WO2007035993A1

WO2007035993A1 - Channel tracking for mimo receivers

Info

Publication number: WO2007035993A1
Application number: PCT/AU2006/001422
Authority: WO
Inventors: Michael Faulkner; Leon Gor
Original assignee: Prescient Networks Pty Ltd
Priority date: 2005-09-29
Filing date: 2006-09-29
Publication date: 2007-04-05

Abstract

A method of processing a signal (yw) encoded on a channel (Hw) in a wireless communications network is disclosed. The method includes, determining (204) an initial channel estimate, represented by an initial channel estimation matrix (H), on the basis of one or more pilot symbols received on the channel. Next an initial channel estimation matrix (H) is decomposed (206) into a product of at least two canonical form matrices. Payload data symbol are recovered (210) from a corresponding received data symbol on the basis of at least one or more of the canonical form matrices. Then the recovered payload data symbol are encoded (212) to generate (216) a feedback data symbol. An error value is calculated on the basis of a feedback data symbol and its corresponding received data symbol and is compared (220) to a threshold. In the event that the error value is greater than the threshold (218), a further matrix decomposition is performed to generate at least one updated canonical form matrix. These steps can be repeated using an updated canonical form matrix.

Description

CHANNEL TRACKING FOR MIMO RECEIVERS Field of the invention

The present invention relates to improvements in methods for performing channel tracking in a wireless communications network. In the preferred embodiment the invention is described in connection with a system having non-frequency selective channels such as a system employing orthogonal frequency division multiplexing, however it should be noted that the present invention is not limited to that exemplary embodiment.

Background of the invention Transmissions made over a wireless communications channel in a telecommunications network are subject to corruption by the channel over which they are transmitted. Estimation of the effect of the channel is complicated by movement of the user's receiver within the network or by changes in its surrounding environment, both of which cause the effect of the transmission channel on the transmitted data to change over time.

In order to be able to remove the effect of the transmission channel on the transmitted data a training sequence can be inserted into a data packet which can be used by the user's handset to accurately estimate the effect of the channel. However, in circumstances where the effect of the channel is changing rapidly, such as when a car goes past a stationary user or the direction of travel of a mobile user changes, the channel estimate derived from the so-called training sequence becomes out of date before the next training sequence is received.

One way of overcoming this problem is to reduce the size of the data packet or to increase the number of training sequences sent within a data packet. Both of these approaches, however decreases the amount of message data transmitted in each packet and accordingly decreases data throughput in the network.

These problems are further exacerbated in Multiple Input Multiple Output (MIMO) antenna systems as many channels need to be estimated simultaneously and therefore the training sequences added to the received data packets need to be longer than in single antenna systems. One possible solution for estimating or determining the effect of the transmission channel (herein denoted by H) between training sequences is to employ channel tracking to generate a continuously updated channel estimate which can then be used by an equaliser to extract data symbols from the received signal. Channel tracking algorithms typically need a reference signal representative of the channel to which the algorithm is to converge toward. The reference signal can be generated by feeding back previously decoded data which is assumed to be nearly error free, a method called decision directed or blind adaptation. Another way is to use known reference pilot signals transmitted so as to be interspersed amongst the wanted data. This is called pilot assisted channel tracking. To implement these methods an adaptive algorithm is required to calculate a new channel estimate from an old channel estimate and the reference signal. There are many types of algorithm which are suitable for this purpose, e.g. LIvIS, RLS, Kalman etc. The LMS algorithm is considered a low complexity and robust algorithm. There are many alternative techniques for implementing equalization of the channel response in a radio telecommunications receiver, including zero forcing, Minimum Mean-Square Error (MMSE) and sub-optimum equalizer algorithms. V-Blast is a pioneering equalization technique aimed to exploit high capacity gains promised by MIMO systems. However, a limitation of this technique is the high computational complexity (about 90% of the total processing power of the receiver) that is required to implement it. In the above cases the equaliser structures will typically involve a complex matrix inversion to determine H, the Channel Estimate. This matrix inversion can be simplified by decomposing the matrix into more easily invertible sub-matrices. Decompositions such as QR, SV (singular value) and LU (lower upper) are three of many such decompositions.

One preferred method to avoid high complexity in the V-Blast algorithm is to restate the algorithm in terms of the QR decomposition (QRD) resulting in a QR detection receiver. QRD can be performed in hardware using a number of CORDIC iterations. However, QR decomposition is still a computationally demanding operation and has to be used for every new channel estimate. When performing channel tracking in a situation in which a new channel estimate is obtained for every symbol, the recurring use of the QR decomposition results in an increased number of operations per symbol, and therefore results in high power consumption in the signal processing circuits.

Accordingly, it would be advantageous to have a method of channel tracking for a non- frequency selective channel, such as that found in narrow band systems, Orthogonal Frequency Division Multiplex (OFDM) systems, and wideband systems employing frequency domain equalisers, that may reduce the number of executed instructions or the power consumed in processing a received signal.

Summary of the invention The present inventors have determined that in an equalisation method using QR decomposition the frequency of QR decomposition can be decreased whilst continuing to track the channel H or another parameter representative of H. In this regard, the Q and R matrices can be held for a number of symbol periods between adjacent QR decompositions which results in a reduction in the amount of processing performed. In certain embodiments rather than tracking the channel H only the upper triangular matrix need be tracked while holding the unitary matrix Q fixed. This embodiment has been shown to allow a further decrease in the frequency of QRD compositions.

In further aspects, the inventors have also discovered an alternative algorithm for implementing the Givens rotation technique for performing matrix decomposition which in some embodiments may be implemented more efficiently than the prior art CORDIC algorithm.

In a first aspect of the present invention there is provided a method of processing a signal encoded on a channel in a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality of payload data symbols, the method including:

(a) determining an initial channel estimate, represented by an initial channel estimation matrix, on the basis of one or more pilot symbols received on the channel;

(b) decomposing the initial channel estimation matrix into a product of at least two canonical form matrices; (c) recovering a payload data symbol from a corresponding received data symbol on the basis of at least one or more of the canonical form matrices;

(d) encoding the recovered payload data symbol to generate a feedback data symbol;

(e) determining an error value calculated on the basis of a feedback data symbol and its corresponding received data symbol;

(f) comparing the determined error value to a threshold, and in the event that the error value is greater than the threshold;

(g) performing a further matrix decomposition to generate at least one updated canonical form matrix; and (h) repeating steps (c) to (f) using the at least one updated canonical form matrix.

In the event that, in step (f), the error value is not greater than the predetermined threshold the method can include, repeating steps (c) to (f) without performing step (g).

The method can also include determining an updated channel estimate using a tracking algorithm. Preferably the feedback data symbol and corresponding payload data symbol are used as an input to the tracking algorithm, and wherein the tracking algorithm is configured to output an updated channel estimate on the basis of said inputs.

The tracking algorithm is configured can use LMS, RLS, or Kalman filtering to determine the updated channel estimate: The tracking algorithm used is preferably configured to minimise the value of the following expression to determine the new channel estimate:

y -H*s , wherein y is the received data symbol, s is the feedback data symbol and H

is the channel estimate.

The error value in step (e) can be determined using any one or more of the following:

determining a Mean Squared Error between S and S

determining a Mean Squared Error between H* S and Y determining a difference in path metrics in the decoding process

determining a difference between a current channel estimate and the most recently updated channel estimate

determining a set sample time delay, on the basis of a measured or assumed Doppler frequency and a modulation scheme employed on the channel or a target signal to noise ratio.

