GB2468721A - Determining the position of OFDM pilot subcarriers based upon channel covariance matrix - Google Patents

Abstract

An OFDM transmission system 100 comprises a transmitter for transmitting N sub-carriers, a channel through which the N sub-carriers propagate and a receiver for receiving the N sub-carriers. The invention comprises: receiving the N sub-carriers at the receiver; processing the N sub-carriers to determine a channel covariance matrix and setting the position of thePpilot sub-carriers amongst the N sub-carriers based on channel covariance matrix of the channel. The rank of the covariance matrix can affect the specific calculation and signal to noise ratio (SNR) can affect stability at high SNR.

Description

A method of determining theposition of pilot sub-carriers in an orthogonal frequency division multiplexing transmission system This invention relates to a method of orthogonal frequency division multiplexed (OFDM) transmissions. In particular, this invention relates to a method of determining the position of pilot symbol sub-carriers in an OFDM transmission.

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 sub-carrier.

By dividing the data over multiple sub-carriers each sub-carrier 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 sub-carriers 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 sub-carrier has particular advantages where channel conditions are significant. One example 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 sub-carrier signals the reduced data rate of each sub-carrier 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, pilot symbols can be transmitted from the transmitter to the receiver. Pilot symbols are predetermined symbols which are known both at the transmitter and the receiver and which are transmitted in some of the sub-carriers of the OFDM transmission. Since the pilot symbols are known both at the transmitter and the receiver then any distortion of the pilot signal due to channel effects can be determined at the receiver and in this manner the channel characteristics can be estimated.

It is known that if the underlying channel impulse response is constituted of uncorrelated taps, then the optimal placement method is to space the pilot symbols with an equal spacing amongst the sub-carriers. Such a pilot symbol placement scheme is shown in Figure 1. In Figure 1 the OFDM transmission comprises sixteen sub-carriers, four of which are pilot symbol sub-carriers. In this scheme the four pilot symbol sub-carriers are equally spaced amongst the sub-carriers. The locations of the sub-carriers are considered in a cyclic sense here. For example, the first and the last sub-carriers are considered to be adjacent to each other. Different methods of placing the pilots, will generally lead to different perfonnance in the task of channel estimation at the receiver.

Pilot symbol placement techniques for OFDM systems are disclosed in "Pilot tone selection for channel estimation in a mobile OFDM system", R. Negi and J. Cioffi, IEEE Transactions on Consumer Electronics, vol. 44, August 1998, pp. 1122-28, and in "Error probability minimizing pilots for OFDM with M-PSK modulation over Rayleigh-fading channels", X. Cai and G. B. Giannakis, IEEE Transactions on Vehicular Technology, vol. 53, January 2004, pp. 146-155.

Negi et al and Cai et al focus upon scenarios where the underlying channels are subject to uncorrelated scattering, when considered as a function of time lag. In other words, they consider scenarios where the taps of the channel impulse responses vary randomly without any correlation between different channel taps. By uncorrelated scattering, we refer to scattering processes such that the separate channel taps in the impulse response vary in an uncorrelated manner. Whereas with correlated scattering, the individual channel taps can vary in a correlated fashion. It is concluded in these works that for such uncorrelated channels, the optimal method of pilot symbol placement is to place them such that there is an equal spacing between one another, as shown in Figure 1.

"Empirical eigenanalysis of indoor UWB propagation channels", R. Saadane, A. Menouni, R. Knopp and D. Aboutajdine, IEEE Globecom conference, pp. 32 15-19, 2004 and "Ultrawideband channel modelling on the basis of information theoretic criteria", U. G. Schuster and H. Bolcskei, IEEE Transactions on Wireless Communications, vol. 6, PP. 2464-75, July 2007, discuss experimental investigations into wideband short range indoor channels and conclude that there exist scenarios where the underlying channel impulse responses are subject to correlated scattering, i.e., there is a statistical relationship between variations in the separate taps of the impulse response. For such scenarios, an equal spacing between the pilot carrying subcarriers may no longer be optimal. A further mechanism that can cause the channel taps as perceived by the receiver to be correlated is the presence of synchronisation errors in the receiver..

"Non-uniform pilot-symbol allocation for closed-loop OFDM", A. Y. Panah, B. Nosrat-Makouei and R. G. Vaughan, IEEE Transactions on Wireless Communications, vol. 7, July 2008 discloses a pilot symbol placement scheme with perfect channel knowledge at the transmitter. However, a problem with the method described in Panah et al is that obtaining perfect channel knowledge at the transmitter is more onerous than obtaining channel statistics.

"Optimal pilot sequence design for channel estimation in MIMO OFDM systems", D. Hu, L. Yang, Y. Shi and L. He, WEE Communications Letters, vol. 10(1), January 2006, discloses design of pilot symbols when non-equally-spaced pilot placement needs to be employed. However, schemes of placing the pilots in such a non-equally-spaced manner are not discussed.

This invention provides an improved method for determining the placement of pilot symbol sub-carriers. The low complexity of the algorithm makes it attractive for real-time implementation.

