GB2225883A - Method of determining the tap weights of a decision feedback equaliser - Google Patents

Abstract

A decision feedback equaliser DFE comprises a feedforward filter section 10 and a feedback filter section 12 including a decision stage 14. Each of the filter sections includes a transversal filter, the tap weights of which are calculated to overcome the dispersive effects of a communications channel on the received signal. A known method of calculating the tap weights recursively comprises w=R<-1>p where R is the autocorrelation matrix describing the DFE, p is the cross correlation vector of the desired and received samples and w is the vector of the equaliser tap weights. In order to avoid the problem of matrix inversion and to permit the calculations to be done directly in a single operation with a reduced amount of computation and storage requirements the above equation is restated as Rw = p and the autocorrelation matrix R is factorised using Gaussian elimination techniques into upper (U) and lower (L) triangular matrices. Thus ULw = p (or LUw = p). This equation is solved in two stages; (1) Solve Uc = p (or LC = p) where c is the unknown and (2) solve Lw = c (or Uw = c). As the triangular matrices are expressed in terms of the channel impulse response samples, then many of the terms can be entered, without computation, on the basis of an estimate of the channel impulse response. <IMAGE>

Description

DESCRIPTION METHOD OF DETERMINING THE TAP WEIGHTS OF OF A DECISION FEEDBACK EQUALISER The present invention relates to a method of determining the tap weights of a decision feedback equaliser and to a communications equipment including a receiver having a decision feedback equaliser.

In digital radio communications an adaptive equaliser is required in the receiver to cancel unwanted echoes of the transmitted signal. A known equaliser is a decision feedback equaliser which comprises a feedforward section coupled to a feedback section including a decision stage. By this arrangement earlier decisions can be used to influence the current decision.

The feedforward and feedback sections are implemented as respective transversal filters. Each transversal filter comprises a tapped delay line or shift register, the taps or stages of which are connected to respective multipliers to which tap weights or filter coefficients are applied. The products from the multiplers are summed in an adding stage.

The values of the tap weights or filter coefficients can be calculated by using one of a number of adaptive algorithms. These algorithms are recursive, requiring a number of iterations before the coefficients are obtained. Most of the suitable algorithms for obtaining the equaliser tap weights are computationally intensive which implies an implementation consuming significant power.

Known recursive algorithms have been used to solve the equation w = R-lp (1) where R is the autocorrelation matrix of the samples in the feedforward and feedback sections of the decision feedback equaliser, p is the cross correlation vector of the desired and the said samples in the decision feedback equaliser and w is the vector of the equaliser tap weights. The adaptive algorithms solve equation (1) recursively but avoid performing an explicit matrix inversion due to the complexity of the operation. The algorithms assume no initial knowledge of R or p, and they get closer to the true values after each iteration.

An object of the present invention is to be able to calculate the tap weights of a decision feedback equaliser by a method which has less computational steps and consumes less power than known recursive methods.

According to one aspect of the present invention there is provided a method of determining the tap weights of a decision feedback equaliser, comprising deriving an autocorrelation matrix R of the received samples, deriving the cross correlation vector p of the desired and the received samples and solving the equation Rw = p (A) where w is the vector of the equaliser tap weights.

In order to solve equation (A) it may be rewritten by factorising the autocorrelation matrix R into upper (U) and lower (L) triangular matrices so that ULw = p (B) Equation (B) may be solved by firstly solving Uc = p (C) where c is an unknown, and secondly solving Lw = c (D) Alternatively the autocorrelation matrix R may be factorised into lower (L) and upper (U) triangular matrices so that LUw = p (B') and solving Lc = p (C') and then solving Uw = c (D') The factorising of equation (A) may be by a Gaussian elimination technique.

When the pivot points of the R matrix are unity, and the relationship of the terms in the U and L matrices comprise terms from the channel impulse response, these terms can be entered without computation.

the method in accordance with the present invention makes use of some initial knowledge of the system, for example the channel estimation, to calculate the tap weights (or filter coefficients) in a single computationally simple step. Significant savings in the complexity over the recursive methods are achieved together with a low power consumption and reduced storage requirements.

