GB2416967A

GB2416967A - Turbo equalization in MIMO systems using the BCJR algorithm

Info

Publication number: GB2416967A
Application number: GB0416933A
Authority: GB
Inventors: Robert Jan Piechocki; Magnus Sandell; Christophe Andrieu
Original assignee: Toshiba Research Europe Ltd
Current assignee: Toshiba Europe Ltd
Priority date: 2004-07-29
Filing date: 2004-07-29
Publication date: 2006-02-08
Anticipated expiration: 2024-07-29
Also published as: GB0416933D0; GB2416967B

Abstract

In a multi antenna (127) receiver (114), turbo decoding (134) incorporates adaptive Soft Input Soft Output (SISO) equalisation whereby marginal posterior distributions are maintained using the Bahl Cocke Jelinek Raviv (BCJR) algorithm. The posterior transition probabilities f(s', s given y) and three metrics (alpha, beta and gamma) are adaptively calculated based on observed measurements y at different times, labelled by an index (t and t-1 being previous measurements), the probabilities of the states s and s' given the measurement y (which is in turn a function of the transmitted vector x multiplied by the channel impulse response matrix H and noise n, ie. y = Hx + n, Hx being branch metrics calculated for N streams from M antennas), and the mean (mu) and covariance (Sigma) factors (see especially Equation 16). A set of input moments is also used in the comparison of calculated and observed values, representing dependence between the multiple antennas and derived on the basis of a hidden markov model, and a training value for (t-1) used to initiate the iterations.

Description

24 1 6967 Turbo equalization in a MIMO digital wireless wideband system

This invention is concerned with decoding a signal received over a MIMO (multiple input - multiple output) channel in a wireless communications system. In particular, the invention is concerned with methods, apparatus and processor control code for decoding a signa] received over a MIMO channel.

Wireless communication is becoming increasingly prevalent in the installation of networking facilities between electronic devices. In such wireless communication, there is a user demand for data to be transmitted as efficiently as possible, which gives rise to a demand for increased data transmission rates. As data transmission rates are increased to provide a user with further data transmission facility, further uses of wireless communication are devised, giving rise to further demand. One example of this phenomenon is the increased use of multimedia services. In the past, it was considered impractical to transmit data pertaining to video and audio transmissions via a MIMO channel to a wireless device, such as a handheld computer device. Presently, wireless data transmission rates have become sufficiently high that such transmission is conceivable, and thus becomes desirable.

Additional multimedia or other services may be devised in the future which will require further increases in data transmission rates, and existing multimedia services may also be enhanced, such as in terms of error correction and/or video signal resolution. This will give rise to further demand for increased data transmission rates.

Increased data rates, for example for multimedia services, may be achieved by simply increasing the data transmission bandwidth. However, this is inefficient and expensive.

On the other hand, MIMO systems have the capability to increase throughput without increasing bandwidth, the throughput potentially scaling linearly with the number of transmit/receive antennas, for example a for transmit, for receive antenna system potentially providing four times the capacity of a single transmit-receive antenna system. However, receivers for MIMO communications systems are complex, because, in a receive antenna array comprising a number of receive antennas, a single receive antenna in use receives signals from all of the transmit antennas within range. This can give rise to difficulties in decoding, in that a single signal submitted by a single transmit antenna of a transmit antenna array can be difficult to detect.

Figure 1 shows a MIMO data communications system 10 of known construction. The communications system lO comprises a transmitter device 12 and a receiver device 14.

It will be appreciated that in many circumstances, a wireless communications device will be provided with the facilities of a transmitter and a receiver in combination but, for this example, the devices have been illustrated as one way communications devices for reasons of simplicity.

The transmitter device 12 comprises a data source 16, which provides data (comprising information bits or symbols) to a channel encoder 18. The channel encoder 18 in this example comprises a convolutional coder such as a recursive systematic convolutional (RSC) encoder, or a stronger so-called turbo encoder (which includes an interleaves).

More bits are output than are input, and typically the rate is one half or one third.

The channel encoder 18 is followed by a channel interleaves 20 and, in the illustrated example, a space-time encoder 22. The space-time encoder 22 encodes an incoming symbol or symbols as a plurality of code symbols for simultaneous transmission from a transmitter antenna array 24 comprising a plurality of transmit antennas 25. In this illustrated example, three transmit antennas 25 are provided.

