GB2524464A

GB2524464A - Frequency error estimation

Info

Publication number: GB2524464A
Application number: GB1401666.1A
Authority: GB
Inventors: Brian Gaffney
Original assignee: Neul Ltd
Current assignee: Huawei Technologies Research and Development UK Ltd
Priority date: 2014-01-31
Filing date: 2014-01-31
Publication date: 2015-09-30
Also published as: CN104821926B; GB201401666D0; CN104821926A

Abstract

Estimating an unknown error in a carrier frequency, comprising sampling a signal that is a modulated version of the carrier frequency to form a plurality of data blocks, each data block comprising a plurality of samples and each sample incorporating a phase error due to the carrier frequency error, and for each data block: generating a product 103 of each sample with another sample in the data block, the other sample being located in the same position in a reversed version of the data block as the sample is in the data block, to form combined samples that each incorporate substantially the same phase error due to the carrier frequency error; and averaging 104 the combined samples to form an average sample; and estimating 106 the carrier frequency error in dependence on the plurality of average samples generated from the plurality of data blocks. The final step may comprise estimating the frequency tone in the vector of averages 106 using e.g. a Fourier transform. The sampling rate may be the chip rate used to spread the signal. The estimation may comprise a fine and coarse stage (Figure 2) with the complete estimate based on a sum of fine and coarse estimates. The arrangement provides a mechanism of estimating the carrier frequency error without using pilots/preambles or involving decisions.

Description

FREQUENCY ERROR ESTIMATION

This invention relates to methods and apparatus for estimating an error in a carrier frequency.

Narrowband communication systems are very sensitive to carrier frequency error. This is because an error in the carrier frequency results in an unwanted phase rotation between symbols, which degrades receiver sensitivity. The degradation can be even more severe when spreading is used. Typically a receiver would first despread the received data and then sum over the length of the spreading code to obtain a good signal-to-noise ratio (SNR). Due to the frequency error, however, the signal will no longer sum coherently. There is also a phase shift between symbols. The combination of both these effects leads to a degradation in performance.

There are a number of techniques that standard receivers can use to estimate carrier frequency error. Many rely on pilot or synchronisation symbols, which are data sequences known in advance by the receiver. The receiver can estimate the carrier frequency error by comparing the phase of the received symbols with those it expected to receive. Having to transmit a synchronisation sequence lowers the achievable data rate, however. Decision directed techniques are another option. They estimate the phase error in received symbols by looking at the difference in phase between the received symbol and the nearest point in the modulation constellation. This mechanism relies on a decision that is typically very error prone at SNRs of interest.

The angle operation is also very noise sensitive and can result in large errors. The additional noise degradation caused by the imperfect coherent combining makes this even worse.

Therefore, there is a need for an improved mechanism for estimating carrier frequency error.

According to one embodiment, there is provided a method for estimating an unknown error in a carrier frequency, comprising sampling a signal that is a modulated version of the carrier frequency to form a plurality of data blocks, each data block comprising a plurality of samples and each sample incorporating a phase error due to the carrier frequency error, for each data block: generating a product of each sample with another sample in the data block, the other sample being located in the same position in a reversed version of the data block as the sample is in the data block, to form combined samples that each incorporate substantially the same phase error due to the carrier frequency error; and averaging the combined samples to form an average sample; and estimating the carrier frequency error in dependence on the plurality of average samples generated from the plurality of data blocks.

The method may comprise averaging the combined samples by summing them and dividing by the number of samples in the data block.

The signal may be a version of the carrier frequency that has been modulated by spreading, and the method may comprise sampling the signal using a sampling rate that is equal to the chip rate used to spread the signal.

The signal may be a version of the carrier frequency that has been modulated by spreading, and the method may comprise forming the data blocks to each comprise a number of samples equal to the spreading factor.

The method may comprise estimating the carrier frequency error using a maximum likelihood estimator.

The method may comprise estimating the carrier frequency error by performing a Fourier transform on the plurality of average samples, and selecting, as the carrier frequency error, the frequency with the highest value in the transform.

The method may comprise for multiple antennas, for each signal received by the multiple antennas, generating a plurality of average samples, on each set of average samples, performing a Fourier transform, and combining the multiple transforms obtained by squaring and summing.

The estimated carrier frequency error may incorporate an ambiguity, and the method may comprise a fine estimation stage, in which a fine estimate of the carrier frequency error is obtained, and a coarse estimation stage, in which the ambiguity in the fine estimate is resolved.

