GB2412826A

GB2412826A - OFDM system with peak predistortion to reduce peak to average power ratio, wherein phase information is transmitted in pilot subcarriers

Info

Publication number: GB2412826A
Application number: GB0407476A
Authority: GB
Inventors: Yanyan Wu
Original assignee: Samsung Electronics Co Ltd
Current assignee: Samsung Electronics Co Ltd
Priority date: 2004-04-01
Filing date: 2004-04-01
Publication date: 2005-10-05
Anticipated expiration: 2024-04-01
Also published as: GB0407476D0; GB2412826B

Abstract

Phase distortions are applied to a multi-carrier signal at a transmitter to reduce PAPR. The receiver can reverse the phase distortions provided it knows the value of the phase distortion coefficient p . The receiver deduces the value of this coefficient by mathematically combining the phases of a plurality of pilot sub-carriers, preferably using equation 9. The pilot sub-carriers used for this purpose must be carefully selected to ensure that mathematical terms representing carrier frequency offset, W f, cancel, and terms representing sampling clock offset, h , also cancel. Consequently the pilot sub-carriers used for this purpose are non-evenly spaced. Carrier frequency offset and sampling clock offset can be deduced independently using a second set of pilot sub-carriers.

Description

241 2826 Frequency Division Multiplexing The present invention relates to

frequency division multiplexing ((FDM), preferably orthogonal frequency division multiplexing (OFDM)) systems, transmitters and receivers, and to methods of transmitting a signal in an orthogonal frequency division multiplexing (OFDM) system.

Orthogonal Frequency Domain Multiplexing (OFDM) is a multi carrier modulation technique that has been adopted for many wireless systems, such as wireless local-area networks (WLAN) and digital video broadcasting (DVB). OFDM distributes the data over a large number of carriers spaced apart at precise frequencies in a particular relationship. The spacing is the minimum possible whilst maintaining orthogonality, and due to its feature of multiple narrow-band sub-carriers, the data transmission rate on each subcarrier is at a lower rate which is less subject to multi-path delay spread. Therefore, OFDM is one of the most attractive choices for high data rate transmission in future mobile radio systems, and is a study item in 3GPP.

Details of OFDM in the context of satellite broadcasting are disclosed in WO96/19055. Use of OFDM with encoding is disclosed in "Evaluation of error correction block encoding for high speed HE Data" - Brayer, K. and Cardinale, O.; IEEE Transactions on Communication Technology vol. COM No. 3, June 1967, "A combined coding and modulation approach for communication over dispersive channels" - Chase, D.; IEEE Transactions on Communications vol. COM - 21 No. 3, March 1973. More recent examples are US 4881241, US 4884139, US 5191576, US 5197061, US 5228025, US 5274629, and US 5307376.

Our earlier application GB 0328895.8, filed on 12 December 2003, and incorporated herein by reference, discloses an OFDM technique having reduced Peak to Average Power Ratio (PAPR - also known as the Crest Factor - which is defined as the ratio of the peak value of the OFDM waveform to its Root Mean Square (RMS) value. This is achieved by applying, on a symbol-by-symbol basis, a phase pre-distortion to each subcarrier, each phase distortion being selected so as to be related to the index of the subcarrier, based on a coefficient which is the same across all subcarriers. That invention relies on the inventive realisation that the application of subcarrier phase corrections in a predetermined relationship, characterised by one (or perhaps a small number of) coefficients, will give an improved performance where the adjustments for each subcarrier are related to the index (or frequency) of the subcarrier. It has been found, for example, that by using a series of phase corrections which are a quadratic function of the subcarrier number and a single coefficient, and by deriving a suitable coefficient value, a good reduction in PAPR can be obtained.

It is unnecessary to transmit this coefficient value where the FDM signal includes a pilot tone (as is common in OFDM) since the coefficient can be derived from the pilot tone, so that no supplementary information has to be transmitted in order be able to recover the transmitted data on the receiver side.

In one aspect, the present invention concerns the use of selected appropriate subcarriers to carry the pilot tone, and to apparatus for transmitting and receiving OFDM signals using those pilot subcarriers.

In known OFDM systems, the pilot tones are on subcarriers which are evenly spaced in frequency, so as to minimise the effects of deep fades. For example, in the WEAN system, numbering subcarriers outwards from a central subcarrier 0, pilots may be on subcarrier numbers {+21, +7}.

In the present invention, however, a set of subcarriers are selected using criteria based on the relationship used for phase distortion. They are spaced apart, but in general not evenly.