The encoded transmitted signals propagate through a MIMO channel 28 defined between the transmit antenna array 24 and a corresponding receive antenna array 26 of the receiver device 16. The receive antenna array 26 comprises a plurality of receive antennas 27 which provide a plurality of inputs to a space-time (and/or frequency) decoder 30 of the receiver device 16. In this specific embodiment, the receive antenna array 26 comprises three receive antennas 27.

The decoder 30 has the task of removing the effect of the encoder 22. The output of the decoder 30 comprises a plurality of signa] streams, one for each transmit antenna 25, each carrying so-called soft or likelihood data on the probability of a transmitted symbol having a particular value. This data is provided to a channel de-interleaver 32 which reverses the effect of the channel interleaver 20, and the convolutional code output by this channel de-interleaver 32 is then presented to a channel decoder 34, in this example a Viterbi decoder, which decodes the convolutional code.

Typically, the channel decoder 34 is a SISO (soft-in soft-out) decoder, that is operable to receive symbol (or bit) likelihood data and to provide similar likelihood data as an output rather than, say, data on which a hard decision has been made. The output of channel decoder 120 is provided to a data sink 36, for further processing of the data in any desired manner.

In this example, turbo (otherwise known as iterative) decoding is employed in which a soft output from the channel decoder 34 is provided to a channel interleaves 38, corresponding to the channel interleaver 20, which in turn provides soft (likelihood) data to the space-time decoder 30 for iterative space-time (and/or frequency) and charnel decoding. It will be appreciated that in this arrangement the channel decoder 34 provides complete transmitted symbols to the space-time decoder 30, that is, for example, including error check bits.

The turbo, or iterative, equalization (MIMO, SISO etc.) technique requires equalizers that exchange so called soft information. The softinput soft-output equalization problem is solved optimally by the forward/backward algorithm (also known as BCJR).

In the standard BCJR (Bahl, Cocke, Jelinek, and Raviv) algorithm as used in MIMO soft-in soft-out (SISO) equalization, the posterior transition probability is expressed as: f (s,s|Y)a: f (s BY,,-) (slS)f (Yogis s)f (Y(+I THIS) Equation I

A_

a/ r/ pi where s is a state of the.. ., and s' is an adjacent state into which a transition can be effected, where y is a corresponding observation of the received data. The posterior transition probability is factored and calculated recursively for numerical efficiency; the outer factors as: (s) = a (s)y (s,s) Equation 2 and: (s) = (s)7 (s, s) Equation 3 s These are the familiar forward and backward vector recursions associated with the BCJR algorithm and are known in the art as alpha and beta metrics respectively. A detailed disclosure of the BCJR algorithm is set out in "Turbo equalization" (Koetter, R., Singer, A.C., Tuchler, M., Signal Processing Magazine, IEEE, Volume 21, Issue 1, Jan. 2004, Pages 67-80) and "A tutorial on hidden Markov models and selected applications in speech recognition" (Rabiner, L.R., Proceedings of the IEEE, Volume 77, Issue 2, Feb. 1989, Pages 257-286) The middle factor (known as the gamma metric in the art) is calculated as: y')(s',s) cX exp( 2 |Y,-FIX,,] | )| (xn) Equation 4 where x is the transmitted vector corresponding to the received observation y.

It will be appreciated that in the above equation the full MIMO channel matrix is used.

However, the complexity of the BCJR algorithm is O(TK No-) ), where T is the data packet length, K is the cardinality of the signalling scheme, N is the number of transmit antennas in use, and L is the channel length measured in taps. The order of complexity of the BCJR algorithm thus has an exponential nature; so if one or more of L, N and K becomes large, the algorithm can become highly complex - performance can become disproportionately time consuming given Mite processing capability. For particularly large systems, the BCJR algorithm becomes substantially computationally intractable decoding cannot be performed without an impractical amount of processing power in a real-time system.

There is thus a need to provide a reduced complexity algorithm for turbo equalization exchanging soft information, and various reduced complexity algorithms for the single antenna case have been proposed. Some of these have been extended to the multi- antenna (MIMO) case, including, for example, reduced states sequence estimation (RSSE), T and M algorithm, particle filters/smoothers, and filter based equalizers.

However, these have not provided a consistently acceptable performance in a range of conditions.

It is an aim of the present invention to provide improvements to the above reduced complexity algorithms for exchanging soft information in turbo equalization in a MIMO system.

According to one aspect of the invention, a receiver for receiving a MIMO signal comprises means for turbo equalization comprising an equalizer, the equalizer being operable to perform probabilistic data association and BCJR (Bahl, Cocke, Jelinek, and Raviv) in combination.

