reasonable error correcting properties. Section 6.5 describes a solution to this problem.
`
`A different approach to the PAP problem is to use the fact that because large
`PAPratios occur only infrequently, it is possible to remove these peaks at the cost of a
`slight amount of self-interference. Now, the challenge is to keep the spectral pollution
`of this self-interference as small as possible. Clipping is one example of a PAP
`reduction technique creating self interference. In the next sections, two other techniques
`are described which have better spectral properties than clipping.
`
`123
`
` log(CDF)
`
`PAPR[dB]
`
`Figure 6.4 Cumulative distribution function of the PAPR for a number of subcarriers of (a) 32, (b) 64,
`(c) 128, (d) 256, and (e) 1,024. Solid lines are calculated; dotted lines are simulated.
`
`6.3
`
`CLIPPING AND PEAK WINDOWING
`
`The simplest way to reduce the PAPratio 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
`distorting the OFDM signal amplitude, a kind of self-interference is introduced that
`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
`
`138
`
`138
`
`

`

`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.
`
`Original signal
`
`Multiplication
`
`Amplitude
`
`0
`
`50
`
`_ 100
`150
`Time in samples
`
`200
`
`250
`
`300
`
`Figure 6.5 Windowing an OFDM timesignal.
`
`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 PAPratio is
`decreased to 6 dB. When peak windowingis 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 PAPratio.
`
`139
`
`139
`
`

`

`
`
`125
`
`=P
`
`-60
`
`60
`30
`0
`-30
`Frequency/subcarrier spacing
`
`PSD[dBr]
`PSD[dBr]
`
`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/subcarrier spacing
`
`Figure 6.7
`
`Frequency spectrum of an OFDM signal with 32 subcarriers 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, (f) 13, and (g) IS
`samples. Curve(h) is the ideal OFDM spectrum.
`
`140
`
`140
`
`

`

`
`
`Figure 6.8 Packet error ratio versus £,/N, for 64 byte packets in AWGN. OFDM signal is clipped to a
`PAPratio of (a) 16 ( = no distortion), (b) 6, (c) 5, and (d) 4 dB.
`
`E/N. (dB)
`
`126
`
`PER
`
`PER
`
`
`
`
`10°
`
`
`
`
`
`10°
`
`10
`
`10
`
`Figure 6.9 PER versus £,/N, 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/N. {dB}
`
`141
`
`141
`
`

`

`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 powerlevel, 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]:
`
`BAj=—_4__.
`(14.4? ppp
`
`(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 outputlevel.
`
`value
`
`Output
`
`oO
`
`0
`
`4
`0.2
`
`it
`04
`
`onl.
`06
`
`L
`
`i.
`
`4
`
`i
`
`O08
`
`1
`
`12
`
`14
`
`16
`
`18
`
`Input value
`
`Figure 6.10 Rapp’s model of AM/AM conversion.
`
`142
`
`142
`
`

`

`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 muchif thoseparts are
`distorted. After peak windowing or any other PAP reduction technique, however, a
`significant part of the signal samples are close to the maximum PAPratio (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
`PAPratio.
`
`
`
`
`
`PSD[dBr]
`
`
`
`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.
`
`Frequency / bandwidth
`
`143
`
`143
`
`

`

`129
`
`Figure 6.12 shows OFDM spectra for a more realistic amplifier model with
`p= 3. The target for undesired spectrum distortion has now beenset 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.
`
`
`
`
`
`02 04 06 J1
`
`
`
`_PSD[dBr]
`
`
`
`-411-0.6_1| 0
`
`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.
`
`144
`
`144
`
`

`

`
`
`a=_aL
`
`L
`
`J ==
`
`130
`
`PSD[dBr]
`
`STTI|)
`
`-2
`
`=1,.5
`
`-1
`
`-0.5
`
`oO
`
`0.5
`
`1
`
`1.5
`
`2
`
`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.
`
`Frequency/Bandwidth
`
`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 reducethis 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 dB’.
`Further, assume that the probability of a PAP ratio larger than 10 dB is 10°. Then, the
`error probability of the peak cancellation technique is 1-3")}10-y'a—10-8)1" =
`
`2-10°'°, which is much less than the 10° in case no forward-error correction coding is
`used.
`
`Although such a low symbolerror probability may be good enoughfor 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
`
`' 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.
`
`145
`
`145
`
`

