FR2783119A1 - Digital equalizing method for signals received over radio channel, using zero forcing method to produce intermediate signal, then obtaining estimates of symbols by application of Viterbi algorithm - Google Patents

Abstract

The W roots of the Z transform of the channel impulse response of W+1 coefficients are determined by producing an intermediate signal, equalizing the received signal by a zero forcing method based on an impulse response whose Z transform is a polynomial in z<-1> of degree W-p, having as roots those of W which are furthest from the circle of unity. Symbol estimates are then obtained by Viterbi equalization of the intermediate signal based on an impulse response whose Z transform is a polynomial in z<-1> of degree p, having as roots those of W which are closest to the circle of unity. (NOTE - The technical content of this patent is identical to that of FR2783120A1, which claims this patent as priority). An Independent claim is included for a radio receiver implementing the method of the invention.

Description

DIGITAL EQUALIZATION PROCESS, AND RECEIVER

OF RADIOCOMMUNICATION IMPLEMENTING SUCH A PROCESS

The present invention relates to the digital equalization of signals. It finds an important application in the field of radiocommunications. The method applies when a signal from a transmitter is received via a transmission channel between transmitter and receiver, the response of which is known or has been estimated beforehand. A main problem which then arises is that of the compromise between

the performance of the equalizer and its complexity.

A complete estimation, according to the maximum likelihood, of all the discrete symbols composing the transmitted signal is possible, for example by employing the Viterbi algorithm (see GD Forney Jr .: "The Viterbi Algorithm", Proc. Of the IEEE, Vol. 61, No. 3, March 1973, pages 268-278). However, as soon as the impulse response of the channels becomes long or the number of possible discrete values of the symbols becomes large, the exponential complexity of these methods makes them difficult.

makes it impractical.

We consider the case of a radiocommunication channel serving for the transmission of a signal composed of successive sequences or frames of n symbols dk (l <k <n). The dk symbols have discrete values: binary (1) in the case of a BPSK (binary phase shift keying) type modulation, quaternary (1 j) in the case of a modulation

QPSK type (quaternary phase shift keying) ...

After conversion to baseband, digitization and adapted filtering, a vector Y of the received signal reflecting the symbols transmitted over the duration of a frame has the following expression: 2 - rO 00 0 d1 YN, 1 Y l À d Y2 rl r0 0 d2 YN, 2 r1 r0 0 rO

= = = A.D + YN (1)

Y Yk rw r1 dk + YN, k 0 rw 0 0 rw 0YL 0 0 rW dn YN, L o W + 1 is the length, in number of bits, of the estimated impulse response of the channel, r = (r0, rl, ..., rw) is the estimated impulse response of the channel, the rq being complex numbers such that rq = O if q <0 or q> W, Yk is the k-th complex sample received with l <k <L = n + W, and YN is a vector of size L composed of additive noise samples YN, k. The estimated impulse response r takes into account the propagation channel, the shaping of the

signal by the transmitter and reception filtering.

The matrix A of size Lxn has a Toeplitz type structure along its main diagonal, i.e. if ai.j denotes the term located at the i-th row and at the j-th column of the matrix A, then ai + l, j + l = ai, j for 1 <i <L-1 and l <j <n-1. The terms of the matrix A are given by: a lj = 0 for l <j <n (A therefore has only zeros above its main diagonal); i, 1 = 0 for W + 1 <i <L

(matrix-band structure); and ail = ril for 1 <i_ <W + l.

The matrix relation (1) expresses that the received signal Y is the observation, affected by an additive noise, of the product of convolution between the impulse response of the channel and the symbols emitted. This convolution product can also be expressed by its transform in Z:

