WO2008052573A1 - Residual carrier frequency offset estimation and correction in ofdm multi-antenna systems - Google Patents

Abstract

Method, system and computer program for estimating a residual carrier frequency offset of an OFDM signal, the method comprising the steps of: receiving at at least one receiving antenna at least one OFDM signal, said OFDM signal comprising a preamble and a data frame, said data frame comprising a plurality of OFDM symbols; sampling said plurality of OFDM symbols; acquiring a channel estimate from said OFDM signal; wherein for each group of samples of each OFDM symbol, performing a Discrete Fourier Transform; from the result of each Discrete Fourier Transform, selecting the samples at the pilot subcarriers; from said samples at the pilot subcarriers and said channel estimate, calculating an estimate of the phase shift between consecutive OFDM symbols due to said residual carrier frequency offset. Correction of residual carrier frequency offset.

Description

RESIDUAL CARRIER FREQUENCY OFFSET ESTIMATION AND CORRECTION

IN OFDM MULTI-ANTENNA SYSTEMS

FIELD OF THE INVENTION

The present invention relates to the frequency offset estimation and correction in multicarrier systems. More particularly, the present invention relates to the residual frequency offset estimation and correction in Orthogonal

Frequency Division Multiplexing (OFDM) systems with multiple antennae at both the transmitter and the receiver

(multiple-input-multiple-output, MIMO) .

STATE OF THE ART

Orthogonal Frequency Division Multiplexing (hereinafter referred to "OFDM") is a promising technique for efficiently transmitting information over a fading channel.^' Its fundamentals are based on the division of the transmission bandwidth into a set of frequency subchannels, also called subcarriers or carriers, each of them used for transmitting a different stream of information in parallel. If the channel is frequency-selective but the time impulse response of the channel is shorter than the length of the cyclic prefix, then each of the frequency subchannels can be seen as a flat fading channel, thus facilitating the equalization process. Besides, the modulation and demodulation procedures are implemented by means of inverse discrete Fourier transformations (IDFT) and discrete Fourier transformations (DFT), respectively, which, when p.ossible, could be implemented by means of inverse fast Fourier transformations (IFFT) and direct fast Fourier transformations (FFT), respectively, thus allowing for a low computational load. Currently, this modulation technique is being used in some broadcasting systems such as terrestrial digital video and audio broadcasting (DVB-T and T-DAB) , and some wireless local area networks such as Hiperlan/2, WiMax, IEEE 802.11a, 802. Hn, among other standards. A complete reference on OFDM ca be found in Z. Wang and G. B. Giannakis, "Wireless Multicarrier Communications", IEEE Signal Processing Magazine, vol. 17, no. 3, pp. 29-48, May 2000.

Figure 1 represents an OFDM signal in the temporal domain. The transmitted temporal signal in a data frame

(DF) is constructed as the concatenation of n_L data OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), each of them consisting of N payload samples ( n = 0,...,N-l ) and L cyclic prefix (CP) samples [n = -L,...,-\) , which are equal to the last L samples of each OFDM symbol. Usually, a preamble

(PRE) is transmitted before the data frame (DF) ; this preamble is known at the receiver (the existing standards define this preamble) .

Besides, the quality and the rate of the communication through a wireless channel can be increased combining OFDM with spatial diversity techniques, i.e., with systems that exploit multiple antennae at the transmitting end and/or the receiving end. Depending on the number of antennae, there are different types of channels: classical single antenna transmission (SISO: single-input-single-output) , multiple antennae only at the receiver (SIMO: single-input- multi-output), multiple antennae only at the transmitter (MISO: multi-input-single-output), and the most general case with multiple antennae at both sides (MIMO: multi- input-multi-output) . Figure 2 shows a schematic representation of the wireless communications MIMO system.

There are many techniques which exploit the multiple antennae channel, depending on the quantity and the quality of the channel knowledge available at the transmitting end. An concrete example is Orthogonal Space-Time Block Coding (OSTBC) , which does not require any channel information at the transmitter. The particular case for two transmitting antennae is known as Alamouti's code and is described in the paper "A Simple Transmit Diversity Technique for Wireless Communications", IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, October 1998, by S. M. Alamouti.

Although the previous techniques are very promising solutions, their practical deployment may encounter some problems, such as the ones described by R. Narasimhan, in

"Performance of Diversity Schemes for OFDM Systems With

Frequency Offset, Phase Noise, and Channel Estimation

Errors", IEEE Trans, on Communications, vol. 50, no. 10, pp. 1561-1565, October 2002. Particularly, the OFDM modulation is very sensitive to mismatches between the sampling clocks and the frequency of the local oscillators at the transmitting and receiving ends. This frequency mismatch can lead to inter-carrier interference between the frequency subchannels, thus increasing the symbol error rate (SER) , and thus, reducing the efficiency of the communication . The frequency mismatch, also called carrier frequency offset (CFO), can be compensated in two stages. Firstly, a coarse CFO correction is performed in an acquisition stage. Normally this coarse CFO correction is not enough, resulting in a residual CFO that has to be compensated using a second tracking stage.

The receiver usually carries out a coarse carrier frequency offset estimation and correction, jointly with the channel estimation. These operations can be performed using the preamble (PRE) (having the same function as a training sequence) before the n_L data OFDM symbols (SYM#0,

SYM#1, ..., SYM#1, ..., SYM#n_L-l) in the data frame (DF) . The preamble is composed of several known OFDM symbols that are transmitted before each data frame (DF) , and which are specified by the standard and are therefore known at the receiver side.

There is a vast literature on techniques for estimating and correcting the carrier frequency offset

(CFO) for systems with a single transmitter and receiver antenna (SISO) . One example of these techniques is disclosed in J. Lei, T-S. Ng, "A Consistent OFDM Carrier

Frequency Offset Estimator Based on Distinctively Spaced Pilot Tones", IEEE Trans, on Wireless Communications, vol.

3, no. 2, pp. 588-599, March 2004. In this case, a coarse

CFO correction is performed in the acquisition stage, for which the preamble of the OFDM signal is used.

Some proposals can also be found which use the cyclic prefix in the tracking mode in SISO systems (see J-J van de Beek, M. Sandell, P. 0. Bόrjesson, "ML Estimation of Time and Frequency Offset in OFDM Systems", IEEE Trans. on Signal Processing, vol. 45, no. 7, pp. 1800-1805, July 1997, in which this is used assuming a flat fading channel) .

Recently, the pilot subcarriers have been exploited in an ad hoc manner to estimate the residual CFO in SISO systems, as described in WO 2005/062564 Al and US 2005/0129135 Al.

However, there are very few methods in which the carrier frequency offset (CFO) is estimated and corrected in systems with multiple antennae (i.e. MIMO systems), in spite of the increasing importance of these systems in the new wireless standards. Some examples are the use of blind techniques, as proposed in R. Ambati, U. Tureli, "Experimental Studies on an OFDM Carrier Frequency Offset Estimator", Proc . ICC (IEEE International Conference on Communications), vol. 3, pp. 2056-2060, May 2003. In J. Li, G. Liao, Q. Guo, "MIMO-OFDM Channel Estimation in the Presence of Carrier Frequency Offset", EURASIP Journal on Applied Signal Processing, vol. 2005, no. 4, pp. 525-531, 2005, a technique for joint estimation of the channel and the CFO using the preamble is presented. The preamble, combined with an adaptive technique based on LMS for a MIMO system, is used for compensating the carrier frequency offset (CFO) in US 2005/0141658 Al. Its transmission structure is constrained to the particular case of beamforming .

If the techniques for the coarse estimation of the carrier frequency offset are good enough, it can be assumed that the phase variation due to the residual carrier frequency offset is low within the samples of a single OFDM symbol in the data frame (DF) , which is realistic, as deduced from practical measurements. However, the phase variation between two OFDM symbols may be not negligible, specially when the number (n_L) of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) in the data frame (DF) is high. This is specially damaging in systems with a plurality of transmitting antennae. None of the prior art documents exploit the whole capabilities of the information contained in the OFDM frame, such as the pilot subcarriers, in MIMO systems. Therefore, the estimation and correction of the residual carrier frequency offset (CFO) is not efficiently achieved.

SUMMARY OF THE INVENTION

In order to solve this problem, the pilot subcarriers are used and exploited in order to estimate and correct the residual carrier frequency offset in MIMO systems with a low complexity.

It is therefore an object of the present invention to provide a method for estimating the residual carrier frequency offset (CFO) in the tracking stage of both single-antenna and multiple-antenna systems. The method of the present invention exploits the whole capabilities of the information contained in the OFDM frame. In particular, it not only exploits the preamble of the data frame, but also the presence of pilot subcarriers in all the OFDM symbols in the frame, as established in all the current standards which use OFDM. Thus, the effects of the residual carrier frequency offset (residual CFO) are corrected by means of exploiting the pilot subcarriers. The method can be applied for any system using any kind of modulation format for the useful data subcarriers.

