OFDM Wireless Multimedia
`Communications
`
`i
`
`«
`
`Richard van Nee
`Ramjee Prasad
`
`zivemzlPersonal Communications
`this book.
`
`‘
`‘
`
`EH
`Arte C h H 0 USe
`Boston 0 London
`
`Page 1 of 38
`
`Ericsson v IV
`
`ERIC-1009
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`ERIC-1009
`Ericsson v IV
`Page 1 of 38
`
`

`
`
`
`Library of Congress Cataloging—in-Publication Data
`Nee, Richard van.
`
`OFDM for wireless multimedia communications / Richard van Nee, Rarnjee Prasad
`p. cm. — (Artech House universal personal communications library)
`Includes bibliographical references and index.
`ISBN O-89006-530-6 (alk. paper)
`1. Wireless communications systems. 2. Multimedia systems. 3. Multiplexing.
`Prasad, Ramjee. II. Title III. Series
`
`I.
`
`TK5103.2.N44 2000
`621/3845~dc21
`
`99-052312
`’ CIP
`
`
`
`All rights reserved. Printed and bound in the United States of America. No part of this book
`may be reproduced or utilized in any form or by any means, electronic or mechanical, includ-
`ing photocopying, recording, or by any information storage and retrieval system, without per—
`mission in writing from the authors.
`All terms mentioned in this book that are known to be trademarks or service marks have
`been appropriately capitalized. Artech House cannot attest to the accuracy of this informa-
`tion. Use of a term in this book should not be regarded as affecting the validity of any trade-
`mark or service mark.
`
`British Library Cataloguing in Publication Data
`Nee, Richard van
`OFDM wireless multimedia communications. — (Artech House
`universal personal communications library)
`l.Wireless communications systems 2. Multimedia systems
`I. Title II. Prasad, Ramjee
`6213,82
`
`ISBN 0—89006-530—6
`
`Cover design by Igor Vladman
`
`© 2000 Richatd van Nee and Ramjee Prasad
`
`International Standard Book Number: 0-89006530-6
`Library of Congress Catalog Card Number: 99—O52312
`10987654321
`
`
`
`ERIC-1009 I Page 2 of 38
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`ERIC-1009 / Page 2 of 38
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`

`
` CHAPTER 6
`
`The Peak Power Problem
`
`6.1
`
`INTRODUCTION
`
`
`
`
`
`An OFDM signal consists of a number of independently modulated subcarriers, which
`can give a large peak—to—average power (PAP) ratio when added up coherently. When N
`signals are added with the same phase, they produce a peak power that is N times the
`average power. This effect is illustrated in Figure 6.1. For this example, the peak power
`is 16 times the average value. The peak power is defined as the power of a sine wave
`with an amplitude equal to the maximum envelope value. Hence, an unmodulated I
`carrier has a PAP ratio of 0 dB. An alternative measure of the envelope variation of a
`signal is the Crestfactor, which is defined as the maximum signal value divided by the
`rms signal value. For an unmodulated carrier, the Crest factor is 3 dB. This 3 dB
`difference between PAP ratio and Crest factor also holds for other signals, provided
`that the center frequency is large in comparison with the signal bandwidth.
`
`A large PAP ratio brings disadvantages like an increased ‘complexity of the
`analog-to—digital and digital—to—analog converters and a reduced efficiency of the RF
`power amplifier. To reduce the PAP ratio, several techniques have been proposed,
`which basically can be divided in three categories. First, there are signal distortion
`techniques, which reduce the peak amplitudes simply by nonlinearly distorting the
`OFDM signal at or around the peaks. Examples of distortiontechniques are clipping,
`peak windowing and peak cancellation. The second category is coding techniques that
`use a special forward—error correcting code set that excludes OFDM symbols with a
`large PAP ratio. The third technique is based on scrambling each OFDM symbol with
`different scrambling sequences and selecting that sequence that gives the smallest PAP
`ratio. This chapter discusses all of these techniques, but first makes an analysis of the
`PAP ratio distribution function. This will give a better insight in the PAP problem and
`will explain why PAP reduction techniques can be quite effective.
`
`{
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`
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`ERIC-1ioo9j Pagéfs of 3.3
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`ERIC-1009 / Page 3 of 38
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`