Additionally the calculation of the tap weights is carried out in a single operation which provides stability which is of importance when receiving data signals. Adaptive training techniques may in contrast lead to instability which is undesirable in a receiver.

According to another aspect of the present invention there is provided a communications equipment including a receiver having a decision feedback equaliser comprising a feedforward filter section and a feedback filter section including a decision stage, each said filter section comprising a transversal filter, the tap weights of which are determined by the method in accordance with the present invention.

The present invention will now be described, by way of example, with reference to the accompanying drawings, wherein: Figure 1 is a block schematic diagram of a decision feedback equaliser (DFE) having five feedforward taps and four feedback taps, Figure 2 is an autocorrelation matrix, R, of the decision feedback equaliser shown in Figure 1, Figure 3 is a tabular summary illustrating how the energy V received at each sample period is made up of portions of a number of dispersed symbols, Figure 4 illustrates the vector multiplication which will go to produce the autocorrelation matrix of a transversal filter which is used in the feedforward section of a DFE, Figure 5 is a representation of the autocorrelation matrix R after the first b pivots have been processed, Figure 6 is the matrix shown in Figure 5 after simplification of the autocorrelation terms, Figure 7 is the U matrix after b pivots have been processed, Figure 8 illustrates diagrammatically the matrix times vector calculation for deriving the values of c in the equation Uc = p, Figure 9 illustrates diagrammatically the simpification which occurs when solving the equation shown in Figure 8, Figure 10 illustrates diagrammatically the matrix times vector calculation for deriving the values of w in the equation Lw = c, and Figure 11 illustrates the alternative arrangement of the R, w and p matrices in the equation Rw = p which is solved by initially factorising the R matrix so that LUw = p.

The DFE shown in Figure 1 comprises a feedforward filter section 10 and a feedback filter section 12 including a decision stage 14. The feedforward filter section 10 comprises a transversal filter formed by a tapped delay line 16 having five taps. The taps are connected to respective multipliers 18 to 22 in which the signal samples present on the taps are multiplied by respective weighting coefficients wO to w4. The outputs of the multipliers 18 to 22 are summed in an addition stage 24. The sum signal from the stage 24 is applied to one input of an addition stage 26, a second input of which is connected to the output of the feedback stage 12. The addition stage 26 functions as a subtractor. The decision stage 14 is connected between an output of the addition stage 26 and an input to another transversal filter formed by a tapped delay line 28. The input signals to the tapped delay line 28 comprise decisions made by the decision stage 14, these decisions are also provided on an output 30. Respective output signals on the four taps of the delay line 28 are coupled to respective multipliers 31 to 34. Each of the multipliers 31 to 34 multiplies the signal on its input by a tap weighting coefficient w5 to w8, respectively. The outputs of the multipliers 31 to 34 are summed in an addition stage 36 whose output is connected to the addition stage 26.

The basic operation of the illustrated DFE is known and can be summarised by saying that the feedforward filter section accepts the received input signal samples and produces a weighted sum of these input signal samples. This sum includes energy from the current input symbol and from the preceding and following symbols.

The feedback filter section 12 is driven by the preceding data decisions and in so doing eliminates the effects of the preceding symbols on the current decision by the decision stage 14.

The tap weights of the feedforward and feedback filter sections 10, 12 are adjusted so that ideally the portion of the signal which has come from the wanted symbol is as large as possible relative to the portions coming from the subsequent signals.

In the case of a substantially static communications channel, such as a telephone cable, the channel impulse response remains substantially unchanged and hence the tap weights once set do not need to be updated. However in a mobile radio environment, the channel impulse response changes and in consequence the tap weights have to be recalculated. Variations in the channel impulse response can be determined in a number of ways. For example the signal as transmitted includes a special training sequence or a sequence, such as preamble or synchronisation, which can be used for retraining the equaliser.

The weighting coefficients for a DFE can be calculated by solving the known equation w = R-1p (1) as defined by S. Haykin in his book "Introduction to Adaptive Filters", McMillan Publishing Company, London.