Y (Z) = R (Z) .D (Z) + YN (Z) (2)

o D (Z), Y (Z), R (Z) and YN (Z) are the respective Z - 3 - transforms of the symbols emitted, the signal received, the impulse response of the channel and the noise: E INCORPORATE Equation .2 ECO (3) O INCORPORATE Equation.2 WLE (4) E INCORPORATE Equation.2 OW] (5) A classic solution to solve a system such as (1) is the so-called zero forcing method ), according to which one determines the vector E INCORPORER Equation.2 [LIOZF with n continuous components which minimizes the quadratic error E INCORPORER Equation.2 ED .. A discretization of the components of the vector E INCORPORER Equation.2 ELEZF relating to each channel occurs then, often through a channel decoder. The solution E INCORPORATE Equation. 2 lDzF in the sense of least squares is given by: DzF = (AHA) -IAHY, o AH denotes the conjugated transposed matrix of A. We are then brought back to the problem of the inversion of the definite Hermitian matrix positive AHA. This inversion can be carried out by various classical algorithms, in a direct way (methods of Gauss, Cholesky ...) or by iterative techniques (algorithms of Gauss-Seidel, of the gradient ...). The estimation error D-DzF is equal to (AHA) -lAHYN, which shows that the solution obtained is affected by a variance noise: Y2 = E (| D-DZF) = N0x Trace [(AHA) 1] (6) o No is the power spectral density of the noise. We see that noise enhancement occurs when the AHA matrix is poorly conditioned, that is to say when it has one or more eigenvalues.

close to 0.

This noise amplification is the main drawback of conventional resolution methods. In practice, cases of poor conditioning of the AHA matrix are frequent, particularly in the presence of

multiple propagation paths.

A relatively simple way is known to partially remedy this drawback, by accepting an interference residue in the solution, that is to say by adopting not the optimal solution in the sense of least squares, but the solution: DMMSE = (AHA + No0) -lAHY, where N0 denotes an estimate of the spectral density of the noise, which the receiver must then calculate. This method is known under the name of MMSE (minimum mean square error) It makes it possible to reduce the estimation variant compared to the "zero forcing" method, but in

introducing a bias.

The "zero forcing" and analogous methods amount to operating an inverse filtering of the signal received by a filter, modeling the transfer function 1 / R (Z), calculated by a certain approximation (quadratic in the case of "zero forcing") . When one or more roots of the polynomial R (Z) (equation (5)) are located on the unit circle, the theoretical inverse filter has singularities such that it cannot be estimated by a satisfactory approximation. In the case of the quadratic approximation, this corresponds to the divergence of the variance of the error a2 when the

AHA matrix has a zero eigenvalue (relation (6)).

This problem is not encountered in methods such as the Viterbi algorithm which inherently take into account the discrete nature of symbols, but which require very high computing power.

superior for large systems.

The object of the present invention is to propose an equalization process providing a good compromise between the

reliability of estimates and complexity of the equalizer.

Another object is to obtain an equalizer requiring a reasonable computing power and capable of processing, with performances comparable to those of a Viterbi equalizer, signals whose symbols have a relatively high number of states and / or signals received following a relatively spread impulse response channel. The invention thus proposes a digital equalization method, for estimating discrete symbols of a signal transmitted from digital samples of a signal received via a transmission channel represented by a finite impulse response of W + 1 coefficients, W being an integer greater than 1. This method comprises the following steps: determining the W roots in the complex plane of the Z transform of the impulse response of the channel; - distribute the W roots into a first set of Wp roots and a second set of p roots, p being an integer greater than 0 and smaller than W, the roots of the second set being closer to the unit circle than those of the first set according to a distance criterion determined in the complex plane; - obtain an intermediate signal by applying to the signal received a first equalization method on the basis of a finite impulse response whose Z transform, formed by a Z-1 polynomial of degree Wp, has for roots the Wp roots of first set; and obtaining estimates of the discrete symbols of the transmitted signal by applying to the intermediate signal a second equalization method on the basis of a finite impulse response whose Z transform, formed by a Z-1 polynomial of degree p, has for roots p roots

of the second set.

The "first equalization method" will generally be chosen so as to treat the unknown symbols as continuous variables. She then drives

- 6 -

to an operation similar to an inverse filtering whose transfer function would be of a form approaching the expression 1 / RS (Z), o RS (Z) denotes the polynomial in Z-1 of degree Wp having for roots the roots Wp furthest from the unit circle. It can in particular be of the “zero forcing” type. This operation only generates reduced noise amplification, since the roots of the associated Z-transfer function are relatively

away from the unit circle.

1Q For the p roots closest to the unit circle, measures are adopted which make it possible to overcome or limit the incidence of the problem of noise amplification. An MMSE or the like method can be chosen as the "second equalization method". However, this second method will advantageously take into account the discrete nature of the unknown symbols. It may in particular be based on a trellis algorithm, such as the Viterbi algorithm, the implementation of which is common in channel equalizers when the size of the

system is not too large.