`

`131
`
`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 oneto the seed after
`every transmission. Further, the length of the scrambling sequence has to be in the
`order of the numberof bits per OFDM symbolto 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 packetis
`10°, the probability that it does not come through within two transmissionsis 10°!*.
`
`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 sinc function is that it has an infinite support. Hence, for practical
`use, it has to be time-limited 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 sinc 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 betweenless
`complexity of the peak cancellation calculations and some increase of the out-of-band
`power. The peak cancellation method wasfirst published in [3], while later it was
`independently described in [4].
`
`146
`
`146
`
`

`

`
`
`132
`
`0.8
`
`0.6
`
`0.4
`
`0.2
`
`
`
`SignalValue
`
`“8
`
`6
`
`-4
`
`-2
`
`0
`
`2
`
`4
`
`6
`
`8
`
`Figure 6.14 Sinc reference function, windowed with a raised cosine window.
`
`Relative time in Nyquist samples
`
`Peak cancellation can be done digitally after generation of the digital OFDM
`symbols. It involves a peak power (or peak amplitude) detector, a comparatorto 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
`cancellation. Incoming data are first coded and converted from a serial bit stream to
`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 + Ng samples. After
`parallel-to-serial conversion, the peak cancellation procedure is applied to reduce the
`PAPratio. 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.
`
`147
`
`147
`
`

`

`133
`
`Cyclic prefix &
`
`
`
`windowing
`
`
`
`Peak
`cancellation
`
`:
`
`Figure 6.15 OFDM transmitter with peak cancellation.
`
`In the previous figures, the peak cancellation was done after parallel-to-serial
`conversion ofthe signal. It is also possible to do the cancellation immediately after the
`IFFT, as depicted in Figure 6.16. In this case, the cancellation 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 lowpassfilter in the frequency domain. In Figure
`6.16, for each OFDM symbol, it is detected which samples exceed some predefined
`amplitude. Then, for each signal peak, an impulse is generated whose phaseis equal to
`the peak phase and whose amplitude is equal to the peak amplitude minusthe desired
`maximum amplitude. The impulses are then lowpass filtered on a symbol-by-symbol
`basis. Lowpass filtering is achieved in the frequency domain by taking the FFT, setting
`all outputs to zero whose frequencies exceed the frequency of the highest subcarrier,
`and then transforming the signal back by an IFFT.
`
`
`
`Figure 6.16 Peak Cancellation using FFT/IFFT to generate cancellation signal.
`
`Figure 6.17 shows an example of the cyclic reference function that is used in all
`methods that apply cancellation before adding the cyclic prefix and windowing.In fact,
`this reference signal itself is a valid OFDM signal, which is obtained in the case of an
`all-ones input to the IFFT.
`
`148
`
`148
`
`

`

`134
`
`
`
`EnvelopeValue
`
`0.8
`
`0.6
`
`0.4
`
`0.2
`
`
`0.2
`0.4
`0.6
`0.8
`I
`
`0
`
`Time in Symbol Intervals
`
`Figure 6.17 Envelope of cyclic reference function.
`
`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 abovethe rmsvalue; see Figure 6.19.
`As an example of the peak cancellation technique, Figure 6.20 shows simulated
`powerspectral densities for an OFDM system with 32 carriers. Without clipping or
`peak cancellation, the worst case PAPratio 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, a 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.
`
`149
`
`149
`
`

`

`135
`
`
`
`EnvelopeValue
`
`
`0
`0.2
`0.4
`0.6
`0.8
`1
`
`Time in Symbol Intervals
`
`Figure 6.18 (a) OFDM symbol envelope, (b) cancellation signal envelope.
`4
`
`3.5
`
`25
`
`
`
`EnvelopeValue
`
`0
`
`0.2
`
`0.4
`
`0.6
`
`0.8
`
`1
`
`Time in SymbolIntervals
`
`Figure 6.19 (a) OFDM symbolenvelope, (b) signal envelope after peak cancellation.
`
`150
`
`150
`
`