According to a first aspect of the invention there is provided a method of determining positions P { 1,2,..., N} of P pilot sub-carriers in an orthogonal frequency division multiplexing, OFDM, transmission system, the system comprising a transmitter for transmitting N sub-carriers, a channel through which the N sub-carriers propagate and a receiver for receiving the N sub-carriers, the method comprising: receiving the N sub-carriers at the receiver; processing the N sub-carriers to determine channel statistics of the channel; and setting the position P of the P pilot sub-carriers amongst the N sub-carriers based on the channel statistics of the channel.

The channel statistics may comprise a channel covariance matrix, R, of the channel given by: E(hhH) = R, where E(.) is the expectation; h = (h1, h2, ..., h)T is the channel impulse response; L is the number of taps in the channel impulse response; T is the transpose; and His the Hermitian transpose.

The step of setting the position of the P pilot sub-carriers may be based on a rank, L', of the channel covariance matrix, R, which may be calculated by eigendecomposing the channel covariance matrix, R, into the form: R = IJDUH, where U is an L x L' matrix having orthonormal columns; and D is an L x L' diagonal matrix with positive real diagonal elements..

When the rank, L of the channel covariance matrix, R, is unity, the position of the P pilot sub-carriers may be set by: computing vector u, where R uuH; computing vector f, where f = ONXLU where 0NxL is the N x L discrete Fourier transform matrix with the (n, l)th element being e N; identifying the P largest magnitude components off; and setting P as the indices of the P largest magnitude components of f.

When the channel covariance matrix, R, is of full rank, i.e., when L' = L, the position of the P pilot sub-carriers may be set by equally spacing the P sub-carriers amongst the N sub-carriers.

When the rank of the channel covariance matrix, R, is equal to P, i.e., when L' = P, , the position of the P pilot sub-carriers may be set by: computing the N x' matrix W, where W = �NXLU, and 0NxL is the N x L discrete Fourier -(n-)(!-I) transform matrix with the (n,1)th element being e N such that f = ONXLh; where (J, f2' *** f) is the frequency domain fading coefficients with j being the fading coefficient on sub-carrier i,. evaluating det(W) for all W where det(.) is the determinant, W,, is the matrix formed from the set of rows of W indexed by P; and Pc {1,2,...,N} is the candidate set of pilot sub-carrier placements; selecting the position of the P pilot sub-carriers for the case where det(W) is maximised.

Alternatively the position of the P pilot sub-carriers may be set by: (a) initialising a set of positions of the P pilot sub-carriers, 7, as an empty set; (b) initialising a set V0 as {l,2,...,N}; (c) initialising the noise variance, N0, as N0 = max(N0, NIh), where NIh is a threshold value of the noise variance; v-Uyf (d) initialising matrix - N0' NXL / (e) initialising matrix E0D; (f) setting i as 1; kEV1 (g) calculating for all H' Vk E:_lvk 1 + V'E11VJ where Vk is the k th column of V; (h) choosing k as the value of k for which the expression in (g) is maximised; (i) including as a member of and renaming this set as (j) removing k from the set V1 and renaming this set as E. vvE.

(k) setting E. as E.1 - k 1 + v E11v (1) incrementing i by 1; (m) repeating steps (g) to (1) until i exceeds P; and (n) taking P,, as the pilot sub-carrier positions.

Alternatively when the signal to noise ratio, SNR, is higher, e.g., greater than 0dB, preferably greater than 10dB and further preferably greater than 15dB, the position of the P pilot sub-carriers may be set by: (a) initialising a set of positions of the P pilot sub-carriers, , as an empty set; (b) initialising a set V0 as {1,2,...,N}; V=(� U)H (c) initialising matrix NxL (d) initialising matrix Eo=nI; (e) setting i as 1; keD1 (f) calculating for all V'E!Vk 1 + v'E1_1v k where Vk is the k th column of V (g) choosing k as the value of k for which the expression in (f) is maximised; (h) including k as a member of 7 and renaming this set as T; (i) removing k from the set V,1 and renaming this set as L; E. v.vE.

(j) setting E. as E_1 -k 1 1+v E1_1v (k) incrementing i by 1; (1) repeating steps (f) to (k) until i exceeds F; and (m) taking 1 as the pilot sub-carrier positions.

The channel statistics comprise the signal to noise ratio, SNR, of the channel.

According a second aspect of the invention there is provided an orthogonal frequency domain multiplexing (OFDM) transmission system comprising: a transmitter for transmitting an OFDM transmission over N sub-carriers; a receiver for receiving the OFDM transmission; and means for determining positions 1' ç {1, 2,..., N) of P pilot sub-carriers amongst the N sub-carriers, the means for determining being configured to operate according the method of any one of the preceding claims.

The means for determining form may part of the receiver of they may form part of the transmitter.