`
`120
`
`..'
`
`
`
`qPAP
`
`0
`
`2
`
`4
`
`6
`
`8
`Time/T’
`
`10
`
`12
`
`14
`
`16
`
`Figure 6.1 Square root of peak—to-average power ratio for a 16-channel OFDM signal, modulated with
`the same initial phase for all subchannels.
`
`6.2
`
`DISTRIBUTION OF THE PEAK-TO-AVERAGE POWER RATIO
`
`For one OFDM symbol with N subcarriers, the complex baseband signal can be written
`as
`
`x(t) = —‘/lwrgan exp(ja)nt)
`
`(6.1)
`
`Here, a,, are the modulating symbols. For QPSK, for instance, an E {—1,1,j,—j}.
`From the central limit theorem it follows that for large values of N,
`the real and
`imaginary values of x(t) become Gaussian distributed, each with a mean of zero and a
`variance of 1/2. The amplitude of the OFDM signal
`therefore has a Rayleigh
`distribution, while the power distribution becomes a central chi—square distribution with
`two degrees of freedom and zero mean, with a cumulative distribution given by
`
`F(z) =1— e”
`
`(6.2)
`
`
`
`
`
`ERIC-1009 / Page 4 of 38
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`

`
`log(P(PAPR>abscissa))
`
`
`
`PAP ratio in dB
`
`
`
`Figure 6.2 PAP distribution of an OFDM signal with (a) 12, (b) 24, (c) 48 and (d) an infinite number of
`subcarriers (pure Gaussian noise). Four times oversampling used in simulation, total number
`of simulated samples = 12 million.
`
`Figure 6.2 shows the probability that the PAP ratio exceeds a certain value. We
`can see that the curves for various numbers of subcarriers are close to a Gaussian
`distribution (d) until the PAP value comes within a few dB from the maximum PAP
`level of 10logN, where N is the number of subcarriers.
`
`What we want to derive now is the cumulative distribution function for the peak
`power per OFDM symbol. Assuming the samples are mutually .uncorrelated——which is
`true for non—oversaInpling——the probability that the PAP ratio is below some threshold
`level can be written as
`
`P(PAPR s z) = F(z)” = (1— exp(— z))”
`
`(6.3)
`
`This theoretical derivation is plotted against simulated Values in Figure 6.3 for
`different values of N.
`
`:" .
`
`ER|C-1009_/ Page 5 of 38
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`ERIC-1009 / Page 5 of 38
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`122
`
`
`
`
`
`6
`
`8
`
`10
`
`
`12
`
`14
`
`PAPR [dB]
`
`Figure 6.3 PAP distribution without oversampling for a number of subcarriers of (a) 16, (b) 32, (c) 64,
`(d) 128, (e) 256, and (f) 1024 (dotted lines are simulated).
`
`The assumption made in deriving (6.3) that the samples should be mutually
`uncorrelated is not true anymore when oversampling is applied. Because it seems quite
`difficult to come up with an exact solution for the peak power distribution, we propose
`an approximation by assuming that the distribution for N subcarriers and oversampling
`can be approximated by the distribution for ocN subcarriers without oversampling, with
`oc larger than one. Hence, the effect of oversampling is approximated by adding a
`certain number of extra independent samples. The distribution of the PAP ratio is then
`given by
`
`P(PAPR s z) = (1 — exp(— z))"‘”
`
`(6.4)
`
`In Figure 6.4, the PAP distribution for different amounts of carriers is given for
`or = 2.8. The dotted lines are simulated curves. We see in Figure 6.4 that Equation (6.4)
`is quite accurate for N>64. For large values of the cumulative distribution function
`close to one (>O.5), however, (6.3) is actually more accurate.
`From Figure 6.4, we can deduce that coding techniques to reduce the PAP ratio
`may be a viable option, as reasonable coding rates are possible for a PAP ratio around 4
`dB. For 64 subcarriers, for instance, about 106 of all possible QPSK symbols have a
`PAP ratio of less than 4.2 dB. This means that only 20 out of a total of 128 bits would
`be lost if only the symbols with a low PAP ratio would be transmitted. However, the
`main problem with this approach is to find a coding scheme with a reasonable coding
`rate (2 1/2)
`that produces only these low PAP ratio symbols and that also has
`
`ERIC-1009 I Page sass. ,
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`ERIC-1009 / Page 6 of 38
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`
`
`‘Y‘""‘W"4"(’~”3».“"="""3'-wsrv-.,,..,...__,,
`
`
`
`A different approach to the PAP problem is to use the fact that because large
`it is possible to remove these peaks at the cost of a
`PAP ratios occur only infrequently,
`ow, the challenge is to keep the spectral pollution
`slight amount of self—interference. N
`e. Clipping is one example of a PAP
`of this self—interference as small as possibl
`ence. In the next sections, two other techniques
`reduction technique creating self interfer
`are described which have better spectral properties than clipping.
`
`
`
`PAPR [dB]
`
`nction of the PAPR for a number of subcarriers of (a) 32, (b) 64,
`Figure 6.4 Cumulative distribution fu 24. Solid lines are calculated; dotted lines are simulated.
`(c) 128, (d) 256, and (e) 1,0
`
`6.3
`CLIPPING AND PEAK WINDOWING
`The simplest way to reduce the PAP ratio is to clip the signal, such that the peak
`amplitude becomes limited to some desired maximum level. Although clipping is
`definitely the simplest solution, there are a few problems associated with it. First, by
`rid of self-interference is introduced that
`distorting the OFDM signal amplitude, a ki
`degrades the BER. Second, the nonlinear distortion of the OFDM signal significantly
`increases the level of the out-of—band radiation. The latter effect can be understood
`easily by viewing the clipping operation as a multiplication of the OFDM signal by a
`rectangular window function that is 1 if the OFDM amplitude is below a threshold and
`smaller than 1 if the amplitude needs to be clipped. The spectrum of the clipped OFDM
`signal is found as the input OFDM spectrum convolved with the spectrum of the
`Window function. The out-of—band spectral properties are mainly determined by the
`
`\ ERIC-1009 I Page 7 of 38
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`ERIC-1009 / Page 7 of 38
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`