The CFO estimation of the method is based on the non- ad hoc maximum likelihood (ML) criterion, which is shown to be asymptotically efficient. This method can be used for any channel configuration, (i.e. SISO, SIMO, MISO or MIMO) and exploits the OFDM pilot subcarriers with any coding strategy for the useful data carriers (i.e. space-time coding, beamforming, etc.). Besides, an efficient implementation of the estimator is proposed based on a reduced or truncated fast Fourier transformation (truncated-FFT) , said implementation being specially appropriate for systems in which the cost and complexity become especially important.

In particular, it is an object of the present invention a method for estimating a residual carrier frequency offset of an OFDM signal, the method comprising the steps of: receiving at at least one receiving antenna at least one OFDM signal, said OFDM signal comprising a preamble and a data frame, said data frame comprising a plurality of OFDM symbols; sampling said plurality of OFDM symbols; acquiring a channel estimate from said OFDM signal; wherein for each group of samples of each OFDM symbol, performing a Discrete Fourier Transform; from the result of each Discrete Fourier Transform, selecting the samples at the pilot subcarriers; from said samples at the pilot subcarriers and said channel estimate, calculating an estimate φ_q of the phase shift between consecutive OFDM symbols due to said residual carrier frequency offset.

In a particular embodiment, the step of calculating an estimate φ_q of the phase shift due to said residual carrier frequency offset comprises finding a value φ_q of the phase shift between two consecutive OFDM symbols which maximizes a certain expression, said value being comprised within a range [ - φ_maκ , «p_max ] , wherein φ_mx is a previously determined value.

Said calculation of the estimate is preferably based on the maximum likelihood criterion.

In a particular embodiment, said certain expression takes the form of Re [ e^~iaφ'ψ^q)H -T^(q)H -xf ] , wherein T^(<?) is a matrix which contains the joint contribution of the transmitted symbols at the pilot subcarriers and the channel estimate; φ^(?) is a column vector whose elements represent, for a receiving antenna q, the phase shift φ_η due to the residual carrier frequency offset at each of the plurality of OFDM symbols; x⁽ _Λ ^?) is a column vector resulting from the columnwise stacking of the column vectors which comprise the received samples at the pilot subcarriers for

each of the n_L OFDM symbols in the data frame; a =

2(N +L) wherein N is the number of samples in the payload of each OFDM symbol, L is the number of samples in the cyclic prefix of each OFDM symbol and a is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated with the preamble (PRE) and the first OFDM symbol in the data frame (DF) . Said value of the phase shift between two consecutive OFDM symbols φ_q is found by carrying out an exhaustive search over a set of M+l possible values of phase shifts within said range [ - #>_max , φ_max ] .

In a particular embodiment, the step of finding said value of the phase shift which maximizes a certain expression comprises calculating a K-points Discrete Fast

Fourier Transform (DFT). Said K-points Discrete Fourier

Transform (DFT) takes preferably the following expression:

"'^"' -//*— 2 • π

Υ_rl^q) e ^κ , wherein: k = φ_q ;

/₌o K is- k = {0,l,...,k_ma,K-l,K-2,...,K-k_mx}; £_max=-—φ_mm , n_L is the number λ •π of OFDM symbols in a data frame (DF); η^(q) (0</<n_L-l) are the n_L components of the column vector r^<<7) , wherein r^(?) = T^(q)Hx^(q) , wherein T^(<?) is a matrix which contains the joint contribution of the transmitted symbols at the pilot subcarriers and the channel estimate and x⁽/' is a column vector resulting from the columnwise stacking of the column vectors which comprise the received samples at the pilot subcarriers for each of the n_L OFDM symbols in the data frame. When K is a power of 2, the DFT can be implemented using a Fast Fourier Transform (FFT) which requires a lower computational load. In a particular embodiment, said channel estimate is acquired from the preamble of said OFDM signal.

In a particular embodiment, a further step of carrying out a coarse CFO estimation and correction from the preamble of said OFDM signal (200) prior to the residual carrier frequency offset estimation is done.

In a particular embodiment, a further step of correcting said residual carrier frequency offset is done.

Said correction can be done in the time domain. Said time domain correction is done by sample-by-sample multiplying the sampled OFDM symbols in the time domain by e^~'^{{ q}"^{+a q)} , wherein a is a real number as explained before, φ and ώ — , wherein N and L are the number of samples in

' N+L the payload and the cyclic prefix of each OFDM symbol, respectively.

Alternatively, said further step of correcting the residual carrier frequency offset can be done in the frequency domain. Said correction is applied to the data subcarrier outputs of the DFT blocks whose inputs are the payload samples of each OFDM symbol in the data frame (DF) . Said correction is performed by multiplying the outputs of said DFT applied to OFDM symbol "1" by

where

_ N + 2a a = , wherein N and L are the number of samples in

2(N +L) the payload and the cyclic prefix of each OFDM symbol, respectively, and a is a real number as explained before. Said OFDM signal can be received by a plurality of receiving antennae. The estimate of the phase shift between consecutive OFDM symbols due to said residual carrier frequency offset can be either the same for all the antennae of said plurality of receiving antennae or different for at least two antennae of said plurality of receiving antennae.

Said received OFDM signal can previously be transmitted by a plurality of transmitting antennae.

It is another object of the present invention to provide a receiver comprising means adapted for carrying out the steps of the method. In particular, it is another object of the present invention to provide a receiver for an OFDM system, comprising: means adapted for receiving at at least one receiving antenna at least one OFDM signal, said OFDM signal comprising a preamble and a data frame, said data frame comprising a plurality of OFDM symbols; means adapted for sampling said plurality of OFDM symbols; means adapted for acquiring a channel estimate from said OFDM signal; for each group of samples of each OFDM symbol, means adapted for performing a Discrete Fourier Transform; means for selecting the samples at the pilot subcarriers; means adapted for calculating, from said samples at the pilot subcarriers and said channel estimate, an estimate of the phase shift between consecutive OFDM symbols due to the residual carrier frequency offset. Preferably, the receiver comprises also means adapted for correcting said residual carrier frequency offset from said estimate of the phase shift. It is another object of the present invention to provide a computer program comprising computer program code means adapted to perform the steps of the method when said program is run on a computer, on a digital signal processor, a field-programmable gate array, an application- specific integrated circuit, a micro-processor, a microcontroller, or any other form of programmable hardware.

The advantages of the proposed invention will become apparent in the description that follows.

BRIEF DESCRIPTION OF THE DRAWINGS

To complete the description and in order to provide for a better understanding of the invention, a set of drawings is provided. Said drawings form an integral part of the description and illustrate a preferred embodiment of the invention, which should not be interpreted as restricting the scope of the invention, but just as an example of how the invention can be embodied. The drawings comprise the following figures:

Figure 1 represents an OFDM signal frame and an OFDM symbol in the temporal domain.

Figure 2 is a schematic representation of a wireless communications system with multiple transmitting and receiving antennae according to the present invention. Figure 3 represents the transform of a set of N subcarriers from the frequency domain to the samples of the corresponding OFDM symbols in the time domain.

Figure 4 is a schematic representation of a transmitter for an OFDM system.

Figure 5 shows a transmitting system with two transmitting antennae in which Alamouti's OSTBC is applied for the modulation of useful data in a per subcarrier basis .

Figure 6 is a schematic representation of a receiver for an OFDM system.

Figure 7 represents how the phase varies with time in an OFDM data frame (DF) due to the carrier frequency offset and its approximation.

Figures 8A and 8B show the constellation of the soft symbols after the space-time block decoding for the quadrature and the in-phase components without correcting the residual carrier frequency offset. This is shown for the cases of one and two transmitting antennae.

Figure 9 is a schematic representation of the whole process of estimating the residual CFO, correcting it and detecting the useful data symbols from an OFDM signal according to the present invention. Figure 1OA is a schematic representation of the processing and estimation of the residual CFO according to the present invention.

Figure 1OB is a schematic representation of the estimation of the residual CFO according to an embodiment of the present invention.

Figure 1OC is a schematic representation of the estimation of the residual CFO according to another embodiment of the present invention.

Figure 11 shows an example of truncated FFT-based scheme .

Figures 12A, 12B, 12C and 12D represent a comparison of the complexity load of the algorithm based on grid search and the algorithm based on the truncated FFT.

Figure 13A is a schematic representation of the correction of the residual CFO in the time domain according to a particular embodiment of the present invention.