This allows pairs of subcarriers to be selected such that the effects of the phase distortion on the pair or pairs can cancel each other out with appropriate processing at the receiver, allowing normal use of the pilot tones for frequency and clock error tracking, as well as for detection of the coefficients (applied at the transmitter) characterising the intentional phase distortion.

Aspects of the invention are defined in the claims. Other preferred features will be apparent from the description and drawings, with advantages which will be apparent therefrom.

An embodiment of the invention will now be described with reference to the accompanying drawings, of which Figure I is a block diagram of an OFDM transmitter according to an embodiment of the invention; Figure 2 is a flow diagram showing the process of deriving a phase adjustments in the embodiment of Figure 1; Figure 3 is a block diagram of an OFDM receiver according to the embodiment of Figure 1; Figure 4 is a diagram showing amplitude against frequency to illustrate the OFDM subcarriers; Figure 5 is a graph of Bit Error Rate (BER) against Signal to Noise ratio (SNR) for the embodiment, also showing the theoretical performance if the embodiment is not used; and Figures 6a and 6b illustrate a mobile telephone incorporating the invention.

Transmitter Figure 1 illustrates a block diagram of an OFDM transmitter I according to an embodiment of the invention. It is conventional except for the phase synthesis and pilot tone usage described below, and suitable digital signal processing hardware is known and widely available. It may for example be implemented as one or more suitably programmed commercially available Digital Signal Processor chips, or as custom- designed integrated circuits.

Transmission binary data is mapped into complex symbols in a baseband modulation mapping block 2 using QPSK (Quadrature Phase Shift Keying), 64QAM (64-state Quadrature Amplitude Modulation) (or any other suitable modulation technique).

Successive frames of Nc symbols from the modulation mapped transmission data streams are fed to a serial to parallel converter block 3 and thereby multiplexed into NC parallel data streams So..sNc-. at a lower data rate of 1/NC. Among the NC streams there are pilot tones (i.e. a predetermined symbol known to the receiver and carrying no data) on a plurality of predetermined subcarriers as discussed below.

The parallel data streams are then fed to a quadratic phase synthesis block 4 where they undergo a phase synthesis process, by which each symbol is multiplied by a phase rotation function, which has the effect of adding a corresponding phase shift to each symbol.

These data streams are then frequency-converted and combined for transmission as described below.

The phase rotation function cm is defined as a quadratic phase function of the number of the subcarrier m (running from the first, m=0 to the total number Nc-l): j2'r P my Cm(p) = e No for m = 0,.. NC -I P[O,NC-1] p, the coefficient characterising the function, is chosen among [0, NC -1] at the transmitter to obtain the lowest PAR. The transmission block is a time sequence, given in brackets below: N-1 j2'r-0 N-1 j2'r-1 N-1 j2'r- (N-l) ECmSme ECmSme cmSme m O m=0 n=() The relationship between the phase rotations is therefore similar in form to the "Newman" phase relationship described in Boyd. However, Boyd is describing the relationship is between the phases of the signals on each subcarrier, whereas in the present embodiment correction factor by which the original phase of each subcarrier signal is added. The phases of the subcarrier signals produced by the present embodiment are therefore not in the relationship described by Boyd or Newman.

Process of determining phase adjustments The value of p is the variable which governs the effectiveness of the correction applied by the embodiment. It can take one of multiple different values, and an appropriate value is selected afresh every OFDM symbol, by the phase synthesis block 4.

A local CF calculation block 7 is provided. It is arranged to calculate the crest factor (CF) of the output OFDM symbol which would be produced, if a test value of p were employed. It is operable to test the CFs of several different test values of p and feed back the results to be used to select an appropriate one of the test values, in an iterative search process.

Referring to Figure 2, for the first test value (step 102) the calculation block calculates a CF value and passes the value back to the phase synthesis block 4 (step 104).

The phase synthesis block 4 then tests (step 106) whether the CF value is low enough to lie underneath a threshold value Th. If not (step 108) the phase synthesis block selects a next test value of p end the CF calculation block 7 once more returns the corresponding CF value.

The process continues until a value of CF which is low enough to lie underneath a threshold value Th is located. This value is then selected (step 1 10) to transmit the OFDM symbol.

Subsequently, each of the data streams is modulated with a different OFDM subcarrier frequency (modulation block 5), combined (combiner block 6) and then up-converted to a radio frequency carrier (not shown) for transmission. The modulation may be achieved using the inverse Fast Fourier Transform (IFFT) since the frequencies of OFDM are in the harmonic relationship used by the IFFT.