The fine estimation stage may comprise generating the fine estimate in accordance with the method of claim 1, and by using a first sampling frequency to generate the data blocks; and the coarse estimation stage may comprise resolving the ambiguity by generating a coarse estimate in accordance with the method of claim 1, and by using a second sampling frequency to generate the data blocks, the second sampling frequency being lower than the first sampling frequency.

The coarse estimation stage may comprising generating the coarse estimate using a different technique for estimating the carrier frequency error from the technique used by the fine estimation stage.

The method may comprise, for the coarse estimation stage, generating the coarse estimate of the carrier frequency error may comprise: mixing the plurality of average samples with two or more frequencies, each of said frequencies being a multiple of the ambiguity, determining a signal power comprised in each of the mixed signals and selecting the frequency that generated the mixed signal that comprises the highest signal power as the coarse estimate of the carrier frequency error.

The method may comprise summing the estimated carrier frequency error with the coarse estimate of the carrier frequency error to form a complete estimate of the carrier frequency error in which the ambiguity is resolved.

The signal may be a version of the carrier frequency that has been modulated using phase shift keying.

According to a second embodiment, there is provided an apparatus configured to estimate an unknown error in a carrier frequency by implementing a method as claimed in any of claims ito 12.

According to a third embodiment, there is provided an apparatus for estimating an unknown error in a carrier frequency, comprising a sampling unit configured to sample a signal that is a modulated version of the carrier frequency to form a plurality of data blocks, each data block comprising a plurality of samples and each sample incorporating a phase error due to the carrier frequency error, a product unit configured to, for each data block, generate a product of each sample with another sample in the data block, the other sample being located in the same position in a reversed version of the data block as the sample is in the data block, to form combined samples that each incorporate the substantially the same phase error due to the carrier frequency error, and average the combined samples to form an average sample, and a tone estimator configured to estimate the carrier frequency error using the plurality of average samples generated from the plurality of data blocks.

The present invention will now be described by way of example with reference to the accompanying drawings. In the drawings: Figure 1 shows an example of a method for estimating a carrier frequency error; Figure 2 shows an example of a method for implementing fine and coarse estimation; Figure 3 shows an example of a method for obtaining a coarse frequency estimation; and Figure 4 shows an example of an apparatus for estimating a carrier frequency error.

To avoid the problems with current techniques for estimating carrier frequency error, an estimator is proposed which avoids the need for decisions, a known preamble or pilots and reduces the problem to an easier tone detection problem.

In this method, an unknown error in a carrier frequency is estimated by sampling a signal that is a modulated version of the carrier frequency to form a plurality of data blocks. Each data block comprises a plurality of samples, and each sample incorporates a phase error due to the carrier frequency error.

Each data block is then processed by generating a product of each sample in the data block with another sample in the same block. The other sample is located in the same position in a reversed version of the data block as the sample is in the data block. The result is a vector of combined samples that each incorporate substantially the same phase error due to the carrier frequency error. The combined samples are then averaged, and the plurality of average samples obtained by performing these steps across the plurality of data blocks are used to estimate the carrier frequency error.

In one example the data is modulated using phase-shift keying, specifically binary phase shift keying (BPSK). The techniques described herein may be particularly advantageous in communication systems that employ a form of modulation in which data is encoded by means of phase modulation; it is modulation techniques that rely on the phase of the received signal that suffer particular performance degradation when that phase is altered due to carrier frequency error. This is just one example, however, and it should be understood that other modulation methods may be used.

The techniques described herein may also be particularly advantageous in communication systems that employ spreading. Performance degradation resulting from unwanted phase rotations due to carrier frequency error can be particularly severe in systems that employ spreading because the signal no longer sums coherently. This deficiency may be addressed by the techniques described herein because they enable the unwanted phase rotations to be removed once the carrier frequency error has been estimated. Thus, while the estimation techniques are described with reference to spread signals: it should be understood that this is merely to illustrate the advantages of these techniques.

An example of a method for estimating a carrier frequency error is shown in Figure 1.

The method starts by separating the received signal into data blocks (step 101). Each data block comprises a plurality of samples. The data blocks may be the same size or different sizes. In one example the data blocks each comprise a number of samples equal to the spreading factor, N5. Other sizes of data block may also be used (as is apparent from the coarse frequency estimator described below).