According to another aspect of the invention, a turbo equaliser for use in a receiver, for processing MIMO signal information received in the receiver, comprises an equaliser operable to process data in accordance with the probabilistic data association algorithm and the BCJR algorithm in combination.

According to another aspect of the invention, a decoder for use in a MIMO system comprises equa]isation means operable in accordance with a substantially BCJR algorithm, wherein the equalisation means comprises computation means operable to compute alpha, beta and gamma metrics for use in said BCJR algorithm, said computation means being operable to compute gamma metrics on the basis of probabilistic data association.

According to another aspect of the invention, a decoder for use in a MIMO system comprises a turbo equaliser operable to iteratively equalise multiantenna data on the basis of previously received data, the equaliscr storing a set of marginal posterior distributions describing the probability of data symbols received in the signals having certain values from an available set of symbol values, said marginal posterior distributions being proportional to each of said alpha, beta and gamma metrics, the decoder comprising means for determining a gamma metric operable on the basis of probability data association to associate observation data relating to previous channel estimates to transmission information describing the likelihood of transition from one symbol to another in received data.

According to another aspect of the invention, a turbo decoder is provided which is operable to determine a set of marginal posterior distributions describing probable values of symbols from observation data received in a receiver based on values of sequences of symbols derived from previously received observation data, the decoder comprising maintaining means for maintaining said marginal posterior distributions, said maintaining means comprising storage means for storing historic state transition information describing the likelihood of a symbol following another symbol in received information, means for computing alpha metrics and beta metrics for use in determining a set of posterior transition probabilities by means of a BCJR algorithm, means for computing a gamma metric for use in said BCJR algorithm, said gamma metric computing means being operable to determine the difference between observation data on the one hand and branch metrics and a set of input moments on the other, the input moments representing the extent of dependence between antenna elements of the multi- antenna transmission, and being derived from previously received information by means of probabilistic data association and means for performing, on the basis of information stored in said storage means, and received information, a BCJR algorithm using said alpha, beta and gamma metrics, to determine and to store updated historic state transition information for use in determining the likely identity of symbols in received information.

Preferably the decoder comprises input moment calculating means for determining said set of input moments on the basis of a set of output moments describing the probability distribution of branch metrics.

The input moment calculating means may be operable to calculate the set of input moments on the basis of each input moment, corresponding with a particular transmission antenna, being a sum of output moments except for the output moment corresponding with the same transmission antenna.

The input moments may comprise expectation and deviation values to describe the corresponding probability distribution.

The decoder may be operable to conduct said determination of marginal posterior distributions a predetermined number of iterations for a received symbol.

A further aspect of the invention provides a method of decoding observation data received in a receiver to determine a set of marginal posterior distributions describing probable values of symbols from observation data received in a receiver based on values of sequences of symbols derived from previously received observation data, the method comprising maintaining said marginal posterior distributions, said maintaining step comprising storing historic state transition information describing the likelihood of a symbol following another symbol in received information, computing alpha metrics and beta metrics for use in determining a set of posterior transition probabilities by means of a BCJR (Bahl, Cocke, Jelinek, and Raviv) algorithm, computing a gamma metric for use in said BCJR algorithm, including determining the difference between observation data on the one hand and branch metrics and a set of input moments on the other, the input moments representing the extent of dependence between antenna elements of the multi-antenna transmission, and being derived from previously received information by means of probabilistic data association.

The method may comprise determining the set of input moments on the basis of a set of output moments describing the probability distribution of branch metrics in the hidden markov model.

The step of calculating input moments may include calculating the set of input moments on the basis of each input moment, corresponding with a particular transmission antenna, being a sum of output moments except for the output moment corresponding with the same transmission antenna.

The method may include iterating the step of determining marginal posterior distributions a predetermined number of iterations for each received signal containing a symbol of data.

Further aspects and advantages of the present invention will become apparent from the following description of specific embodiments of the invention, provided by way of example only, with reference to the accompanying drawings, in which: Figure I shows an example of a MIMO space-time coded communications system; Figure 2 shows a MIMO space-time coded communications system including a receiver in accordance with a specific embodiment of the invention; Figure 3 shows a flow diagram of an embodiment of an equalization procedure perfonned by a SISO equaliser of the receiver illustrated in Figure 2; Figure 4 shows a flow diagram of an embodiment of a method of updating marginal posterior distributions performed in the equalization procedure of Figure 3; and Figure 5 shows a flow diagram of an embodiment of a gamma metric determination procedure of the method illustrated in figure 4.