Figure 13B is a schematic representation of the correction of the residual CFO in the frequency domain according to a particular embodiment of the present invention.

Figures 14A and 14B show the evaluation in terms of symbol error rate (SER) of the estimation and correction of CFO using the 4 pilot subcarriers of the 4 and 8 first consecutive OFDM data symbols. Figures 15A and 15B show the effects over the signal constellation after OSTBC decoding under a residual CFO of

2 KHz in two different systems (a system with one transmitting antenna and another system with 2 transmitting antennae) without any CFO correction.

Figures 16A and 16B show the effects after estimation and correction of the residual CFO over the signal constellation after OSTBC decoding in two different systems (a system with one transmitting antenna and another system with 2 transmitting antennae) .

Figures 17A and 17B represent an example corresponding to the results of the simulated mean-square error corresponding to the phase shift estimation obtained from the method of the present invention, compared to the theoretical Cramer-Rao lower bound for the case of the IEEE 802. Hn standard.

DETAILED DESCRIPTION OF THE INVENTION

NOTATION

The notation used through the description of the present invention is the one shown in the following table:

Number of OFDM symbols in a frame containing information, i.e., without including the preamble .

Number of pilot subcarriers in each OFDM symbol . k_p, p = \,...,n_f Positions of the pilot subcarriers.

S^ip)[k,l] Complex symbol transmitted through the pth antenna and the kth subcarrier during the Ith OFDM symbol.

Temporal signal samples corresponding to the Ith transmitted OFDM symbol through the pth antenna .

4"(1I) Transmitted signal samples for the whole frame corresponding to the pth transmit antenna .

,(<?) (n) Received signal samples at the qrth receive antenna . h^(qp){n) Time impulse response of the channel between the pth transmit and the qrth receive antennas .

X^[k,l] Received signal sample in the frequency domain at the kth. subcarrier at the qrth antenna during the Ith OFDM symbol in the frame .

Φa Incremental phase between consecutive samples at the qrth receive antenna due to the residual carrier frequency offset.

Ψa Incremental accumulated phase or phase shift between consecutive OFDM symbols at the qrth receive antenna due to the residual carrier frequency offset.

Furthermore, in the context of the present invention, the term "around" should be understood as indicating values very near to those which accompany the aforementioned term. That is to say, a deviation within reasonable limits from an exact value should be accepted, because the expert in the technique will understand that such a deviation from the values indicated is inevitable due to measurement inaccuracies, etc.

In this text, the term "comprises" and its derivations (such as "comprising", etc.) should not be understood in an excluding sense, that is, these terms should not be interpreted as excluding the possibility that what is described and defined may include further elements, steps, etc .

MIMO SYSTEM

Figure 2 shows a wireless communications system having a plurality of transmitting antennae (10-1, 10-2, ..., 10-n_τ) and a plurality of receiving antennae (20-1, 20-2, ..., 20-q, ..., 20-n_R) . Figure 2 therefore represents a multiple-input- multiple-output (MIMO) system with n_τ transmitting antennae and n_R receiving antennae. Communication is carried out through a wireless MIMO channel (30) . The system of figure 2 comprises the particular cases of a single-input- multiple-output (SIMO) system, a multiple-input-single- output (MISO) system or to a single-input-single-output (SISO) system, by simply reducing the number of transmitting and/or receiving antennae.

DESCRIPTION OF THE TRANSMITTED OFDM SIGNALS In Orthogonal Frequency Division Multiplexing (OFDM) , as represented in figure 1, the temporal samples corresponding to the lth transmitted OFDM symbol through the pth transmitting antenna are:

where S^ip)[k,l] is the symbol transmitted through the kth subcarrier and the pth antenna during the 2th OFDM symbol. Figure 3 shows an implementation of the generation of the temporal samples using an IDFT (Inverse Discrete Fourier Transform), as will be detailed later. An Inverse Fast Fourier Transform (IFFT) can be used instead of the IDFT when N is a power of 2. The transmitted signal is then obtained as the concatenation of the time samples corresponding to the n_L OFDM symbols:

The symbols S^ip)[k,l] may be data symbols or pilot symbols. The pilot subcarriers are fixed, i.e., they are specified in the standard and can be used to assist in the channel estimation or the frequency offset correction. As a matter of example, the proposal of standardization for IEEE

802. Hn defines 4 pilot subcarriers at each OFDM symbol, the position of said 4 pilot subcarriers being fixed. The value of the pilot subcarriers are repeated cyclicly every

4 OFDM symbols. This definition has been used when necessary in the simulations which accompany the present invention, although the method according to the present invention is generic and, therefore, valid for any definition of the pilot subcarriers, not being limited to the definition of pilot subcarriers proposed by IEEE 802. Hn. The value of these subcarriers and their positions are summarized below:

π,=4 it, =7, Ar₂ = 21, k₃ = 43, Ic₂ =57

For example, in a system with 4 transmitting antennae Transmitting antenna 1

Figure 3 represents the transform of a set of N subcarriers from the frequency domain to the samples of an OFDM symbol in the time domain. In figure 3, six OFDM symbols (0, 1, 2, 3, 4, 5) are illustrated, each of them comprising N frequency subcarriers (from 0 to N-I) . As can be appreciated, four of these N subcarriers are crossed. These crossed subcarriers represent the 4 pilot subcarriers. The non-crossed subcarriers represent the data subcarriers. An inverse Discrete Fourier transform (IDFT) is applied to each of the frequency domain symbols (0, 1, ...) , each one with its corresponding N subcarriers, thereby arriving at N time domain samples (0, ..., N-I) , each of the samples being dependent on each of the N frequency subcarriers. An Inverse Fast Fourier Transform (IFFT) can be used instead of the IDFT when N is a power of 2. L of these N samples are added at the beginning of the N samples, forming a cyclic prefix (CP) . The L+N samples form an OFDM symbol (in figure 3, OFDM SYM#0 represents the L+N time domain samples corresponding to the IDFT (or IFFT) applied to the N subcarriers of OFDM symbol 0 and the inclusion of the CP) .

EXAMPLE: ORTHOGONAL SPACE-TIME BLOCK CODING (OSTBC)

The OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) are generated depending on the modulation strategy specified by the standard. A possible example for the modulation of the useful data subcarriers is given by the application of orthogonal space-time block coding (OSTBC) in a per subcarrier basis. This coding scheme is useful for systems with multiple antennae. This example has no limiting intention. Just to give an idea and an example of the generation of OSTBC signals for the subcarriers corresponding to the useful data, let us assume the particular case of having n_τ=2 transmitting antennae and Alamouti's code, which is the OSTBC corresponding to this case. As previously stated, the OSTBC is applied in a per subcarrier basis. Let us assume that i[k,l] is the sequence of complex information symbols to be transmitted through the kth subcarrier. In Alamouti's case, the symbols are grouped in groups of two consecutive symbols and encoded together over two periods of time, i.e., two OFDM symbols, in the following way, as represented in figure 5:

S^m[k,2l] Sⁱ²⁾[k,2l] i[k, 21} i[k, 2l + l]

S[k, 21] = S^w[k,2l + 1] S⁽²⁾[£,2/ + l] i'[k,2l + l] -C[k,2l]

DESCRIPTION OF AN OFDM TRANSMITTER

Figure 4 is a schematic representation of a transmitter for an OFDM system. At the transmitter side, data bits (41) are mapped with an M-QAM constellation (42) (depending on the transmission rate) , giving inphase and quadrature components at its output. Then, pilot carriers are inserted (43) at their positions (specified a priori by the standard in use) and multiplexed with data mapped symbols (44). Before transmitting data, a preamble is usually inserted (45) at the beginning of each frame (46).

This preamble is also known at the receiver and is normally used for synchronization and channel estimation purposes.

When multiple antenna systems are used, spatial encoding is done (47), mapping each symbol in space and time according, for example, to an OSTBC scheme. After spatial processing, each antenna performs an IDFT calculation (51) to transform data and pilots to the time domain, where a cyclic prefix

(CP) is inserted to avoid inter-symbol interference at the receiver (52). Once pulse shaping is done (53), digital up- conversion is performed (54) and the signal goes through a digital to analogue converter (DAC) (55) , to obtain the analogue signal. After filtering (61) and amplifying the signal at intermediate frequency (IF) (62), it is mixed (63,64) and then amplified at RF (65), before being sent throughout the antenna (10-i). A general case in which the transmitter comprises multiple antennae is considered. Note that the method of the present invention is not limited to the transmitter of figure 4. This transmitter is to be regarded as an illustrative example and not a limitative one .