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`
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`124
`
`
`wider spectrum of the two, which is the spectrum of the rectangular window function.
`This spectrum has a very slow rolloff that is inversely proportional to the frequency.
`To remedy the out—of—band problem of clipping, a different approach is to
`multiply large signal peaks with a certain nonrectangular window. In [1], a Gaussian
`shaped window is proposed for this, but in fact any window can be used, provided it
`has good spectral properties. To minimize the out—of—band interference, ideally the
`window should be as narrowband as possible. On the other hand, "the window should
`not be too long in the time domain, because that implies that many signal samples are
`affected, which increases the BER. Examples of suitable window functions are the
`cosine, Kaiser, and Hamming windows. Figure 6.5 gives an example of reducing the
`large peaks in OFDM with the use of windowing.
`20
`
`18
`
`1 6
`
`Original signal
`
`14
`
`
`12
`
`Amplitude
`
`Multiplication
`si ,-, nal
`
`O
`
`50
`
`100
`
`150
`
`200
`
`250
`
`300
`
`Time in samples
`
`Figure 6.5 Windowing an OFDM time signal.
`
`In Figure 6.6, the difference between clipping the signal and windowing the
`signal can be seen. Figure 6.7 shows how the spectral distortion can be decreased by
`increasing the window width.
`Figure 6.8 shows PER curves with and without clipping, using a rate 1/2
`convolutional code with constraint length 7. The simulated OFDM signal used 48
`subcarriers with 16-QAM. The plots demonstrate that nonlinear distortion only has a
`minor effect on the PER; the loss in SNR is about 0.25 dB when the PAP ratio is
`decreased to 6 dB. When peak windowing is applied, the results are slightly worse; see
`Figure 6.9. This is caused by the fact that peak windowing distorts a larger part of the
`signal than clipping for the same PAP ratio.
`
`ERIC-1009 I Page 33
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`3:3;
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`ERIC-1009 / Page 8 of 38
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`

`
`Undistorted
`
`spectrum
`
`
`
`-60
`
`0
`
`Frequency/subcarrier spacing
`
`Figure 6.6 Frequency spectrum of an OFDM signal with 32 subcarriers with clipping and peak
`windowing at a threshold level of 3 dB above the rms amplitude.
`
`
`
`Frequency spectrum of an OFDM signal with 32 subcaniers with peak windowing at a
`threshold level of 3 dB above the rms amplitude. Symbol length is 128 samples (4 times
`oversampled) and window length is (a) 3, (b) 5, (c) 7, (d) 9, (e) 11, (i) 13, and (g) 15
`samples. Curve (h) is the ideal OFDM spectrum.
`
`PSD[dBr]
`
`-60
`
`60
`30
`0
`-30
`Frequency/subcarrier spacing
`
`Figure 6.7
`
`ERIC-1009 I Pagep9 of 38
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`ERIC-1009 / Page 9 of 38
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`