The second equalization method is generally more complex to implement than the first. In each particular case, the choice of the number p makes it possible to search for the best compromise between the reliability of the estimations, which makes the high values of p prefer, and the complexity of the equalizer, which makes the low values of p prefer. Another aspect of the present invention relates to a radio communication receiver, comprising: - conversion means for producing digital samples from a radio signal received via a transmission channel represented by an impulse response finite of W + l coefficients, W being an integer greater than 1; - means for measuring the impulse response - 7 - of the channel; means for calculating the W roots in the complex plane of the Z transform of the measured impulse response; - means for distributing the W roots into a first set of Wp roots and a second set of p roots, p being an integer greater than O and smaller than W, the roots of the second set being closer to the unit circle than those the first set according to a distance criterion determined in the complex plane; - a first equalization stage to obtain an intermediate signal by applying to the signal received a first equalization method on the basis of a finite impulse response whose Z transform, formed by a Z-1 polynomial of degree Wp, has for roots the Wp roots of the first set; and - a second equalization stage for obtaining estimates of discrete symbols of a signal transmitted on the channel by applying to the intermediate signal a second equalization method on the basis of a finite impulse response whose Z transform, formed by a polynomial in Z-1 of degree p, has for roots the p roots

of the second set.

Other peculiarities and advantages of this

invention will appear in the following description

non-limiting exemplary embodiments, with reference to the appended drawings, in which: FIG. 1 is a block diagram of an exemplary radiocommunication receiver according to the invention; FIG. 2 is a flowchart showing an embodiment of the method according to the invention; and - Figure 3 is a diagram illustrating the

process performance.

The receiver shown in FIG. 1 comprises a radio stage 1 which receives the radio signal picked up by the antenna 2 and converts it into baseband. The baseband signal is digitized by an analog-to-digital converter 3, then supplied to a reception filter 4. The filter 4 provides filtering suited to the shaping of the signals by the transmitter. It delivers a digital signal at the rate of a complex sample per

symbol issued.

This digital signal is supplied to a demodulator comprising on the one hand a synchronization module 6 and

channel estimation, and on the other hand an equalizer 7.

The synchronization and the channel estimation are for example carried out in a conventional manner using a synchronization sequence included by the transmitter in each signal frame. The detection of this sequence, known to the receiver, makes it possible on the one hand to synchronize the receiver with respect to the temporal structure of the frames transmitted, and on the other hand to estimate the impulse response r = (r0, r1,

., rw) of the channel on which the frames are transmitted. The calculated impulse response..DTD: by module 6 is supplied to equalizer 7.

The equalizer 7 operates for example in accordance with the flowchart shown in FIG. 2 to process each synchronized frame of the received signal, having (Y1 in the form of a vector Y = y, with L = n + W taking up

previous notations.

The channel estimation module 6 having provided the W + 1 complex coefficients rq of the estimated impulse response of the channel, the first step 10 consists in finding the W roots of the Z transform of this

impulse response, given by equation (5).

Various classical methods of finding complex roots of a polynomial can be used in step 10. In this regard, see the work by 9 - E. DURAND: "Numerical Solutions of Algebraic Equations; Tome I: Equations of Type F (x) = 0 ",

Editions Masson, 1960.

The W complex roots thus found al, a2, ...., W are then ordered so as to be able to divide them into two sets, one containing the Wp roots furthest from the unit circle, and the other the p roots the

closer to the unit circle.

For this, a distance 8q is calculated in step 11 for each of the roots aq (l <qW). This distance is advantageously obtained as follows: 6 Oj s (ql <(7) q {1 - l1 / lq if ql> 1 In step 12, the roots aq of the transfer function R (Z) are sorted in the order of decreasing distances: 81 2 82 2 ...> &. We then separate the first roots Wp al, ..., Caw_p, which are the most distant

of the unit circle, of the remaining p roots awp + l, ..., aw.

In step 13, equalizer 7 develops a polynomial in z-1 defined by: w-p W-p RS (z) = H (1- q.Z1) = E sq. q (8) q = lq = 0 This makes it possible to determine the sq coefficients of the transfer function RS (Z) associated with the impulse response s = (s0, sl, ..., swp) of a virtual channel, which would correspond to the estimated transmission channel with elimination of the contributions closest to the zones

of singularity.

