GB2478604A - Wireless communications channel estimation - Google Patents

Abstract

This invention relates to a communication system 100 and method using orthogonal frequency division multiplexing (OFDM) and multiple transmit antennas 201. The transmissions are selected such that only a single antenna 201 is allowed to occupy any given subcarrier (ATSA). The invention proposes optimisation of cost functions different to the mean squared error estimator for the purpose of channel estimation, including choosing between a minimax function and a minimum expected Mean squared error ("miniexp") cost function. Some of the proposed estimation techniques rely on a Gauss-Markov approximation of the wireless communication channel 400 in the frequency domain.

Description

WIRELESS COMMUNICATIONS METHOD AND APPARATUS

This invention relates to a method of estimating channel characteristics of a channel in a wireless orthogonal frequency division multiplexed (OFDM) transmission system using multiple transmit antennas. In particular, this invention relates to a method of selecting and optimising cost functions for the purpose of estimating transmission characteristics of channels corresponding to the subcarriers of an OFDM transmission system.

OFDM is a frequency division multiplexing scheme where a signal is divided into a plurality of data streams and each data stream is carried on its own subcarrier. By dividing the data over multiple subcarriers each subcarrier has a transmission rate which is less than the transmission rate would be for a single carrier. OFDM effectively converts a rapidly-modulated wideband signal into a plurality of narrowband signals.

The subcarriers are transmitted at orthogonal frequencies which are spaced from each other such that cross-talk between the channels is eliminated.

The reduced data rate of each subcarrier has particular advantages where channel conditions are significant. One exampte of the effect of channel conditions on the transmission is where the transmitted signal suffers from multipath. Multipath occurs when a signal propagates from a transmitter to a receiver over a plurality of routes due to, for example, reflection or refraction, which means that the length of each path of the signal varies resulting in different propagation times for the different paths of the signal.

By splitting the signal into multiple subcarrier signals the reduced data rate of each subcarrier reduces potential degradation of the signal quality due to the multipath effect.

To facilitate coherent detection at the receiver, knowledge of the channel characteristics is desirable. To estimate the channel characteristics, a preamble can be transmitted from the transmitter to the receiver. A preamble is predetermined symbols which are known both at the transmitter and the receiver and which are transmitted in the subcarriers of the OFOM transmission. Since the preamble is known both at the transmitter and the receiver then any distortion of the preamble due to channel effects can be determined at the receiver and in this manner the channel characteristics can be estimated.

This invention relates to OFDM transmissions with multiple transmit antennas, as shown schematically in Figure 1. Figure 1 shows an OFDM wireless system 100 comprising an OFDM wireless transmitter 200 and an OFDM wireless receiver 300.

The OFDM wireless transmitter 200 has nr transmit antennas, of which the first antenna 201-1 and the nith antenna 201-flT are shown. Signals transmitted by the antennas of the transmitter 200 propagate through a transmission channel 400 to the receiver 300.

There exist a number of transmission schemes which use multiple transmit antennas for wireless communications. Examples include space-time coding methods and transmit precoding methods. Transmit precoding requires channel knowledge at the transmitter so that the transmissions can be modified appropriately to improve system performance. Transmit antenna selection is a popular transmit precoding methodology in which, depending on the channel conditions, transmissions are completely switched to one or few transmit antenna. With the use of OFDM transmissions, one variation of antenna selection used for the present invention is "per-subcarrier transmit antenna selection".

The "per-subcarrier transmit antenna selection" transmission system of the present invention is such that at any one time only a single antenna is active for each transmitted subcarrier. This arrangement is shown in Figure 2, where for each of the 16 subcarriers only one transmit antenna is active at any one time. Such systems are attractive due to the diversity benefits they offer and have been considered in, for example, C. M. Vithanage and S. C. J. Parker and M. Sandell, "Antenna Selection with Phase Precoding for High Performance UWB Communication with Legacy WiMedia multi--band OFDM devices", Proc. IEEE International Conference on Communications, pages 3938-3942, 2008 and in H. Zhang and R. U. Nabar, "Transmit antenna selection in MIMO-OFDM systems: bulk versus per-tone selection", Proc. IEEE International Conference on Communications, pages 4371-4375, 2008.

Vithanage et al discloses a transceiver design for improving the robustness and mean throughput when communicating with ultra wide band devices. Multiple antennas are used in conjunction with per subcarrier antenna selection based on channel state information at the transmitter.

Zhang et al discloses a diversity gain analysis for transmit antenna selection in a multiple input, multiple output OFDM system with linear receivers.

In prior art methods, coherent receivers are employed. That is, prior to the detection of transmitted symbols, practical receivers estimate the underlying channels. For OFDM transmissions, channel estimation is considered in J. J. van de Beek, 0. Edfors, M. Sandell, S. K. Wilson and P. 0. Borjesson, "On channel estimation in OFDM systems', Proc. IEEE Vehicular Technology conference, pages 815-819, 1995 which discloses a method of channel estimation based on time-domain channel statistics, using minimum mean squared error (MMSE) and least square (LS) estimators.

An employment of the per-subcarrier antenna selection technique for wideband communication requires the knowledge of the antenna to subcarrier assignment (ATSA) information at the receiver for the channel estimation. This means that ATSA information has to be conveyed to the receiver reliably over the noisy channel, or by means of a separately transmitting the ATSA information from the transmitter to the receiver, for example via a control channel.

Errors in the ATSA information can lead to a poor performance of the standard MMSE channel estimator (by giving rise to a floor, as described below) for high signal to noise ratios (SNRs). If strong error-correction techniques are not to be used to protect ATSA information, then other methods are required at the receiver side to address this problem.

One such method is to use joint channel-state and parameter estimation. This technique does not require separately sent ATSA information. Instead, it estimates both the ASTA and the channel coefficients in the frequency domain at one time.