`

`136
`
`
`
`PSD[dBr}
`
`
`
`
`L
`-10
`
`1
`0
`
`1
`10
`
`i
`20
`
`L
`30
`
`40
`
`Frequency / Subcarrier Spacing
`
`Figure 6.20 Power spectral density for (a) undistorted spectrum with 32 subcarriers, PAP = 15dB, (b)
`spectrum after peak cancellation to PAP = 4 dB, and (c) clipping to PAP = 4 dB. Reference
`cancellation function has a length equal to %4 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 encodethe 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.
`
`151
`
`151
`
`

`

`137
`
`
`
`
`
`.PER
`
`E,/N, [dB]
`
`Figure 6.21
`
`PER versus E,/N, 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 byfiltering. If a sampled OFDM signal x(n)is
`clipped to reduce the PAPratio, the output signal r(m) can be written as
`
`r(n) = x(n)— } a,e/" 5(n—-T,)
`
`(6.6)
`
`Here, a;, 9;, and T; 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 supposethe clipped signal is filtered by an ideal
`lowpassfilter with an impulse response of sinc(mm7), 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
`
`(n)=x'(n)— > a,e/"'sine(xT(n-7,))
`
`(6.7)
`
`to a peak cancellation operation, with the only
`This expression is identical
`exception that with peak cancellation, a sum of sinc, functions 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
`
`152
`
`152
`
`

`

`138
`
`removealiasing 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 techniquein [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 PAPratio
`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
`windowingis slightly worse than peak cancellation.
`
`
`
`PER
`
` 6
`
`
`
`
`f
`
`8
`
`9
`
`10
`
`11
`
`12
`
`Figure 6.22
`
`Packet error ratio versus E;/N, for 64-byte packets in AWGN.PAPratio is reduced to 5
`dB by(a) clipping, (b) peak cancellation, and (c) peak windowing.
`
`E/N, (dB)
`
`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 powerratio. 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 rate
`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 PAPratio of 3 dB. The
`results in [6] are based on an exhaustive search through all possible (QPSK)
`
`153
`
`153
`
`

`

`139
`
`codewords. Unfortunately, these results only tell us that there exists a large numberof
`code words; it does not say if there exists a structured way of encoding and decodingto
`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 wayto 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 whenthe
`codes are used to modulate an OFDM signal. Based onall 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 WANDproject [12].
`Fundamental studies on the coding properties of Golay sequences appeared in [13—16],
`proving codeset sizes, distance properties, the relation to Reed-Muller codes, and many
`moreinteresting details.
`
`A sequence x of length WN is said to be complementary to another sequenceyif
`the following condition holds on the sum of both autocorrelation functions:
`
`Nl
`es + YM) = 2N,
`k=0
`
`i =0
`
`=0,
`
`i#0
`
`(6.8)
`
`By taking the Fourier transforms of both sides of (6.8), the above condition is
`translated into
`
`IX) +P)?=2N
`
`(6.9)
`
`Here, Ix | is the power spectrum of x, which is the Fourier transform ofits
`autocorrelation function. The discrete Fourier transform X(f) is defined as
`
`X(f)= Yi xe
`
`¥
`
`k=0
`
`(6.10)
`
`Here, 7, is the sampling interval of the sequence x. From the spectral condition
`(6.9), it follows that the maximum value of the power spectrum is bounded by 2N:
`
`Ix(f)|’ <2N
`
`(6.11)
`
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`140
`
`Because the average power of X(f) (6.10) is equal to NV, assuming that the power
`of the sequence x is equal to 1, the PAP ratio of X(f) is bounded as
`
`_
`
`2N
`
`PAPratio < =~ =2
`
`(6.12)
`
`In an OFDM transmission, normally the IFFT is applied to the input sequence x.
`However, because the IFFT is equal
`to the conjugated FFT scaled by I/N,
`the
`conclusion that the PAP ratio is upper bounded by 2 is also valid when X(/f) is replaced
`by the inverse Fourier transform of the sequence x. Hence, by using a complementary
`code as input to generate an OFDM signal, it is guaranteed that the PAP ratio does not
`exceed 3 dB. Figure 6.23 shows a typical example of an OFDM signal envelope when
`using a complementary sequence. For this case of 16 channels, the PAP ratio is reduced
`by approximately 9 dB in comparison with the uncoded case of Figure 6.1.
`
`1.4
`
`1.3
`
`1.2
`
`+4
`
`vPAP
`
`0.4
`
`0
`
`2
`
`4
`
`6
`
`8
`
`10
`
`12
`
`i4
`
`16
`
`Time/T
`
`Figure 6.23 Square root of peak-to-average powerratio for a 16-channe] OFDM signal, modulated with
`a complementary code.
`
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`
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`
`