The equation can be solved recursively by using adaptive algorithms which avoid performing an explicit matrix inversion due to the complexity of operation. The algorithms assume no initial knowledge of the autocorrelation matrix, R, of the received samples or the cross correlation vector, p, of the desired samples and the received samples, but the calculated values of R and p get closer to the true values after each iteration.

By using some calculation, an estimate of the channel can be obtained and the elements of the matrix R and the vector p can be calculated from this knowledge. Equation (1) can then be restated as Rw = p (A) As R and p are known, w can be found by solving a set of simultaneous equations.

Figure 2 illustrates an autocorrelation matrix R of the DFE shown in Figure 1, which matrix has been partitioned into four sections I to IV. The order of the matrix, that is the number of columns and rows, is determined by the number of taps of the two filter sections 10, 12 shown in Figure 1. The partitioning of the matrix into the 4 sections is indicated by the broken lines 38, 40 which coincide with the boundaries of the two filter orders. The number of taps in the feedforward filter section is f, where f = 5 in this example, and the number of taps in the feedback filter section is b, where b = 4 in this example. A constant representing the noise variance on the received samples, is added to the rO term so that a non-trivial solution to the DFE tap weights is obtained. The value of the on is not too critical, a typical value being 0.3.

The four sections or sub-matrices are as follows: Section I is an f x f sub-matrix which holds the autocorrelation terms of the channel impulse response, its symmetrical form is termed a Toeplitz matrix and Hermitian if the channel is complex. In the present example the order of section I is equal to the number of terms comprising the autocorrelation of the channel. However, if the order of section I is greater than the number of terms then section I will be a banded matrix with the terms outside the band being zero.

Section II; is an f x b sub-matrix. It contains the part of the channel impulse response which is to be cancelled by the feedback filter section 12 in Figure 1.

Section III is a b x f sub-matrix which is a mirror image or transpose matrix of section II as the matrix is symmetrical.

Section IV is a b x b identity sub-matrix. This sub-matrix assumes that the average power level of the transmitted signal is unity, as is generally the case.

It was said in connection with equation (A) above that assuming R and p are known, w can be found by solving a set of simultaneous equations. In one embodiment of the method in accordance with the present invention, the simultaneous equations are solved by first factorising the auotocorrelation matrix R into two triangular matrices by using a method known as UL factorisation. More particularly the technique of UL factorisation is to decompose the autocorrelation matrix R into two triangular matrices, one upper triangular matrix U and one lower triangular matrix L. This is achieved by applying Gaussian eliminlation to the autocorrelation matrix. Equation (A) can now be written as ULw = p (B) where UL = R.

The procedure for obtaining a solution for w is done in two stages, firstly by solving Uc = p (C) for the vector c, and secondly by solving Lw = c (D) for the desired tap weight vector w.

As the U and L matrices are in triangular form the solutions for the unknown vectors c and w are easily found by back substitution.

In general it requires approximately N3/3 operations to obtain w by Gaussian elimination, where N is the order of the autocorrelation matrix R. By virtue of the fact that the autocorrelation matrix R is banded then it is well suited to this form of efficiently calculating the tap weights of the DFE in one (non-recursive) operation.

Before discussing factorisation using Gaussian elimination, it may be helpful to the understanding of the present invention if consideration is given to the dispersion of the symbols as transmitted. The channel impulse response can be represented by a transversal filter which in this example has five taps, weighted h1, h2, h3, h4 and h5. Thus a transmitted symbol dk has its energy spread over five sample periods when received so that the proportions received in successive periods are dkh1, dkh2, dkh3, dkh4 and dkh5. However, due to intersymbol interference, at any one sampling period the received samples comprise the sum of samples of five symbols plus noise, n.For ease of reference these have been tabulated in Figure 3, where dk represents the currently transmitted symbol, dk1 to dk-4 Preceding symbols and dk+1 to dk+4 the following symbols.