M. Sandell and J. P. Coon, "Per-subcarrier antenna selection with power constraints in OFDM systems", IEEE Transactions on Wireless Communications, 8(2):673-677, 2009 discloses a scheme that allocates an equal number of subcarriers to all antennas in a multi-antenna OFDM system, by using integer optimization.

F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory", IEEE Transactions on Information Theory, vol. 39, no. 4, pp 1293- 1302, 1993 discloses a treatment of certain complex random variable and processes to show their usefulness in statistical communication theory.

S. Dey and J. B. Moore, "Risk-sensitive filtering and smoothing via reference probability methods", Proc. American control conference Seattle Washington, pages 129-133, 1995 discloses risk sensitive filtering and smoothing problems for discrete-time nonlinear and linear Gauss-Markov state-space models.

W. James and C. Stein, "Estimation of quadratic loss", Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, pp 361-379, 1961 discloses optimum properties or failure of optimum properties of the natural estimator in certain special problems.

WO-A-20Q6062308 discloses a cell search device using a preamble for a downlink of a cellular system using an orthogonal frequency division multiplexing access (OFDMA) stheme.

EP-A-1438800 discloses a method for combining pilot symbols and Transmit Parameter Signalling (TPS) channels within an OFDM frame. The method uses Differential Space-Time Block Coding to encode a fast signalling message at an OFDM transmitter. At an OFDM receiver, the encoded fast signalling message can be decoded using differential feedback to recover information about the channel responses that would normally be carried by pilot symbols. In wireless data transmission employing adaptive modulation and coding, an instantaneous channel quality measurement, independent of the origin of interference for example, neighbouring-cell interference, white thermal noise, or residual Doppler shift is provided. Using the correlation between a signal which has been symbol de-mapped, and one which has also been soft decoded and re-encoded, a channel quality indicator is produced.

EP-A-1290845 discloses a method for synchronising OFDM symbols during radio transmissions, said method being used to facilitate strong and efficient frame and frequency synchronisation. Additional pilots are added to the OFDM symbols by the transmitter in order to form pilot pairs. A sequence is modulated on said pilot pairs and then extracted by the receiver in order to produce a measure for each OFDM symbol by comparing the extracted sequence and a stored sequence. The OFDM symbol with the largest measure is recognised as being the first OFDM symbol in a frame. The extracted sequence and the stored sequence are compared by means of cross correlation, said cross correlation providing the measure. The integer frequency error can thus also be determined.

US-A-2007/0217552 discloses a system including a differential demodulation module and a correlation module. The differential demodulation module differentially demodulates modulated signals to generate differentially demodulated signals. The correlation modules correlate the differentially demodulated signals with derived preamble sequences and generates correlation values.

The content of each of the above documents are incorporated in their entirety herein by reference.

According to a first aspect of the invention there is provided a method of estimating channel characteristics of a channel in a wireless orthogonal frequency division multiplexed, OFDM, transmission system having a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, to a receiver, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the method comprising: estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: Qmirnmax = argmin max MSE(Q,model = S,) L'{°" N) where: MSE(Q, model = S0) = Tr[S0RjS0H ( -1)11 (Q -1)1 + N0Tr (QQ"); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an NxnN selection matrix describing the ATSA information; n7. is the number of transmit antennas; Tr(.) denotes the trace of a matrix; R is the nTNxnTN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the NxN identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: Qminiexp = argmin[E,{O! N)MSE(Qmode1 = S)] where E.E{o, N) denotes the expectation over the possible set of models.

A set of (2U+1) subcarriers around the position n forming a subset of subcarriers may be considered, where U is an integer and (2U + 1) < N. According to a second aspect of the invention there is provided a method of estimating channel characteristics of a channel in a wireless orthogonal frequency division multiplexed, OFDM, transmission system having a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, to a receiver, the channel being made up of individual channels from each transmit antenna to the receiver, each subearrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the method comprising: estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.

The estimation may also be based on a predetermined number of following estimates.

The approximation may be based on a model defined by: Sk+I = Ak Sk + B Wk÷I where k is a Gauss-Markov model of channel coefficients; Ak is a known, possible erroneous, state transition matrix; w is white Gaussian noise; and B is the matrix which determines the power of the term Bwk÷l and the estimation error cost-functions is: f() := E[p(s -i)] where E is expectation; k p(.)__exp[p>IeiI2]; is a positive constant; and e, =s1-1 A risk-sensitive estimator may be used for the estimation. The risk-sensitive estimator may be smoothed.

The state transition matrix may not be known and the approximation may be based on a Gauss-Markov model defined by: = where x an augmented state containing not only the channel coefficients but also unknown entries of the state transition matrix Ak; F(xk) is a function of Xk given by t(x) := [AS]; a k+1 Sk is a Gauss-.Markov model of channel coefficients; ak is the row vector obtained by concatenating the rows of Ak matrix [Bwkl w is white Gaussian noise given by Wk:= ; and B is the matrix which determines the power of the term Bw and the estimation error cost-functions is: f(i):= E[p(x-i)1 where E is expectation; p(.)=e,2; and e, -1, An extended Kalman estimator may be used for the estimation.

According to a further aspect of the invention there is provided an orthogonal frequency division multiplexed, OFDM, transmission system, comprising: a plurality of transmit antennas, a, associated with one or more transmitter, for transmitting an OFDM transmission over a channel using a plurality N of subcarriers, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment; a receiver for receiving the OFDM transmission, the channel being made up of individual channels from each transmit antenna to the receiver, the receiver comprising: means for estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: Qmininiax = arg mm [ max MSE (Q, model S1)] where: MSE(Q,model = S0) Tr[S0RjS0'(Q_I)' (Q_I)]+N0Tr(QQ11); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an N x nTN selection matrix describing the ATSA information; n is the number of transmit antennas; Tr(.) denotes the trace of a matrix; Rf is the nN x nN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the NxN identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: Qmirnexp = argmjn[EIE{O,l N}MSE(Q,model = S.)] where EIEtOI N} denotes the expectation over the possible set of models.