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

It is possible to consider an exactly optimal method of placing P pilot sub-carriers among N sub-carriers. However, such an approach would incur a complexity in the order of: (N N! pJ (N-P)!P! which is clearly unattractive for practical implementation for large N. The invention given here performs close to such an optimal approach. But, importantly, the complexity of implementing this invention is only linear in N. Thus it is able to handle correlations within the underlying channels and still decide on the pilot symbol placement to optimise channel estimation at the receiver, in a practically feasible manner.

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. An 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 ROM, magnetic tape device or solid state memory device.

The invention will be described by way of example with reference to the accompanying drawings in which: Figure 1 shows an OFDM sub-carrier scheme having equally spaced pilot carrying sub-carriers amongst the data sub-carriers; Figure 2 shows an OFDM sub-carrier scheme having non-equally spaced pilot carrying sub-carriers amongst the data sub-carriers; Figure 3 is a schematic representation of an orthogonal frequency domain multiplexing (OFDM) wireless system according to an embodiment of the invention; Figure 4 is a block diagram of a transmitter of the system of Figure 3; Figure 5 is a block diagram of a receiver of the system of Figure 3; Figure 6 is a flow chart showing the overall method for determining the position of pilot sub-carriers according to an embodiment of the invention; Figure 7 is a flow chart showing an alternative method for determining the position of pilot sub-carriers according to an embodiment of the invention; Figure 8 is a flow chart showing sub-routine 4 of Figure 6; and Figures 9, 10 and 11 are graphs showing numerical simulations of the performance of the method of the invention; Figure 3 is a schematic representation of an orthogonal frequency domain multiplexing (OFDM) wireless system 100 according to an embodiment of the invention. The system 100 comprises an OFDM wireless transmitter 200 and an OFDM wireless receiver 300. Signals transmitted by the transmitter 200 propagate through a transmission channel 400 to the receiver 300. 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 the system 100 described in this application pilot symbol sub-carriers are interspersed among data sub-carriers. The skilled person will understand that in practice, such a pilot symbol insertion is required only when the channel needs to be estimated at the receiver. Once it is estimated, such a pilot symbol insertion can be avoided and the sub-carriers previously used for carrying pilot symbols can be used for carrying data whilst the transmission channel 400 remains static.

We assume that the transmitter 200 knows the statistics of the channel 400 to the receiver 300. Note that this is a much looser requirement than knowing the exact channels, since channel statistics vary slower over time, thus enabling their accurate tracking at the transmitter.

Figure 4 is a block diagram of the transmitter of the OFDM system 100 shown in Figure 3. A frame of data symbols, which contains the data to be conveyed, and pilot placement position indications 1' are received by a pilot symbol intersperse module 204 which outputs a frame of symbols to be transmitted. The pilot symbol intersperse module 204 also receives instructions in relation to the method of pilot symbol placement within the sub-carriers. In the pilot symbol intersperse module 204 a set of P pilot symbols are interspersed with the data symbols to build up a frame of N symbols or sub-carriers to be transmitted.

The N symbols are serial-to-parallel converted in a serial to parallel converter module 205. The inverse fast Fourier transform (IFFT) of the result is taken in a multicarrier modulator 206. The output of the multicarrier modulator 206 is input to a cyclic prefixer and parallel to serial converter module 207 where a cyclic prefix is added and the result is parallel-to-serial converted. The resultant baseband signal is converted to an analog signal in a digital to analog converter (DAC) 208 and thereafter upconverted to the transmission frequency in an RF upconversion and transmission module 209 and then transmitted via the antenna 210.

The position of the pilot sub-carriers is determined in a pilot symbol placement decision module 211 which feeds its decision into the intersperse module 204 as described above. This pilot symbol placement decision module 211 bases its decision on statistics of the channel 400 as described in detail below. One method of obtaining the channel statistics is to estimate them at the receiver and feedback the channel statistics to the decision module 211. Alternatively, one can use channel reciprocity to estimate these statistics directly at the transmitter 200. This second approach can become feasible when the forward and reverse transmissions from one communication device to another occupy the same frequency bandwidth.

Let us denote the pilot symbol placement as decided upon by decision module 211 to be given by the set 7, where the candidate set of 7 is {1,2,..., N}, i.e., the indices of the sub-carriers that can be selected to transmit the pilot symbols. For example if 1 and 65 are elements of 7', then the and 65th sub-carriers are used to transmit pilot symbols. The receiver 300 also needs to know this placement for the differentiation between known pilot symbols and data symbols. Thus, 7 is fed forward to the receiver as shown by the thick line at the bottom of Figure 4.

P is dependent on the channel statistics rather than the individual channel realisations. Thus it is expected that such transfer of feedback and feed forward information (represented by the thick lines in Figure 4) needs to be performed at a much slower rate compared to the rate of information transfer. Also, when the receiver 300 collects the channel statistics, it is possible for it to compute the pilot symbol placement P itself by running the algorithm described below. In such a case, the feed forwarding of 7' can be avoided.