In some implementations the at least one updated canonical form matrix may not strictly conform to its corresponding canonical form.

The method further includes determining at least one updated canonical form matrix using a tracking algorithm.

Preferably the feedback data symbol and corresponding payload data symbol are used as an input to the tracking algorithm, and the tracking algorithm is configured to output an approximated canonical form matrix on the basis of said inputs.

In the event that an updated canonical form matrix does not conform to its canonical form the method can further include forcing the updated canonical form matrix into a corresponding canonical form to enable channel equalisation using said matrix. The further matrix decomposition performed in step (g) can be performed on an updated canonical form matrix that does not comply with its corresponding canonical form.

In some embodiments of the method the canonical form matrices generated in step (b) include either an upper triangular or lower triangular matrix. In this case the upper triangular or lower triangular matrix can subsequently be used for channel equalisation and the inverse to said upper triangular or lower triangular matrix is determined by back- substitution.

The canonical form matrices generated in step (b) can include an orthogonal unitary matrix and/or a diagonal matrix.

In embodiments of the method the decompositions performed can include QR decompositions, SV decompositions. In an embodiment using QR decompositions the Q and R matrices can be evaluated by an LU transform.

The method is preferably implemented for a plurality of received channels.

The method can be implemented in a receiver having more than one antenna for receiving said channels.

The communications network can be a multi-subcarrier system or a network having a single carrier frequency.

In one embodiment the method can be implemented in a receiver employing a frequency domain equaliser structure in which equalisation is applied on a bin by bin basis.

In a second aspect the present invention provides a method of tracking a channel in a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality of payload data symbols, the channel tracking method including: (a) determining an initial channel estimate, represented by a initial channel estimation matrix, on the basis of one or more pilot symbols received on the channel; and

(b) decomposing the initial channel estimation matrix into a product of at least two canonical form matrices, wherein for each received data symbol the method includes: (c) recovering a payload data symbol from a corresponding received data symbol on the basis of at least one or more of the canonical form matrices;

(e) generating a new channel estimate and an error value on the basis of the feedback data symbol and its corresponding received data symbol; and (f) comparing the determined error value to a threshold, wherein in the event that the error value is greater than the threshold; the method further includes:

Preferably, in the event that, in step (f), the error value is not greater than the predetermined threshold the method includes repeating steps (c) to (f) without performing step (g). In a third aspect of the present invention there is provided a method of obtaining a data symbol encoded on a channel in a wireless communications network which includes processing said data symbol using a method according to the first aspect of the invention.

In a fourth aspect of the present invention there is provided a method of obtaining a data symbol encoded on a channel in a wireless communications network, including tracking the channel using a method according to the second aspect of the invention.

In a fifth aspect of the present invention there is provided a method of performing a decomposition of a matrix M in a data processing device, said method including:

(a) defining, within the matrix M, an arbitrary vector v = with tan(α) = — , where a y x represents an angle between the vector and the positive x-axis, and wherein tan(α) can

M * 2^ey M be represented in the floating point binary format tan(or) = — = — - * 2^(e-⁾"^"as0 , where

^• M_x *!** M_x

My and M_x are mantissas and e_y and e_x are the exponents of y and x respectively;

(b) moving the input vector v into a first octant by inverting signs and flipping x and y magnitudes of one or more of the elements of matrix M; (c) determining an exponent difference, a, equal to (e_y-e_x);

(d) rotating the input vector v by 2^a to perform a micro-rotation of the form

to generate a new vector v_i+1; and

(e) in the event that y_j+i does not meet a predetermined accuracy threshold the method further includes repeating steps (c) and (d), and wherein in the event that y_i+1 meets the predetermined accuracy threshold, the method includes; (f) applying a scaling factor; and

(g) readjusting signs of x_{i+1 and} y_i+1.

In one embodiment the scaling factor can be determined using a look up table.

Step (c) preferably includes selecting a from a range of possible values. Most preferably step (c) includes selecting a as the central value in a range of possible values for a.

Preferably the method is used to perform either QR or SVD decomposition.

In a further aspect of the present invention there is provided a method of processing a signal received on a channel in a wireless communications network according to an embodiment of the first aspect of the present invention and in which at least one matrix decomposition is performed using a method of according to an embodiment of the fifth aspect of the present invention.

In yet another aspect of the present invention there is provided a method of tracking a channel in a wireless communications network according to an embodiment of the second aspect of the present invention and in which at least one matrix decomposition is performed using a method according to an embodiment of the fifth aspect of the present invention.

In yet another aspect of the present invention there is provided a signal processing device adapted to process a signal received on a channel of a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality of payload data symbols, the device including: a channel estimation stage configured to determine at least an initial channel estimation; a matrix decomposition stage configured to decompose a matrix into a product of at least two canonical form matrices; a symbol decoding stage configured to process a received data symbol to obtain a payload data symbol on the basis of at least one or more of the canonical form matrices generated by the matrix decomposition stage; a feedback coding stage configured to process a recovered payload data symbol to generate a feedback data symbol; error determination means for determining an error value calculated on the basis of a feedback data symbol and its corresponding received data symbol; and threshold detection means to compare the determined error value to a threshold and in the event that the error value is greater than the threshold, trigger the matrix decomposition stage to perform a matrix decomposition to generate at least one updated canonical form matrix for use by the symbol decoding stage to decode at least one subsequent symbol. The signal processing device can further include a matrix tracking stage configured to determine, at least intermittently, an updated channel estimation using a tracking algorithm.

The matrix tracking stage preferably determines an updated channel estimation for each received data symbol. The matrix tracking stage can use LMS, RLS, or Kalman filtering to determine the updated channel estimate.

The signal processing device can alternatively include a matrix tracking stage configured to track least one updated canonical form matrix using a tracking algorithm. In this case the device can additionally include a matrix forcing stage configured, in the event that an updated canonical form matrix does not conform to its canonical form, to force an updated canonical form matrix into its corresponding canonical form to enable channel equalisation to be performed using said matrix.

The symbol decoding stage can preferably include an equalisation stage.

Preferably the matrix decomposition stage performs a QR decomposition. The matrix decomposition stage may perform a SV decomposition.

Preferably the matrix decomposition stage of the signal processing device is configured to perform a matrix decomposition method according to an embodiment of the fifth aspect of the invention.

In another aspect of the present invention there is provided a receiver for use in a wireless communications network which includes a signal processing device of the type disclosed above. The receiver may be configured to receive signals on a plurality of channels. The receiver can include a plurality of receiving antennas.

The receiver can be configured to operate in either a communications network using multiple sub-carriers or a single carrier frequency. The receiver can include a frequency domain equaliser structure in which equalisation is applied on a bin by bin basis.

Preferably the matrix decomposition stage of the receiver is configured to perform a matrix decomposition method according to an embodiment of the fifth aspect of the invention. .

For clarity, in the present specification the following notation conventions will be followed:

All variables marked with ' ^Λ ' are estimated parameters.