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`
`
`.
`
`.
`
`(a) 5
`
`
`
`12
`
`Figure 6.8 Packet error ratio versus E,/No for 64 byte packets in AWGN. OFDM signal is clipped to a
`PAP ratio of (a) 16 ( = no distortion), (b) 6, (c) 5, and (d) 4 dB.
`
`Eb/No [C113]
`
`Figure 6.9 PER versus E,/N0 for 64—byte packets in AWGN. Peak windowing is applied with a window
`width of 1/16 of the FFT duration. The PAP ratio is reduced to (a) 16 ( = no distortion), (b)
`6, (c) 5, and (d) 4 dB.
`
`
`
` E:/No [dB]
`
`10
`
`—I
`11
`
`i11
`
`;
`I
`'
`9
`
`.
`
`1o
`
`PER
`
`PER
`
`
`
`V
`
`‘
`‘E
`2.5,ya
`
`VF
`
`ERIC-1009 / Page 10 of 38
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`

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`
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`127
`
`6.3.1 Required Backoff with a Non-Ideal Power Amplifier
`The previous section demonstrated that peak windowing is very effective in reducing
`the PAP ratio. This does not immediately tell us, however, what backoff is required for
`a practical power amplifier to attain an acceptable level of out-of—band radiation. The
`backoff is defined here as the ratio of the output power and the maximum output power
`(saturation power) with a sinusoidal input signal. Another definition that is frequently
`used in the literature uses the power at the 1—dB compression point instead of the
`saturation power. Because the 1—dB compression point is typically 1 to 3 dB lower than
`the maximum power level, depending on the amplifier transfer function, the backoff
`values according to the latter definition are 1 to 3 dB smaller than the values mentioned
`in this section.
`To simulate a power amplifier, the following model is used for the AM/AM
`conversion [2]:
`
`gm) =
`
`
`A
`,
`(1+ A2” )5
`
`(6.5)
`
`The AM/PM conversion of a solid—state power amplifier is small enough to be
`neglected. Figure 6.10 gives some examples of the transfer function for various values
`of p. A good approximation of existing amplifiers is obtained by choosing p in the
`range of 2 to 3 [2]. For large values of p, the model converges to a clipping amplifier
`that is perfectly linear until it reaches its maximum output level.
`
`1.8
`
`1.6
`
`1.4
`
`
`
`Outputvalue
`
`1.2
`
`1
`
`' 0.8
`
`0.6
`
`0.4
`
`0.2
`
`
`
`0
`
`
`0.8
`1
`1.2
`0
`0.2
`0.4
`0.6
`
`1.4
`
`1.6
`
`1.8
`
`Input value
`
`Figure 6.10 Rapp’s model of AM/AM conversion.
`
`ERIC-1,009 I Page 11 of 38
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`ERIC-1009 / Page 11 of 38
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`