A system 100 is illustrated in figure 2, in which a transmitter 12 is provided, equivalent to the transmitter illustrated in figure 1, and described above. A receiver 114 in communication with the transmitter 12, in accordance with a specific embodiment of the invention, comprises an antenna array 127 comprising antennas 126, the array feeding an observation vector to a SISO equaliser 130 which is operable, in accordance with the invention, to apply a variant of the BCJR algorithm to the observed data, with gamma metrics determined on the basis of probabilistic data association.

The SISO equaliser 130 outputs equalised data to a channel de-interleaver 132, of known construction and substantially functionally equivalent to the channel de- interleaver 32 described in the receiver 14 of figure 1. Further, the receiver l 13 comprises a channel decoder 134, a channel interleaver 138 and a data sink 136 corresponding to the same elements of the receiver 14 of figure 1. For the purposes of describing the operation of a specific embodiment of the present invention, it will be appreciated that the channel decoder 134, the channel interleaver 138 and the data sink 136 can be of conventional construction and require no special functionality to enable performance of the present invention.

It will be appreciated that, in this communication system, both the channel coding and the space-time coding provide diversity and thus this diversity is subject to the law of diminishing returns in terms of the additional signal to noise ratio gain which can be achieved. Thus when considering the benefits provided by any particular space- time/frequency decoder these are best considered in the context of a system which includes channel encoding.

A summarised description of the operation of decoding according to the specific embodiment of the invention now follows with reference to figure 3 of the drawings.

The illustrated system is a STBC communication system with N transmitting and M receiving antennas (in this example, M= N= 3). The system signals over a wideband channel, which for the purpose of decoding a received signal, is modelled as an equivalent multidimensional (M x N) FIR filter with L taps.

The process is iterative and recursive, and therefore certain variables need to be assigned with initial conditions in order to take into account the absence of 'prior' information at commencement. In fact, as the algorithm requires a number of iterations before it is fully trained and capable of providing suitable approximations, further initialization is required. Thus, in step S1-2, the algorithm commences by assigning an initial value to the posterior transition probability which is the basis of the BCJR algorithm used in the performance of this embodiment of the invention. The assignment of initial values will be described in further detail below, in the context of the detailed description of the method which follows.

The process then enters a for-next loop in step S 1-4, for a predetermined number of iterations of following steps, and an embedded for-next loop in step S1-6 for each of the N streams of data extracted for the N transmit antennas 25.

Then, in step S1-8, the method calculates prior mean and covariance vectors for the stream under consideration. These prior mean and covariance vectors are determined in preparation for performance of the variant BCJR algorithm, and comprise output moments at each time instant t: flnt)ou, = E{Hnx} = Hnx5 sf(s',s|y) Equation 5 and Y - E)(H X(S S) XH X(S S) (I) )H Bout n n,out n 11n,out = (Hnx( )-,ilnt)ou (Hi X(5 S) _ lI(t) )I]f( | ) Equation 6 s,s From these, input moments are determined. These comprise the mean value input moment fun in for the n'th stream:

N

Un,n = lout Equation 7 J=l,J=n and the variance value input moment for the n'th stream:

N

Nan = J,out Equation 8 J=.J=n These input moments will be used in subsequent processing steps as will be described in due course.

Then, in step S 1-10, the process updates a set of marginal posterior distributions {f(xn Ylf(Xn IY)' f(xn IY)} via the variant BCJR algorithm. This set relates to all T symbols sent from the n'th transmit antenna. In doing so, the variant BCJR algorithm exchanges the set of input moments (A n in n in) and the set of output moments (11 n,out n,o't)' In this signal processing step, the observation, or vector of received signals on the received antennas 27 at time t is expressed as: y ') = Hx i') + n 6'), Equation 9 where y is the received signal (size Mx 1), H is the channel (impulse response) matrix, and n is the noise (size N x 1). H is expressed as: h T h T H = . . . , Equation 10 h M h T where: Hn =(h,nth2,n, Than) Equationll and: hm,n =(hm,, ohms then)) Equationl2 xn, the transmitted stream of symbols transmitted from the n transmit antennas is expressed as: xn (xn,xn, ,xn)) Equation 13 where the symbols x" at a specific time t are expressed as: X ((X} ) ,(X2) ,-'',(XN) Equation 14 Each element of x belongs to a constellation S. consisting of Q symbols.