R denotes the strictly upper triangular matrix while AR and represents a matrix that is approximately upper triangular in form, but which has lost, or loses, its upper triangular structure in processing. Brief description of the drawings

Preferred embodiments of the present invention will now be described by way of non- limiting example only with reference to the accompanying drawings, in which

Figure 1 shows a simplified 2 x 2 base band model of a MIMO system;

Figure 2A depicts a MIMO receiving stage using QR detection and channel tracking according to the first embodiment of the present invention;

Figure 2B is a flowchart illustrating a method channel tracking implemented by the receiving stage illustrated in figure 2;

Figure 3 shows a decision directed tracking and QR decomposition stage used in an embodiment of the present invention; Figure 4 depicts a second embodiment of the decision directed tracking and QR decomposition stage in accordance with an alternative embodiment of the present invention;

Figure 5 depicts schematically the least mean square configuration employed in the decision directed tracking and QR decomposition stage of Figure 4; Figure 6 is a graph depicting tracking performance of three alternative tracking schemes;

Figure 7 depicts a threshold detection component used in of an embodiment of the present invention; Figure 8 is a graph showing tracking performance of several embodiments of the present invention;

Figure 9 depicts a procedure for CORDIC-based QR decomposition of a 2X2 complex matrix to obtain upper triangular matrix R and unitary matrix Q;

Figure 10 is a schematic representation of the known CORDIC algorithm; Figure 11 illustrates a method performing matrix decomposition using Given's rotations according to an embodiment of the present invention; and

Figure 12 depicts a flow chart illustrating a method of finding the difference between exponents of two fixed point numbers that can be used in an embodiment of the present invention.

Detailed description of the embodiments

For convenience only, examples of the present invention will now be described in the context of a system that employs non frequency selective channels or sub-channels. These conditions can be found, for example, in narrowband systems, OFDM systems or wideband systems employing frequency domain equalising techniques. However the other embodiments of the present invention may be applicable to different network types.

In a first embodiment the frequency of matrix decompositions is decreased while continuing to track the channel H. During equalisation the decomposed matrices (e.g. Q and R if using QR decomposition) are held fixed between adjacent decompositions for a number of symbol periods. Advantageously this results in a complexity reduction, due to the reduced number of decompositions that need to be performed, at the expense of a growing equalization error caused by the use of the outdated decomposed matrices during equalisation. As long as the error does not exceed a suitable threshold then performance degradation can be managed. Figure 1 depicts a simplified 2X2 base-band model of an exemplary MIMO-OFDM structure 100. In figure 1 , serial input data d is channel coded, interleaved and then converted to parallel streams s1 and s2 in block 102. An inverse fast Fourier transform (IFFT) is applied in blocks 104 and 106. The signals are then transmitted over a wideband channel H_w across multiple paths 108 to a pair of antennas 110 and 112. In the system 100, the IFFTs, wideband channel H_w corrupt the encoded data, and white Gaussian noise is added at the receivers 110 and 112.

Figure 2A depicts a receiver 200 operating in a system such as that depicted in figure 1. Figure 2B is a flowchart illustrating a method 250 of channel tracking implemented by the receiver of Figure 2A. At the receiver 200 the received wideband signal is represented by y_w, which can be represented as y_w = H_w s + /?. First, in FFT blocks 202 an N-point fast Fourier transform (FFT) is performed on each received wideband signal y_wi and y_W2 to transform the received wideband channel, H_w, into N groups of 2X2

narrowband flat fading channels, H. y - is the output signal of one such group. y2

In a system using packet transmission, the data transmission packet will typically consist of a group of known pilot symbols, known as a training sequence, and a data payload of encoded symbols. In the present embodiment, in step 252 of the method 250, each training sequence is used to perform an initial channel estimation in block 204 and for setting the initial conditions of the tracking adaptive filter. In step 254, the initial channel estimation H, which has been made in block 204,

Λ Λ undergoes QR decomposition by block 206 and the resultant Q and R matrices are then used by the equalisation stage 208 to equalize the channel for the received data symbols y, in step 256.

For subsequent received symbols, a decision directed algorithm tracks the channel H using y as its reference signal. This step is performed in the "DD track and QRD" block 206. Many channel tracking algorithms can be used for MIMO-OFDM. The least mean squares (LMS) algorithm is used in this embodiment and has been found by the inventors to perform well in a slow changing environment. Moreover it has low complexity and is robust. As will be appreciated by those skilled in the art, QR equalization is based on the identity

Q^H*y = R*S (1 )

When using QR decomposition to equalise data symbols y in the equalisation stage

Λ Λ 208, since R is an upper triangular matrix symbol recovery of the symbol estimate s is possible via back substitution using equation (1). Finally, de-interleaving, decoding and decision are performed by block 210 to arrive the original data d.

In step 258, the mapping coding block 212 re-codes the data d. As will be appreciated by those skilled in the art additional IFFT and FFT blocks, 214 and 216 respectively, are necessary in implementations using frequency domain equalisers.

The recoded data s is provided as a feedback data symbols to the channel tracking algorithm of block 206.

An expansion of the DD track and QRD block 206 of Figure 2A is shown in figure 3. The DD track and QRD block 206 has three main sub blocks, a LMS channel tracking block 300, a QR decomposition block 302, and a memory block 304. The LMS channel tracking block 300 implements a LMS tracking algorithm which uses a decision directed structure to track the channel matrix H. The recoded and remapped signal, s forms the input to the LMS algorithm and the received signal y (appropriately delayed) is used as

reference. The LMS algorithm adjusts the coefficients of the channel estimate H to

minimize the error signal y -H*s

The matrix H^ represents the channel estimate matrix at the i^th time instance. The QR

decomposition of H^ can be expressed,

H® = Q(!)**(f) (2)

from which the term Q^H y can be obtained for use in the equalisation stage 208. In preferred embodiments, the QR decomposition of Hu\ is not performed every time instance, i.e. for every received symbol, to reduce computational complexity. So the old

Q and R are used for equalisation which results in the matrices Q and R becoming

outdated. Outdated Q and R values will cause an error in the equalisation performed

by stage 208 that increases as Q and R get older.

The error matrix, er#(φ that represents the error introduced into the system by the

aging Q and R estimates can be written as follows

Λ Λ er_H{i) = H^ -Q_{i__nγR{i-_n) (3)

Here H(^ is the actual channel matrix at the time instance i, while β₍.__B) and R(i-_tt) are unitary and upper triangular matrices obtained n time instances earlier from the LMS

estimate #₍,-_„₎ ■ The most recent value of #₍,-_„₎ that is determined by QR decomposition is stored in the memory block 304. When n = 0 (QR decomposition is performed every time instance) er#(_z) represents the tracking error of the LMS algorithm, which is calculated in step 260. When n=i the QR decomposition is performed only once, directly after the training sequence, and so er_Hφ represents the error due to Doppler variations in the channel. The average error power per element of er_Hu\ is the ensemble average of the squared magnitude of the frobenious norm

Where L denotes number of ensemble repetitions and M_TMR is the number of paths (elements) in H. MR is the number of receive antennas and Mγ is the number or transmit antennas. In step 262, if the detected error exceeds a predetermined threshold, the threshold detector block 220 enables the trigger line 218 to cause the QR decomposition block to perform a further decomposition in step 264. In the present example the current channel

estimate H^ is decomposed in step 264 by the QR decomposition block 302, and

Q₍i-_n) and i?_(;__Λ) are updated (n=0 at the moment of updating) for use in equalising subsequent symbols, until either the error exceeds the predetermined threshold or a new training sequence is received. In a practical situation the actual channel matrix

Λ

H(Λ of equation 3, is not available and so the current tracked channel estimate Hu\ is

Λ used instead. The trigger line 218 also updates the value of #(_/_„) stored in the

Λ memory block 304, so that it stores the most up-to-date channel matrix H^ obtained at the time of the QR decomposition. As will be discussed below, this value is used as the input into a threshold detector 220 to determine the current error level.