`
`
`
`‘.1
`l
`E,
`[H‘N?
`l
`
`',
`
`128
`
`Figure 6.11 shows the output spectra of an undistorted OFDM signal and the
`spectra of two distorted signals, assuming a highly linear amplifier model (p = 10 in
`(6.5). The backoff relative to the maximum output power was determined such that any
`significant distortion of the spectrum is at least 50 dB below the in—band spectral
`density. In this case, peak windowing gives a gain of almost 3 dB in the required
`backoff relative to clipping. This difference in backoff is much less than the difference
`in PAP ratio at the input of the power amplifier; without peak windowing, the PAP
`ratio is about 18 dB for the OFDM signal with 64 subcarriers. With peak windowing,
`this PAP ratio is reduced to approximately 5 dB. Hence, for the latter case, it is clear
`that the backoff of a highly linear amplifier must be slightly above this 5 dB to achieve
`a minimal spectral distortion. It is not true, however, that without peak windowing, the
`backoff must be in the order of 18 dB for the same amount of distortion as with peak
`windowing. The reason is that there is little energy in the signal parts that have a
`relatively large PAP ratio, so it does not affect the spectrum that much if those parts are
`distorted. After peak windowing or any other PAP reduction technique, however, a
`significant part of the signal samples are close to the maximum PAP ratio (e.g., 5 dB);
`in this case, any distortion of samples that is a dB or so below this maximum produces
`more spectral distortion than clipping the original OFDM signal at 10 dB below its
`maximum PAP level, simply because for the latter, a much smaller fraction of the
`signal is affected. Thus, the lower the PAP ratio is made by PAP-reduction techniques,
`the less tolerant the signal becomes against nonlinearities in the area of its maximum
`PAP ratio.
`
`PSD[dBr]
`
`Frequency / bandwidth
`
`
`
`p '_—.
`levc
`
`pea
`pol
`
`91$LT»-]
`
`Figure 6.11 (a) Ideal OFDM spectrum for 64 subcarriers, (b) spectrum after highly linear amplifier
`(Rapp’s parameter p = 10) with 8.7-dB backoff, (c) spectrum using peak windowing with
`5.9—dB backoff.
`
`ERIC-1009)/,’tPlalge 1i2lof 33. g
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`=2
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`«
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`ERIC-1009 / Page 12 of 38
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`

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`129
`
`
`Figure 6.12 shows OFDM spectra for a more realistic amplifier model with
`p = 3. The target for undesired spectrum distortion has now been set to a less stringent
`level of 30 dB below the in—band density. The difference in backoff with and without
`peak windowing is now reduced to 1 dB. This demonstrates that the more spectral
`pollution can be tolerated, the less gain can be achieved with PAP reduction techniques.
`
`
`B£”'E65:’E)a'..m'(fis:>WFnc6l§‘i‘*'U<"oO.‘:L-k<::cI:
`
`PSD[dBr]
`
`
`
`
`-1
`-0.8 -0.6 -0.4 -0.2
`O
`0.2
`0.4
`0.6
`0.8
`1
`
`Frequency/bandwidth
`
`Figure 6.12 (a) Ideal OFDM spectrum for 64 subcarriers, (b) plain OFDM with 6.3-dB backoff and
`Rapp’s parameter p = 3, (c) peak windowing with 5.3-dB backoff.
`
`Figure 6.13 shows similar plots as Figure 6.12, but now for 256 subcarriers.
`This demonstrates that the required backoff with or without peak windowing is almost
`independent from the number of subcarriers, as long as this number is large compared
`with 1. In fact, the difference in backoff with and without peak windowing reduced
`slightly to 0.8 dB by going from 64 to 256 subcarriers.
`
`ERIC-61009] Page 13 of 38
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`ERIC-1009 / Page 13 of 38
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`

`
`130
`
`
`PSD[dBr]
`
`Frequency/B andwidth
`
`Figure 6.13
`
`(a) Ideal OFDM spectrum for 256 subcarriers, (b) plain OFDM with 6.3-dB backoff and
`Rapp’s parameter p = 3, and (c) peak windowing with 5.5—dB backoff.
`
`6.3.2 Coding and Scrambling
`
`A disadvantage of distortion techniques is that symbols with a large PAP ratio suffer
`more degradation, so they are more vulnerable to errors. To reduce this effect, forward-
`error correction coding can be applied across several OFDM symbols. By doing so,
`errors caused by symbols with a large degradation can be corrected by the surrounding
`symbols. In a coded OFDM system, the error probability is no longer dependent on the
`power of individual symbols, but rather on the power of a number of consecutive
`symbols. As an example, assume that the forward—error correction code produces an
`error if more than 4 out of every 10 symbols have a PAP ratio exceeding 10 dB1.
`Further, assume that the probability of a PAP ratio larger than 10 dB is l0'3. Then, the
`Ill
`error probability of the peak cancellation technique is 1—§{1l)}10‘3)"(1—1O‘3)1°"
`2-104°, which is much less than the 10‘3 in case no forward—error correction coding is
`used.
`
`i=0
`
`1
`
`Although such a low symbol error probability may be good enough for real—time
`circuit~switched traffic, such as voice, it may still cause problems for packet data. A
`packet with too many large PAP ratio symbols will have a large probability of error.
`Such packets occur only Very infrequently, as shown above, but when they occur, they
`
`l The simplifying assumption is made here that 4 symbols with reduced power always result in an error,
`while in reality there is always a certain error probability < 1, depending on the SNR.
`
`\x.
`
`‘,
`
`-
`
`.\
`
`-
`
`.",.‘~,;,;
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`‘ ERIC-1:009 /,Page 14 ores j
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`
`ERIC-1009 / Page 14 of 38
`
`