The successive inputs V can be represented as a vector and the autocorrelation matrix can be obtained by multiplying the two vectors together as shown in Figure 4. The representation of the autocorrelation matrix can be simplified as follows: di x dj = 1 if i = j di x dj = O if i does not equal j Also the autocorrelation terms are defined as follows:

(where * denotes conjugation if the samples are complex) for example rO = h12 + h22 + h32 + h42 + h52 + cn when synthesising the rO term from the channel impulse response a value corresponding to the noise term n should be included.

for example ri = h1h2+h2h3+h3h4+h4h5.

for example r2 = hlh3+h2h4 + h3h5 similarly r3 = h1h4 + h2h5, and r4 = hlh5.

For convenience the values of r have been written in above the product terms in Figure 4. Expressed in terms of r the autocorrelation matrix shown in Figure 4 is the same as section I in Figure 2.

Section IV of the autocorrelation matrix R shown in Figure 2 is determined by the autocorrelation of the feedback filtering section of the DFE and is compiled by multiplying a 4 by 1 matrix by a 1 by 4 matrix. In this case the matrices are formed by the previous decisions dk-1 to do~4.

From the foregoing explanation it can be deduced that Figure 2 is formed by multiplying 9 x 1 matrix by a 1 x 9 matrix, the terms in each of these matrices being Vk+4 to Vk, dk1 to dk4.

The factorisation of the autocorrelation matrix R shown in Figure 2 using Gaussian elimination will for convenience commence in section IV because of the presence of zeros which reduces the complexity. A UL factorisation is obtained rather than an LU one by this approach. As will become apparent from the subsequent explanation the first b elimination steps can be predicted which allows an intermediate factorisation stage to be formed directly.

This amounts to reducing the factorisation from an order f + b to an order f which corresponds to the order of the feedforward filtering section.

The factorisation process comprises reducing the matrix R to to a U matrix comprising the multipliers and an L matrix comprising the residues. The terms above the diagonal line 42 (Figure 2) will be reduced to zero during the factorisation. For convenience, the rows and columns will be numbered 1 to 9 respectively, from the top left hand corner. Also a position on any one of the matrices will be identified by the letter of the matrix, for example R, U or L, followed by the numbers of the row and column in that order.

In order to factorise these terms, one considers each pivot point in turn. The pivot points are the rO and 1 terms lying immediately along the diagonal 42 at the positions R1,1 to R9,9.

Commencing at the pivot point R9,9, looking up column 9 there is a non-zero term h5 in row 5. In order to eliminate this term, a multiple of this pivot when subtracted from the non-zero term (h5) must reduce the term to zero, that is h5 - lx = 0 therefore x = h5 In this instance, as well as with all the cases where the pivot point is 1, the multiple will be the term to be eliminated.

The non-zero terms in the row 9 containing the pivot point are multipled by h5 and subtracted from the corresponding term in row 5. Thus, in row 9 the h5 term at R9,5 is multiplied by h5 so that the product is h52 and this is subtracted from rO at R5,5 so that this term becomes rO - h52. Likewise, the multiple of the pivot point 1 is h5 and h5 is subtracted from h5 in R5,9 to reduce this term to 0. The multiple h5 is inserted at the location U5,9 in the newly created U matrix (see Figure 7).

Proceeding to the next pivot point at R8,8 which equals 1, it will be noted that column 8 contains two non-zero terms namely h4 at position R5,8 and h5 at position R4,8. Each term is reduced to zero in separate operations. Consider first the elimination of h4 at position R5,8. Because the pivot point has a value I then the multiple is h4 and the non-zero terms in row 8 are multiplied by h4 so that one obtains h4h5 from position R8,4, h42 from position R8,5 and h4 from position R8,8 which are subtracted from the terms in the corresponding columns of row 5, thus the term at position R5,4 becomes r1 - h4h5, the term at position R5,5 becomes rO-h52-h42 and term at R5,8 becomes 0. The multiple h4 is entered at position U5,8 of the U matrix.The second non-zero term h5 in column 8 becomes the next multiple of the non-zero terms in row 8 which become h52, h4h5 and h5. These product terms are subtracted from the corresponding terms in row 4 so that the term at position R4,4 becomes rO-h52, the term at position R4,5 becomes rl-h4h5 and the term at position R4,8 becomes 0. As before the multiplier h5 is entered into the U matrix at position U4,8 (see Figure 7). The process is repeated for successive pivot points with the result that the L matrix comprises a matrix based on a modified R matrix but with all the terms above the diagonal 42 (Figure 2) reduced to zero; the modifications to the R matrix resulting from the Gaussian elimination. A U matrix is formed simultaneously with the factorisation of the R matrix and comprises non-zero terms above a diagonal.