The receiver 300 shown in Figure 5 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 analog to digital converter 302 and converted to a digital signal. The digital signal is fed into a cyclic prefix removal and parallel to serial converter module 303 where the cyclic prefix, 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 pilot symbols.

The output of the parallel to serial converter 305 is passed to a channel estimator 306, where the knowledge of the pilot symbols including their location P is used to estimate the channel 400 between the baseband of the transmitter 200 and the baseband of the receiver 300. The positions P of the pilot symbols could have been fed forward by the transmitter 200 as described above and as illustrated by the thick line at the top of Figure 5. Alternatively, the receiver 300 itself can calculate the pilot symbol placement P by running the algorithm presented below. The estimated channel is thereafter utilised in a symbol detector 307 to detect the data symbols, which were transmitted on the data carrying sub-carriers.

The receiver 300 further comprises a channel statistics estimator 308 for estimating the channel statistics at the receiver 300. The input to the channel statistics estimator 308 may be the channel estimates provided by the channel estimator 306 or the raw baseband signals produced by the cyclic prefix removal and parallel to serial converter module 303. The estimated channel statistics can then be fedback to the transmitter 200, to facilitate the symbol placement decision module 211 to determine the best pilot symbol placement 1'. As stated above, the estimation of channel statistics needs to be performed (or updated) only periodically since the channel statistics are static for a longer duration compared to the rate at which the actual channels themselves change. Hence, the feedback of channel statistics needs to be performed at a much lower rate compared to the rate of information transfer. In the case where the transmitter 200 exploits channel reciprocity to obtain the channel statistics, feedback of the channel statistics from the receiver 300 to the transmitter 200 can be avoided altogether.

The thick lines in Figure 4 and Figure 5 denote transfer of control parameters between the two communicating devices 200, 300. In practice, these control signals themselves need to be multiplexed with the information bearing signals and transmitted and received via the antennas 210, 309 of the transmitter 200 and receiver 300.

The method of determining the position of the pilot symbols is described below.

In the following equations the superscripts T and H respectively denote the matrix transpose and conjugate/Hermitian transpose. L is the length (number of taps) of the time domain channel impulse response; L is chosen at the receiver to be less than the number of sub-carriers, N. The time domain channel impulse response between the transmitter 200 and the receiver 300 is given by: h-_(/,h2,...,hL)T.

The frequency domain channel fading coefficients are represented by the vector: f__(f,,f2,...,fN)T with j being the fading coefficient on sub-carrier i. If we let �NxL be the N x L -(n-)(1--.1) discrete Fourier transform (DFT) matrix with the (n, l)th element being e N, we have f�NXLh Suppose, is the P x L sub-matrix of 0NxL' which is composed of the rows of JNXL corresponding to the set 1' c {l, 2,..., N}. If yp, x, and n are respectively the length P received signal, transmitted signal and the additive noise realisations on the sub-carriers where the pilots were transmitted, we can write: y=X�h+n where X,, diag(x), where the diag(x) operation produces a diagonal matrix with x as the diagonal.

Let the elements of the noise vector be independent, and zero mean circularly symmetric complex Gaussian distributed with variance N0. We also assume that each element of x7, has a constant amplitude, for simplicity. (When the elements of x, are selected from some alphabet containing elements with differing amplitudes, the expressions for the mean-squared error in the channel estimation derived below is still valid in an average sense.) In relation to the probability distribution from which the channels materialise, we model the channel impulse response, h = (h1, ha,..., hL)T to have a complex Gaussian distribution with mean zero (extension to the scenario of having a non-zero mean is trivial) and covariance matrix: E(hh'1)=R where E(.) denotes expectation. In general, R is an arbitrary positive semi-definite matrix. Since R is a positive semi-definite matrix it can be eigendecomposed into the form of: R = Assuming that the rank of R is L ( L), then D is an L x L' diagonal matrix with positive real diagonal elements and U is an L x' matrix with orthonormal columns. With such a description, we can also define an L x L' matrix square root for R, which we denote by R2 as: ! ! R2 =UD2 Assuming the receiver knows the channel statistics, which is essentially the knowledge of the parameters R and N0, we can write down the linear minimum mean squared error channel estimator as: h= ROX (xeR�x + N0I)' where I, is the P x P identity matrix. This equation is based on chapter 12 of "Fundamentals of statistical signal processing: Estimation theory" by S. T. Kay, Prentice-Hall, 1993, the contents of which are incorporated herein by reference.

Furthermore, from this book the error due to such an estimator, can be shown to be distributed as a Gaussian distribution with mean zero and covariance matrix:

I H I H

R =R2 I R2 R2

E L N

As shown above, the channel estimation error at the receiver is determined by the pilot symbol placement 2. Therefore, we can state the core task that is considered in this invention as follows.

We need to select a set P c {i, 2,..., N} with P elements in order to minimise the mean squared error (MSE) in the channel estimation,

I H H

a,, = trace[R [IL +-R2�OR2) R21 where trace(M) is used to denote the sum of the diagonal elements of the square matrixM.