`
`
`
`:.«u-.—.—._«..
`
`may never come through, because every retransmission of the packet has the same large
`error probability. To solve this problem, standard scrambling techniques can be used to
`ensure that the transmitted data between initial transmission and retransmissions are
`uncorrelated. To achieve this, the scrambler has to use a different seed for every
`transmission, which can be realized for instance by simply adding one to the seed after
`every transmission. Further, the length of the scrambling sequence has to be in the
`order of the number of bits per OFDM symbol to guarantee uncorrelated PAP ratios for
`different seeds. Different scrambling in every transmission will
`then guarantee
`independent PAP ratios for the OFDM symbols in retransmissions and hence,
`independent error probabilities. For example, if the probability of a worst case packet is
`106, the probability that it does not come through within two transmissions is 1042.
`
`6.4
`
`PEAK CANCELLATION
`
`The key element of all distortion techniques is to reduce the amplitude of samples
`whose power exceeds a certain threshold. In the case of clipping and peak windowing,
`this was done by a nonlinear distortion of the OFDM signal, which resulted in a certain
`amount of out—of-band radiation. This undesirable effect 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
`reduces the peak power of at 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. One example of a suitable reference signal
`is a sinc function. A
`disadvantage of a sine function is that it has an infinite support. Hence, for practical
`use, it has to be time—lirnited in some way. One way to do this without creating
`unnecessary out—of-band interference is multiplication by a windowing function; for
`instance, a raised cosine window. Figure 6.14 shows an example of a reference
`function, obtained by multiplication of a sine function and a raised cosine window. If
`the windowing function is the same as used for windowing of the OFDM symbols, then
`it is assured that the reference function has the same bandwidth as the regular OFDM
`signals. Hence, peak cancellation will not degrade the out—of-band spectrum properties.
`By making the reference signal window narrower, a tradeoff can be made between less
`complexity of the peak cancellation calculations and some increase of the out—of-band
`power. The peak cancellation method was first published in [3], while later it was
`independently described in [4].
`
`ckoff and
`
`lo suffer
`forward-
`
`oing so,
`'ounding
`it on the
`lSCCLltlVC
`iuces an
`
`10 dB1.
`.‘hen, the
`-3 10-’
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`I
`
`:oding is
`
`real—time
`: data. A
`of error.
`
`cur, they
`
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`
`
`
`ER|C-1009_/ Page 15 of 38
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`ERIC-1009 / Page 15 of 38
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`

`
`1.
`
`l
`
`-6
`
`-4_
`
`2
`0
`-2
`Relative time in Nyquist samples
`
`4
`
`6
`
`8
`
`Figure 6.14 Sinc reference function, windowed with a raised cosine window.
`Peak cancellation can be done digitally after generation of the digital OFDM
`symbols. It involves a; peak power (or peak amplitude) detector, a comparator to see if
`the peak power exceeds some threshold, and a scaling of the peak and surrounding
`samples. Figure 6.15 shows the block diagram of an OFDM transmitter with peak
`coded and converted from a serial bit stream to
`-- cancellation. Incoming data are first
`blocks of N complex signal samples. On each of these blocks, an IFFT is performed.
`Then, a cyclic prefix is added, extending the symbol size to N + N5 samples. After
`parallel—to—seria1 conversion, the peak cancellation procedure is applied to reduce the
`PAP ratio. It is also possible to do peak cancellation immediately after the IFFT and
`before the cyclic prefix and windowing. Except for the peak cancellation block, there is
`further no difference with a standard OFDM transmitter. For the receiver, there is no
`difference at all, so any standard OFDM receiver can be used.
`
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`ERIC-1009 / Page 16 of 38
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`