A second embodiment of the present invention will now be described with reference to figures 4 and 5 of the accompanying drawings. In this embodiment rather than tracking the channel estimate the upper triangular matrix R resulting from QR decomposition of the channel estimate H is tracked while holding the unitary matrix Q fixed.

The inventors have discovered that by tracking the upper triangular matrix a slower equalization error growth occurs than when tracking the H matrix, thus a lower rate of preforming decompositions can be tolerated for the same error level. The second illustrative embodiment uses the same overall processing structure as the first embodiment, with the exception of the DD track and QRD block 206. The DD track and QRD block of this embodiment is depicted in Figure 4. In this embodiment the Decision Directed tracking and QR decomposition block 400 has the same inputs as the previous embodiment, namely the input data signal y, feedback signal s representing a

Λ re-encoded output data symbol d, and the initial channel estimation matrix HV₀) obtained from received pilot symbols. The DD track and QRD block 400 includes an LMS channel tracking block 402 and a QR decomposition block 404. To begin channel Λ tracking the initial channel estimate H(_o) '^s decomposed into Qⁿ and A_R by the QR

Λ decomposition block 404. Q^H represents the unitary matrix corresponding to the initial

Λ channel estimate and is held constant while the channel is being tracked. Q^H is only

Λ Λ updated when a new decomposition is performed. A_R is the equivalent of R of the previous embodiment and is initially an upper triangular matrix but in this embodiment

Λ

A_R becomes non triangular with time. Accordingly, other than immediately following

Λ decomposition, A_R is only an approximation of an upper triangular matrix. In this example, the LMS algorithm of the LMS channel tracking block 402 tracks the A_R matrix only. At any time instance the channel matrix H can be presented as

H{ή = Q{i-n) *M) ^)

Here H® is the actual channel matrix at the i^th time instance, <2(_z__w\ is a unitary matrix obtained n time instances ago (from H(i-_n)), and A_R{^ is generally a non-upper triangular matrix. A_Ru\ is only strictly upper triangular form when n=0, i.e. immediately after a QR decomposition. As n increases Q(i-_n\ becomes more outdated hence the output of the LMS algorithm, A_R(Λ has to change to compensate for the changing channel. As a

result, non-zero components are introduced into A_R(Λ below the main diagonal as well as non-real components on the main diagonal.

As can be seen, the reference signal for the LMS algorithm is now taken after the Q¹ processing block 406. Figure 5 depicts schematically the LMS configuration for tracking the AR matrix. In the LMS tracking block, an estimate of an upper triangular matrix AR at

Λ the time instance / can be represented as A_R(ή . In this case the error eru_\ due to

tracking and noise can be expressed as er^ - Q₍ι-_n) ^H-* y_(ή - A_R^* s^. Now, substituting from equation 5

This error signal is minimised by the LMS tracking algorithm to update the estimate of

Λ

A_R(Λ for the next data symbol processed. However, as A_Rfy is typically non triangular it is not suitable for the back substitution process used in the R^"1 equalisation block 208

Λ of Figure 2A. Therefore a modification to , is needed.

In this example, all the non-zero imaginary elements on the main diagonal and all the elements below the main diagonal are set to zero by the forcing block 408 in the following manner

J + j ImI A_m

+yo forced

The forced upper triangular matrix Rf is then used for back-substitution by the equalisation block 208.

The forcing to zero operation performed by the forcing block 204, however, creates an

Λ additional error. The effective channel estimate, with all errors included (i.e.

those caused by zero-forcing and tracking) is defined as

The new error matrix becomes

Expanding (8) using equation (5) gives erR(i) = Q{i-n) * A_R(i) - β(,-__B)* Rf ^ (9)

Λ

when considering the power of the error.

When the power of this error grows beyond a threshold, set by the designer in block

Λ 704, QR decomposition is performed by the QRD block 404 on A_RIΛ to bring it back to upper triangular form and update the unitary matrix Q as shown below

In (10)

is the new upper triangular

Λ matrix that updates the A_R value in the LMS tracking block 402 of figure 4.

A variation of this embodiment could be implemented using svd decomposition, in which case the diagonal matrix D is tracked. Moreover, rather than using a LMS algorithm to do the tracking, other algorithms such as RLS, Kalman filter etc could be used.

It should be noted that the present embodiment is described as applied to a zero forcing QR equaliser, but that the present invention should not be considered limited to this type of equalisation, alternatively a MMSE QR equaliser or a sub-optimum equaliser could be used.

Figure 6 depicts the MSE performance of the two embodiments described above as a function of the number of data symbols in a packet. Two plots are shown for each embodiment, the first with QR decomposition performed every symbol, and the second

Λ with QR decomposition performed on the initial channel estimate H(_o) only. In this example a 2X2 normalized channel matrix is used, at a Doppler frequency of 48Hz. Such a high Doppler frequency is chosen to show the asymptotic behaviour of the curves. There is no noise in the system. As can be seen from plot 600, both embodiments, i.e. tracking H or A_R , perform equally when QR decomposition is performed every time instance. In both cases the error floor is due to the tracking errors and misadjustment of the LMS algorithm.

The error growth in the A_R tracking case, plot 602, is due to the conversion of A_R to

Rf for equalization using back substitution. For the case of a 2X2 matrix:

Then the instantaneous error matrix due to introducing zeros in A_R can be written

The elements in (14) are reset to zero every time QR decomposition is performed on

A_R . Given that A_R is normalized to have an average total power of unity (equivalent to the channel), the maximum average power of this error matrix reaches -3dB of the normalized power, which is the upper bound plot 602 of figure 6.

Plot 604 of Figure 6 shows the difference between the current channel estimate, Hh\ ,

and the original current estimate, H(_X__M) taken n instances earlier for the first embodiment when QR decomposition is only performed once. In this case, the error matrix can be presented as

\\r +J^dh \\i dh ₂lr +jdh ₂li

_l2r + jdh _m dh _22r + jdh ₂u _

Where (15)

and ^₇ can be either real or an imaginary and 1 < i ≤ 2, l ≤ j ≤ l Assuming all four complex coefficients of (15) have identical statistics and the channel matrix is again normalised to have an average power of unity, the average power per element in e#(_w) is given by:

3dB of the normalized power (16)

which is the upper bound for plot 604 of figure 6.

Figure 7 depicts an exemplary threshold detector block such as the block 220 of Figure 2A. The threshold detector block 700 of Figure 7 controls the frequency of QR decompositions to maintain the average mean square error (MSE) below a predetermined threshold in order to minimize the power consumption. In the first embodiment described above, the inputs 310 to the threshold detector block 220 are

Hφ and #(;_„) which are obtained from the DD tracking and QR Decomposition block 206. In the second embodiment, the inputs 410 to the threshold detector block 220 are

A_Rφ and Rf_{t-_n) which are obtained from the block 400. The error obtained from the difference between the two inputs is averaged in the ensemble averaging stage 702 to get a Mean Squared Error. Finally, a comparison block 704 compares the resulting MSE with the preset threshold. If the threshold value is exceeded, 704 invokes the trigger line

218 that enables a QR decomposition in blocks 400 or 206 and updates the value of

H{i-n) stored in memory 304.

Figure 8 presents MSE comparisons of five scenarios, details of which are given below, from both tracking schemes in the presence of additive white Gaussian noise of -3OdB. The pilots have the same SNR as the data and the environment is slowly changing with a Doppler frequency of 6Hz as specified in the channel model.