One can decide on the pilot symbol placement P optimally, simply by enumerating all the possible subsets of {i, 2,..., N) of size P and then by selecting the subset which minimises o7,. Although optimal, such an approach requires the evaluation of N = N! number of candidate solutions. Thus such an L) (N-P)!P! approach is clearly not feasible for practical implementation, even for small N. Below we will compare the performance of the method of this invention with the optimal approach for some examples.

The overall procedure for the selection of the pilot symbol placement P is described in the flow chart of Figure 6. The inputs required for the procedure are the channel covariance matrix R, noise variance N0, and the number of pilot symbols to be placed among the sub-carriers, P. The channel covariance matrix R is calculated using known methods, by applying the equation R = E(hh") to samples of the channel impulse response at the receiver 300.

The inputs R, N0 and P are taken in step S402. The channel covariance matrix is eigendecomposed in the form R UDUH in step S403 to derive the L x L' matrix U with orthonormal columns and the L x L' diagonal matrix D with positive real diagonal elements. Here, L' is the rank of the channel covariance matrix R, the maximum value of which is L. In step S404, the algorithm has the freedom to move on directly to step S41 1. Otherwise it will go to step S405, which is rank test 1.

In rank test 1, it is checked if the rank L' of the channel covariance matrix R is unity. If it is, sub-routine 1 is followed in step S406 to produce the pilot symbol placement P. If the channel covariance matrix R does not have rank of one, i.e., if != 1, the algorithm moves to step S407, which is rank test 2.

In rank test 2, it is checked if the rank L' of the channel covariance matrix R is of full rank, i.e., if L' L. If it is, sub-routine 2 is followed in step S408 to produce the pilot symbol placement P. If the channel covariance matrix R does not have full rank, i.e., if L' < L, the algorithm moves to step S409, which is rank test 3.

In rank test 3, it is checked if the rank L' of the channel covariance matrix R is equal to the number of pilot symbols, i.e., if P = L'. If the number of pilot symbols is found to be equal to the rank of the channel covariance matrix R, then sub-routine 3 is followed in step S4 10 to produce the pilot symbol placement P. If the number P of pilot symbols is found not to be equal to the rank L' of the channel covariance matrix R, then sub-routine 4 is followed in step S41 1 to produce the pilot symbol placement P. Thus through one of sub-routines 1 to 4, the algorithm outputs its decision on the best pilot symbol placement at step S412 and then stops at step S413.

Sub-routines 1 to 4 are described below.

Sub-routine 1 Sub-routine 1 is followed when the rank of the covariance matrix R is one (i.e. when L' =1). In such a case, suppose u is a vector such that R = uu''. In this sub-routine firstly the N x 1 vector f is computed as f = ONXLU. Thereafter, the P largest magnitude components of f are selected. The locations of these selected P components are taken to be the locations where the pilots should be placed. That is, P is simply the indices of the P components of I that have the largest magnitudes.

Example 1

When N=1 6, P4, L=1 2, and L'=l, the vector u is calculated and from u the 16 >< 1 vector f is calculated. If the P=4 elements of vector f with the largest magnitude are the 2nd 5th 7th and 12th then P is set as {2,5,7,12}.

Sub-routine 2 This sub-routine is used when the channel covariance matrix R is of full rank, i.e., L = L'. In this case the P pilot sub-carriers are equally spaced among the N sub-carriers of the OFDM transmission so that pilots are spaced as evenly as possible across the sub-carriers, as shown in Figure 1. The relative positions of the sub-carriers need to be considered in a cyclic manner. Thus the first and the last sub-carriers need to be considered as being next to each other. Note that such a placement is the conventional method of pilot symbol placement. However, this invention goes beyond the prior art in being able to calculate instances where such an equally-spaced placement is suitable, depending on the channel statistics. A suitable symbol placement when P is a factor of Nis: for some

Example 2

When N=16, P=4, L=12=L', P is set so that the pilot sub-carriers are equally distributed, i.e., {{1,5,9,13},{2,6,10,14},{3,7,11,15},{48,l2l6}} Sub-routine 3 When the channel covariance matrix R is not of full rank but where the rank L' of the channel covariance matrix R is equal to the number of pilot symbols, F, i.e., P = L', sub-routine 3 followed. In this sub-routine an N x L' matrix W is computed as W = �NXLU. Suppose W,, denotes a sub-matrix of W which is composed of the rows of W indexed by the members of P. The algorithm finds the best pilot symbol placement to be the one which maximises det(W), where det(W) denotes the determinant of matrix W. To reduce the number of calculations required to calculate each value of det(W), a known low complexity algorithm can be used to find sub-matrices of larger matrices, which maximises the determinant. Such a low complexity algorithm is disclosed, for example, in "How to find a good submatrix" S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov and N. L. Zamarashkin, preprint 08- 10, ICM-HKBU (Hong Kong), November 2008, the contents of which are incorporated herein by reference.