`

`141
`
`6.5.1 Generating Complementary Codes
`
`In [7-9], several coding rules are given for generating a set of complementary
`sequences, based upon some
`starting complementary pair,
`the kernel.
`For
`complementary sequences of length 2, for instance, a possible kernelis the pair 1, 1 and
`1, —1. The basic coding rules for generating complementary codes from this kernel are
`(7, 9]:
`
`EkBee
`
`Interchanging both codes,
`Reversing and conjugating second code,
`Phase-rotating second code,
`Phase-rotating elements of even order in both codes,
`Phase-rotating first code, and
`Reversing and conjugating first code.
`
`Whenrules 1 to 4 are applied, the following 16 different length 4 codes can be
`obtained for the case of 4 phase modulation (see Table 6.1):
`
`Table 6.1
`Length 4 complementary codes
`
`
`
`The number of codes can be extended to 64 by applyingthe fifth and sixth rules,
`which gives the same result as applying 4 different phase shifts to the 16 codes. Hence,
`these 4-symbol codes can easily be generated by using a 16-word-long lookup table to
`encode 4 bits, followed by a phase rotation to mapatotal of 6 bits onto all possible
`complementary codes.
`Unfortunately, as the previous example already indicated, the six coding rules
`do not unambiguously produce all complementary sequences. This makesit difficult to
`find the size of the code set and to find a systematic way to produce complementary
`sequences. Thus, some other algorithm has to be found to generate complementary
`codes.
`
`[9] showed that from one set of complementary sequences, others can be
`generated by multiplying the original sequences with columnsof the discrete Fourier
`transform matrix. Although [9] only mentions this method to generate sets with longer
`
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`
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`

`142
`
`code length by using the Kronecker product, it can also be used to generate different
`sequences with the same length, by multiplying an original sequence elementwise with
`columnsof the DFT matrix. It is easy to show that such multiplications do not change
`the correlation properties. Each DFT column is a delta function in the frequency
`domain. Because multiplication in the time domain is equivalent to a convolution in the
`frequency domain, the power spectrum of a complementary sequence multiplied by a
`DFT column remains the same. Hence, also its correlation function, which is the
`Fourier transform of the power spectrum, remains the same, so that the outcome again
`is a complementary sequence.
`
`is that complementary sequences can be
`[9]
`Another interesting remark in.
`multiplied by columns of the binary Walsh-Hadamard matrix, without losing their
`complementary characteristics. Further,
`it
`is stated in [9]
`that “if the code is an
`expansion of shorter lengths, an arbitrary phase angle can be addedto all elements in
`any orthogonal subset.” These operations turn out to be very useful
`in generating
`distinct codes.
`
`The coding algorithm for generating complementary sequences is now given by
`the following steps:
`
`from which all other
`is, one complementary pair
`that
`Make a kernel;
`1.
`complementary sequences can be derived. For lengths equal to a powerof two, kernels
`can easily be formed by using Golay’s rule for length expansion. Starting with the
`length 2 sequence A; B;, where A; = 1 and B; = 1, longer length codes can be formed
`by making A, B, with A, = Ay.) Ba and By = A,-; —Bn.1. In this way, codes of length
`2"*! are formed from the codes of length 2". For example, the following codes up to
`length 16 can be obtained:
`
`length2 : A, B,;=-11
`
`length4 : A,sBo= 111-1
`
`length8 : A3;Bz;= Ll1i-l 11-11
`
`(6.13)
`
`length 16: AgBg= 111-1 11-11 111-1 -1-11-1
`
`Determine the numberof orthogonal subsets. For length N codes, formed by
`2.
`the length expansion method described above, there are logs orthogonal subsets, all of
`which can be given an arbitrary phase offset. The orthogonal subsets within a code are
`formed byall single elements, pairs, quads, and so forth, which are of even order. Thus,
`a length 16 code has 4 orthogonal subsets, being all even elements, pairs, quads, and
`one octet. All of these can be given a different phase without changing the
`complementary characteristics of the code. Further,
`it is also possible to apply an
`arbitrary phase shift to the entire code. Hence, a complementary code set based upon
`the kernel of (6.13) can be written as:
`
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`
`157
`
`