According to a further aspect of the invention there is provided an orthogonal frequency division multiplexed, OFDM, transmission system, comprising: a plurality of transmit antennas, a, associated with one or more transmitter, for transmitting an OFDM transmission over a channel using a plurality N of subcarriers, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment; a receiver for receiving the OFDM transmission, the channel being made up of individual channels from each transmit antenna to the receiver, the receiver comprising: means for estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.

According to a further aspect of the invention there is provided a receiver for receiving a wireless orthogonal frequency division multiplexed, OFDM, transmission and for estimating channel characteristics of a channel over which the OFDM transmission is transmitted, the OFDM transmission being transmitted over a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the receiver comprising: means for estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: Qminimax = argmin[ max MSE(Q,model = where: M5'E(Q,model = S0) = Tr[S0R 1S0H (Q-I)11 (Q_I)]+N0Tr(QQ''); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an N x nTN selection matrix describing the ATSA information; n. is the number of transmit antennas; Tr(.) denotes the trace of a matrix; R1 is the nTNxnTN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the N x N identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: Qminiexp = arg mm [E N}MsE(Q,model = S.)] where E16{01 N) denotes the expectation over the possible set of models.

According to a further aspect of the invention there is provided a receiver for receiving a wireless orthogonal frequency division multiplexed, OFDM, transmission and for estimating channel characteristics of a channel over which the OFDM transmission is transmitted, the OFDM transmission being transmitted over a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the receiver comprising: means for estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.

According to a further aspect of the invention there is provided a carrier medium carrying computer readable code for controlling a microprocessor to carry out the method described above.

This invention proposes several channel estimation approaches which show satisfactory performance for both low and high SNRs, regardless of the ATSA errors.

The present invention can be implemented in any convenient form, for example using dedicated hardware, or a mixture of dedicated hardware and software. The present invention may be implemented as computer software implemented by one or more networked processing apparatuses. The network can comprise any conventional terrestrial or wireless communications network, such as the Internet. The processing apparatuses can comprise any suitably programmed apparatuses such as a general purpose computer, personal digital assistant, mobile telephone (such as a WAP or 3G-compliant phone) and so on. Since the present invention can be implemented as software, each and every aspect of the present invention thus encompasses computer software implementable on a programmable device. The computer software can be provided to the programmable device using any conventional carrier medium. The carrier medium can comprise a transient carrier medium such as an electrical, optical, microwave, acoustic or radio frequency signal carrying the computer code. Art example of such a transient medium is a TCP/IP signal carrying computer code over an IP network, such as the Internet. The carrier medium can also comprise a storage medium for storing processor readable code such as a floppy disk, hard disk, CD ROMP magnetic tape device or solid state memory device.

The invention will be further described by way of example with reference to the accompanying drawings in which: Figure 1 is schematic representation of an orthogonal frequency domain multiplexed (OFDM) wireless system according to an embodiment of the invention; Figure 2 is a representation of a subcarrier allocation in a two antennas system according to an embodiment of the invention; Figure 3 is a block diagram of a transmitter of the system of Figure 1; Figure 4 is a block diagram of a receiver of the system of Figure 1; Figure 5 is a representation of a set of models to consider at the receiver, when the antenna to subcarrier assignment information is possibly erroneously received; Figure 6 is a graph showing mean squared error improvement when using Minimax and Miniexp estimators in a thirty two subcarrier system with two transmit antennas embodying the invention; Figure 7 is a graph showing mean squared error improvement when using estimators for one hundred subcarriers and two transmit antennas embodying the invention; Figure 8 is a graph showing mean squared error improvement when using estimators for one hundred subcarriers and four transmit antennas embodying the invention; Figure 9 is a graph showing mean squared error improvement when using estimators for four hundred subcarriers and two transmit antennas embodying the invention; and Figure 10 is a graph showing mean squared error improvement when using estimators for eight hundred subcarriers and two transmit antennas embodying the invention; Figure 1 is a schematic representation of an orthogonal frequency domain multiplexed (OFDM) wireless transmission system according to an embodiment of the invention.

The system 100 comprises an OFDM wireless transmitter 200 having n. transmit antennas and an OFDM wireless receiver 300 having a single receive antenna 309.

Wireless communication takes place between the transmitter 200 and the receiver 300 via a channel 400.

Figure 3 is a block diagram of the transmitter 200 of the OFDM system 100 shown iii Figure 1. A frame of data symbols, in this case a preamble stream for determining the channel characteristics, are received by an antenna to subcarrier assignment module 202. The antenna to subcarrier assignment module 202 allocates each suboarner to an antenna such that at any one time only a single antenna is active for each transmitted subcarrier, as shown in Figure 2.

The symbols are passed from the antenna to subcarrier assignment module 202 to the assigned transmitter path and are serial-to-parallel converted in a serial to parallel converter module 203. The inverse fast Fourier transform (IFFT) of the result is taken in a multicarriér modulator 204. The output of the multicarrier modulator 204 is input to a guard interval and parallel to serial converter module 205 where a guard interval is added and the result is parallel-to-serial converted. The resultant baseband signal is converted to an analogue signal in a digital to analogue converter (DAC) 206 and thereafter upconverted to the transmission frequency in an RF upconversion and transmission module 207 and then transmitted via the respective n,. transmit antennas 201.

The receiver 300 shown in Figure 4 comprises an antenna 309 for receiving wireless transmissions from the transmitter 200. The signal detected by the antenna 309 is fed into an RF downconversion module 301 which downconverts the signal to baseband.

The baseband signal is passed to an analogue to digital converter 302 and converted to a digital signal. The digital signal is fed into a guard interval removal and parallel to serial converter module 303 where the guard interval, which was inserted at transmission, is removed and the resulting signal is serial-to-parallel converted. The output is passed to a multicarrier demodulator 304 where it is fast Fourier transformed and the output is passed to parallel-to-serial converter 305 where it is parallel-to-serial converted. The output of the parallel to serial converter 305 comprises a channel corrupted version of the transmitted frame of data and preamble symbols.