In each case a QR decomposition is taken immediately after receipt of a training sequence. Plot 800 - "H no QRD" represents the case where H is tracked by the LMS tracking stage and no further decompositions are made, i.e. no further updates of the Q and R matrices are made. In this case the MSE steadily increases with time. The second curve 802 - "Ar no QRD" is the equivalent case except that the upper triangular matrix is tracked. Ia this case no further decompositions are performed, but the partial tracking of the R Matrix by the zero forcing of the upper triangular matrix initially improves the MSE for the first 15 data symbols before the error growth in the zeroed elements (14) starts to dominate. The best results come from plot 804 - "H or Ar QRD", which represents the situation where the QRD is executed after every symbol. This plot 804 forms a lower bound to the MSE. This curve represents the tracking floor of the LMS algorithm (assuming negligible QR decomposition errors). In practice there is some compromise here, since the LMS feedback coefficient, μ, can be adjusted to lower this floor, but at the expense of slower convergence of the algorithm. These three lines are identical to the curves on Figure 6, except for the more realistic lower Doppler Frequency suggested by the IEEE802.11n channel model.

The last two curves show the threshold mode of operation of the illustrative embodiments described above with the threshold set 6dB below the operating noise power level, N₀. This means that the implementation loss associated with channel

estimation errors is bounded at -10*log₁₀ °— - «1dB. As will be appreciated, a

N_n

"' * ^■ 4 reduction in the number of QRDs performed occurs after MSE drops below the threshold of -36dB. The oscillatory motion of the MSE is constrained between the predetermined threshold set and by plot 804.

It can be observed that after the LMS algorithm has converged, it takes about 10 symbols longer for plot 806 - "AR threshold QRD" which represents the second of the illustrative embodiments, to reach the error threshold, than for plot 808 - "H threshold QRD" which represents the first embodiment described above.

The slower divergence of AR allows a reduced number of QR decompositions to be used, resulting in a reduced number of operations per symbol and consequent reduction in power consumption without an additional sacrifice in the performance of the system. If the AR tracking and H tracking scenarios require n and m symbols respectively to

degrade to a given MSE, the ratio — will express the complexity gain of the AR tracking n scheme over the channel (H) tracking scheme. Table 1 presents the gain in the complexity of the A_R tracking scheme for the various SNR, and MSE threshold cases. In this example the IEEE 802.11n channel. "F" model is used.

Table 1 Number of symbols between QR decompositions during which the target MSE is reached for both tracking schemes at various SNR levels

The first column of table 1 gives the SNR level, and the error threshold MSE is shown in the second column. The last column shows the reduction in the complexity that can be achieved by tracking the upper triangular matrix over the channel H tracking embodiment. In this example is possible to have up to 1.5 times less operations per symbol without sacrificing performance in the upper triangular matrix tracking case.

In the example embodiments described above, and in general, matrix decompositions can be performed in a number of ways. The Givens Rotation technique, the Householder technique and the Gramm-Schmidt technique are some of the known ways that are often used to implement qr and svd decompositions. The Givens Rotation Technique, implemented using the CORDIC algorithm is a preferred means of hardware implementation due to its relatively low complexity. Typically the CORDIC algorithm is set up to have a fixed number of iterations, or micro- rotations, the number of which are dependent on the computation accuracy required. This fixed number of micro-rotations represents a certain complexity, which a further aspect of the present invention seeks to reduce.

In broad concept this is achieved in the illustrative embodiments by modifying the CORDIC algorithm to remove unnecessary micro-rotations. The number of micro- rotations removed can be calculated using the calculations set out as described below.

As will be come apparent from the following description, implementation of the A_R tracking case, using an embodiment of this aspect of the invention, may be particularly advantageous as the initial angles by which complex elements and column vectors of A_R are rotated are often small when using the CORDIC algorithm.

Details of the vectoring and rotation modes of the CORDIC algorithm can be ascertained from M. H. Dawid.H, "CORDIC Algorithms and Architectures," in Digital Signal Processing for Multimedia Systems, N. T. Parhi.K.K, Ed.: Marcel Dekker, 1999, pp. 623-655.

In the CORDIC algorithm, in the vectoring mode an arbitrary vector

with tan(α) = — , where α represents the angle between the vector and the positive x- x axis, is rotated towards the x-axes using a set of rotation matrices, called micro- rotations. This algorithm is presented in follows:

d=sign(x_m)*sign(y_m) (17)

In the above, q_m is an orthogonal matrix that rotates the vector v_m by tan^"1(2^"m) angle. Since 2^"m, 0 < m ≤ K, m e N is a converging sequence, v_m is bound to align with the x- axes to within a 'set angular tolerance' given by tan^"1(2^"κ). After K micro-rotations, the algorithm is stopped.

Symbol d in (17) indicates the direction of the next micro-rotation. The square root term in front of the matrix in (17) is known as the scaling factor, and quickly converges to 1.0 and there is no need to calculate it after nt_>/2 iterations, where % is the processing wordlength, since its value drops below the quantization noise floor. The product of all the scaling factors can be pre-calculated, stored, and applied only once after the final iteration. in the rotational mode, the vector v is rotated by a preset angle.

d = sign(z_m) (18) z_m+, = z_m -d *a_m, α_m = tan-¹(2-^M)

In this case, z-z₀ is the preset angle. In (18), z_m+i contains residual angle that must be driven to zero, and d indicates the direction of the next micro-rotation. When z_m+-_/ reaches a 'set angular tolerance' of tan^"1(2^"κ), the algorithm is stopped.

In order to better understand the exemplary implementation of QR decomposition presented below is useful to consider the typical method of matrix decomposition using Givens Rotations. Consider firstly the 2X2 real matrix,

The idea of Givens rotations is to find such a unitary rotation matrix Q so that when it multiplies column vector Ri it rotates it in the following way.

Multiplying R by Q gives

In (21 ) R is an upper triangular matrix with Q as its unitary basis.

A procedure for CORDIC-based QR decomposition of a 2X2 complex matrix to obtain upper triangular matrix R and unitary Q is shown in figure 9. At the end of the procedure

900 shown on figure 9, R and Q are obtained. Since R is a complex matrix there are a total of 8 rotations needed to obtain the upper triangular matrix, with half of them in the vectoring mode. There are four rotations to generate real coefficients in the first column, three to perform Givens rotations and a final rotation to make the diagonal real. There are 8 additional rotations required to obtain Q with all of these rotations performed by the CORDIC in a rotational mode. For the CORDIC based QR decomposition, the vectoring mode of CORDIC is used to introduce zeros in appropriate places while creating R, as shown on figure 9. The Rotational mode of the CORDIC is used to update the rest of the vectors. If the sequence of the signs used during the vectoring mode is saved, it then can be used in (18) for the CORDIC in rotational mode to update the vector (updating r_l2e^}{βn>! to r_ne^j(βl2~θn^ for instance). The middle two steps of (18) are not needed, reducing algorithm complexity.

Turning now to the complexity of QR decomposition based on CORDiC, two additions and two shifts are required to implement a micro-rotation by the orthonormal matrix in the first line of (17) and (18). The total scaling factor resulting from the multiplication of individual scaling factors is stored separately. Vector v gets multiplied by K orthogonal matrices and then by the stored scaling coefficient.