In this example the signal has been spread, so the received chips are multiplied by the corresponding spreading value 5m (step 102). The element-wise product of the vector represented by the data block with the reversed version of itself is then calculated (step 103). Each element of the resulting vector incorporates substantially the same unwanted phase rotation due to the carrier frequency error. Consequently, the elements can be summed largely coherently when averaged (step 104). This sequence of steps is performed for each data block over the measurement period of L data symbols (step 105). The steps may be performed in series, so that each data block is processed in turn, or may be efficiently processed in parallel. L does not need to be the length of the data sequence and can be chosen as a trade-off between accuracy and error variability. The resulting vector of length L is the sum of an aliased tone of frequency 2fe and noise. The final step is to estimate the tone frequency present in that vector to obtain an estimate of the carrier frequency error (step 106).

Due to aliasing, the estimated carrier frequency error has an ambiguity equal to a multiple of. A coarse frequency estimation stage may resolve this ambiguity by performing a similar process to that shown in Figure 1 after fine frequency correction.

This is described in more detail below.

The method shown in Figure 1 will now be described in more detail with reference to a signal that is modulated using binary phase shift keying (BPSK) and spread. This is for the purposes of example only. As explained above, the invention is limited to neither a particular form of modulation or spreading.

The binary data bm is spread by some binary sequence Sk and then mapped to a BPSK constellation point dk E {-1, -l-1}. The data can also be differentially encoded if required. The transmitted signal is then filtered by some pulse (e.g. a root raised cosine), p(t). The transmitted signal is therefore k b[k/N5jskp(t -kT) (1) where T is the chip time (i.e. the inverse of the 3dB bandwidth R) and N is the spreading factor.

The received signal experiences a frequency error due to crystal error, channel effects and frequency error of the transmitted signal. This frequency error is written as fe and can vary with time. The explanation below assumes that the frequency error is constant over the burst used to estimate the error. The principles of the invention may, however, also be applied to a system encountering changing channel conditions, in which fe may vary (which is described in more detail below).

The received signal can be written as: r(t) = Ek b[k/N JSkP(t -. kT)eJ2Tct + n(t) (2) The signal is match filtered and decimated to the chip rate. The discrete signal can be written as: --j21rfm I -U[k/WjSme m This assumes that the receiver has knowledge of the timing, either due to timing alignment on the transmission of the bursts or from some pre-processing stage.

Equation 3 ignores a term to take into account the breaking of the Nyquist pulse structure by the frequency error. This will lead to intersymbol interference, but only at very high frequency errors. Hence it can be ignored in the analysis.

The processing for a single channel symbol k is described first. The received vector for this channel symbol is written as: dkeJ/)k[soe,sleJ1e... swg_ie_i211_] + n (4) where Øk5 the initial phase for the kth symbol.

The spreading is removed by multiplying by the spreading symbols Sm. Since these can be either ±1, the resulting received sequence can be written as: = dke_i1c[e0, eJ2!e... e_i2Th1e(ts_1)I + 11 (5) Element-wise multiplications are then performed on this sequence and a time-reversed version i: = deJ2'Pk [e°, e12Th!e... e_12Th1e(Ws_1)] * e_12ff1e(ws_2) ... + nil = deJ21'k [e0_12 JiJ1), ... e_i2wt(1_1)+0I + nil = cite I2'Pk [e j27r!e(Ws1) e -J2irf(W5-1) e + nE (6) For BPSK modulation, d = 1 for all k. This gives: = e_J2k[e_J21(1Vs_1), J2I(/s1) e_121(Ns_1)] nil (7) As shown in equation 7, multiplying each sample with its corresponding number in the reversed version of the data block generates combined samples that all have the same phase. (In reality, these phase components may not be exactly the same; there may be relatively small variations due to a fluctuating frequency error and channel across the data block). The combined samples are then averaged. Any form of averaging might be used, e.g. mean, median, mode. A preferred option is to calculate the mean: = E = e_J2ke_J21e(Ns_l) + 1]k (8) Because the combined samples have substantially the same phase, they are summed largely coherently when calculating the mean.