DESCRIPTION OF AN OFDM RECEIVER

Figure 6 is a schematic representation of a receiver for an OFDM system according to an embodiment of the present invention. At the receiver side, an OFDM signal is received at each antenna (20-q) (considering a general case in which the receiver comprises multiple antennae) and amplified with a low noise amplifier (LNA) (66) . In the representation of figure 6, only one (20-q) of the plurality of receiving antennae has been shown. Then, the signal is filtered (67) and mixed (68) to an intermediate frequency IF. Before being digitized, the signal is again filtered (69) and amplified at the intermediate frequency IF (70) . Analogue to Digital conversion (ADC) (71) and digital down-conversion (72) is consequently performed before starting base-band (73) processing. The first baseband block performed at each of the receiving antennas of an OFDM system is synchronization. As mentioned, there exist two different synchronization stages; the first one performs the initial synchronization and is called acquisition (74) . This block performs the following: it uses the preamble at the beginning of each data frame, known a priori also by the receiver, to synchronize frames in the time domain (74-3); it performs a first coarse carrier frequency offset estimation (74-2) and it estimates the channel response (74-1). The second synchronization stage (75) uses the pilot subcarriers that have been multiplexed with data in the OFDM symbols in order to track some of the synchronization parameters such as residual CFO or channel variations. After a synchronization correction unit (76) , where timing and CFO is corrected, Cyclic prefix removal (77) and Discrete Fourier Transform (DFT) (78) are performed. Once DFT (78) is performed for each receiving antenna (20-q) , spatial decoding (79) is done taking into account the information received and the channel estimated at each antenna. Finally, demapping (80) is performed, resulting its output the decoded bits. Note that the method of the present invention is not limited to the receiver of figure 6. This receiver is to be regarded as an illustrative example and not a limitative one. DESCRIPTION OF THE RECEIVED OFDM SIGNALS

In an ideal case in which no carrier frequency offset

(CFO) exists, the received signal at the qth receiving antenna (20-q) can be written in terms of the transmitted signals through all the transmitting antennae (10-1, 10-2,

..., 10-n_τ) as:

Ji) {n) = ∑h^(q'^p){n)*x¥\n)+w^(q\n), q =\,...,n_R

where w^(q)(n) is the noise at the qth receiver and h^(q'^p)(ή) is the time impulse response of the channel between the pth transmitting antenna and the qth receiving antenna. Based on this received signal, the receiver applies the OFDM demodulation by means of the DFT, obtaining the following sample at the kth subcarrier of the lth OFDM symbol in the frame :

k = 0,...,N-\, q = \,...,n_R, l = 0,...,n_L-\

If the channel order is equal to or shorter than the length of the cyclic prefix (CP) , the following relationship is fulfilled:

X_R ^q)[k,l] = ∑H^iq'^p)[k]S^{p)[k,l]+W^{q)[k,l] p=\

where W^Ul)[k,l] =-^₌∑w"(n+l(N+L))exd-j—kn

VN n=0 V N DESCRIPTION OF THE RECEIVED OFDM SIGNALS WITH RESIDUAL CFO

Nevertheless, in real systems, a residual carrier frequency offset (residual CFO) always exists. Note that although currently there are techniques that, using, for example, the preamble, are able to perform coarse carrier offset estimation and correction, a residual CFO always remains. Just to give an illustrative example, from experimental studies and measurements with the parameters of the standard IEEE 802. Hn, after the coarse carrier frequency offset estimation and correction, a residual CFO of around 3 kHz remains, which is equivalent to a phase variation of around 4.3° between consecutive OFDM symbols in the data frame (DF) .

This residual CFO can be modeled as a complex exponential signal that multiplies the useful signal at the receiver. This residual CFO can be different at different receiving antennae (20-1, 20-2, ..., 20-n_R) if the necessary synchronization and channel estimation stages are not performed jointly at all the receiving antennae (20-1, 20- 2, ...,20-q, ..., 20-n_R) , which is the usual approach. Note, however, that the residual CFO can be common for all the receiving antennae (20-1, 20-2, ..., 20-q, ..., 20-n_R) . The method according to the present invention encompasses both cases. The time domain received signal can be represented as follows:

q = \,...,n_R

wherein exp(j(φ_qn +aφ_q)) represents the^' residual CFO, φ_q is the incremental phase between two consecutive samples due to the residual CFO and aφ_q is the initial phase, which depends on the time difference between the beginning of the payload samples in the OFDM frame and the period in which the channel estimation h^iq'^p)(n) was performed, at which the incremental phase is considered to be zero. This is illustrated in figure 7, which represents how the phase varies with time (n) in an OFDM data frame (DF) . As explained before, or is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated with the preamble (PRE) and the first OFDM symbol in the data frame (DF) .

If the phase variation over the samples corresponding to a single OFDM symbol is analyzed, it can be seen that the phase difference between the first and the last sample (from -L to N-I) is usually low when considering the residual CFO. For example, using the parameters of the IEEE 802. Hn standard, and after some experimental results, it has been seen that the order of magnitude of the residual frequency offset may be around 3 kHz, which corresponds to a phase shift of only 4.3° between the first (-L) and the last sample (N-I) in the same OFDM symbol. The main consequence of this is that the intercarrier interference (ICI) can be neglected. As a consequence, a good approximation is to assume that the phase shift is constant within a single symbol, and, consequently, only changes between consecutive OFDM symbols need to be considered. This is illustrated in figure 7. As shown in figure 7, the phase during a single OFDM symbol (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), whose duration is L+N samples, is assumed to be equal to the instantaneous phase shift at the centre of the N samples (payload part) of the OFDM symbol, i.e., excluding the cyclic prefix (CP) . Based on this, the incremental phase φ_q between two consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) due to the residual CFO is equal to φ_q=(N+L)φ_g.

According to this, the received samples in the frequency domain can be approximated and expressed as:

wherein: φ(l)-aφ +lφ =aφ +l(N + L)φ and a =

2(N +L)

This residual carrier frequency offset (CFO) can be due to the residual effects which remain after a coarse carrier frequency offset (coarse CFO) correction or, under good conditions, as a result of having local oscillators at the receiver almost perfectly synchronized to the frequency of the OFDM signal, without requiring any coarse CFO correction .

The performance of the detection of OFDM signals, independently from the modulation format which has been used, is very sensitive to this residual CFO, specially in MIMO systems, as is shown in the example that follows. EXAMPLE: EFFECTS OF THE RESIDUAL CFO OVER THE DETECTION OF

OSTBC SIGNALS

If, as a matter of example, we go back to the transmission of OSTBC signals, the effects of the residual

CFO in the symbol constellation obtained after the OSTBC decoding at the subcarriers corresponding to the useful data can be appreciated.

As an illustrative example, a scenario in which there are two transmitting antennae and in which Alamouti's coding scheme is applied is analyzed (see figure 5), where two consecutive data symbols are space-time processed and transmitted over the two active antennae using two consecutive periods of time (full-rate).

The frequency-domain symbols received at the gth antenna in two consecutive periods of time can be expressed as :

X^(q)[k,2l] = Φ_R ^{q) (2l)S [k,2l]H^(q)[k] + W^(q)[k,2ll

W⁽⁹⁾[A:, 2/]

where the effects of residual carrier frequency offsets are introduced following the assumption explained above. This expression can be modified in order to use a channel matrix H^[A:] which models the equivalent space- time processing between the transmitted information symbols l[&,2/] at the kth subcarrier and the received samples in the frequency domain:

X⁽;> [k, 21] = Φ«⁹⁾ (2/) H⁽;» [k] I [k, 21] + W<⁹⁾ [k, 21],

If the residual CFO modeled by means of the matrix Φ_A ^ιq)(2l) is not corrected, the OSTBC decoding process implies that at the gth receiving antenna, the channel effects are compensated in the frequency domain by applying

H (q)H to the symbols received in the two consecutive periods of time. Hence, the soft symbols, i.e., the symbols before taking hard decisions over them, at the output of the space-time decoder, neglecting the noise terms, can be represented as:

i [k,2i] = H*;'" [k] x⁽;> [k] = Wf [k, 2i] i [k, 21] ,

M^<;^>μ,2/] = _Hr WΦr (2/)H⁽;^>μ]

If we analyze the structure of M^ [&,2/] , it is observed that the different accumulated phases between symbol 2/ and 2/+1 impact directly not only on the phase, but also on the magnitude of the decoded symbols:

This can be appreciated in figures 8A and 8B, which show the soft symbols after the space-time block decoding for the quadrature and the in-phase components. In particular, figure 8A shows the case of a single transmitting antenna and figure 8B shows the situation corresponding to two transmitting antennae. In both cases, QPSK modulation is assumed for the data subcarriers.

We have therefore shown that in OSTBC it is specially important to correct the residual CFO, even if it might seem low, since its impact on the detection of the signal is extremely negative.