Example 3

When N=16, L=12, P=4z=L', W is a 12x4 matrix and W,, is a 4x4 sub-matrix of W composed of the rows of W indexed by 4 members of 7', (7, Pa, P and P4), where the candidate set of 7' is {l,2,. ..,16}. The determinant of sub-matrices W,, is calculated by choosing 4 rows of W at a time and the set of 4 rows with maximises the determinant of W, is chosen. For example, if the 7th 9th 12th and 16th rows of sub-matrix W maximise the determinant of the possible W,, sub-matrices then P is set as {7,9,12,16}.

Sub-routine 4 A fourth sub-routine is reached either when the rank tests of steps S405, S407 and S409 have failed, i.e., the channel covariance matrix R is neither of full rank nor has rank equal to the number P of pilot sub-carriers nor unity rank, or because it has been decided not to consider one or more of these rank tests. Sub-routine 4 works for any channel covariance matrix R and any number of pilot symbols P to be placed, hence the freedom exists to forgo one or more of the tests S405, S407 and S409 and revert to this general algorithm. Hence, subroutine 4 is a general pilot symbol placement method and a method of determining the position of pilot symbols without applying rank tests is shown in the flowchart of Figure 7.

Figure 8 is a flowchart showing the steps of sub-routine 4. In the initialisation stage S502 of sub-routine 4 the set, which is the set of positions of the pilot sub-carriers chosen from the set {1,2,...,N}, is initialised to be the empty set; and the set V0 is initialised to be the set {l,2,...,N}, i.e., the set of available positions amongst the OFDM sub-carriers. To improve the numerical stability of the algorithm the noise variance N0 is checked to ensure that it is greater than some positive value NIh. If it is not greater than NIh then the noise variance N0 is set as NIh for the subsequent steps of the algorithm. A suitable value for N1 h can be 0.1. I.e., N0 is initialised as N0 = max(NO,N,h).

An L'xN matrix: -\Jii(ONLU) and an Lx L' matrix E0 = I) are also computed during the initialisation stage S502.

Thereafter the algorithm goes into a P stage recursion. In step S503 the quantity VEIV, is calculated for all k selected from the set V,., where Vk is the kth column of V. In step S504 the value of k which maximises Q is selected and this value of k is denoted as k.

In step S505 the value k is removed from D, to build the new set D1 and the new set 7 is set as 1 incorporating the value k. Thereafter a new matrix E, is computed using the chosen value k, as: E. v.vE.

E -E -I-I k k -i - 1 + v7E1_1v In step S506 i is incremented and in step S507 it is determined if i is greater than P. If i is not greater than P then steps S503 to S506 are repeated. During each recursion the chosen value of k is removed from the set V1 after each step and the value of E* is recalculated after k has been identified for that stage of the recursion.

After completing the P stages of the recursion, the set is output as the best pilot symbol placement scheme P amongst the N sub-carriers in step S508.

Example 4

When N=1 6, L1 2, P4 and L'=5; since L' != L and L' P and L' != 1 then sub-routine 4 is followed.

In the initialisation stage: is initialised to be the empty set {O,O,O,O}; D0 is initialised to be the set {1,2,...,16}; V is initialised as a 5 >< 16 matrix as described above; and E0 is initialised as a 5 x 5 matrix as described above.

In the first stage of the recursion, the quantity: V'EVk 1+v'EOvk is calculated for each of the sixteen vectors of Vk, where Vk is a 5 xl vector k th column of V. If the fifth column of V gives the highest value for Q, then Q is maximised when k=5 and the value 5 becomes the first member of P. The value 5 is removed from V0 and the set V is built as V1 = {l,...,4,6,...,16}. The new set = {5} is built. A new matrix E1 is computed using the chosen value, ==5 as: E =E -E0v5vE0 1 l+vE0v5 In the second stage of the recursion Q is calculated for the fifteen vectors v, where k is chosen from the set V = {l,...,4,6,...,16}. If k=9 give the highest value of Q the new sets = {5,9} and V2 = {l,...,4,6,...,8,l0,...,16} are built and E2 is computed as described above.

This process is repeated a further two times with the chosen value of k being removed from the set V1 after each step and the value of E being recalculated after k has been identified for that stage of the recursion. After completing the four stages of the recursion, the set 7 is output as the best pilot symbol placement scheme P amongst the 16 sub-carriers.

For some scenarios, when the signal to noise ratio (SNR) becomes large, e.g., greater than 0dB, preferably greater than 10dB and further preferably greater than 15dB, it can be seen that the quantities dealt with in the generic method described immediately above can become small and hence can lead to numerical issues. This is especially the case with finite precision implementations, as would be required in practical hardware.

One simple solution to this problem was already given in the initialisation stage of the generic method by not allowing the initialised noise variance N0 to go below some suitable value NIh. In a further embodiment of the invention an alternative initialisation algorithm is described below.

As before the set is initialised to be the empty set; and the set V0 is initialised to be the set {1,2,...,N}. The L'x N matrix V is initialised as V = (�NXLU)" , which is independent of the noise variance N0; and the L'xL' matrix E0 is initialised as I, the identity matrix.