`
`
`
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`133
`
`Figure 6.15 OFDM transmitter with peak cancellation.
`e after parallel—to—serial
`cellation was don
`gures, the peak can
`In the previous fi
`diately after the
`do the cancellation imme
`also possible to
`conversion of the signal. It is
`the cancellation is done on a symbol—by—
`16. In this case,
`IFFT, as depicted in Figure 6.
`ancellation signal without using a
`symbol basis. An efficient way to generate the, c
`the frequency domain. In Figure
`stored reference function is to use a lowpass filter in
`les exceed some predefined
`6.16, for each OFDM symbol, it is detected which samp
`ated whose phase is equalto
`litude. Then, for each signal peak, an impulse is gener
`itude minus the desired
`amp
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`hose amplitude IS
`symbol
`the peak phase and w
`The impulses are then lowpass filtered on a symbol—by—
`domain by taking the FFT, setting
`maximum amplitude.
`achieved in the frequency
`basis. Lowpass filtering is
`quency of the highest subcarrier,
`frequencies exceed the fre
`all outputs to zero whose
`k by an IFFT.
`and then transforming the signal bac
`
`9
`
`Figure 6.16 Peak Cancellation using FFTIIFFT to
`lic reference function that is used in all
`ample of the cyc
`ic prefix and windowing. In fact,
`before adding the cycl
`ich is obtained in the case of an
`alid OFDM signal, wh
`
`generate cancellation signal.
`
`Figure 6.17 shows an ex
`methods that apply cancellation
`this reference signal itself is a V
`all—ones input to the IFFT.
`
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` ERIC-1 009 Page 17 of 33
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`ERIC-1009 / Page 17 of 38
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`

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`Figure 6.18 shows an example of the signal envelopes of one arbitrary OFDM
`symbol and the corresponding cancellation signal.
`In this particular case,
`the
`cancellation signal actually consists of two separate sinc functions, because one sinc
`function is not wide enough to reduce the peak in this example. After subtraction, the
`peak amplitude is reduced to a maximum of 3 dB above the rms value; see Figure 6.19.
`As an example of the peak cancellation technique, Figure 6.20 shows simulated
`power spectral densities for an OFDM system with 32 carriers. Without clipping or
`peak cancellation, the worst case PAP ratio of this system is 15 dB, and the undistorted
`spectrum is depicted by curve (a). If the signal is clipped such that the PAP ratio
`reduces to 4 dB, 21 significant spectral distortion is visible; see curve (c). When peak
`cancellation is applied (b), a negligible distortion is present for the same PAP ratio of 4
`dB.
`
`0.4
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`0.6
`0.8
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`Time in Symbol Intervals
`
`Figure 6.17 Envelope of cyclic reference function.
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`ERIC-1009 / Page 18 of 38
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`

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`Figure 6.18 (a) OFDM symbol envelope, (b) cancellation signal envelope.
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`
`ERIC-1009 / Page 19 of 38
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`

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`Frequency / Subcarrier Spacing
`
`Figure 6.20 Power spectral density for (a) undistorted spectrum with 32 subcarriers, PAP = l5dB, (b)
`spectrum after peak cancellation to PAP = 4 dB, and (c) clipping to PAP = 4 dB. Reference
`cancellation function has a length equal to ‘A of the length of an OFDM symbol.
`
`The effect of the peak cancellation on the PER is depicted in Figure 6.21. A rate
`1/2, constraint length 7 convolutional code is used to encode the input bits. The coded
`bits are then modulated onto 48 OFDM subcarriers using 16—QAM. The curves show an
`SNR degradation of about 0.6 dB in AWGN when peak cancellation is used to reduce
`the PAP ratio to 6 dB.
`
`
`
`ERIC-1009 / Page 20 of 38
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`

`
`
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`(b)
`ICC
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`E1/Na [dB]
`
`Figure 6.21 PER versus Eb/N0 for 64-byte packets in AWGN. Peak cancellation is applied to reduce
`the PAP ratio to (a) 16 ( = no distortion), (b) 6, (c) 5, and (d) 4 dB.
`
`At first sight, peak cancellation seems to be a fundamentally different approach
`than clipping or peak windowing. It can be shown, however, that peak cancellation is in
`fact almost identical to clipping followed by filtering. If a sampled OFDM signal x(n) is
`clipped to reduce the PAP ratio, the output signal r(n) can be written as
`
`r(n) = x(n) -2 a,.eJ'<”»' 5(n—r,.)
`
`(6.6)
`
`Here, a,-, go,-, and 17,- are the amplitude, phase, and delay of the correction that is
`applied to the ith sample in order to reach the desired clipping level. Hence,
`it is
`possible to describe clipping as a linear process, even though this is not the way
`clipping is performed in practice. Now suppose the clipped signal is filtered by an ideal
`lowpass filter with an impulse response of sinc(7mT), where T is chosen such that the
`filter bandwidth is equal or larger than the bandwidth of the OFDM signal. The filtered
`output is given by
`
`
`
`r'(n) = x'(n) —2 a,.ef‘Pi 'sinc(7zT(n—1,.))
`
`(6.7)
`
`This expression is identical to a peak cancellation operation, with the only
`exception that with peak cancellation, a sum of sinefunctions is subtracted from the
`unfiltered OFDM signal x(n), while in (6.7) we see a filtered signal x’(n). In practice,
`however, also for peak cancellation, the OFDM signal needs to be filtered anyway to
`
`ERICA-1.009 I Page 21 of 33
`
`ERIC-1009 / Page 21 of 38
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`