A new method for estimating the residual CFO with a low complexity before detecting the transmitted symbols is described. Once it is estimated, the residual CFO can be corrected.

ESTIMATION OF THE RESIDUAL CARRIER FREQUENCY OFFSET

In order to analyze this estimation problem, we refer again to the assumption that the phase variation from the first (-L) to the last (N-I) sample of one OFDM symbol (see figures 1 and 7) is very low, and thus, no ICI is considered in the model. With this assumption, the problem solved by the present invention is the procurement of an estimate of the phase corresponding to the residual CFO, i.e., of φ_q , exploiting the received samples at the pilot subcarriers during some or all the n_L OFDM symbols in the data frame (DF) ; if all the n_L OFDM symbols in the data frame (DF) are used and the resulting complexity is too high, the complexity can be reduced by using less OFDM symbols in the data frame (DF) for the estimation of the phase shift φ_q .

Once the OFDM signal is received, the residual CFO is estimated and corrected. Afterwards, the useful information carried in the data subcarriers is extracted by means of conventional demodulation, equalization and detection procedures, i.e., as if no CFO was present.

Next, the processing of a received OFDM frame at a receiving antenna q and the estimation of the residual CFO will be explained in terms of mathematical formulae and matrixes. Then, the same processing will be explained in a more graphical way, based on figure 1OA.

MATHEMATICAL FORMULATION OF THE PROCESSING OF THE

RECEIVED OFDM FRAME

A vector x^eC"'"'"¹ (vector of complex numbers) with n_L-n_P rows and 1 column is constructed, with all the frequency samples at all the pilot subcarriers for all the OFDM symbols in the data frame (DF) . The following signal model is thus obtained:

x⁽/» = exp(jaφ_q)T^(tι)φ^{q) + w^(?)

wherein :

q is a natural number which represents the qth receiving antenna (20-1, ..., 20-n_R) ;

N + 2a a = , wherein N is the number of samples in the

2(N +L) payload of an OFDM symbol, L is the number of samples in the cyclic prefix and a is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated with the preamble (PRE) and the first OFDM symbol in the data frame (DF) ;

φ_q is the incremental phase or phase shift between two consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) and is equal to φ_q = (N +L)φ_q ;

and wherein:

_v(<7)

W^(<7)

wherein the notation " e C" (alternatively "e.R") denotes that the vector or matrix to which it refers belongs to the group of complex numbers (alternatively, real numbers), and wherein the expression which accompanies the symbol "C" (alternatively, "R") indicates the size of the corresponding vector or matrix;

and wherein vector X^ contains all the frequency samples at all the pilot subcarriers for all the OFDM symbols in a data frame, vector x^ [1] contains all the frequency samples at all the pilot subcarriers for the OFDM symbol "1", vector w^(?) represents the noise samples in the frequency domain at all the pilot subcarriers for all the OFDM symbols in a data frame, vector w^<<7) [1] represents the noise samples in the frequency domain at all the pilot subcarriers for the OFDM symbol "1", the matrix T^(<7) is a block diagonal matrix, where each of the n_L blocks is a column vector t^(<?) [1] whose elements represent the samples that would have been received in the frequency domain at the qth receiving and all the pilot subcarriers in case that no CFO existed, and vector φ⁽'^;) contains complex numbers representing the phase shift for each of the n_L OFDM symbols in the DF due to the residual CFO.

Vector φ^U) contains the value of the phase shift φ which must be estimated and is therefore the only part of x⁽/> =exp(jάφ_q)Υ^(q)ψ^(q) + w^(<l) which is unknown, together with the noise vector w^(<7) .

Matrix T^(<?) contains the joint contribution of the symbols transmitted at the pilot subcarriers and the channel frequency response. Since the channel is known based on an estimate calculated during the transmission of the preamble (PRE) and also the symbols transmitted at the pilot subcarriers, this matrix T^(<?) can be constructed before estimating the phase shift.

Once matrix T^(<7) is built, the estimation of the residual carrier frequency offset, represented by φ_q , must be carried out, using the received samples at the pilot subcarriers during all the OFDM symbols in the data frame (DF) and collected in vector x^⁹⁾ . If it is assumed that the noise is white and Gaussian, the maximum likelihood estimator of the residual carrier CFO (i.e., the one that asymptotically leads to the Cramer- Rao lower bound of the variance of the error in the estimation) leads to the minimum mean square error criterion, formulated as follows:

=argmin[x⁽;⁾"x⁽;⁾-expO^'^)^"^V*-exp(-y^z>₇)φ⁽⁹⁾"l^<?)//x⁽;⁾+φ^(?)//T⁽⁹⁾"T^<V'⁷⁾]

wherein argmin and argmax indicate the value of φ which φ_q ψ_q ^q respectively minimizes and maximizes the expression which follows .

The last equality is obtained taking into account that does not depend on φ_q and that T^(q)HT^(q) is a diagonal matrix with non-negative entries and, therefore, _φ<<7^)tf _T ⁽,^)// _T ⁽ ₉y_? ⁾ _{does not} depend on φ_q either, because the

dependence of the vectors φ^(<7) and φ^(?)// on φ_q is counteracted in the expression φ^(?)//T^<9)//T^(<7)φ^(?) ) ■ In fact, since the matrix T^(<7)"T⁽⁹⁾ is diagonal, _φ ⁽*>"τ^(<?)//T⁽⁹⁾φ⁽⁹⁾ has a

"L 2 closed-form expression given by ∑ t^(?)[/] .

/=o

The signal model which has been presented corresponds to the situation in which a different residual CFO is considered at each receiving antenna. In the case in which a joint coarse CFO correction is previously performed among all the receiving antennae, then the residual CFO is the same at all the receiving antennae. In this situation, the method according to the present invention can also be applied directly, by using the following signal model resulting from the column-wise stacking of all the received samples at all the receiving antennae:

x_Λ = exp(yα^)Tφ+ w

where

eC"

In this model, φ is the common phase shift due to the residual CFO at all the receiving antennae and

T^T ₌ ^χ^(?)^χ^(?) _1S found to be diagonal. Consequently, the

current method can be applied in a direct and straightforward way without any variation.

ILLUSTRATED FORMULATION OF THE PROCESSING OF THE RECEIVED OFDM FRAME Figure 1OA shows in detail how a received OFDM frame at a receiving antenna q is processed according to the present invention. Reference 200 shows a received OFDM signal in the temporal domain, comprising a preamble (PRE) and n_L OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), with the same structure as those represented in figure 1. Each OFDM symbol comprises a cyclic prefix (CP) and N samples .

Box 300 represents a block in which channel estimation h^{q'^p)(n) is performed, that is to say, the time impulse response of the channel between the pth transmiting antenna and the qrth receiving antenna is estimated. This channel estimation is out of the scope of the present invention, and is obtained from the information available at the preamble PRE of the received OFDM signal (200) . Methods for estimating the channel from the preamble of an OFDM signal are well-known and have been introduced in the "State of the art" of this specification. With this estimate of the channel time impulse response, H^(q'^p)[k] is calculated as the frequency response of the channel for each pair of transmitting and receiving antenna at each subcarrier k (k=0,...,N-l) .

Additionally, at this block 300 a coarse CFO estimation and correction can be performed. This coarse CFO estimation and correction is neither object of the present invention. Methods for making a coarse CFO estimation and correction are also well-known and have been introduced in the "State of the art" of this specification. As previously stated, a coarse CFO correction is usually applied, since local oscillators at the receiver are normally not well synchronized at the received signals.

From these channel estimates H^<q'^p)[k], together with the transmitted pilot complex symbols ( S^ip)[k,l] being the complex symbol transmitted through the pth antenna and the kth pilot subcarrier during the lth OFDM symbol) , matrix

T^(q) is built at block 400.

Returning to the received OFDM signal 200, the cyclic prefix (CP) is extracted from each of the OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), and a Discrete Fourier Transform (DFT) is performed at each of the n_L groups of N samples, each n_L group corresponding to the payload of an OFDM symbol. This is represented in figure 1OA by references 500-0, 500-1, ..., 500-n_L-l. A person skilled in the art will understand, without exercising inventive capabilities, that a Fast Fourier Transform (FFT) can be performed instead of a Discrete Fourier Transform (DFT) in situations which so allow (when N is a power of 2) -

Once these DFTs or FFTs are carried out, the pilot subcarriers present at each of the OFDM symbols are selected. These pilot subcarriers are represented in figure

1OA by x™ [0] , x⁽;> [1] , ..., X^ [n_L-l] .