For the CORDIC that uses K micro-rotations, it takes 8*2*K shifts + 8*2*K additions to form R and the same number of shifts and additions to form Q. The CORDIC in the above examples needed K=11 micro-rotations to reach the -41dB tracking error bound of Figure 8. Two multiplications are required for the scaling coefficient at the end of each of the 16 rotations. Hence there are 352 shifts and 352 additions and 32 multiplications needed to perform the QR decomposition of a complex 2X2 matrix.

A preferred method of performing matrix decomposition will now be described in connection with the second channel tracking embodiment described above. In this example, when tracking the upper triangular matrix, the interference elements below the main diagonal and imaginary elements on the main diagonal start from zero.

The first complex element r-n will have very small phase angle at the beginning. The second complex element r₂i will have both elements very small at the beginning, with a uniformly ^"distributed phase angle between 0° to 360°. The initial angle between the vector and the x-axes is also likely to be very small. Since the QR decomposition

is performed on this nearly upper triangular matrix as explained above, the present inventors have determined that is it advantageous to use a smaller number of micro- rotations than the conventional CORDIC algorithm for the smaller angles. x For an input vector defined as , tan(or) = — , the operation of the conventional y x

CORDIC algorithm can be presented as in figure 10. As can be seen in Figure 10 the CORDIC algorithm goes through a rotation sequence from 2° to 2^"3 irrespective of the input angle. However the present inventors have determined that this conventional CORDIC algorithm becomes inefficient when the input angle is small, since the first few micro-rotations cause large overshoots in the phase, which causes divergence rather than convergence. In this case processing power is wasted on the initial big angular swings.

In preferred embodiments of this aspect of the invention, the number of micro-rotations is reduced by choosing the shortest distance to the residual angle out of the set of all the possible micro-rotations for the next iteration. Figure 11 schematically illustrates this idea. In this case the number of micro-rotations has been reduced from four to only one.

Techniques for finding the shortest distance is to use a method of approximate rotations is disclosed in Gotze.J, "An efficient Jacobi-like algorithm for parallel eigenvalue computation," Transactions on Computers, vol. 42, pp. 1058-1065, 1993 and Gotze.J, "Iterative Version of the QRD for Adaptive RLS filtering', Proc. SPIE Advanced Signal Processing: Algorithms, Architectures and Implementations V, vol.2296, pp.438-449,

1994. Alternatively, the number of iterations for the small angles of the A_R matrix can be reduced using the following technique.

Firstly, representing tan(α) in the floating point binary format

M * 2^ey M tan(α) = —- ^y = — *- * l<*^>-*> (23)

Af_x * 2^β M_x ^V '

Where M_y and M_x are mantissas and e_y and e_χ are the exponents of y and x respectively. Since l ≤ M_y ≤ 2, Λ ≤M_X ≤ 2 then ₂-^] < ^iL < 2¹ (24) M,

With 2° = 2^iey~ex) , used as an approximate rotation and taking (24) into account, the set of possible choices for the initial rotation is narrowed down to

|₂(_β-i)_j2(_β)₎₂(_β+i)j (25)

Preferably the middle of the three options 2^(a) is used, regardless of which of the three

choices is closer to — . However other selection techniques, such as that taught in x

Dickson, K, "QRD and SVD processor design based on an approximate rotations algorithm," SIPS, Signal Processing Systems, pp. 42-47, 2004. Statistically, the probability that the middle value is the correct choice is 50%. For the other 50% of occurrences the system still converges, but in the worst case it will activate all the following available micro-rotations.

In the preferred embodiment the value of V₁ rather than the 'set angular tolerance' is used to terminate the rotating process of the vectoring CORDIC. This will tend to equalise the residual errors in the AR matrix coefficients after the QRD has been completed. It will also eliminate unnecessary rotations.

The vectoring mode algorithm of an embodiment of the present invention can be described in pseudo-code as follows

1. Move the input vector, , into the first octant by inverting signs and flipping the x and y

magnitudes as appropriate

2. Obtain the exponent difference (e_y-eχ) =a

3. Rotate with 2^s to perform the approximate micro-rotation

4. Check the stop condition

If (y, < accuracy floor), apply scaling factor from LUT, readjust signs and stop. Else next iteration (go to 2) The sign and flipping information, sequence of a's and the scaling factor are used in the rotational part of the illustrative example of the inventive algorithm to update the other vectors.

Figure 12 illustrates a method of obtaining the exponent difference a in a preferred embodiment of the present invention. The flowchart 1200 depicts the steps in an algorithm that finds the exponent difference between two positive fixed-point numbers A and B where 0<A<1 and 0<B<1. Initially in step 1202 the |A| and |B| are successively multiplied by 2 until one of them is greater than or equal to 1. If A becomes greater than or equal to 1 B is multiplied by 2 until it is greater than or equal to 1 in step 1204. In this case the difference in exponents is calculated as dif - 1 , where 'dif is given by dif=exp(B)-exp(A). Conversely, if B becomes greater than or equal to 1 first, A is multiplied by 2 until it is greater than or equal to 1 in step 1206. In this case, the difference in exponents is calculated as dif + 1 , where 'dif is given by dif=exp(B)-exp(A).

There is an additional computational overhead in the proposed algorithm that is associated with obtaining the exponential difference (step 2) when the algorithm is implemented on fixed-point hardware. Also, a look up table (LUT) containing all possible combinations of the scaling factors will typically be needed since the rotational sequence is no longer fixed. The number of scaling factors (equal to 2^πb/2) determines the LUT size. When a QR decomposition is performed some vectors are rotated by the sequence obtained earlier from the vectoring mode as shown in figure 9.

Notwithstanding this, the inventors have determined that the overall complexity of the calculations performed can be reduced in certain embodiments of the present invention, by using the above decomposition scheme.

For example, Table 2 shows a comparison in the complexity of the conventional CORDIC algorithm and an implementation of a decomposition algorithm according to an embodiment of the present invention. It was found that the conventional CORDIC required 11 micro-rotations to guarantee convergence to the LMS tracking floor depicted in Figure 8. Similarly, the decomposition algorithm according to an embodiment of the present invention was required to meet yι<2^'9 before termination. Because the operation of the decomposition algorithm according to an embodiment of the present invention is data dependent, the average number of operations for this comparison have been obtained from uniformly distributed values of x and y between 0 and 1.

Table 2 Complexity comparison between standard CORDIC of 11 micro- rotations and illustrative example of inventive algorithm with termination condition Iy₁ 1< 2^"9

It can be observed from the table 2 that the overhead increases the complexity of the illustrative example of the inventive algorithm in the vectoring mode, but reduces the complexity in the rotational mode. However, there is a net complexity reduction when the illustrative example of the inventive algorithm is used to perform QR decomposition. Overall for the 2x2 case, the illustrative example of the inventive algorithm uses only 0.53 times the number of shifts and 0.47 times the number of additions compared to the conventional CORDIC. These performance savings may improve as the matrix size increases.

For the AR tracking scenario described in the second embodiment above, even larger savings may be realised as the rotation angles on average are smaller, resulting in fewer micro-rotations. In particular the inventors have found that larger savings in the number of shifts and additions per symbol are realised in situations where QR decomposition are performed more frequently, but that the savings become less as the QR decomposition frequency decreases. At diminished signal to noise ratios the LMS tracking floor as depicted in figure 8 rises which reduces the accuracy requirements on the QR decomposition. This allows a larger terminating value for y, in the decomposition algorithm and so a lower number of micro-rotations needs to be performed.

Table 3 shows the required y, termination levels for the illustrative example of the inventive algorithm as well as the number of micro rotations for the conventional CORDIC for various SNRs.

