finland, pp. 1090-1094, September 1997.[9] 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, November 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 , Volume: 18 Issue: 11 , Nov. 2000 Page(s): 2270 –2277. [11] Wulich, D.; Dinur, N.; Glinowiecki, A,“Level clipped high-order OFDM,” Communications, IEEE Transactions on , Volume: 48 Issue: 6 , June 2000 Page(s): 928 –930.


References[12] S. Müller and J. Huber, “OFDM with Reduced Peak-to-Average Power Ratioby Optimum Combination of Partial Transmit Sequences, ” Electronics Letters, vol. 33, no. 5, pp. 368-369, February 1997. [13] S. Müller, R. Bäuml, R. Fischer, and J. Huber, “OFDM with Reduced Peak-to-Average Power Ratio by Multiple Signal Representation,” Annals of Telecommunications, vol. 52, no. 1-2, pp. 58-67, February 1997.[14] 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, November 1997.[15] M. Breiling, S. Müller-Weinfurtner, and J. Huber, “SLM Peak-Power Reduction without Explicit Side Information,” In IEEE Communications Letters, vol. 5, no. 6, pp. 239-241, June 2001. [16] Ochiai, H. and Imai H. ,“On the distribution of the peak-to-average power ratio in OFDM signals,”Communications, IEEE Transactions on , Volume: 49 Issue: 2 , Feb 2001 Page(s): 282 –289.[17] Hyo-Joo Ahn, Yoan Shin and Sungbin Im, “A block coding scheme for peak-to-average power ratio reduction in an orthogonal frequency division multiplexing system,” Vehicular Technology Conference Proceedings, 2000. VTC 2000-Spring Tokyo. 2000 IEEE 51st , Volume: 1 , 2000 Page(s): 56 -60 vol.1.[18] Pingyi Fan; Xiang-Gen Xia, “Block coded modulation for the reduction of the peak to average power ratio in OFDM systems,” Consumer Electronics, IEEE Transactions on , Volume: 45 Issue: 4 , Nov. 1999 Page(s): 1025 -1029 [19] Fernando, W.A.C.; Rajatheva, R.M.A.P. “Performance of turbo and trellis coded OFDM for LEO satellite channels in global mobile communications “Communications, 1998. ICC 98. Conference Record. 1998 IEEE International Conference on , Volume: 1 , 1998 Page(s): 412 -416 vol.1


References[20] Rohling, H.; Grunheid, R. “ Performance of an OFDM-TDMA mobile communication system ”Vehicular Technology Conference, 1996. Mobile Technology for the Human Race., IEEE 46th , Volume: 3 , Page(s): 1589 -1593 vol.3, 1996.[21] Schmidt, H. and Kammeyer, K.-D., “Reducing the peak to average power ratio of multicarrier signalsby adaptive subcarrier selection,” Universal Personal Communications, 1998. ICUPC '98. IEEE 1998 International Conference on , Volume: 2 , Page(s): 933 -937 vol.2, 1998.

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