After the pilot subcarriers are selected, they are stored (700) in a vector x⁽ _Λ ^?) . As explained before, this vector contains all the frequency samples at all the pilot subcarriers for all the OFDM symbols in a data frame. Finally, block 800 carries out the residual CFO estimation φ_q . For carrying out this residual CFO, block

800 needs as inputs the matrix T^<q) and vector x^' .

Following, two possible ways of finding out the value of φ_q are shown.

METHOD BASED ON GRID SEARCH

According to a particular embodiment of the present invention, one possible way of finding out the value of φ_q is by carrying out an exhaustive search over the margin of possible values of phase shifts which have been previously stored in the system (i.e. receiver) φ_q e [-^_013x,^_max] • The storage of a plurality of possible values of phase shifts is out of the scope of the present invention and is carried out by means of any conventional way of storing information. φ_ma]i is a design parameter which represents the maximum phase deviation. The value of this <p_max depends on the phase shift due to the CFO associated to the received signal or on the phase shift which remains after a coarse CFO correction, if such a coarse CFO correction has been made. If M+l points are to be calculated in this margin [ - φ_nmx , φ_max ] , then, the value of the function

has to be calculated at these M+l

points, i.e., at the points: φ = -φ_^ +n—ψ-, n = 0,...,M ■

M Figure 1OB illustrates how this is carried out: Block 800-B represents the block 800 introduced in figure 1OA for the particular embodiment in which an algorithm based on grid search is utilized. As can be seen in figure 1OB, using as inputs the matrix T^(q) and vector x⁽/' , a representation of the real part of the following expression is performed:

∞p{-j-a-φ_q)'V™^H-T^q)H-xf , where φ^(?) = [l e^]ψ- ... e^A"'^'l)ψ" ]^τ ,

for each of the M+l points of φ_q <≡[-ζZ>_max,<p_max] .

Once these M+l values of the real part of the previous expression have been calculated, the point φ_q e [~φ_max,φ_m3X] at which the maximum value of the expression is achieved is selected (810-B), this point being the desired φ_q .

SIMPLIFIED METHOD BASED ON THE USE OF DFT

According to a second particular embodiment of the present invention, the previous strategy of finding out the value of φ_q based on the exhaustive or grid search can be simplified in terms of computational load by taking advantage of the particular structure of the expression to be maximized, i.e., of Re[exp(-jα^)φ^(9)//T^(9)Wx⁽ _Λ ⁹⁾] . Thus, vector

r^<<7) is defined as:

The function to be maximized can subsequently be rewritten as

The sum in the expression above has the same structure as a Discrete Fourier transform (DFT) and, therefore, the available efficient algorithms for calculating DFT can be applied. If it is assumed that fC-points DFT is applied, then the expression of the sum is given by:

which corresponds to the evaluation of the sum at the phase

shift k— . If the search of the phase shift is to be done K over the margin φ_q e[—φ_^^,^_13x], then the values of the index k in the DFT should be k = {θ,l,...,k_tm,K-l,K-2,...,K-k_πm} , where

φ * max . The number of K p_*-oints of the DFT is chosen

taking into account the higher or lower resolution desired for the estimator. The resolution in the phase estimation

is qiven by — . K This method is shown in figure 1OC, which illustrates how this is carried out: Block 800-C represents the block 800 introduced in figure 1OA for the particular embodiment in which a simplified algorithm based on the DFT is utilized. As explained before, a person skilled in the art will understand, without exercising inventive capabilities, that a Fast Fourier Transform (FFT) can be performed instead of a Discrete Fourier Transform (DFT) when K is a power of 2. In this case, a Fast Fourier Transform (FFT) is especially useful because the mathematical expression of the DFT allows for an algorithmic implementation based on a butterfly architecture that reduces extremely the computational load. As can be seen in figure 1OC, using as inputs matrix T^(q> and vector x⁽ _Λ ⁹⁾ , a vector r⁽⁽⁷⁾ is defined and a truncated FFT is calculated (810-C) . In this embodiment, the estimate of the phase variation φ_q is

calculated as , where k is the index corresponding

to the active output of the truncated FFT (810-C) for which the real part of this output multiplied (820-C) by f o Λ exp\-jak— is maximum (830-C). V K)

It is important to notice that in this FFT, only n_L entries out of K are active and only 2&_niax+l outputs out of the K possible ones have to be calculated (this is the reason why this method is called "truncated-FFT") . That means that if an efficient implementation of the DFT, by means of the butterfly architecture commonly used for FFT, is exploited, many of the sub-FFT blocks will be inactive and, therefore, the computational load can be decreased very importantly. Besides, the sub-FFT blocks that have to be used depend only on K, n_L and k_ma!i and, therefore, the optimum architecture could be implemented once off-line.

In order to illustrate the characteristics and complexity saving mechanisms of the proposed DFT-based algorithm, figure 11 shows an example of truncated FFT- based scheme, in which only the necessary butterflies needed to estimate the residual CFO have been marked. Complexity is therefore reduced. The example of figure 11 has been carried out with K=64 , n_L=5 and k_max=3 , which represents a simple example, where said values have been selected low enough so as to obtain a figure that can be represented. In figure 11, we can see the butterfly processing units necessary to obtain the results, which results in a significant reduction of the computational load, even for simple cases. For realistic values of the residual carrier frequency offset and the resolution of the estimator, the reduction of complexity gets clearer, as will be evaluated and commented in what follows.

Figures 12A, 12B, 12C and 12D represent a comparison of the complexity load of the algorithm based on grid search and the algorithm based on the truncated FFT. The X- axis represents the number of symbols used (that is to say, n_L) , while the Y-axis represents the amount of complex multiplications (xlO⁴). It is illustrated the complexity load of the proposed truncated FFT-based algorithm in comparison with the brute force grid search algorithm previously described. The computational complexity of the total DFT (that is to say, for the case of activating all the butterfly sub-blocks of the algorithmic implementation even if some of them may be not necessary) is also shown. "Brute force" refers always to the grid search algorithm, while "total FFT" and "truncated FFT" refer always to the algorithm based on the FFT. From the figures, it is concluded that the proposed method is an efficient solution when the maximum value of the residual CFO is above a previously fixed threshold, while maintaining the resolution constant. Depending on the magnitude of the maximum carrier frequency offset to be estimated, that depends on the tolerance allowed in the standard and on the coarse CFO estimation algorithms (if performed in acquisition mode), the proposed algorithm is an efficient solution. It can also be shown that the amount of reduction in the implementation complexity is increased when more resolution is used, i.e. the number of points K of the DFT or FFT is increased.

CORRECTION OF THE RESIDUAL CARRIER FREQUENCY OFFSET

Once the residual phase shift (or residual CFO) has been estimated by any of the means described in accordance with figures 1OB and 1OC, this estimation is applied to correct the CFO and detect the received signals.

The present invention provides with two different ways of correcting the carrier frequency offset which has been previously estimated.

According to a particular embodiment, this is done by applying the following operation to the signals in the time domain, and afterwards applying the classical OFDM demodulation based on the DFT and the decoding procedure, which depends on the modulation format applied to the useful data subcarriers :

This is illustrated in figure 13A: The residual CFO estimate φ_q serves as input of block 900, in which the phase shift between consecutive time samples is calculated, thus obtaining a sample phase shift φ_q .

With this value of φ_q , a sample-by-sample multiplication (1100) between the received samples in the data frame (DF) x_R ^(q)(n) (1200) and the phase shift correction

expl-j(φ_qn + aφ_q IJ (1000) is performed.

Afterwards, a process of demodulating, equalizing and detecting the received OFDM signal is carried out: The cyclic prefix (CP) from each of the corrected OFDM symbols is extracted, and a Discrete or Fast Fourier Transform (DFT or FFT) is performed for each of the n_L. groups of N samples, each n_L group corresponding to the payload of a corrected OFDM symbol. This is represented in figure 13A by references 1500-0, 1500-1, ..., 1500- n_L-l.

Once these DFTs (or FFTs) are carried out, the data subcarriers present at each of the corrected OFDM symbols are selected. After the data subcarriers are selected, equalization and detection of the information are carried out (1800) . Figure 9 is a schematic representation of the whole process of estimating the residual CFO, correcting it and detecting the useful data symbols from an OFDM signal according to the present invention. In figure 9, the blocks which represent corresponding stages as those shown in figures 1OA, 1OB, 1OC and 13A (400, 500, 700, 800, 900, 1000, 1100, 1200, 1300, 1500, 1800) are identified by corresponding reference numbers (400', 500', 700', 800', 900', 1500", 1800' ' ) .