The output of the parallel to serial converter 305 is passed to a channel estimator 306, where the antenna to subscriber assignment information received by the receiver 300 is used to estimate the channel 400 between the baseband of the transmitter 200 and the baseband of the receiver 300, as is explained below. The ATSA information is transmitted by the transmitter 200 but the ATSA information received at the receiver 300 may be erroneous, the consequences of which are discussed below.

The skilled person will understand that the transmitter 200 and receiver 300 may also respectively comprise receiver and transmitter portions, i.e., both of the transmitter 200 and receiver 300 shown may form part of two transceivers. The skilled person will also understand that on the transmitter side of the system of Figure 3 there may be a plurality of transmitters 200.

In embodiments of the invention orthogonal frequency division multiplexed (OFDM) modulations are utilised with N subcarriers and on each subcarrier only a single transmit antenna is allowed to transmit, i.e., per-subcarrier transmit antenna selection is employed. Such systems are described in the two Vithanage et al referenced documents and in Zhang et al. The channel impulse response (CIR) from antenna a E {i,..., n,. } to the receiver is given by h0 = (hQl,h02,...,haL)T, where L is the largest spread in the individual CIRs.

For simplicity, the channels from different antennas are assumed to be uncorrelated, although the following can be generalised to account for antenna correlations.

Taking the partial Fourier matrix 0 to be the N x L matrix with the (n,I) th element (for n=1,2,...,N; l=1,2,...,L) as e N, the frequency domain channel fading coefficients from antenna a are given by f0 Oh0. Suppose the n th element of f0 is f0p,, which denotes the frequency domain channel fading coefficient in the nth subcarrier from ath antenna.

In a typical system employing per-subcarrier antenna selection as described above, for the n th subcarrier, the transmit antenna is selected to maximise the resultant received signal power, f2. Other practical constraints might lead to a different antenna to subcarrier assignment, for example as described in Sandell et al. Suppose a set of subcarriers, r c {1,2,...,N} is selected for each antenna a and that all Jr the subcarriers are utilised, i.e., L.J'a ={1,2,...,N} the union of all sets of subcarriers for each antenna is the set of subcarriers. Since only one antenna is allowed to transmit on each subcarrier, then a = 0 for a!=b, i.e., for a pair of different antennas, the set of subcarriers for each antenna do not intersect.

Now suppose X0 is an N x N diagonal matrix representing the transmit signals from antenna a on its diagonal. Specifically, the (n,n)th element denotes the signal transmitted on subcarrier n (for n = 1,2,...,N), which is zero if a' l.e, if there is a transmission on a subcarrier n for a particular antenna then the diagonal value on column n of row n of the matrix Xa is the transmitted signal and if there is no transmission on subcarrier n for a particular antenna then the diagonal value on column n of row n of the matrix X is zero.

Due to the properties of OFDM transmissions, which allow for a convenient inter-symbol interference-free frequency domain signal analysis, the received signals on the set of subcarriers can be expressed by an N xl vector y: YXa1a' where w represents the complex Gaussian additive noise realisation, which is modelled to have a zero mean and covariance matrix NOIN, where N is the N x N identity matrix, It is assumed that each non-zero element of X0 is of unit magnitude (since in this analysis we are considering preamble symbols of unit magnitude, although the model can be extended to preamble symbols of different amplitudes) and we take the system signal-to-noise ratio (SNR) as J_. The punctured nature of X0 (i.e., that a signal is only transmitted on some of the subcarriers) enables the above signal model to be re-expressed as: (1) where X=X0 and $e/ is the Nxl frequency domain resultant channel fading coefficient vector whose nth element is J'0 when nET0 i.e., when there is a transmission on a subcarrier n for that antenna.

If it is assumed that the non-zero transmitted signals are the unit symbol, then the received signal is Yfsei+W (2) The frequency correlations, within the resultant channel fading coefficient vector f, are now considered. First, let the frequency domain channel fading coefficient vector of dimenstions nN xl, f, for the n7. transmit antennas be f = (f1T fT f and the frequency domain channel correlation matrix of dimensions n7N x nTN, be Rf =E1(ff"), where Ea denotes expectation value and H is the Hermitian or conjugate transpose.

A selection matrix S0 of dimensions N x nTN is defined such that f, = S0f. It is apparent that S0 = [diag (d1) diag (d2). . . diag (d,27)], where each d0 is a length N vector with the n th element being I if n E J, and 0 otherwise for each n = 1,2,..., N. Thus the frequency correlations in the channels after per-subcarrier antenna selection are given by the frequency domain resultant channel correlation matrix: Rf, = E1, (1sei1'se/') SORJS' (3) For general situations, the above correlation matrix is not known explicitly. An approximation to be used in the subsequent algorithms is: R pS�RfS' (4) "Ti where p is some selected scalar, e.g. p = 1 or p =-1=1 The importance of the latter value of p is that this value along with equation (4) produces the exact frequency domain resultant channel correlation matrix R1, when the underlying channel impulse responses are of length 1.

Linear estimators for frequency domain channel estimation are described below, i.e., where the receiver utilises the N x N matrix Q to estimate channels from the received signals as: c=Qy where, is the estimated value of f.