The phase increases by 211fe(Ns -1) between consecutive symbols. Therefore, = cPIC + 27Tf(JVs -1). This implies: Zk+l = e_j24hjc+ie_j2te(tTs_1) --Th<+ = e_J2k_J21 se_]27t1e(Ns_1) + 11k+1 (9) Substituting O as an initialising phase gives: Zk+l = e_JOoe_iTh2te.c< +7/k+1 (10) The average samples thus form a sequence in which the carrier frequency error causes a change in phase from one sample to the next. It is clear from equation 10 that the vector of data z is therefore a tone of frequency 21e* The problem has now been reduced to a tone frequency estimation problem. There are multiple methods available to estimate the frequency of this tone, and it is possible to apply any tone estimation algorithm to the vector z. Preferably a maximum likelihood estimator is used to estimate the tone. A practical way of implementing this may be via a Fourier transform. For example, an L point FFT may be performed on vector z. The bin with the maximum absolute value is identified. The frequency corresponding to this bin is determined to be the tone of frequency 2f. This is close to a maximum likelihood estimator, with the only difference between this and a true maximum likelihood estimator being the discrete nature of the bins.

In the case of multiple receive antennas, the result of the FET's for each of the receive antennas can be squared and summed to combine the results. This is just one example of how the results from multiple receive antennas might be combined and it should be understood that the invention covers the use of any suitable method.

It is also possible to assume a changing carrier frequency error. The frequency error is preferably constant over the length of the spread symbol. The maximum permissable frequency error deviation from the mean is ±. The mean can be any value in the range. The methods described herein estimate the mean over the length of the spread symbol burst. After this is corrected for, any deviating frequency error should be small and can be tracked by known tracking schemes (e.g. by using Costas loop phase tracking).

The frequency capture range is dependent on the sampling frequency. In the example described above, this means that the frequency capture range is dependent on the spreading factor N. This introduces aliasing with the Nyquist (or folding) frequency being half the sampling frequency. The carrier frequency error estimate output by the

SW

process described above is in the range ± The extra factor of two on the denominator is because the tone in vectorz is of frequency 2f. The frequency capture range may be limiting, particularly for larger spreading factors.

A limited capture range means that the estimated error may include an ambiguity. This may result in a correction made to remove the unwanted phase rotations in the received data being wrong. A coarse frequency estimation stage may therefore be applied. This is shown in Figure 2, in which the coarse stage (step 202) is applied after the fine stage (step 201). The frequency ambiguity output by the coarse estimation stage is added to the fine estimate to generate a complete estimate of the carrier frequency error (step 203).

The ambiguity can only take certain values. The range of values are ± for / = -1,0, +1 and so on. The number of valid / values is a design choice based on the expected range of frequency errors to deal with. For the purposes of explanation only, a method for solving this ambiguity is described below with reference to 1 = -1,0, +1.

The basic principle of the coarse stage is to create a new tone signal using the same technique outlined for fine frequency estimation. This is illustrated in Figure 3. The fine frequency error compensation is first applied to the data to remove the fine estimate of the error(step 301). The coarse frequency estimation is likely to use data blocks of different lengths from the fine frequency estimation to obtain a larger frequency capture range. Steps 302 and 303 are thus essentially repetitions of steps 101 and 102 but for a different sampling rate. In one example, the N chips for each channel symbol are split into a pair of samples. The coarse estimation then proceeds with the same processing steps as in Figure 1 (steps 304 to 306) but for a different sampling frequency to get a new tone signal 1. In the example where each channel symbol is split into a pair of samples, this essentially results in repeating the processing laid out in equations 4 to 10 above but with N5 = 2. The tone signal i is noisier than z but we already have knowledge of where we expect the tone to be, i.e. around± for! = -1,0, +1. It is unlikely to be exactly at any of those frequencies if there is a small error in the fine frequency error, which has already been estimated.

The coarse frequency estimation stage deviates from the fine frequency estimation stage when it comes to estimating the tone. The tone signal I is likely to be noisy, but the tone's frequency need only be identified to the extent necessary to identify which of the valid I values is the correct one. Therefore, a different estimation technique is likely to be applied for the coarse estimation stage from the fine estimation stage. In the example of Figure 3, 1 is first mixed by ±L for the chosen £ values (step 307).

The resulting signals are then passed through a low pass filter (step 308). The bandwidth of this filter may be set based on the expected residual frequency error and Doppler spread of the channel. For example, a simple moving average filter can be used. The power contained in the resulting signal is calculated to obtain a power value Pi (step 309). Steps 306 to 308 are repeated for each chosen 1 value (step 310). Finally the coarse frequency is estimated by picking the value £ value that gives the largest power value Ri (step 311). Other selection methods may be used. This approach, however, is both straightforward to implement and close to a maximum likelihood estimator.