Thereafter the algorithm follows the same P stage recursion as described above.

Namely, at the i th stage of this recursion, an element k is selected from the set V in order to maximise the quantity: = 1+vEI_lvk where, Vk is the kth colunin of V. Let the selected element of k which maximises Q be denoted as k. The value k is removed from V,1 to build the new set V1 and the new set 1 is set as 7-incorporating the value k. Thereafter a new matrix E, is computed using the chosen value k as: E. v-vE.

E -E -" k k 1-1 - 1 + v7E11v This process is repeated for all F, with the chosen value of k being removed from the set V1 after each step and the value of E being recalculated after k has been identified for that stage of the recursion. After completing the P stages of the recursion, the set Pd, is output as the best pilot symbol placement scheme P amongst the N sub-carriers.

It has been demonstrated that this alternative initialisation procedure produces near-optimal pilot symbol placements at high SNR. This alternative embodiment does not depend on the noise variance N0 (i.e. the operating SNR), and hence does not possess SNR related numerical stability issues. Furthermore, the actual channel statistics utilised by this new algorithm is just the L x L' matrix U. Thus the overhead needed to communicate channel statistics for the purpose of generating the pilot symbol placement is also reduced by following this procedure. (The previous method requires the knowledge of L x L matrix R, where L �= L', in addition to the operating SNR.) Non-uniform pilot symbol placements produce improved performance over equally-spaced pilot symbol placement when correlations exist in the underlying channel impulse responses. In practice, the channels perceived by the baseband at the receiver can have correlated channel taps due to several possible causes. For example, the underlying wireless channel might induce a correlated scattering process. Also, another possible mechanism for inducing channel tap correlations is the presence of synchronization errors at the receiver. When there are synchronization errors, it is obvious to a skilled person that there will be inter symbol interference induced at the receiver. Such interference caused by synchronisation errors would induce channel taps that are correlated." The performance of the methods describe above will now be illustrated by means of numerical simulations. We consider an N -32 sub-carrier system and evaluate various combinations of the parameters L, L' and P. For each given combination of L and L', we numerically simulate channel covariance matrices as follows.

Firstly the L x L matrix B is obtained by drawing each of its elements with a complex Gaussian distribution with mean zero and a variance of 0.5 in each dimension.

The matrix C is then computed as C = BBH. Suppose the eigendecomposition of matrix C is given by C = TSTH. Here T is a unitary matrix. Due to the method of drawing the random matrix B, matrix C is almost surely of full rank. Thus the matrix S has positive real values in all of its L diagonal positions.

Next, we set the smallest (L -L') of these diagonal elements to zero to build the new matrix SL.. Finally the channel covariance matrix is taken to be R = /cTSLT", where. is a signal normalisation factor which ensures that trace(R) = 1, for each random realisation. By this procedure we are able to generate random L x L matrices as the trial covariance matrices such that their rank was L'. Note that this procedure for channel covariance matrix generation does not form an embodiment of the invention, but is used to evaluate average performance of the invention.

The channel estimation performance of a pilot symbol placement 7' is evaluated using the mean squared error (MSE), u, in the channel estimation computed as described above: 1 H H = trace 1R2 [IL +j-R2O�pR2J R2) In, Figures 9, 10 and lithe averaged MSF for the drawn channel covariance matrices are plotted for various combinations of the parameters L, L' and P. In Figure 9, since L' L, sub-routine 2 is selected by the algorithm to place the pilot symbols with an equal spacing. It can be seen that the average MSE performance of such a scheme is near optimal. We note that a slightly better MSE could have been achieved bad sub-routine 4 been executed to find the pilot symbol placement P. However, the possible improvement is small in this case since the equally-spaced placement performs well.

In Figure 10, since L' < L and L' F, sub-routine 3 is selected by the algorithm. In this case, it is seen that one can achieve an averaged MSE much better than that produced by the conventional equally-spaced placement. The near-optimal performance of this method is clearly shown.

In Figure 11, the generic sub-routine 4 is executed by the algorithm. Once again, it can be seen that the proposed invention performs near optimally, which is not the case with an equally-spaced placement. As stated above sub-routine 4 can be performed in all possible cases to obtain near-optimal performance in the channel estimation.