Thus, the observation and the input and output moments are then applied to an algorithm which, in accordance with this specific embodiment of the invention, is a variant of the standard BCJR algorithm as described in the published art. This process is illustrated in further detail in figure 4.

As a preliminary step, in step S2-2, the branch metricsHnxn and the posterior probabilities f(s',s|y)oc a,,l3,y, are pre-calculated as part of the standard procedures of a BCJR algorithm. The use of the 'proportional to' relation in the equations noted herein reflects the fact that a scaling factor will be required to take account of the fact that the area under the graph of a probability density function should equal 1.

The form of the variant BCJR algorithm remains in the form: f(s's|y)x a, Ay, Equation 15 where the alpha metric a, and the beta metric,l], are determined as in the algorithm described in Equation 2 and Equation 3 in the introduction in steps S2-4 and S2-6 respectively, but the gamma metric y, is calculated in step S2-8 to take into account non-zero mean noise and non-diagonal covariance. The determination of the gamma metric is now described with reference to figure 5. The determination of a gamma metric is to be made in accordance with the following relation: y' (s, s) oc exp( - (y (')-H n X no s)-lI n')n) (A (fin) (Y (')-H n XnS S)- nt) ))f (x) Equation 16 To calculate this factor, in step S3-2 the SISO equaliser 130 calculates output moments at each time instant I: 1ln)u' = E{HnX} = HnX<S s)f(stslY) Equation l7 s,s and E() - EIH (s's) - (I) YH (s',s) _ (I) )H n,our 0 Chin n,ou/ J nXn 11n,out) = (H n x no)-I1 n)OII' XH I' x (n) - 11 n) Ut) f (5 SIY) Equation 18 s,s From these, input moments are determined in step S3-4. These comprise the mean value input moment,u,, m for the n'th stream:

N

IUn,n = ',ou' Equation 19 J=I,J;en and the variance value input moment for the n'th stream:

N

in = rout Equation20 /=I,J=n These are fed back into the expression of y, set out in Equation 16, in step S3-6. The process then ends.

The detailed description ofthe initialization step described in step Sl-2 now follows.

Initial values of the function f (5', sty) need to be provided to the algorithm. For the first turbo iteration, the algorithm is provided with an initial value of f (st,sly)=-, 5 is defined as the number of states in the reduced trellis (one per stream), and K as the number of symbol constellation points.

This is required because no 'previous' values of a<'-" and,ll<'" are available, which are required to enable use of the algorithm of Equation 15. Thereafter, once earlier values of f(xn), the 'prior' distributions, become available, a training value can be provided. In this example, f (s',s|y)= I(s',s) f ( ) is used with early iterations, where the value obtainable from the algorithm itself may diverge considerably from the later available trained value and will thus not necessarily provide a practical estimate for use in the decoding process. I(s', s) is an indicator function that is of value I if xn is compatible with the state transition s' s or O otherwise.

Following execution of the modified BCJR algorithm for the stream of data under consideration, the process, in accordance with the embedded fornext loops, firstly in step S1-8 restarts the process from step S1-4 with consideration ofthe next stream of data until all streams of data have been processed and then, in step S1-10 repeats the process from step S 14 with regard to the first stream of data until all of the predetermined number of iterations have been completed.

Thereafter, the process ends, the equalised data being output to the channel de- interleaver 132 to be processed further in the conventional manner.

It will be appreciated that, while the receiver described above is specifically configured to operate in accordance with the invention, a general purpose receiver can be configured to operate in this manner by introduction of suitable software to be executed by a computer apparatus of the receiver. To that end, an aspect of the invention comprises a product, storing computer executable instructions in a computer readable form, which in use causes a computer with suitably configurable hardware components, to operate substantially in accordance with the invention as exemplified by the described embodiment. The product may comprise a storage medium such as an optical disk, a magnetic storage medium or a storage medium of any other technology, an active component such as a removable ROM unit or other memory device such as a memory card, or may comprise a signa] such as could be received in a download, the signal bearing data defining such computer readable instructions as to establish a computer executable program product.

The product could also comprise an application specific integrated circuit which, when installed in a suitably configured general purpose receiver device, renders the resultant system operable in accordance with any of the aspects of the invention exemplified by the described embodiments.