If there are multiple receive antennas, the respective Ri values for each £ value may be generated by summing the respective Ri values from each receive antenna for a given £ value.

An apparatus for implementing the invention is likely to be implemented as part of a receiver for a wireless communication system. Such a receiver might typically include RF circuitry well known for implementing wireless communications, including e.g. an antenna system, an RF transceiver, one or more amplifiers, a tuner, one or more oscillators, a digital signal processor, a memory and so forth.

An example of an apparatus is shown in Figure 4. The apparatus, shown generally at 401, is configured to receive a signal r (402). Although the "received" signal may be a signal directly from an antenna, it may also have undergone some pre-processing before being input into the apparatus.

The apparatus comprises a first sampling unit 403 for forming the data blocks. The apparatus also comprises a product unit 404 for forming the vector of averages z. Together these are referenced as step I in Figure 4, corresponding to steps 101 to 105 in Figure 1. The product unit is followed by a tone detector, which is configured to compute the fine frequency error f. This is referenced as step II in Figure 4 and corresponds to step 106 in Figure 1. The fine frequency error is output to a summation unit 405.

The apparatus may also be configured to perform coarse frequency estimation, although this functionality may be disabled if the frequency capture range is deemed sufficient for coarse frequency estimation to be unnecessary.

The apparatus comprises a mixer 406 configured to mix the received signal with the estimated fine frequency error fe The apparatus further comprises second sampling and product units (407 and 408), which are configured to output another vector of averages to a group of parallel processing paths for estimating the ambiguity in the fine estimate. Each path comprises an ambiguity mixing unit 409. These are connected to low pass filters 410 and power computation units 411. Finally the apparatus comprises a maximum identification unit, configured to receive the power values output by the power computation units and identify the frequency associated with the maximum of those power values as the coarse frequency estimate. The maximum identification unit is configured to output the coarse frequency estimate to summation unit 405.

The parallel processing paths illustrated in Figure 4 might equally be implemented serially. Similarly it should be understood that others of the components might perform some of all of their processing in parallel (e.g. the product calculations performed across the plurality of data blocks by product units 404, 408).

The structures shown in Figure 4 are intended to correspond to a number of functional blocks. This is for illustrative purposes only. Figure 4 is not intended to define a strict division between different parts of hardware on a chip or between different programs, procedures or functions in software. In some embodiments, some or all of the algorithms described herein may be performed wholly or partly in hardware. In many implementations, at least part of the communication apparatus may be implemented by a processor acting under software control. Any such software is preferably stored on a non-transient computer readable medium, such as a memory (RAM, cache, hard disk etc) or other storage means (USB stick, CD, disk etc).

The methods described herein allow for accurate carrier frequency offset estimation even in very low signal to noise environments, particularly for BPSK systems that employ spreading. The methods are applicable both to systems which experience flat fading and also to multipath channels. The frequency capture range is also increased over current estimation algorithms, approaching chip frequency rather than symbol frequency. The methods may also be applied to multiple antenna systems.

The methods described herein may be applied to a communication network configured for Internet of Things (loT) communication. An example would include a network configured to operate according to the WeightlessTM protocol (although the methods described herein may be readily implemented by networks configured to operate according to a different protocol, such as e.g. LTE, Bluetooth, WiFi, V0IP). Typically the network will consist of a number of communication devices (e.g. base stations) that are each configured to communicate with a large number of geographically spaced terminals. The communication apparatus described herein may be implemented by just such a communication device or terminal. The network may be a cellular network, with each communication device being responsible for over the air communications with terminals located in a respective cell. The communication devices suitably communicate via a wired or wireless interface with a core network and may act, at least partially, under the core network's control.

In one example, the communication apparatus described herein may be configured to operate in accordance with the WeightlessTM loT specification. WeightlessTM uses a cellular WAN architecture, with protocols optimised for the requirements of an loT system (low terminal cost, low terminal duty cycles and hence low power consumption, and scalability to very low data rates). It was originally designed to operate in TV Whitespace spectrum from 470 to 790 MHz, but the PHY is generalised to operate in licensed, shared licensed access and license-exempt bands of varying bandwidths.