From S. M. Kay, Fundamentals of statistical signal proceesing: Estimation Theory", Prentice Hall 1993, the contents of which are incorporated herein by reference, the mean square error (MSE) due to such an estimation is given as: MSE(Q, model S0) = E -c1) = , ([t, ]H[. f 1) ([f]H[fJ) = Tr[Rj, (Q_I)hi(QI)]+NoTr(QQt) In the above, Tr(.) denotes the trace (i.e., the sum of the elements on the main diagonal) of a matrix. Using the correlation matrix approximation of equation (4), with the scalar p set to 1 for example, we have: M5E(Q,mode1 = S0) = Tr[S0R1S01(Q_1)1 (Q-I)]+N0Tr(QQ') (5) For a receiver equipped with perfect knowledge about the channel correlations, the optimal linear transformation to minimise the above cost function is given by the following minimum mean square error (MMSE) estimator: Q = argmin[MSE(Q model = s0)] = SORJ-SOH (s0R1S0' + N0I)' (6) The fact that QM is the minimiser for the MSE function given in equation (5) is known, for example from S. M. Kay referred to above. The present invention lies in the utilisation of the frequency domain resultant channel correlation matrix of equation (4) in the MSE function of equation (6), since the approximation of the correlation matrix allows the MMSE estimator of equation (6) to be evaluated.

This MMSE estimator optimally takes into account the channel correlation information when they are known. On the other hand the least squares (LS) channel estimator does not take any correlation information into account and for this system is given by: Q N (7) where N is the N x N identity matrix.

The MMSE estimator above in equation (6) is optimal when the channel correlation information is known at the receiver. From equation (4), one can see that channel correlations R1 are composed of two elements: the correlations of the individual channels in the frequency domain from different antennas, R1, and the antenna to subcarrier assignment (ATSA) information, S0. It has been observed that estimation performance is usually not very sensitive to errors in knowing correlations of the individual channels from different antennas, R1 at the receiver. However, estimation performance is seen to be significantly sensitive to errors in the ATSA information, S0, at the receiver.

An object of embodiments of this invention is to robustify the channel estimation against errors in the ATSA information at the receiver. This may be done by optimising alternative cost functions, rather than equation (5) at the receiver. In other words, a scheme is chosen whereby the channel estimate is solved for each possible value and the solution with the lowest (or highest) value is selected, i.e., the "best available" value of channel characteristics is selected given a defined domain. These alternative cost functions are selected such that the robustness of the resultant estimators is improved.

In one embodiment of the invention, a "Minimax" approach is used. Minimax is a technique for minimising the worst case estimation error and is described in P. J. Huber, "Robust Statistics', Wiley 1981, the contents of which are incorporated herein by reference.

Considering a case where there are two transmit antennas, we assume that the possible errors in the ATSA information transfer to the receiver is such that the receiver receives at most one antenna in error for the set of subcarriers. (Extension of the following to having more than two transmit antennas and the possibility for more errors at the receiver is straightforward.) For this case, when the receiver obtains the ATSA information as represented by the model S0, it can only be certain that the correct model is one in the set {SO,SJ,...,SN}. Here, Sa represents the perturbation of S0 such that the antenna to be transmitting on the ath subcarrier is changed. For example, as illustrated in Figure 5, when the receiver has received the antenna assignment on the 5th subcarrier in error, then to get the actual model it would need to change the active antenna for the 5th subcarrier. In other words, the correct model is S5. However, the difficulty is that the receiver does not know which of the received information was in error and therefore cannot identify the required correction(s).

In this minimax approach, the channel estimator is selected such that the maximum MSE over the possible set of models is minimised. This is a conservative approach and attempts to ensure that model errors (i.e., errors in the ATSA information) at the receiver do not lead to catastrophic performance degradations. Mathematically, the estimator is given as: Qminimax argmQin[J11ax}MsE(Qmode1 = si)] (8) It can be shown that equation (8) is a convex optimisation problem, which ensures the existence of numerical solutions with polynomial complexity. To see this, note that solving equation (8) is equivalent to solving the following optimisation problem: mm t :;. Tr[S,RjS,H (Q -j)H (Q -I)]+ N0Tr(QQH) «=t Vi = It is apparent that this problem is a convex optimisation problem since the objective function and constraints are all convex functions.

In another embodiment a Miniexp" approach is used. Here the cost function to be minimised is the expected MSE over the possible set of models. Mathematically, the estimator is given as: Qminiexp arg mm [E a{o,i N}MSE (Q, model = S.)] (9) Here, E,E{OI N} denotes the expectation over the possible set of models. The main attraction of the Miniexp estimator over the Minimax estimator is that the solution is available in closed form. To see this, rewrite the expectation E1{01 N) as: Eie(OIN)MSE(Q,model =S1) = {Tr[SRJSH (Q -j)H (Q -i)]+ NoTr(QQ)} = Tr[E.01,, {s1R1s/'}(Q -I)" (Q-i)]+ N0Tr(QQ") Similar to the minimiser for equation (5), the minim iser to the above is found in closed form as: Qmuuexp E11 N}IRJSI (EIE{oI N}IRfSI}+ N0I,,)' (10) The fact that equations (10) minimises the relevant cost function can be understood from observing that equation (6) minimises equation (5).

Above, estimators of the form: are considered, where y is the full set of received signals and the n th element of seJ gives the estimate of the fading on subcarrier n. With such full signal estimators, the computation of the N x N matrix Q can be burdensome when N is large, for example since matrix inversions are necessary, as shown in equations (6) and (10).

Hence, when estimating the fading on a subcarrier n, the received signals on a reduced set of subcarriers around the position n can be considered. For example, the received signals on subcarriers {n-U,n--U+1,...,n+U---1,n+U} can be considered, where U is an integer such that the number of subcarriers in the reduced set is (2U+1) and (2U+1)< N. When such a reduced set is considered, since the vector y is of much shorter length (being of dimensions ((2U+1)xl)) than the number of subcarriers N, the computations become simpler.

Further estimators are described below which are iterative in their nature. This means that the estimate of a frequency domain channel fading coefficient fan depends only on the estimates of selected number of previous coefficients fO,_1'fO,2'**.'fO,fl_M but not on all of the coefficients. (Below, the notation for channel coefficients does not contain a'for the sake of brevity.) This is achieved by introducing a Gauss-Markov assumption for the channel coefficients ç,. Here, M represents the order of the Gauss-Markov process.