`

`c= {ei +02 +P3+P4) se I(P; +P3+P4) erm +P2 +94) a ertPa) 3
`ICQ +P2tP3) PtP) Pit P2)
`219)
`é
`e
`e
`.
`3
`5€
`}
`
`143
`
`(6.14)
`
`Notice that this code is actually implemented in a 20-Mbps OFDM modem for
`the Magic WANDproject [12]. It is also used in the 11-Mbps IEEE 802.11 wireless
`LANstandard [17]. The latter is not an OFDM system, but here the benefit from using
`complementary sequences is in its good aperiodic autocorrelation properties, which
`makesit easier to build a receiver with sufficient robustness to multipath.
`
`An alternative code description is to write the code phases as
`
`0
`111 1
`0,
`6
`10 1
`i
`0,
`1101 of|”
`@
`?
`1001 1{|°?
`0,
`1110 of]®
`A,
`1010 0||*7t
`0,
`6, 11001
`8
`@
`10000
`
`=
`
`(6.15)
`
`The output code is given by exp(G-27@/M), where 6; is the coded phase and M is
`the size of the phase constellation. For BPSK (M=2), the codeset is equal to the Walsh-
`Hadamard codes, which is offset by the kernel—defined by the fourth columnin (6.15).
`
`Finally, a transformation can be applied that unfortunately cannot be
`a
`described by simple multiplications or phase rotations. Instead, it can be described as
`an interleaving operation on the underlying shorter length codes that are used to make a
`longer length code [14]. For a length 8 sequence, for instance, two new length 8 codes
`can be generated by interleaving the first and second half of the original code.
`Interleaving the code three times reproducesthe original code. In general, a code with a
`length of 2” can be interleaved n—1 times before reproducing itself. The following
`example showsthree different codes out of a length 8 code produced by interleaving:
`
`O Dl+: bd=La4
`
`I: L111 id-I-l1
`
`2: 111-1 i-1l1i1
`
`(6.16)
`
`For a length 16 code, it turns out that except for four different codes that can be
`producedby interleaving the first and second half of the code, more codes can be made
`by simultaneously interleaving the quarters of the code, giving a total of 3 - 4 = 12
`different codes. The described coding rules can now be usedto determine the size of
`
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`
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`
`

`

`144
`
`complementary code sets. For an N-length code with M possible phases, the kernel can
`be multiplied by 1 + logsN modified Walsh-Hadamard rows with M different phases.
`This gives a code set size of M’*”8N_ The amount of bits per codeword can be
`expressed as (1 + log2 N)log2 M. For instance, a length 8 code with four possible phases
`gives 8 bits per code word. The above numbers did not yet take into account the
`interleaving rule, which adds another logs([(log2/V)!]/2) bits to the total numberofbits
`per symbol (for N > 4 and N being a power of 2). Notice that the interleaving rule does
`not necessarily produce an integer numberof bits per encoded symbol.
`
`6.5.2 Minimum Distance of Complementary Codes
`
`In OFDM. systems, the effects of multipath are mitigated by error correction coding
`over the various subchannels. Thus, when using a PAP-reduction code, it would be very
`desirable if you could use this code also for forward-error correction. Otherwise, a
`separate code would be required, with the disadvantage of additional complexity and a
`reduction in the overall coding rate and spectral efficiency.
`
`to what minimum distance the above
`the question arises as
`Therefore,
`mentioned complementary sequences have. Looking at (6.15), we can state that if this
`is the only generating rule used, then N/2 + 1 correctly received symbols are always
`sufficient
`to calcul