The applicant hereby discloses in isolation each individual feature described herein and any combination of two or more such features, to the extent that such features or combinations are capable of being carried out based on the present specification as a whole in the light of the common general knowledge of a person skilled in the art, irrespective of whether such features or combinations of features solve any problems disclosed herein, and without limitation to the scope of the claims. The applicant indicates that aspects of the present invention may consist of any such individual feature or combination of features. In view of the foregoing description it will be evident to a person skilled in the art that various modifications may be made within the scope of the invention.

Claims

CLAIMS1. A method for estimating an unknown error in a carrier frequency, comprising: sampling a signal that is a modulated version of the carrier frequency to form a plurality of data blocks, each data block comprising a plurality of samples and each sample incorporating a phase error due to the carrier frequency error; for each data block: generating a product of each sample with another sample in the data block, the other sample being located in the same position in a reversed version of the data block as the sample is in the data block, to form combined samples that each incorporate substantially the same phase error due to the carrier frequency error; and averaging the combined samples to form an average sample; and estimating the carrier frequency error in dependence on the plurality of average samples generated from the plurality of data blocks.
2. A method as claimed in claim 1, comprising averaging the combined samples by summing them and dividing by the number of samples in the data block.
3. A method as claimed in claim 1 or2, wherein the signal is a version of the carrier frequency that has been modulated by spreading, the method comprising sampling the signal using a sampling rate that is equal to the chip rate used to spread the signal.
4. A method as claimed in any preceding claim, wherein the signal is a version of the carrier frequency that has been modulated by spreading, the method comprising forming the data blocks to each comprise a number of samples equal to the spreading factor.
5. A method as claimed in any preceding claim, comprising estimating the carrier frequency error using a maximum likelihood estimator.
6. A method as claimed in any preceding claim] comprising estimating the carrier frequency error by: performing a Fourier transform on the plurality of average samples; and selecting, as the carrier frequency error, the frequency with the highest value in the transform.
7. A method as claimed in claim 6, comprising, for multiple antennas: for each signal received by the multiple antennas, generating a plurality of average samples; on each set of average samples, performing a Fourier transform; and combining the multiple transforms obtained by squaring and summing.
8. A method as claimed in any preceding claim, wherein the estimated carrier frequency error incorporates an ambiguity, the method comprising a fine estimation stage, in which a fine estimate of the carrier frequency error is obtained, and a coarse estimation stage, in which the ambiguity in the fine estimate is resolved.
9. A method as claimed in claim 8, in which: the fine estimation stage comprises generating the fine estimate in accordance with the method of claim 1, and by using a first sampling frequency to generate the data blocks; and the coarse estimation stage comprises resolving the ambiguity by generating a coarse estimate in accordance with the method of claim 1, and by using a second sampling frequency to generate the data blocks, the second sampling frequency being lower than the first sampling frequency.
10. A method as claimed in claim 9, the coarse estimation stage comprising generating the coarse estimate using a different technique for estimating the carrier frequency error from the technique used by the fine estimation stage.
11. A method as claimed in any of claims 8 to 10, the method comprising, for the coarse estimation stage, generating the coarse estimate of the carrier frequency error by: mixing the plurality of average samples with two or more frequencies, each of said frequencies being a multiple of the ambiguity; determining a signal power comprised in each of the mixed signals; and selecting the frequency that generated the mixed signal that comprises the highest signal power as the coarse estimate of the carrier frequency error.
12. A method as claimed in any preceding claim, the method comprising summing the estimated carrier frequency error with the coarse estimate of the carrier frequency error to form a complete estimate of the carrier frequency error in which the ambiguity is resolved.
13. A method as claimed in any preceding claim, wherein the signal is a version of the carrier frequency that has been modulated using phase shift keying.
14. An apparatus configured to estimate an unknown error in a carrier frequency by implementing a method as claimed in any of claims ito 12.
15. An apparatus for estimating an unknown error in a carrier frequency, comprising: a sampling unit configured to sample a signal that is a modulated version of the carrier frequency to form a plurality of data blocks, each data block comprising a plurality of samples and each sample incorporating a phase error due to the carrier frequency error; a product unit configured to, for each data block, generate a product of each sample with another sample in the data block, the other sample being located in the same position in a reversed version of the data block as the sample is in the data block, to form combined samples that each incorporate the substantially the same phase error due to the carrier frequency error, and average the combined samples to form an average sample; and a tone estimator configured to estimate the carrier frequency error using the plurality of average samples generated from the plurality of data blocks.
16. A method substantially as herein described with reference to the accompanying drawings.
17. An apparatus substantially as herein described with reference to the accompanying drawings.