To implement iterative robust recursive estimators, the channel coefficients are first approximated by the Mth order Gauss-Markov stochastic process in the frequency domain. For M = 1 the model is given by: [.fR,k+l1 [fR.kl I I+Bwk+j LJI,k+1 J LJ1,k J where fk+l = fR,k÷1 + ff, and fk = fR,k + if,k are (k + 1)th and kth complex channel coefficient, respectively and A is the state transition matrix and the matrix B determines the power of the second term Bwk+J. The real and imaginary part of a coefficient have to be separated since the real versions of the robust filters are employed. Here, it can be seen that the (k +1) th coefficient only depends on the k th coefficient which is in accordance with the assumption M = 1 (i.e., an iterative Gauss-Markov model based on only the previous value). The method of determining the matrix A (the state transition matrix) is explained below, and it is based on Neeser at al. The matrix BE R2X2 is a 2x2 real matrix and can be determined by tuning for a particular system at hand. R is the set of real numbers. Also, w:= {wk}k»=O is a white Gaussian noise (where the notation:= denotes "is by definition equal to"), where Wk E R2XI is a 2x1 real random vector having Gaussian distribution (each Wk is a zero-mean 2x1 identity matrix).

With reference to R. J Elliott, L. Aggoun and J. B. Moore, "Hidden Markov models: Estimation and control", Springer-Verlang, 1995, the content of which is incorporated herein by reference and Neeser et al, in general the Mth order Gauss-Markov stochastic process which approximates channel coefficients f denoted by s {sk}k»=O is given by: Ak Sk + B Wk+j (11) where Sk E R2MXI, Ak e R2M>(2M, B e R2MX2M and w: {wk}k»=O is a white Gaussian noise, Wk e R2MI (each w is zero-mean having the covariance matrix Zx which is 2M x 2M identity matrix), and M is the order of the approximation. The vector = [ss MRkM1k] is related to the channel coefficients f in the frequency domain by 5,Rk = fR.k+iI and S,1 k fj,k+11' 1 «= i M, where fk÷I_1 = fR,k÷a_1 + J!1,k+I_I is the (k + 1)th complex channel coefficient in the frequency domain.

The recursive formula (11) is also called the state equation, and in the general case it could have a complex field representation. The A matrix is of the form: A 1-I2M21 (12) k [AkL where the zero block is a (2M-2)x2 matrix, "2M-2 is the (2M-2)x(2M---2) identity matrix, and AkL is the 2x2M matrix whose entries are deduced from the correlation matrix of the resultant channel coefficients of equation (4) by using the following identities, which are derived in Neeser et al: E[fRkfJkI= ° E[fRkff,]= -E[fR,fjkl E [fR,k JR.,] = E [fjJj,] Re {R1 (k, i)} / 2 E[fRkfII}= -E[JR,fjkj= Im{Rj (l,k)}/2 for k!=1.

Thus, the received signal y:= {Yk} can be expressed by: yk_Csk+')vk (13) where Yk E R2M)l, C=I2M, D ER2MX2M, and v:= {vk}k»=1 is a white Gaussian noise, E R2M (each Vk is zero-mean with the covariance matrix which is 2M x 2M identity matrix). Here, equation (13) is also called the observation equation.

The previous model captures the case when the entries of Ak are known (and possibly erroneous). When the matrix Ak is not known in advance, it is possible to estimate from the received signal y, in addition to the estimation of the channel coefficients s.

To do so, another form of the Gauss-Markov model is used. For this, an augmented state x:= {xk}k»=O contains not only the channel coefficients but also unknown entries of the matrix Ak. This new state is defined by: = AkSk BWk+I (14) a÷1 where a is the row vector obtained by concatenating the rows of AkL matrix and where Xfr: a,, f. When the following substitutions are introduced [Asi [Bw,,1l b(x):=[] wk+I:=[ + j, equation (14) becomes: The received signal can now be described by: Yk(Xk)+')'k (15) where h (xk) is a substitution for Cs of equation (13).

To measure the performance of estimators, the estimation error cost-function is introduced as: f(I):=E[p(x-i)] (16) where p is a strictly convex function which is bounded from below, attaining global minimum at zero and whose second derivative is positive and bounded from below.

Two choices for p(S) which are used in this invention are p0= Iei2 and p(.)= exp[pIeiI2] where e1 = x, -i, as described in Elliott et al. Here, 1= is the estimator, and bL is a positive constant. The optimal estimator is defined by: î0=argmjnf(j) (17) The Extended Kalman Estimator (EKE) described in Elliott et al is a technique for dealing with uncertain or unknown slowly time-varying communication channels. The estimator achieves the robustness in the following way. The EKE does not rely on the erroneous correlation matrix of the resultant channel correlation R1 to compute the channel coefficient estimates. Rather, it computes the correlation matrix estimate as well as the estimate of the channel coefficients from the received signal y at the same time. The finite dimensional EKE, which optimizes the p(c) = el2 cost-function is given by: k =Ik_l)+Lk(yk -h(cI(ikI))) (iSa) Lk = Rk_lH' (1 8b) k HkRk_IH +Vk (18c) RkIk = (6M -LkHk)RkIk_I (18d) Rk_l =FkRklIk_lF +Wk (18e) i0=E[x0] (18f) Here, Lk is the optimal Kalman gain, RkIk_1 is the predicted estimate covariance matrix, RkIk is the updated estimate covariance, «=»=k is the innovation covariance, Wk = BTB is the covariance of the process noise (the additive noise in the recursive equation for the state x given by equation (14)), and Vk = DTD is the covariance of the observation noise (the additive noise in the received signal y given by equation (15)). The matrices Fk and Hk are defined by: F:= H:= (19) Xk_I kIk-I where XkIkl 1(ik_I).