Table 3 Termination y, value for various operating SNRs

Finally, table 4 demonstrates the complexity savings for various SNRs with MSE thresholds set 6dB below these SNR values. Table 4 shows that the illustrative example of the inventive algorithm reduces the complexity of the AR tracking scheme even further when compared to the results listed in Table 1. The complexity of the AR tracking scenario may be reduced to around 20% of the shifts and about 18% of the additions compared to the original channel tracking scenario.

Table 4 Complexity comparisons for a subset of the SNR values listed in Table 3 A third illustrative embodiment of the present invention takes advantage of complexity savings of both the second tracking scheme described above and the decomposition implementation described above. In this embodiment the receiver uses the A_R matrix tracking method described above in combination with the second embodiment and performs QR decomposition using the algorithm described. This embodiment takes advantage of the smaller average phase angles of the elements in the tracked upper triangular matrix (AR) matrix compared to the elements in the H matrix, and as such exploits the benefits of inventive matrix decomposition algorithm. As will be appreciated, in alternative embodiments the above algorithm can be easily modified to perform other types of decomposition, such as when decomposing the diagonal matrix for svd systems.

It should be understood that the illustrative embodiments described herein should not be considered limiting on the invention, and that various modifications and adaptations to the methods and systems described herein can be made without departing from the scope of the invention. For example the methods described herein can be readily applied to a case in which the number of transmit antennas is not equal to the number of receive antennas. Additionally it is possible to modify the above to apply the techniques to the [H;σ_n I] matrix, with σ_n defined as additive white Gaussian noise standard deviation, rather than H to implement a MMSE QR equalizer for tracking the H matrix. For example, by applying the upper triangular (A_R) tracking to QR MMSE systems. An explanation of QR MMSE can be found in Bohnke.R, "Reduced complexity MMSE detection for BLAST architectures," Global Telecommunications Conference, vol. 4, pp. 2258-2262, 2003.

H

In this case, the extended channel matrix H = of (NT⁺NR) X NT dimensions can σ_n *I be decomposed as H = Q * R , Where Q of (Nτ+N_R) x (NT⁺NR) dimensions forms a unitary basis for R of (NT⁺NR) X NT dimension.

Noisy and channel corrupted input data y is adjusted to y = and the LMS

tracking algorithm then tracks R , which becomes A_R . Next zeros are inserted to form R_f and QR decomposition is performed the same way as in the QR zero forcing MIMO case.

Measurement of the MSE threshold that sets off the QR decomposition process can also be made in a number of ways, including but not limited to: • Measuring the SNR as part of the channel estimate at the beginning of the packet.

• Measuring MSE, by differences in path metrics (from Viterbi decoder). • Measuring the average power of the zeroed matrix coefficients (before they are zeroed) in the AR tracking scenario.

• Measuring the data constellation point MSE (error vector magnitude).

It should also be noted that in the exemplary embodiments described herein, low Doppler channels as specified for wireless local area networks (WLAN) have been assumed. These channels generally assume stationary terminals but moving scatterers. The IEEE 802.11n standard has specified six such channels. The most aggressive of these, in terms of Doppler, is channel F, which assumes scatter velocities of 1.2km/hr for all scatterers except for one, which moves at 40km/hr. This channel has been used for all simulations discussed. A complete list of the simulation parameters is shown below:

The IEEE 802.11n type F channel model @ 5.27GHz [13-14]

• A 4usec OFDM symbol period (3.2usec with a cyclic prefix of O.δusec)

• The number of sub-carriers, N=64 • A 20MHz system bandwidth

QPSK modulation

• No hardware imperfections, perfect synchronization and no frequency offset.

It should be noted however that these parameters are used for illustrative purposes only and that the present invention is applicable to other types of channel with different parameters.

It will be understood that the invention disclosed and defined in this specification extends to all alternative combinations of two or more of the individual features mentioned or evident from the text or drawings. All of these different combinations constitute various alternative aspects of the invention. The applicants do not concede that the prior art discussed herein forms part of the common general knowledge in the art in Australia at the priority date of the specification

Claims

We claim:

1. A method of processing a signal encoded on a channel in a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality of payload data symbols, the method including:

(a) determining an initial channel estimate, represented by a initial channel estimation matrix, on the basis of one or more pilot symbols received on the channel;

(e) determining an error value calculated on the basis of a feedback data symbol and its corresponding received data symbol; (f) comparing the determined error value to a threshold, and in the event that the error value is greater than the threshold;

(g) performing a further matrix decomposition to generate at least one updated canonical form matrix; and

(h) repeating steps (c) to (f) using the at least one updated canonical form matrix.

2. A method of processing a signal as claimed in claim 1 , wherein in the event that, in step (f), the error value is not greater than the predetermined threshold the method includes, repeating steps (c) to (f) without performing step (g).

3. A method of processing a signal as claimed in either of claims 1 or 2 wherein the method includes determining an updated channel estimate using a tracking algorithm.

4. A method of processing a signal as claimed in claim 3 wherein the feedback data symbol and corresponding payload data symbol are used as an input to the tracking algorithm, and wherein the tracking algorithm is configured to output an updated channel estimate on the basis of said inputs.

5. A method of processing a signal as claimed in either of claims 3 or 4 wherein the tracking algorithm is configured to use any one of the following tracking algorithms to determine the updated channel estimate:

LMS, RLS, or Kalman filtering.

6. A method of processing a signal as claimed in either of claims 3 or 4 wherein the tracking algorithm used is configured to minimise the value of the following expression to determine the new channel estimate:

wherein y is the received data symbol, s is the feedback data symbol and H is the channel estimate.

7. A method of tracking a channel as claimed in any one of the preceding claims wherein determining the error value in step (e) includes any one or more of the following:

Λ determining a Mean Squared Error between S and S

determining a Mean Squared Error between H* S and Y

determining a difference in path metrics in the decoding process

8. A method of processing a signal as claimed in either of claims 1 or 2 wherein at least one updated canonical form matrix may not strictly conform to its corresponding canonical form.

9. A method of processing a signal as claimed in either of claims 1 , 2 or 8 wherein the method further includes determining at least one updated canonical form matrix using a tracking algorithm.

10. A method of processing a signal as claimed in claim 9 wherein the feedback data symbol and corresponding payload data symbol are used as an input to the tracking algorithm, and wherein the tracking algorithm is configured to output an approximated canonical form matrix on the basis of said inputs.

11. A method of processing a signal as claimed in any one of claims 8 to 10 wherein the tracking algorithm is configured to use any one of the following tracking algorithms to determine the updated canonical form matrix: LMS, RLS, or Kalman filtering.

12. A method of processing a signal as claimed in claim 11 wherein in the event that an updated canonical form matrix does not conform to its canonical form the method further includes, forcing the updated canonical form matrix into a corresponding canonical form to enable channel equalisation using said matrix.

13. A method of processing a signal as claimed in either of claims 1 or 2 or any one of claims 8 to 12 wherein the further matrix decomposition performed in step (g) is performed on an updated canonical form matrix that does not comply with its corresponding canonical form.

14. A method of processing a signal as claimed in any one of the preceding claims wherein the canonical form matrices generated in step (b) include either an upper triangular or lower triangular matrix.

15. A method of processing a signal as claimed in claim 14 wherein the upper triangular or lower triangular matrix is subsequently used for channel equalisation and the inverse to said upper triangular or lower triangular matrix is determined by back- substitution.