Figure 5 shows the R matrix or more accurately the partly formed residue matrix L after the first b (where b = 4 in the example) pivots have been processed. Since the terms rO, rl, r2, r3 and r4 are defined in terms of h, then Figure 5 can be simplified as shown in Figure 6. Figure 7 shows the U matrix after b pivots have been processed. Both the matrices shown in Figures 6 and 7 are incomplete because the remainder of the pivot points have to be processed and since they have values different from unity some computation is required because they are not simple factors.

An inspection of the partly constructed L and U matrices shows that the terms are expressed as samples of the channel impulse response. Also, when factorising the autocorrelation matrix using Gaussian elimination, it is evident that where the pivot points are unity many of the terms in the L and U matrices can be entered without computation thus saving power and time.

Having now created the L and U matrices, the equation (B) has been reached. The solving of equation (C), that is Uc = p, is now implemented, the cross correlation vector p being p H = h5 h4 h3 h2 hl :0000 It will be noted that the last b terms of pH, where H denotes complex conjugate transpose, are always zero. Hence the U matrix can be truncated down to an f x f matrix of section I. In consequence the factors corresponding to the first b pivots are not required. Equation (C) yields the f non-zero elements of c.

More particularly, referring to Figure 8 which shows the equation (C) in matrix form, the calculation commences by calculating c9: U row 9 x c = p9 = 0 c9 : Next calculate c8: U row 8 x c = p8 = 0 C8 = 0 The process is repeated for all rows Urow (f+b) to Urow (f+1), inclusive. The resulting values of c(f+b) to c(f+1) are always zero so that there is no necessity to solve the equations as the answers are known.

Hence section IV of the U matrix (Figure 8) is not required as c(f+1) to c(f+b) are known to be zero.

The next operation is to calculate c5: Urow 5 x c = p5 c5 + h2c6 + h3c7 + h4c8 + h5c9 = p5.

From the foregoing calculations it has been determined that c6 to c9 are zero, so that the equation simplifies to c5 = p5.

It will be noted that the row 5 elements h2 to h5 of section II of the U matrix (Figure 7) have not been used.

The operations continue by calculating c4, Urow 4 x c = p4 thus c4 + U4,4c5 + h3c6 + h4c7 + h5c8 = p4.

Again the elements h3 to h5 of section II of the U matrix are not used so that the calculation simplifies to c4 + U4,4c5 = p4 and c4 = p4-U4,4c5.

The calculations are repeated for c3, c2 and cl and it will be seen that none of section II of the U matrix (Figure 7) is used.

As section III of the U matrix contains all zeros this too need not be formed when creating the U matrix. The U matrix has now been simplified to section I so that Figure 8 is reduced to Figure 9. It should be noted however that the equations can no longer be written in matrix form because of the matrix order mismatch.

Having now obtained the values for cl to c9 and formed the residue matrix L by the Gaussian elimination operation, the final set of calculations is to solve equation (D). This equation is shown in matrix form in Figure 10. One starts by solving wO using the equation L1,1wO = cl hence wO = ci ILl Continuing to row 2: L2,1xwO + L2,2xw1 = c2 hence wl = c2 - L2,1xw1 L2,2 and so on until all the values of w have been calculated.

The foregoing calculations to solve the DFE vector can be summarised as comprising the steps: 1. Initialise the U matrix to a fxf identity matrix, where f is the number of taps in the feedforward filter section.