It would be possible to perform an exhaustive search for the value of p which gave the lowest CF, but we have found that in practice this is not necessary, as we have found that there are many values of p giving low CFs, distributed across the range of p values, as shown in Figure 5.

For example, if QPSK is employed, and if the PAPR threshold Th is set to be 7dB, the number of OFDM symbols is 50,000 and the number of subcarrier signals 64, an average of 2.791 IFFTs (Inverse Fast Fourier Transforms) are required to determine a value of p resulting in a PAPR below 7dB. If instead, 64QAM is employed, then the average number of IFFTs required is 2.736. Generally, we have found that the number of IFFTs required in the search process is about 4% of the number of sub- carriers.

Thus, the search process can be conducted computationally quickly.

Reciever Referring to Figure 3, a receiver comprises demodulation and down conversion stages (not shown), followed by a Fast Fourier Transform (FFT) stage IS which separates the signal into its separate subcarrier channels. At each OFDM symbol, the phase on the pilot channel (in this case channel 0) is measured by a phase measurement device 17. From the measured phase the value of p is calculated, by a phase synthesis block 14, which then calculates for each subcarrier a phase correction using the calculated value of p and the subcarrier number, according to the expression The reception of sub-carrier k gives I j27r-(n) - j2'r-n 1 j2(m-k)n Yk N \ CmSme e = Smcm - e n m _ m n Nit 1 j2 (m k)n = it, Sme -it, e al N n For the received signal it P Yk Ske m = k O mink The other channel corrections conventionally applied to OFDM signal processing are also applied as appropriate. The samples are then converted by a parallel-to-serial convertor back into a time-domain sequence which is output for decoding.

The receiver may be implemented as one or more suitably programmed commercially available Digital Signal Processor chips, or as customdesigned integrated circuits.

Thus far, the apparatus described corresponds to that of our above mentioned earlier UK application. The transmission of pilot tones will now be discussed.

Pilot Subcarriers Pilot tones are conventionally used in OFDM to track the carrier frequency offset (CFO) and the sampling clock offset (SCO) in OFDM systems, for example WEAN. In the present invention, the pilot tones are used also to recover the phase distortions which are introduced to minimise PAPR.

In the recovery of frequency and phase errors (i.e. CFO and SCO errors), the pilot tones on pilot subcarriers are received over several successive symbols, and accumulated over time to calculate the frequency and clock error. These errors are typically small, and hence difficult to detect using only a single symbol period. A small frequency error typically manifests itself as a phase error if only a single frequency period is considered.

On the other hand, according to the present invention, in every symbol period a different phase distortion is provided. Thus, the phase distortions must be calculated from the pilot tones each symbol period: there is no carry over from one symbol period to the next in calculating the phase distortions.

As in the prior art, initial carrier frequency and clock synchronization is achieved by providing a preamble or initialization sequence prior to transmitting data. Thus, it is only necessary in the embodiment to track very small frequency and clock errors.

Referring to Figure 4, the subcarriers are numbered from a central subcarrier given an index number 0. As most OFDM signals are generated using the IFFT, they have a number of subcarriers which is a power of 2 (selected from 2, 4, 8, 16, 32, 64, 128 and so on). In many applications not all subcarriers are used for data and thus the unused subcarriers exist outside the central band of used carriers as shown in Figure 4.

As discussed above, the following phase correction is applied at the transmitter for mth sub-carrier in ith symbols: C I m ( p) = e N Equation 1 At the receiver, when the signal is converted into the time domain, one OFDM symbol after has a total of Ns samples that is equal to N+Ng, (where N are used subcarriers and Ng are extra time samples corresponding to the guard time between OFDM symbols). For example, in WEAN there are Ns = 80 samples corresponding to N=64 data samples from 64 subcarriers, plus Ng= 16 guard samples.

The [th received symbol is j2 Af t I j2 tn-(lNs+Ng)T rl (n) = e N SI,mCl, me + zip (n) Equation 2 where Af = f - f' is the CFO (Carrier Frequency Offset) - one of the parameters to be estimated at the receiver based on the pilot tones.