The Risk sensitive estimator (RSE) described in Elliott et al achieves robustness by optimizing the exponential cost-function p(.)= exp[/.tIeiI2]. The RSE assumes that the correlation matrix of the channel correlation is given (and is possibly erroneous). The estimator employs this matrix to compute the estimates of the channel coefficients:= {k}k»=O by: Sk =Akk_I +Lk(yk -CAkk) (20a) Lk = (Ri' +CTVC)CTV (20b) Rk+I =Wk+l +Ak(R1 +CTV1C_,2I2M)lA (2Cc) 0=E[s0] (20d) Here, Wk and V are defined in the same way as above. The matrix R0 is positive- definite. The positive constant is a design parameter, and it is called the risk-sensitive parameter. By tuning i, it is possible to improve the robustness of the estimator. When tends to zero, the estimator tends to the Kalman estimator.

Equations (20a)-(20d) represent the so-called forward recursive equations. Next, the backward recursive equations are provided to include a possibility of taking into account not only the estimates of the previous frequency coefficients fa.n_l'fa,n_2'"'fa,n_M but also the estimates of the following coefficients fa,n+1'fa,n+2'"'fa,n+M to estimate fan, as described in Dey et al: khik = AWISkPl11k+I +CT Vy /II2MSk (21a) A(W1 -WISkWI)Ak +CTV,1C-pI2M (21b) Sk =(w1+1)' (21c) Here, i:= {Ik} are the channel coefficient estimates based on the future observations.

Combined with the forward recursion (20a)-(20d), the smoothed estimates of the channel coefficients i:= {iik} are computed by: k =i(P'11k +R'yk) (22a) = (p + R1)l (22b) where Yk:= Akk_I.

The following examples compare the performance of the estimators described above with the performance of the LS estimator, MMSE estimator with the perfect and erroneous (uncertain) ATSA information, and the Stein estimator (SE), which is described in James et al. The performance is measured in terms of the MSE. It is demonstrated that some of the proposed techniques behave differently when the parameters are varied.

Figure 6 illustrates the MSE performance of the Minimax and Miniexp estimators. A system with 32 subcarriers, two transmit antennas and a bit error rate (BER) of 0.01 in the control channel used to convey the ATSA to the receiver is considered. To introduce errors in the ATSA information at the receiver, they were first coded in binary form and then sent through a binary symmetric channel with a cross over probability of 0.01. It can be seen that even though the MMSE estimator with perfect ATSA information performs better than the LS estimator, when errors exist the high SNR performance is much worse. The Stein estimator (SE) shows stable performance at high SNR and improved performance at low SNR, when compared to the LS. However, its low SNR performance is inferior to the MMSE with ATSA errors. The proposed Minimax and Miniexp estimators attain best of both worlds in that the MSE is minimised and close to the perfect MMSE performance at low SNR while having performance comparable to the LS performance at high SNR.

Figures 7 to 10 illustrate the performance of the estimators described above based on the Markov channel model. It can be seen from these graphs that there is a large constant gap (»= 12 dB) between the LS method and the MMSE estimator with the perfect ATSA information. The MMSE estimator with the imperfect ATSA information is close to the perfect ATSA case for low SNRs. However, for high SNRs, imperfect ATSA information leads to an unacceptable flooring of the MSE. The estimators described above represent a compromise; whereby at low SNRs the performance is between the performance of the SE and the MMSE estimator with the perfect ATSA information, while at high SNRs, the performance is either equal to the LS estimator or it is slightly worse. Smoothed (SM) estimators show significant improvement as compared to non-smoothed estimators for low SNRs, while this advantage disappears at high SNRs.

Figure 7 shows the mean squared error improvement when using estimators for 100 subcarriers and 2 transmit antennas with a BER of the ATSA channel equal to 0.01. It can bee seen that the Kalman estimator (the risk-sensitive (RS) estimator for a 0) and the EKE perform better than the SE. The RS estimator performs slightly better than the EKE at low and high SNRs, while for the mid-SNRs the MSE is identical. For the low SNR, the RS estimator is about 3 dB better than the SE and 5 dB better than the LS. The risk-sensitive smoother (SMRS) is slightly better than the non risk-sensitive smoother SM, and they are about 5 dB better than the RS estimator for the low SNR. As SNR increases, the gap between the SMRS and the LS diminishes, and it vanishes at around 20 dB. The MMSE estimator with uncertain ATSA experiences an error floor starting at around 15 dB.

Figure 8 shows mean squared error improvement when using estimators for 100 subcarriers and 4 transmit antennas with a BER of 0.01 and depicts a situation similar to that found in Figure 7. The main difference is that the RS filter performs better than the EKE for the whole range of SNR.

Figure 9 shows mean squared error improvement when using estimators for 400 subcarriers and 2 transmit antennas with a BER of 001. It can be seen that the EKE performs slightly better that the RS estimator for medium SNRs. This suggests that EKE shows better characteristics for large bandwidth than the RS estimator.

This can also be seen from Figure 10 which shows mean squared error improvement when using estimators for 800 subcarriers and 2 transmit antennas with a BER of 0.01.

Known approaches of MMSE estimation lead to severe performance degradation at medium to high SNRs, when the ATSA information is not known at the receiver perfectly. As has been shown, the proposed estimators do not show such performance degradation. Furthermore, performance of the proposed estimators has been shown to be better than that of the LS estimator at low to medium SNRs. Hence the proposed estimators provide improved (compared to LS estimation) and robust channel estimations for systems with per-subcarrier transmit antenna selection.

The improvement in the MSE in channel estimation can result in improvements to the system throughput at a given SNR. Furthermore, since the proposed estimators enable ATSA errors to be tolerated, a lower bandwidth channel can be assigned to transmit such information to the receiver. This may in turn, further improve the data throughput of the communication system.

Various modifications will be apparent to those skilled in the art and it is desired to include all such modifications as fall within the scope of the accompanying claims.