2. Form the intermediate L matrix from the channel estimate, add the noise term dn to rO.

3. Complete the factorisation of the fxf section I of the L matrix saving the factors to the U matrix.

4. Solve equation (C), that is Uc = p, for the fxl c vector.

5. Solve equation (D), that is Lw = c, for the (f+b)x1 w vector, where b is the number of taps in the feedback section.

Referring to Figure 2, the feedback filtering section II requires an order b to cancel all 4 of the previously transmitted bits, this is the general case when 5 channel impulses are present. If the channel impulse response has less than 5 impulses or samples, then the matrices will have more zero elements. The number of non-zero terms in the autocorrelation matrix R determines the complexity needed to solve for the tap weights w. Hence the lower the number of impulses in the channel impulse response, the lower the complexity in calculating the tap weights. This relationship shows that the complexity of the method in accordance with the present invention is dependent upon the channel and not purely on the filter orders.The known methods using recursive algorithms do not have this advantage as their complexity is governed by the filter orders and is independent of the channel impulse response.

For the sake of completeness, forming the normal LU factorisation, instead of the UL factorisation already discussed, will be described briefly. In order to do this the R matrix and w and p vectors are rearranged as shown in Figure 11. The factorisation process then proceeds from the normal position of the top right hand element of the R matrix shown in Figure 11.

The equations are now LUw = p (B') solve Lc = p (C') then solve Uw = c (D').

All the savings in processing time and power obtained by applying the UL solution also apply to the LU form.

The method in accordance with the present invention is suited to other environments besides radio in which a matrix similar to the autocorrelation matrix R can be compiled. The efficiency of this method depends upon the special nature of the autocorrelation matrix R.

From reading the present disclosure, other modifications will be apparent to persons skilled in the art. Such modifications may involve other features which are already known in the calculation of tap weights for decision feedback equalisers and in design, manufacture and use of decision feedback equalisers and component parts thereof and which may be used instead of or in addition to features already described herein. Although claims have been formulated in this application to particular combinations of features, it should be understood that the scope of the disclosure of the present application also includes any novel feature or any novel combination of features disclosed herein either explicitly or implicitly or any generalisation thereof, whether or not it relates to the same invention as presently claimed in any claim and whether or not it mitigates any or all of the same technical problems as does the present invention. The applicants hereby give notice that new claims may be formulated to such features and/or combinations of such features during the prosecution of the present application or of any further application derived therefrom.

Claims (11)

1. A method of determining the tap weights of a decision feedback equaliser, comprising deriving an autocorrelation matrix R of the received samples, deriving the cross correlation vector p of the desired and the received samples and solving the equation Rw = p (A) where w is the vector of the equaliser tap weights.

2. A method as claimed in Claim 1, wherein equation (A) is rewritten by factorising the autocorrelation matrix R into upper (U) and lower (L) triangular matrices so that ULw = p (B)

3. A method as claimed in Claim 2, wherein equation (B) is solved by firstly solving Uc = p (C) where c is an unknown, and secondly solving: Lw = c (D)

4. A method as claimed in Claim 1, wherein equation (A) is rewritten by factorising the autocorrelation matrix R into lower (L) and upper tU) triangular matrices so that LUw = p toe')

5. A method as claimed in Claim 4, wherein equation (B') is solved firstly by solving: Lc = p (C') where c is an unknown, and secondly solving: Uw = c (D')

6. A method as claimed in any one of Claims 2 to 5, wherein the factorising of equation (A) is by a Gaussian elimination technique.

7. A method as claimed in Claim 6, wherein when the pivot points of the R matrix are unity, the relevant terms in the U and L matrices comprise terms from channel impulse response which are entered without computation.

8. A method as claimed in Claim 7, further comprising training the decision feedback equaliser by means of a training sequence.

9. A method as claimed in Claim 3 or Claim 5 or Claim 6,7 or 8, when appended to Claim 3 or Claim 5, wherein when solving equation (C) or (C'), zeros are inserted wherever appropriate in response to the presence of zeros in the cross-correlation vector P.

10. A method of determining the tap weights of a decision feedback equaliser, substantially as hereinbefore described with reference to and as shown in the accompanying drawings.

11. A communications equipment including a receiver having a decision feedback equaliser comprising a feedforward filter section and a feedback filter section including a decision stage, each said filter section comprising a transversal filter the tap weights of which are determined by the method in accordance with any one of Claims 1 to 10.