This is sampled with the sampling clock T' In = (INy + Ng)T'+nT'. Denote the relative SFO (Sampling clock offset) by = (T'-T) I T. j2 m( :'±8)r7 j2 n(l T<lNs+Ng)(l+r7)Tej24f(l+7)T 1 Emcee N e +zl(n) m Equation 3 Demodulation using the Discrete Fourier Transform (DFT) at the receiver yields the data symbols in the frequency domain, as follows: bilk,r(n)e =e He s R>(+r777 1 its c j2'Tnr7 j2'rl7(nk+mr7+y(+7)/) (/N,+Ng7 7 1 j2 r;+y( + 7 j2 -+'nr7+y(1+ T j2ry(/N,+Ng)+rj)7 iN - e N) + 1 s c e N N) =e {skC/eN n /,,,' /,m |+n'(k) =e ( a gXl+q)7 |: C jack N g r' :+lCI+n'(k) (/N+Ng) j2'ry (/Ns+Ng)(1+7)7 e' N S/ kC/ k|3+ ICI+n'(k) Equation 4 Here, due to the small value of residual frequency error Af and clock error A, it is generally found that, B is close to 1 and ICI is smaller compared to the Gaussian Noise in the transmission channel.

Considering the effects of frequency selective fading channel, assume that a prefix of cyclic extension is used, and the received path has a delay of with an attenuation factor of as. Hence the above result is modified as A/(IN N)(I+)T j2k-N-E_( -j2N)(e N)S by+ ICI +n,(k) Equation 5 Both CFO and SCO have been estimated and pre-compensated in the acquisition process using pre-ambles. Hence the residual value of Af and are very small. We can neglect the factor of,B (which is close to 1).

The total phase rotation is obtained as (k) = 2 Af (IN s + Ng)(] + y)T + 2k ( s N g + 2p, N +,(k) Equation 6 Where /(k)= kah--2'T N is phase contributed by fading channel given kh -,2'r- In Equation 6, the third term on the right hand side of the equation contains the effect of the pre-distortion which was added to reduce PAPR. On the one hand, it is desired to separate this term and estimate the coefficient p so as to reverse the phase distortion. On the other hand it is desired to remove this term so that it does not affect the calculation of CFO and SCO.

Estimation of the phase distortion coefficient The estimation of the phase distortion introduced for PAPR reduction has to be carried out for each symbol. A number M of subcarriers simultaneously carry a pilot value (e.g. a data symbol of a predetermined value, such as 1+j, known to both the transmitter and the receiver) which is subjected at the transmitter to the same phase distortion as the data values.

Denote the pilot subcarriers k5 (k, e[-N12,N12]) in one symbol. The difference of the rotated phase of the ky, th pilot and k,2 th pilot of the symbol I is obtained by , (Ak) = 2 N. + NR. Ok + 2p, K, - K2 - 2 Equation 7 If two pairs of pilots are chosen which satisfy: (k,'2)mod(N) = 1 and (ks22)mod(N) = 0 Equation 8 e.g.{-24,-1}, and {1,24} in a 64 subcarrier system, the difference of the rotated phase of the pilots of the symbol I is 46, (Akl) = Us, (- I + 24) = 2'T (-1 + 24) + 2 7zpl 2'r-(-1 + 24) 1(2) = , (1 - 24) = 2 ' g (1- 24) + 2 - 2 N (1- 24) The estimation of p, is obtained by P. = (, (Ak, ) + , (Ak2))N / 4 Equation 9 Thus, by selecting two pairs of pilot subcarriers according to the selection rule of Equation 8, the effects of other factors such as the frequency error (CFO) and clock error (SCO) are cancelled out and the coefficient can directly and simply be obtained from the phase differences between those pilot subcarriers.

Estimating CFO & SCO In a slow fading channel, the difference of the rotated phases between two adjacent symbols of the same subcarrier is: 0,(k) = ,(k) - A, ,(k) = 2 Af NsT(1 + R) + 2k Ns + 2 N (P' - P'-') 27r Af N,T + 2k s + 2-(p, - p, ,) Equation 10 Thus, the last term indicates a dependence on the successive values of the phase distortion coefficient applied.

However, if a pair of subcarrier pilots {+ k,. } are chosen such that (ks)mod(N)=0 Equation 11 (in other words, the modulus over N - the number of subcarriers - of the bracketed term is zero) e.g. {-8,8} and/or {24,-24} in a 64 sub-carrier system.

The unbiased estimation of CFO and SCO yields 0,(k5) At = s I 2NsTM 0,(k5)- O,(k5) Cs Ak where C:, = 2riN I N and Ak5 = k - k5.