According to another particular embodiment, the correction of the residual CFO is done by applying the following operation to the signals in the frequency domain. This is illustrated in figure 13B. The outputs, without any CFO correction, of the DFT blocks (1500-0', 1500-1', ..., 1500-n_L-l' ) whose inputs are the payload samples of OFDM symbol "1" (with "1" going form "0" to "n_L-l") in the data frame (DF) are represented, according to the previous signal model, by X^[k,l] . Then, the so-called correction in the frequency domain is done by calculating:

X^[k,l) = X^[k,l]exp(-j(aφ_q +lφ_q)) .

This is illustrated in figure 13B. Once the DFTs (or FFTs) (1500-0', 1500-1', ..., 1500- n_L-l' ) are carried out, the data subcarriers are selected. The residual CFO estimate φ_q serves as input of blocks 1600-0, 1600-1, ..., 1600-n_L-l, wherein with this value of φ_q a sample-by-sample multiplication (1600-0, 1600-1, ..., 1600-n_L-l) between the samples which are the result of the DFT blocks (1500-0', 1500-1', ..., 1500-n_L-l') (corresponding to the data subcarriers) and the shift correction e^'l{"^φ<l) ... _e ^'λ"^φ"^+{"'^~ήφ"^} is performed.

Afterwards, a process of equalizing and detecting (1800') the received OFDM signal is carried out, in a similar way as that described with respect to figure 13A.

Note that the correction in the frequency domain does not require applying again DFT to the payload samples of the OFDM symbols in the data frame (DF) . In other words, the outputs of the DFT (blocks 500 and 500' in figures 1OA and 9, respectively) calculated before the estimation of the phase shift due to the CFO can now be directly used for the correction in the frequency domain (the DFT' s represented by blocks 500 and 500' are the same as the DFT' s represented by blocks 1500' in figure 13B). On the other hand, if the CFO correction is performed in the time domain, DFT (blocks 1500 in figure 13A) has to be applied again to each group of corrected payload samples of each OFDM symbol in the data frame (DF), thus, increasing the computational load. Note, however, that in some particular scenarios, a better performance can be expected if the correction is carried out in the time domain, since it permits to correct inter-carrier interference when the residual CFO is higher than expected.

In comparison to the schematic representation of figure 9, which corresponds to the CFO correction in the time domain, this alternative embodiment for the correction of the residual CFO in the frequency domain is slightly different, as explained in what follows. When the correction of the residual CFO is performed in the frequency domain, the DFTs represented in figure 9 corresponding to block 1500' ' are not necessary, since when the correction of the residual CFO is performed in the frequency domain, these DFTs are performed prior to the correction of the residual CFO, as is apparent from figure 13B. Additionally, when the correction of the residual CFO is performed in the frequency domain, the inputs for the block 900' corresponding to the CFO correction are the outputs of block 500' representing the demodulation of the OFDM signals using DFT, i.e., the received samples in the frequency domain before the CFO correction.

In order to illustrate the effective gains obtained through the estimation and correction of the residual carrier frequency offset, it is important to analyze the improvement achieved in the Symbol Error Rate (SER) curves plotted versus the average received SNR per bit.

As an example, some simulations were performed taking the parameters defined in the IEEE 802. Hn standard. Note that this standard uses OSTBC as the modulation format in the subcarriers corresponding to the useful data. Two cases have been considered: a) a single transmit and receive antenna (SISO) and b) a two transmitting-antennae system with a single receive antenna (MISO) , under different residual carrier frequency offsets.

The channel, with Rayleigh distributed channel coefficients, was considered to be perfectly known at the receiver, and constant along frames of 24 OFDM symbols. In order to show the effect of the number of OFDM symbols used for the estimation, i.e., of the parameter n_L , the results are presented for 4 and 8 OFDM symbols. Different values of residual carrier frequency offset have been also evaluated, being the highest value 20 kHz.

Figure 14A shows the SER vs Eb/N₀ results for SISO- 16QAM with Rayleigh distributed channel coefficients, average SNR per symbol (OdB to 2OdB), residual CFOs (500Hz, IKHz, 2KHz and 20KHz) and estimation and correction of CFO using the 4 pilot subcarriers of the 4 and 8 first consecutive OFDM data symbols.

Figure 14B shows the SER vs Eb/N₀ results for a system using 2 transmitting antennae and Alamouti' s OSTBC with 16QAM modulation, Rayleigh distributed channel coefficients, average SNR per symbol (OdB to 2OdB), residual CFOs (500Hz, IKHz, 2KHz and 20KHz) and estimation and correction of CFO using the 4 pilot subcarriers of the 4 and 8 first consecutive OFDM data symbols.

From both figures 14A and 14B it is concluded that the residual CFO can be efficiently corrected, and that, in case that no correction is applied, the SER can be very high, even in the case of a low CFO of 500 Hz (<p_g=0,72°). Note also, that very high CFO can be corrected, for example see the case of 20 kHz

9° ) , and that, in this case the performance of the correction in the time domain is better than in the frequency domain, as expected. For a broad margin of values of CFO, the performance resulting from the correction method is very close to the case of having no CFO, proving the goodness of the method.

Figures 15A and 15B show the effects on the signal constellation assuming QPSK after OSTBC decoding under a residual CFO of 2KHz (φ_q=2,9°) in a system using one

(figure 15A) and two (figure 15B) transmitting antennae

(Alamouti) and no estimation and correction of the CFO.

Figures 16A and 16B show the effects on the signal constellation assuming QPSK after estimation and correction of the residual CFO and OSTBC decoding in a system using one (figure 16A) and two (figure 16B) transmitting antennae (Alamouti ) .

PERFORMANCE EVALUATION OF THE METHOD

In order to evaluate the performance of the method of estimation according to the present invention, the mean square error in the estimation of the phase shift due to the residual CFO for different SNRs is analyzed and compared to the Cramer Rao Lower Bound (CRB) . In the following, we present the theoretical derivation of the lower bound for the variance of the estimation problem.

We define the log-likelihood function A(φ_q;x^Λ from

the probability density function as

:

Λ(^;x⁽;⁾) = logp(x⁽;⁾;^) = -^« _/) ^« _ilog(^σ²)-^|x⁽;⁾-expαα_n)T⁽V'⁾

From here, we obtain the CRB by taking the first and second derivatives of the log-likelihood function and computing the expectation along the received symbols.

A e C"'""' = diag{jά j(α + l) ... j(α + «_Δ - l)} A² e /?"^tXπ' = -diag{α² (α + l)² ... (α + ^ - l)²}

From the structure of the final expression, it can be noted that the minimum attainable variance of the estimator, that is, the Cramer-Rao bound (CRB) , according to the present invention does not depend on the actual value of the parameter to be estimated φ_q . This can also be concluded from the results shown in the previous SER curves (figures 14A and 14B), in which the standard IEEE

802. Hn has been taken as an example. From figures 14A and

14B, it is seen that after the correction of the residual carrier frequency offset, the attained SER' s are equal for a broad margin of values of the frequency offset.

When the frequency offset increases, it is no longer true the assumption of having no inter-carrier interference, i.e., of having a negligible phase shift between the first and the last samples in a single OFDM symbol, which has been used to derive the estimator. Note, however, that even in this case, i.e., when the residual carrier frequency offset is much higher than expected and usual values, the obtained estimator allows to reduce the SER in a very efficient way, i.e., the obtained estimator can be used to efficiently improve the system performance. (It is to be noted that in the simulation results, the simplification of having no ICI is not introduced and, therefore, the performance results are accurate, fair, and correspond to a real system) .

Figures 17A and 17B represent an example corresponding to the results of the simulated mean-square error of the proposed estimator of the phase shift compared to the theoretical Cramer-Rao lower bound for the case of the IEEE 802. Hn standard. It can be shown that starting from SNRs around 0 dB, the estimator attains the minimum variance.

Figure 17A represents the MSE vs CRB for a system with 2KHz of residual CFO and Rayleigh distributed channel coefficients .

Figure 17B represents the performance gains obtained by increasing the total number of OFDM symbols n_L used for the estimation of the phase shift. It can be noted that important improvements are attained when increasing this number of OFDM symbols from 4 to 8 or 12. Figure 17B shows the MSE vs CRB using different number of OFDM symbols for the estimation in a system with 2KHz of residual CFO and Rayleigh distributed channel coefficients.

In summary, the present invention provides a method for estimating and correcting the residual carrier frequency offset (CFO) of OFDM signals in the tracking stage of both single-antenna and multiple-antenna systems, by exploiting the whole capabilities of the information contained in the OFDM frame. In particular, it exploits the presence of pilot subcarriers in all the OFDM symbols in the frame, as established in all the current standards which use OFDM.

The invention is obviously not limited to the specific embodiments described herein, but also encompasses any variations that may be considered by any person skilled in the art (for example, as regards the choice of components, configuration, etc.), within the general scope .of the invention as defined in the appended claims.