16. A method of processing a signal as claimed in any one of the preceding claims wherein the canonical form matrices generated in step (b) include an orthogonal unitary matrix.

17. A method of processing a signal as claimed in any one of the preceding claims wherein the canonical form matrices generated in step (b) include a diagonal matrix.

18. A method of processing a signal as claimed in any one of the preceding claims wherein the decompositions performed are QR decompositions.

19. A method of processing a signal as claimed in any one of the preceding claims wherein the decompositions performed are SV decompositions.

20. A method of processing a signal as claimed in claim 14 wherein the Q and R matrices are evaluated by an LU transform.

21. A method of processing a signal as claimed in any one of the preceding claims wherein the method is implemented for a plurality of received channels.

22. A method of processing a signal as claimed in any one of the preceding claims wherein the method is implemented in a receiver having more than one antenna for receiving said channels.

23. A method of processing a signal as claimed in any one of the preceding claims wherein the communications network is a multi-subcarrier system.

24. A method of processing a signal as claimed in any one of the preceding claims wherein the communications network has a single carrier frequency.

25. A method of processing a signal as claimed in any one of the preceding claims wherein the method is implemented in a receiver employing a frequency domain equaliser structure in which equalisation is applied on a bin by bin basis.

26. A method of tracking a channel in a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality payload data symbols, the channel tracking method including:

(a) determining an initial channel estimate, represented by a initial channel estimation matrix, on the basis of one or more pilot symbols received on the channel; and

(b) decomposing the initial channel estimation matrix into a product of at least two canonical form matrices; wherein for each received data symbol the method includes: (c) recovering a payload data symbol from a corresponding received data symbol on the basis of at least one or more of the canonical form matrices;

(e) generating a new channel estimate and an error value on the basis of the feedback data symbol and its corresponding received data symbol;

(f) comparing the determined error value to a threshold: wherein in the event that the error value is greater than the threshold; the method further includes:

27. A method of tracking a channel in a wireless communications network as claimed in claim 26, wherein in the event that, in step (f), the error value is not greater than the predetermined threshold the method includes, repeating steps (c) to (f) without performing step (g).

28. A method of obtaining a data symbol encoded on a channel in a wireless communications network, including processing said data symbol using a method as claimed in any one of claims 1 to 25.

29. A method of obtaining a data symbol encoded on a channel in a wireless communications network, including tracking the channel using a method of either one of claims 26 or 27.

30. A method of performing a decomposition of a matrix M in a data processing device, said method including: x

(a) defining, within the matrix M, an arbitrary vector v = with tan(α) = — , where a y x represents an angle between the vector and the positive x-axis, and wherein tan(or) can

M *2^ey M be represented in the floating point binary format tan(or) = — = — - * i^^ , where

M_x *!^** M_x

My and M_x are mantissas and e_y and e_x are the exponents of y and x respectively; (b) moving the input vector v, into a first octant by inverting signs and flipping x and y magnitudes of one or more of the elements of matrix M;

(c) determining an exponent difference, a, equal to (e_y-e_x);

(d) rotating the v by 2^a to perform a micro-rotation of the form

^~x_M to generate a new vector v_!+1; and

(e) in the event that yι+i does not meet a predetermined accuracy threshold the method further includes repeating steps (c) and (d); and wherein in the event that y_i+i meets the predetermined accuracy threshold, the method includes;

(f) applying a scaling factor; and (g) readjusting signs of x_i+i an_d Yi+i

31. A method of performing a decomposition of a matrix as claimed in claim 30 wherein the scaling factor is determined using a look up table.

32. A method of performing a decomposition of a matrix as claimed in either of claims 30 or 31 wherein step (c) includes selecting a from a range of possible values.

33. A method of performing a decomposition of a matrix as claimed in claim 32 wherein step (c) includes selecting a as the central value in a range of possible values for a.

34. A method of performing a decomposition of a matrix as claimed in any one of claims 30 to 33 wherein the method is used to perform either QR or SVD

35. A method of processing a signal received on a channel in a wireless communications network as claimed in any one of claims 1 to 25 in which at least one matrix decomposition is performed using a method of any one of claims 30 to 34.

36. A method of tracking a channel in a wireless communications network as claimed in any one of claims 26 or 27 in which at least one matrix decomposition is performed using a method of any one of claims 30 to 34.

37. A signal processing device adapted to process a signal received on a channel of a wireless communications network, said channel transmitting a series of data symbols including one or more pilot symbols interspersed within a plurality payload data symbols, the device including: a channel estimation stage configured to determine at least an initial channel estimation; and a matrix decomposition stage configured to decompose a matrix into a product of at least two canonical form matrices; a symbol decoding stage configured to process a received data symbol to obtain a payload data symbol on the basis of at least one or more of the canonical form matrices generated by the matrix decomposition stage; a feedback coding stage configured to process a recovered payload data symbol to generate a feedback data symbol; error determination means for determining an error value calculated on the basis of a feedback data symbol and its corresponding received data symbol; threshold detection means to compare the determined error value to a threshold and in the event that the error value is greater than the threshold, trigger the matrix decomposition stage to perform a matrix decomposition to generate at least one updated canonical form matrix for use by the symbol decoding stage to decode at least one subsequent symbol.

38. A signal processing device as claimed in claim 37 which further includes a matrix tracking stage configured to determine, at least intermittently, an updated channel estimation using a tracking algorithm.

39. A signal processing device as claimed in claim 38 wherein the matrix tracking stage determines an updated channel estimation for each received data symbol.

40. A signal processing device as claimed in claim 38 or 39 wherein the matrix tracking stage uses any one of the following tracking algorithms to determine the updated channel estimate:

LMS, RLS, or Kalman filtering.

41. A signal processing device as claimed in claim 37 which further includes a matrix tracking stage configured to track least one updated canonical form matrix using a tracking algorithm.

42. A signal processing device as claimed in claim 41 which further includes a matrix forcing stage configured, in the event that an updated canonical form matrix does not conform to its canonical form, to force an updated canonical form matrix into its corresponding canonical form to enable channel equalisation to be performed using said matrix.

43. A signal processing device as claimed in any one of claims 36 to 42 wherein the symbol decoding stage includes an equalisation stage.

44. A signal processing device as claimed in any one of claims 36 to 43 wherein the matrix decomposition stage performs a QR decomposition.

45. A signal processing device as claimed in any one of claims 36 to 43 wherein the matrix decomposition stage performs a SV decomposition.

46. A receiver for use in a wireless communications network which includes a signal processing device as claimed in any one of claims 36 to 45.

47. A receiver for use in a wireless communications network as claimed in claim 46 which is configured to receive signals on a plurality of channels.

48. A receiver for use in a wireless communications network as claimed in either of claims 46 or 47 which includes a plurality of receiving antennas.

49. A receiver for use in a wireless communications network as claimed in any one of claims 46 to 48 which is configured to operate in either a communications network using multiple sub-carriers or a single carrier frequency.

50. A receiver for use in a wireless communications network as claimed in any one of claims 46 to 49 which includes a frequency domain equaliser structure in which equalisation is applied on a bin by bin basis.

51. A signal processing device as claimed in any one of claims 37 to 45 wherein the matrix decomposition stage is configured to perform a matrix decomposition method as claimed in any one of claims 30 to 34.

52. A receiver for use in a wireless communications network as claimed in any one of claims 46 to 50 wherein the matrix decomposition stage is configured to perform a matrix decomposition method as claimed in any one of claims 30 to 34.