Claims

CLAIMS: 1. A turbo decoder operable to determine a set of marginal

posterior distributions describing probable values of symbols from observation data received in a MIMO receiver based on values of sequences of symbols derived from previously received observation data, ready for further channel decoding in the MIMO receiver, the decoder comprising maintaining means for maintaining said marginal posterior distributions, said maintaining means comprising: storage means for storing historic state transition information describing the likelihood of a symbol following another symbol in received information, means for computing alpha metrics and beta metrics for use in determining a set of posterior transition probabilities by means of a BCJR (Bahl, Cocke, Jelinek, and Raviv) algorithm governed by said alpha metrics and beta metrics and further by gamma metrics; means for computing a gamma metric for use in said BCJR algorithm, said gamma metric computing means being operable to determine the difference between observation data on the one hand and branch metrics and a set of input moments on the other, the input moments representing the extent of dependence between antenna elements of the multi-antenna transmission, and being derived from previously received information by means of probabilistic data association; and means for performing, on the basis of information stored in said storage means, and received information, a BCJR algorithm using said alpha, beta and gamma metrics, to determine and to store updated historic state transition information for use in determining the likely identity of symbols in received information.
2. A decoder in accordance with claim 1 and comprising input moment calculating means for determining said set of input moments on the basis of a set of output moments describing the probability distribution of branch metrics in the hidden markov model.
3. A decoder in accordance with claim 2 wherein the input moment calculating means is operable to calculate the set of input moments on the basis of each input moment, corresponding with a particular transmission antenna, being a sum of output moments except for the output moment corresponding with the same transmission antenna.
4. A decoder in accordance with claim 3 wherein said input moments comprise expectation and deviation values to describe the corresponding probability distribution.
5. A decoder in accordance with any preceding claim and operable to conduct said determination of marginal posterior distributions a predetermined number of iterations for a received symbol.
6. A method of decoding observation data received in a receiver to determine a set of marginal posterior distributions describing probable values of symbols from observation data received in a receiver based on values of sequences of symbols derived from previously received observation data, the method comprising maintaining said marginal posterior distributions, said maintaining step comprising: storing historic state transition information describing the likelihood of a symbol following another symbol in received information, computing alpha metrics and beta metrics for use in determining a set of posterior transition probabilities by means of a BCJR (Bahl, Cooke, Jelinek, and Raviv) algorithm; computing a gamma metric for use in said BCJR algorithm, including determining the difference between observation data on the one hand and branch metrics and a set of input moments on the other, the input moments representing the extent of dependence between antenna elements of the multi-antenna transmission, and being derived from previously received information by means of probabilistic data association.
7. A method in accordance with claim 6 and comprising determining said set of input moments on the basis of a set of output moments describing the probability distribution of branch metrics in the hidden markov model.
8. A method in accordance with claim 7 wherein the step of calculating input moments includes calculating the set of input moments on the basis of each input moment, corresponding with a particular transmission antenna, being a sum of output moments except for the output moment corresponding with the same transmission antenna.
9. A method in accordance with claim 8 wherein said input moments comprise expectation and deviation values to describe the corresponding probability distribution.
10. A method in accordance with any of claims 6 to 9 including iterating said step of determining marginal posterior distributions a predetermined number of iterations for each received signal containing a symbol of data.
11. A receiver for use in a MIMO communications system, including a turbo decoder according to any of claims I to 5.
12. A receiver for use in a MIMO communications system, operable to decode a received signa] in accordance with the method of any of claims 6 to 10.
13. A method of receiving a signal, including a method of decoding a received signal in accordance with any of claims 6 to 10.
14. A decoder substantially as described herein, with reference to the accompanying drawings.
15. A method of decoding a signal as described herein, with reference to the accompanying drawings.
16. A computer readable storage medium storing instructions executable by a computer, operable to cause the computer on execution of the instructions to become configured as a decoder in accordance with any of claims 1 to 5 or claim 14.
17. A computer readable storage medium storing instructions executable by a computer, operable to cause the computer on execution of the instructions to perform the method of any of claims 6 to 10 or claim 15.
18. A signal receivable by a computer, the signal carrying data defining instructions executable by a computer, operable to cause the computer on execution of the instructions to become configured as a decoder in accordance with any of claims 1 to 5 or claim 14.
l 9. A signal receivable by a computer, the signal carrying data defining instructions executable by a computer, operable to cause the computer on execution of the instructions to perform the method of any of claims 6 to 10 or claim 15.
20. A decoder substantially as described herein with reference to the accompanying drawings.
21. A method of decoding data substantially as described herein with reference to the accompanying drawings.