As >0 k, <0 In order to get robust estimation, L consecutive symbols are detected, and result in the following estimation: Af = Af, and 77 = it, l l Equation 12 In other words, the frequency and clock offsets are detected by accumulating successive sample estimates.

Note that in the above equations for estimation of CFO and SCO, there is no error contribution from the third term in Equation 10 - in other words, the phase distortion terms have been cancelled out when the pilot subcarriers are selected according to the selection rules of Equation 11. Hence the performance remains unchanged for the system with phase rotation for PAPR 1 5 reduction.

Some observations will now be made on the selection rules indicated above. Firstly, the subcarriers consist of + pairs. Secondly, for large multiplexes (i.e. large N) there are several different sets of pairs which can be used. For example, for WEAN, for Nc=64, the pilots can be chosen at {+24, +8, +1}, of which two pairs {+8,+24} can be used for CFO and SCO tracking, and the pairs of pilots at {+ k, ,+1} can be chosen for phase recovery for each OFDM symbol. Other examples for WEAN are {+16,+1}, or {+ 16,+8,+1}.

The pilot subcarrier index numbers k are selected based on the total number of subcarriers N. and the relationship between the subcarrier index numbers k and the quadratic form of the phase distortion relationship (which includes k2/N).

Implementation at the receiver Referring back to Figure 3, the pilot subcarrier samples are extracted by extractor circuit 23, and fed to the phase estimation circuit 17, which performs the calculation of Equation 9 above. In parallel, they are fed also to frequency and clock error estimation circuit 25, which performs the calculation of Equation 12 above by accumulating over several sample periods. The clock error is fed back to the clock driving the baseband sampling (not shown) and the frequency error is fed back to the local RF oscillator (not shown) which demodulates the received RF signal.

Mobile Telephone The invention may advantageously be used in a mobile telephone, which includes a transceiver comprising a transmitter according to Figure I and a receiver according to Figure 3.

Referring to Figures 6a and 6b, the general components of a mobile telephone according to a first embodiment of the invention are shown illustratively. It comprises a handset for use with a third generation or similar radio network, and having a user interface consisting of a keypad input device 38, a screen 39, a loudspeaker 34 and a microphone 36. Also provided are an antenna 31 and a radio frequency (RF) modulator-demodulator 32. The modem 32 includes the above-described OFDM circuit.

A SIM interface circuit 33 is arranged to receive a subscriber identity module (SIM) 35. The control circuit 37 (which may in practice be integrated with codec 30) consists of a suitably microprocessor, microcontroller or digital signal processor (DSP) chip or chip set.

In use, it will be understood that the user can set up a voice call using the keypad 38 to indicate the desired number, and then communicate using the loudspeaker 34 and microphone 36. Alternatively, the user may access data services such as WAP services, or may communicate text messages, using the keypad 38. A suitable radio base station (not shown) also includes an identical modem including the above-described OFDM circuit.

Other embodiments and variants It should be noted that the invention is not limited to the above described exemplary embodiment and it will be evident to a skilled person in the art that various modifications may be made without departing from the invention.

For example, the pilot subcarriers used might not be permanently assigned but might vary over time in a predetermined manner, between different allowable sets meeting the above criteria. The pilot sample value may likewise vary.

Where a phase relationship other than a quadratic is used to pre distort, then likewise the selection rules in the equations above will be modified accordingly.

Although transmission and reception are described, the invention may be practiced using recording onto, and reproduction from, a recording medium.

These and any other variants apparent to the skilled person are intended to be covered by the claims. For the avoidance of doubt, protection is sought for any and all novel subject matter contained herein.