Claims

1. Method for estimating a residual carrier frequency offset of an OFDM signal, the method comprising the steps of:

- receiving at at least one receiving antenna (20-1, 20-2, ..., 20-q, ..., 20-n_R) at least one OFDM signal (200), said OFDM signal (200) comprising a preamble (PRE) and a data frame (DF), said data frame (DF) comprising a plurality (n_L) of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l);

- sampling said plurality of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l);

- acquiring a channel estimate (H^(q'^p)[k]) from said OFDM signal (200) ;

characterized by the steps of:

- for each group of samples of each OFDM symbol (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), performing a Discrete Fourier Transform (500-0, 500-1, ..., 500- n_L-l);

- from the result of each Discrete Fourier Transform, selecting the samples at the pilot subcarriers (x⁽ _Λ ⁹⁾[0],

^X _Λ L-LJ i -r ^X _Λ L^L -LJ / ^X _R I '

from said samples at the pilot subcarriers (x⁽ _Λ ^?)[0], x⁽/'[l], ..., x⁽ _Λ ^?) [n_L-l] ; x⁽ _Λ ^?) ) and said channel estimate

(H^(q'^p)[k]), calculating an estimate φ of the phase shift between consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) due to said residual carrier frequency offset.

2. Method according to claim 1, wherein said Discrete Fourier Transform (DFT) is a Fast Fourier Transform (FFT) .

3. Method according to either claim 1 or 2, wherein the step of calculating an estimate φ_q of the phase shift due to said residual carrier frequency offset comprises finding a value φ_q of the phase shift between two consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) which maximizes a certain expression, said value φ_q being comprised within a range [- ζø_max,φ_maκ] , wherein φ_ma)i is a previously determined value.

4. Method according to claim 3, wherein the step of calculating an estimate φ_q of the phase shift due to said residual carrier frequency offset is based on the maximum likelihood criterion.

5. Method according to claim 4, wherein said certain expression is: Re [ e^'J"^ψ^^(q)H -T^(?)// -X^ ] ,

wherein:

T^<<7) is a matrix which contains the joint contribution of the symbols transmitted at the pilot subcarriers and the channel estimate; φ^(<7) is a column vector whose elements represent, for a receiving antenna q, the phase shift φ_q due to the residual carrier frequency offset at each of the plurality (n_L) of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l);

x⁽ _Λ ⁹⁾ is a column vector resulting from the columnwise stacking of the column vectors x⁽/'[0], x⁽ _Λ ^?)[l], ..., x⁽/> [n_L-1 ] , said column vectors comprising the received samples at the pilot subcarriers for each of the n_L OFDM symbols in the data frame (DF) ;

a = , wherein N is the number of samples in the

2(N +L) payload of each OFDM symbol, L is the number of samples in the cyclic prefix (CP) of each OFDM symbol and a is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated and the first OFDM symbol in the data frame (DF) .

6. Method according to claim 5, wherein said value of the phase shift between two consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) φ_q is found by carrying out an exhaustive search over a set of M+l possible values of phase shifts within said range [ - φ_mm , φ_max ] .

7. Method according to claim 5, wherein the step of finding said value of the phase shift φ_q which maximizes a certain expression comprises calculating a K-points Fast Fourier Transform (FFT) .

8. Method according to claim 7, wherein said K-points Fast Fourier Transform (FFT) takes the following expression:

^(q) -e ^κ

I=O

wherein:

k = {0,l,...,k_n^,K-l,K-2,...,K-k_rmii}_;

n_L is the number of OFDM symbols in a data frame (DF) ; η^(q) (0</<n_L-l) are the n_L components of the column vector r^(<7), wherein r^(?) - T^(q)Hx⁽ _R ^q) , wherein

T^iq) is a matrix which contains the joint contribution of the symbols transmitted at the pilot subcarriers and the channel estimate and x⁽/' is a column vector resulting from the columnwise stacking of the column vectors x⁽ _Λ ^?)[0], x⁽ _Λ ⁹⁾[l], ..., x⁽/' [n_L-l] , said column vectors comprising the received samples at the pilot subcarriers for each of the n_L OFDM symbols in the data frame (DF) .

9. Method according to any preceding claim, wherein said channel estimate is acquired from the preamble (PRE) of said OFDM signal (200) .

10. Method according to any preceding claim, further comprising a step of carrying out a coarse CFO estimation and correction using the preamble (PRE) of said OFDM signal (200) prior to the residual carrier frequency offset estimation.

11. Method according to any preceding claim, further comprising the step of correcting said residual carrier frequency offset, which generates a phase shift φ_q between

OFDM symbols in the data frame (DF), of said received OFDM signal at each of said at least one receiving antenna (20- 1, 20-2, ..., 20-q, ..., 20-n_R) .

12. Method according to claim 11, wherein said correction of the residual carrier frequency offset is done in the time domain.

13. Method according to claim 12, wherein said time domain correction of the residual carrier frequency offset is done by sample-by-sample multiplying (1100) the sampled OFDM symbols by _e ^~^→\ wherein a is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated and the first OFDM symbol in the data

frame (DF), and φ = <P«

N+L wherein N is the number of samples in the payload of each OFDM symbol and L is the number of samples in the cyclic prefix (CP) of each OFDM symbol.

14. Method according to claim 11, wherein said correction of the residual carrier frequency offset is done in the frequency domain.

15. Method according to claim 14, wherein said frequency domain correction of the residual carrier frequency offset is done by multiplying the outputs of the DFT (1500-0', 1500-1' , ..., 1500-n_L-l' ) applied to the payload samples of the OFDM symbol "1" by _e-'(^aφ'⁺'^φ<\ wherein 1 is a natural

number and 0</<«,-l, wherein a = , wherein N is the

^L 2(N+ L) number of samples in the payload of each OFDM symbol, L is the number of samples in the cyclic prefix (CP) of each OFDM symbol and a is a real number related to the initial point at which the residual CFO is calculated and that depends on the time difference between the instant at which the channel is estimated and the first OFDM symbol in the data frame (DF) .

16. Method according to any preceding claim, wherein said OFDM signal is received by a plurality of receiving antennae (20-1, 20-2, ..., 20-q, ..., 20-n_R) .

17. Method according to claim 16, wherein the estimate φ_q of the phase shift between consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) due to said residual carrier frequency offset is the same for each antennae of said plurality of receiving antennae (20-1, 20-2, ..., 20-q, ..., 20-n_R) .

18. Method according to claim 16, wherein the estimate φ_q of the phase shift between consecutive OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l) due to said residual carrier frequency offset is different for at least two antennae of said plurality of receiving antennae (20-1, 20-2, ..., 20-q, ..., 20-n_R) .

19. Method according to any preceding claim, wherein said received OFDM signal has been previously transmitted by a at least one transmitting antenna (10-1, 10-2, ..., 10-n_τ) .

20. Method according to claim 19, wherein said received OFDM signal has been previously transmitted by a plurality of transmitting antennae (10-1, 10-2, ..., 10-n_τ) .

21. A receiver for an OFDM system, comprising:

- means adapted for receiving at at least one receiving antenna (20-1, 20-2, ..., 20-q, ..., 20-n_R) at least one OFDM signal (200), said OFDM signal (200) comprising a preamble (PRE) and a data frame (DF), said data frame (DF) comprising a plurality (n_L) of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l);

-means adapted for sampling said plurality of OFDM symbols (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l);

- means adapted for acquiring a channel estimate (H^(q'^p)[k]) from said OFDM signal (200) ;

the receiver being characterised in that it further comprises :

-for each group of samples of each OFDM symbol (SYM#0, SYM#1, ..., SYM#1, ..., SYM#n_L-l), means adapted for performing a Discrete Fourier Transform (500-0, 500-1, ..., 500- n_L-l); -means for selecting the samples at the pilot subcarriers (x_R [UJ, x_R [H, ..., x_Λ [n_L-lJ, x_R ),

-means adapted for calculating, from said samples at the pilot subcarriers {x_R ^q) [0] , X^[I], ..., x^ [n_L-l] ; x⁽/> ) and said channel estimate (H^(q'^p) [k] ) , an estimate φ_q of the phase shift between consecutive OFDM symbols due to the residual carrier frequency offset;

22. A receiver according to claim 21, further comprising means adapted for correcting such residual carrier frequency offset from said estimate φ_q of the phase shift between consecutive OFDM symbols.

23. A computer program comprising computer program code means adapted to perform the steps of the method according to any claims from 1 to 20 when said program is run on a computer, a digital signal processor, a field-programmable gate array, an application-specific integrated circuit, a micro-processor, a micro-controller, or any other form of programmable hardware.