Claims (14)

1. CLAIMS: 1. A method of estimating channel characteristics of a channel in a wireless orthogonal frequency division multiplexed, OFDM, transmission system having a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, to a receiver, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the method comprising: estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: Qrnn,imax = arg min[ max MSE(Q,model = S.)] where: MS'E(Q,model S0) =Tr[S0R1S0"(Q_I)"(Q_I)]+N0Tr(QQ"); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an Nxn7N selection matrix describing the ATSA information; n is the number of transmit antennas; Tr(.) denotes the trace of a matrix; is the nTNxnTN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the N x N identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: = argmin [E101 N}MSE(Q,model = S.)] where EIE{oI N} denotes the expectation over the possible set of models.
2. 2. A method as claimed in claim 1, wherein a set of (2U+1) subcarriers around the position n forming a subset of subearriers is considered, where U is an integer and (2U+1)<N.
3. 3. A method of estimating channel characteristics of a channel in a wireless orthogonal frequency division multiplexed, OFDM, transmission system having a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a to a receiver, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the method comprising: estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.
4. 4. An method as claims in claim 3, wherein the estimation is also based on a predetermined number of following estimates.
5. 5. A method as claimed in claim 3 or 4, wherein the approximation is based on a model defined by: Sk+I = Ak Sk + B Wk+l where Sk is a Gauss-Markov model of channel coefficients; Ak is a known, possible erroneous, state transition matrix; Wk is white Gaussian noise; and B is the matrix which determines the power of the term Bwk÷j and the estimation error cost-functions is: where E is expectation; p(.)= exp[,.zle,I2]; a is a positive constant; and C1 = S! -s,
6. 6. A method as claimed in claim 5, wherein a risk-sensitive estimator is used for the estimation.
7. 7. A method as claimed in claim 6, wherein the risk-sensitive estimator is smoothed.
8. 8. A method as claimed in claim 3 or 4, wherein the state transition matrix is not known and the approximation is based on a Gauss-Markov model defined by: D (; ) + w where 1k an augmented state containing not only the channel coefficients but also unknown entries of the state transition matrix Ak; 1 (xk) is a function of Xk given by D (xk) : [Aksk]. k+isk is a Gauss-Markov model of channel coefficients; a is the row vector obtained by concatenating the rows of Ak matrix rBWkl w k is white Gaussian noise given by Wk:= ; and B is the matrix which determines the power of the term Bw and the estimation error cost-functions is: where E is expectation; p(.) ; and e1 =x-1,
9. 9. A method as claimed in claim 5, wherein an extended Kalman estimator is used for the estimation.
10. 10. An orthogonal frequency division multiplexed, OFDM, transmission system, comprising: a plurality of transmit antennas, a, associated with one or more transmitter, for transmitting an OFOM transmission over a channel using a plurality N of subcarriers, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment; a receiver for receiving the OFDM transmission, the channel being made up of individual channels from each transmit antenna to the receiver, the receiver comprising: means for estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: = argmin[ max MsE(Q,model S.)] where: M5'E(Q,model S0) Tr[S0RjS0" (Q_1)H (Q_I)]+N0Tr(QQ"); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an NxnN selection matrix describing the ATSA information; n2. is the number of transmit antennas; Tr(.) denotes the trace of a matrix; Rf is the nTNxnTN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the N x N identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: Qminicxp argmin[EIO)MSE(Q,mode1 = s1)] where E101 N} denotes the expectation over the possible set of models.
11. 11. An orthogonal frequency division multiplexed, OFDM, transmission system, comprising: a plurality of transmit antennas, a, associated with one or more transmitter, for transmitting an OFDM transmission over a channel using a plurality N of subcarriers, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment; a receiver for receiving the OFDM transmission, the channel being made up of individual channels from each transmit antenna to the receiver, the receiver comprising: means for estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.
12. 12. A receiver for receiving a wireless orthogonal frequency division multiplexed, OFDM, transmission and for estimating channel characteristics of a channel over which the OFDM transmission is transmitted, the OFOM transmission being transmitted over a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the receiver comprising: means for estimating the channel characteristics based on a cost function chosen from: A. a Minimax cost function which considers the maximum mean squared error, MSE, over a possible model set and whereby the estimator, which minimises this, is given by: Qminimax arg mm [ maXMSE (Q, model S1)] where: MSE(Q,model = S0) = Tr[S0R 1S0" (Q-I)" (Q-I)]+N0Tr(QQ"); Q is an NxN estimator matrix used to estimate the resultant channel from the received signal y as Qy; S0 is an N x nTN selection matrix describing the ATSA information; n7. is the number of transmit antennas; Tr(.) denotes the trace of a matrix; Rf is the nN x nN frequency domain channel correlation matrix; H is the Hermitian transpose; I is the N x N identity matrix; and N0 is the noise variation, and B. a Miniexp cost function which considers the expected MSE over a possible model set and whereby the estimator, which minimises this, is given by: Qrniniexp = arg9jn[EI{O,l N)MSE(Q,model S.)] where E1{01 N} denotes the expectation over the possible set of models.
13. 13. A receiver for receiving a wireless orthogonal frequency division multiplexed, OFDM, transmission and for estimating channel characteristics of a channel over which the OFOM transmission is transmitted, the OFDM transmission being transmitted over a plurality of N subcarriers which are transmitted by a plurality of transmit antennas, a, the channel being made up of individual channels from each transmit antenna to the receiver, each subcarrier being transmitted by a single transmit antenna at any one time in accordance with an antenna to subcarrier assignment, ATSA, wherein the ATSA information is not known perfectly at the receiver, the receiver comprising: means for estimating the channel characteristics using an iterative recursive estimator based on a Gauss-Markov stochastic process approximation of the channel in the frequency domain and considering a predetermined number of previous estimates.
14. 14. A carrier medium carrying computer readable code for controlling a microprocessor to carry out the method of any one of claims I to 9.