Claims

Claims 1. Frequency division multiplexing (FDM) modulator apparatus,

comprising means for transforming an input signal into a plurality of sub carrier signals NC making up a multi-carrier signal, and means for applying a predetermined pilot data value to a plurality of said subcarrier signals to provide a plurality of pilot subcarriers; further comprising phase adjustment means which performs, for each FDM symbol, a separate and different phase adjustment to the phase of each of the sub-carrier signals Nc, the phase adjustments being a function of the subcarrier index number k, and of at least one coefficient which is the same across all said subcarriers and is derived for each FDM symbol; in which said pilot subcarriers are so selected that: a first subset thereof can be arithmetically combined for each symbol to give a phase value from which said coefficient can be determined substantially independently of receiver phase and/or clock errors; and a second subset thereof can be arithmetically combined to give a phase value from which receiver frequency and/or clock errors can be determined substantially independently of said coefficient.
2. FDM transmitter apparatus comprising a modulator apparatus according to claim 1, and means for transmitting the modulated signal.
3. Frequency division multiplex (FDM) demodulator apparatus, comprising means for transforming a multiplexed input signal made up of a plurality of sub-carrier signals NC into a time-domain output signal, and means for separating therefrom a plurality of pilot subcarriers each carrying a predetermined pilot data value; further comprising means for performing, for each FDM symbol, a separate and different phase adjustment to the phase of each of the sub-carrier signals Nc' the phase adjustments being a function of the subcarrier index number k, and of at least one coefficient which is the same across all subcarriers for each FDM symbol; in which said pilot subcarriers are so selected that: a first subset thereof can be arithmetically combined for each symbol to give a phase value from which said coefficient can be determined substantially independently of receiver phase and/or clock errors; and a second subset thereof can be arithmetically combined to give a phase value from which receiver frequency and/or clock errors can be determined substantially independently of said coefficient.
4. Apparatus according to claim 3, comprising means for arithmetically combining said first subset to derive, for each symbol, a value of said coefficient.
S. Apparatus according to claim 3 or claim 4, comprising means for arithmetically combining said second subset to derive a value of receiver frequency and/or clock errors.
6. FDM receiver apparatus comprising a demodulator apparatus according to any of claims 3 to 5, and means for receiving the modulated signal.
7. FDM transceiver apparatus comprising FDM transmitter apparatus according to claim 2 and FDM receiver apparatus according to claim 6 in a single housing.
8. Mobile telephone apparatus comprising FDM transceiver apparatus according to claim 7.
9. Apparatus according to any preceding claim, in which the modulation is orthogonal FDM (OFDM).
10. Apparatus according to any preceding claim, in which the phase adjustments are a quadratic function of the subcarrier number.
11. Apparatus according to claim 10, in which the coefficient is a secondorder coefficient.
12. Apparatus according to claim 1 1, wherein the quadratic phase synthesis function is j2'r P m2 Cm(p) = e No for m = 0,.. N -1 pe[O,Nc-l] where m is the subcarrier number and p is the coefficient.
13. Apparatus according to any preceding claim, wherein the signal is modulated using PSK (Phase Shift Keying).
14. Apparatus according to any preceding claim, wherein the signal is modulated using quadrature amplitude modulation (QAM).
15. Apparatus according to any preceding claim, wherein the first subset includes two pairs of subcarriers such as to meet the criteria: (kid 2) mod(N) = l and (k,22) mod(N) = 0 where N is the number of subcarriers, and ks and k52 are the index numbers of the subcarriers about a central subcarrier having index number kS=O.
16. Apparatus according to claim 15, wherein the first subset are subcarriers having index numbers {+1}.
17. Apparatus according to any preceding claim, wherein the second subset includes a pair of subcarriers which meet the criterion: (k2) mod(N) = 0 where N is the number of subcarriers, and k5 is the index number of a subcarrier about a central subcarrier having index number k=0.
18. An FDM signal, comprising, at each symbol, a plurality of subcarriers, in which the phase of each subcarrier is adjusted by a separate and different phase adjustment, the phase adjustments being a function of the subcarrier index number k, and of at least one coefficient which is the same across all subcarriers, in which a plurality of said subcarriers comprise pilot subcarriers which carry predetermined data values, said pilot subcarriers being unevenly distributed in frequency over said signal, said pilot subcarriers comprising a first subset and a second subset, selected so as to separate the effects at a receiver of said phase adjustment from those of frequency and/or clock errors.
19. A method of selecting pilot tones in an OFDM transmission which has been phase distorted for reducing PAPR in accordance with a mathematical relationship, comprising using selected, non-evenly spaced subcarriers corresponding to said mathematical relationship.
20. A method of OFDM transmission, including at the transmitter, preadjusting each subcarrier phase by an amount determined by a quadratic function of the subcarrier index and a predetermined coefficient, and at the receiver, recovering the predetermined coefficient, and reversing the preadjustments to recover the signal and tracking frequency and clock offset errors, characterized by using a plurality of pilot subcarriers, in which the subcarriers comprise a first subset thereof which can be combined for each symbol to give a phase value from which said coefficient can be determined substantially independently of receiver phase and/or clock errors; and a second subset thereof which can be arithmetically combined and accumulated over time to give a phase value from which receiver frequency and/or clock errors can be determined substantially independently of said coefficient.