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`
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`
`
`Figure 6.22
`
`Packet error ratio versus E,/N0 for 64-byte packets in AWGN. PAP ratio is reduced to 5
`dB by (a) clipping, (b) peak cancellation, and (c) peak windowing.
`
`6.5
`
`PAP REDUCTION CODES
`
`As Section 6.2 shows, only a small fraction of all possible OFDM symbols has a bad
`peak-to—average power ratio. This suggests another solution to the PAP problem, based
`on coding. The PAP ratio can be reduced by using a code which only produces OFDM
`symbols for which the PAP ratio is below some desirable level. Of course, the smaller
`the desired PAP level, the smaller the achievable code rate is. Section 6.2, however,
`already demonstrated that for a large number of subcarriers, a reasonable coding rat6
`larger than 3/4 can be achieved for a PAP level of 4 dB. In [6], it was found that for
`eight channels, a rate 3/4 code exists that provides a maximum PAP ratio of 3 dB. The
`results in [6] are based on an exhaustive search through all possible (QPSK)
`
`1 ll l4Il1l
`
`138
` _&
`
`remove aliasing after the digital—to-analog conversion. Hence, for practical purposes, it
`may be concluded that peak cancellation has the same effect as clipping followed by
`filtering, which was proposed as a PAP reduction technique in [5].
`
`As a final comparison of the three described signal distortion techniques, Figure
`6.22 shows the PERS for an OFDM system with 48 subcarriers for which the PAP ratio
`is reduced to 5 dB. In addition to the three PAP reduction technique, the nonlinear
`amplifier model described in section 6.3.1 was applied such that the output backoff of
`the transmitted OFDM signal was 6 dB. We can see from the figure that clipping
`(without filtering) performs slightly better than peak cancellation, and that peak
`windowing is slightly worse than peak cancellation.
`
`PER
`
`E:/N.» [dB]
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`ERIC-1009 / Page 22 of 38
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`

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`139
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`
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`codewords. Unfortunately, these results only tell us that there exists a large number of
`code words; it does not say if there exists a structured way of encoding and decoding to
`generate a large part of these code words, nor what the minimum distance properties of
`the code are. However, [6] did mention the interesting fact that a large part of the codes
`found are Golay complementary sequences, which opened the way to a structured way
`of generating PAP—reduction codes. Golay complementary sequences are sequence
`pairs for which the sum of autocorrelation functions is zero for all delay shifts unequal
`to zero [7—9]. It was already mentioned in [10] that the correlation properties of
`complementary sequences translate into a relatively small PAP—ratio of 3 dB when the
`codes are used to modulate an OFDM signal. Based on all these hints towards Golay
`sequences, [11] presented a specific subset of Golay codes together with decoding
`techniques that combined peak—to—average power reduction with good forward—error
`correction capabilities. Based on this work, Golay codes were actually implemented in
`a prototype 20—Mbps OFDM modem for the European Magic WAND project [12].
`Fundamental studies on the coding properties of Golay sequences appeared in [13—16],
`proving code set sizes, distance properties, the relation to Reed—Muller codes, and many
`more interesting details.
`~
`A sequence x of length N is said to be complementary to another sequence y if
`the following condition holds on the sum of both autocorrelation functions:
`N—l
`
`Z(xkxk+i+ykyk+i)=2N= i=0
`k=0
`
`(6.8)
`
`=0,
`
`i¢0
`
`By taking the Fourier transforms of both sides of (6.8), the above condition is
`translated