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

Claims (16)

1. CLAIMS: 1. A method of determining positions P.c {1, 2,..., N} of P pilot sub-carriers in an orthogonal frequency division multiplexing, OFDM, transmission system, the system comprising a transmitter for transmitting N sub-carriers, a channel through which the N sub-carriers propagate and a receiver for receiving the N sub-carriers, the method comprising: receiving the N sub-carriers at the receiver; processing the N sub-carriers to determine channel statistics of the channel; and setting the position P of the P pilot sub-carriers amongst the N sub-carriers based on the channel statistics of the channel.
2. 2. A method as claimed in claim 1, wherein the channel statistics comprise a channel covariance matrix, R, of the channel.
3. 3. A method as claimed in claim 2, wherein the channel covariance matrix, R, is given by: E(hh')=R where E(.) is the expectation; h = (hl,h2,...,liL)T is the channel impulse response; L is the number of taps in the channel impulse response; T is the transpose; and H is the Hermitian transpose.
4. 4. A method as claimed in claim 2 or 3, wherein the step of setting the position of the P pilot sub-carriers is based on a rank, L', of the channel covariance matrix, R.
5. 5. A method as claimed in claim 4, wherein the rank, L', of the channel covariance matrix, R, is calculated by eigendecomposing the channel covariance matrix, R, into the form: R=UDU'1 where U is an L x' matrix having orthonorrnal columns; and D is an L x L' diagonal matrix with positive real diagonal elements..
6. 6. A method as claimed in claim 4 or 5, wherein when the rank, L of the channel covariance matrix, R, is unity, the position of the P pilot sub-carriers is set by: (a) computing vector u, where R = (b) computing vector f, where f = ON)(LU where 0NxL is the N x L discrete -.(n-)(1-I) Fourier transform matrix with the (n,l)th element being e N (c) identifying the P largest magnitude components off; and (d) setting P as the indices of the P largest magnitude components off.
7. 7. A method as claimed in claim 4 or 5, wherein when the channel covariance matrix, R, is of full rank, i.e., when L = L, the position of the P pilot sub-carriers is set by equally spacing the P sub-carriers amongst the N sub-carriers.
8. 8. A method as claimed in claim 4 or 5, wherein when the rank of the channel covariance matrix, R, is equal to P, i.e., when L' = F, , the position of the P pilot sub-carriers is set by: (a) computing the N x L' matrix W, where W = ONXLU; and °NxL. is the N x L discrete Fourier transform matrix with -(n-)(1-I) the (n,1)th element being e such that f = ONXLh, where f = (fl,f2,...,fN)T is the frequency domain fading coefficients with j being the fading coefficient on sub-carrier i; (b) evaluating det(W) for all W, where det(') is the determinant; W, is the matrix formed from the set of rows of W indexed by 1'; and P c {i, 2,..., N} is the candidate set of pilot sub-carrier placements; and (c) selecting the position of the P pilot sub-carriers for the case where det(W) is maximised
9. 9. A method as claimed in any one of claims 2 to 8, wherein the position of the P pilot sub-carriers is set by: (a) initialising a set of positions of the P pilot sub-carriers, 1, as an empty set; (b) initialising a set V0 as (c) initialising the noise variance, N0, as N0 = max(N0, NIh), where NIh is a threshold value of the noise variance; (d) initialising maix V = (0NXLU); (e) initialising matrix E0D; (f) setting ias 1; kEV1 (g) calculating for all vE1 Vk 1+ V'EI_IVk where Vk is the k th column of V; (h) choosing k as the value of k for which the expression in (g) is maximised; (i) including k as a member of J and renaming this set as (j) removing Ic from the set V,1 and renaming this set as 2),; E. vvE.(k) setting E, as - li 1+v, E11v (I) incrementing i by 1; (m) repeating steps (g) to (1) until i exceeds P; and (n) taking 7, as the pilot sub-carrier positions.
10. 10. A method as claimed in any one of claims 2 to 8, wherein the position of the P pilot sub-carriers is set by: (a) initialising a set of positions of the P pilot sub-carriers, 20, as an empty set; (b) initialising a set V0 as {l,2,...,N}; V=(� U)H (c) initialising matrix NXL (d) initialising matrix E01; (e) setting i as 1; kED1 (f) calculating for all V'ElVk 1 + v'E_1v k where Vk is the k th column of V (g) choosing k as the value of k for which the expression in (f) is maximised; (h) including k as a member of and renaming this set as (i) removing k from the set V. and renaming this set as E. v-vE.-k k I-I (j) setting E. as E1_1 -H 1+v E1v (k) incrementing i by 1; (1) repeating steps (f) to (k) until i exceeds P; and (m) taking P,. as the pilot sub-carrier positions.
11. 11. A method as claimed in claim 10, wherein the method is performed when the signal to noise ratio, SNR, is greater than 0dB, preferably greater than 10dB and further preferably greater than 15dB.
12. 12. A method as claimed in any one of the preceding claims wherein the channel statistics comprise the signal to noise ratio, SNR, of the channel.
13. 13. An orthogonal frequency domain multiplexing, OFDM, transmission system comprising: a transmitter for transmitting an OFDM transmission over N sub-carriers; a receiver for receiving the OFDM transmission; and means for determining positions P ç {i, 2,..., N} of P pilot sub-carriers amongst the N sub-carriers, the means for determining being configured to operate according the method of any one of the preceding claims.
14. 14. A system as claimed in claim 13, wherein the means for determining form part of the receiver.
15. 15. A system as claimed in claim 13, wherein the means for determining form part of the transmitter.
16. 16. A carrier medium carrying computer readable code for controlling a microprocessor to carry out the method of any one of claims ito 12.