A first equalization 14 can then be carried out, amounting to performing an inverse filtering approaching the transfer function 1 / RS (Z). Several implementations

- 10 -

can be used to perform this reverse filtering.

It is in particular possible to carry out an equalization by “zero forcing” as indicated above. About these

methods, we can refer to the work of J.G.

Proakis: "Digital Communications" McGraw-Hill,

2nd edition, 1989.

The inverse filtering 14 produces an intermediate signal in the form of a vector Y 'of L' = n + p samples Y '' ... fyfL. In the case of a "zero forcing" method, the vector Y 'is obtained by the matrix relation: Y' = (A 'H A') - 1 AH y (9) In expression (9), A 'denotes a matrix of n + W rows and n + p columns having a Toeplitz structure, formed from the sq coefficients of the polynomial RS (Z): s OOO S1 So 0 s1 so'. 0 s1 To 0 A = sw-p s1 (10) 0 sw_p 0 0 sw_p 0. 0 oswp Thanks to the sorting of the roots aq, the eigenvalues of

the matrix A 'H A' are relatively far from 0.

As a variant, the inverse filtering could be carried out by cascading W-p filtering cells each corresponding to the inverse of a transfer function Rq (Z) = 1 - caqZ1, for l <q W-p. If q = 1, the filreinvrs d Rq () inverse filter of R (Z) is impractical. If cCq <1, we can expand 1 / Rq (Z) in the form:

- 11 -

I q * 2 -2 m zm - 1 + q.- Z- + Sq. + .. + q. Z + .. (11) Rq (Z) The development (11) is causal, and stable since the domain of convergence contains the unit circle. The inverse filtering cell can therefore be carried out in transverse form or in recursive form. If czq> 1, we can expand 1 / Rsq (Z) in the form: 1 - -2 22 m zm S - -q. Z. 1 + aq. Z + aq.z + ... + q.Z + ... (12) Rq (Z)) This development (12) is anti-causal and stable. For the realization of the inverse filtering cell, the development (12) is truncated and an implementation in transverse form is adopted. Anti-causation causes delay

corresponding to the length of the response selected.

We note that the developments (11) and (12) justify the criterion of distance to the unit circle ôq

used according to relation (7).

In step 15, the equalizer 7 develops a polynomial of degree p in Z-1, the roots of which correspond to the p roots of R (Z) closest to the unit circle, such that

R (Z) = RS (Z) .RI (Z):

w p RI (Z) = ro. H (1 - q z1) = 1 tq.Zq (13) q = W-p + 1 q = 0 The complex coefficients t define the impulse response q of another virtual transmission channel, whose equalization by a method of type "zero forcing" or similar would pose problems

noise amplification.

The intermediate signal Y 'is then subjected to an equalization according to another method, on the basis of the impulse response t = (t0, tl, ..., tp). This second equalization 16 is advantageously carried out using a

- 12 -

Viterbi lattice (see the aforementioned article by G.D. Forney

Jr., or the aforementioned work by J.G. Proakis).

The second equalization stage 16 produces the estimates dk of the symbols of the frame (1 <k <n). These estimates dk forming the output of the equalizer 7 can be supplied to a deinterlacing module 8 then to a channel decoder 9 which detects and / or corrects

possible transmission errors.

FIG. 3 illustrates the performance of the method ia in the case of the transmission of a signal frame according to the format of the European cellular radiotelephone system GSM, by replacing the binary modulation of the GMSK type by

eight-state phase modulation (8-PSK modulation).

The channel impulse response was truncated to five bit times (W = 4). FIG. 3 shows the dependence between the bit error rate BER, expressed in%, and the ratio Eb / N0 between the energy per bit and the spectral density of the noise, expressed in decibels. BER is that observed in symbol estimates after channel deinterlacing and decoding performed according to methods employed in GSM. Curve I shows the results

provided by the pure "zero forcing" method, that is

say in the borderline case o p = 0. Curve II shows the theoretical result that would be obtained by equalizing the channel

purely with the Viterbi algorithm (limit case o p = W).

In practice, the corresponding trellis should have 84 = 4096 states, so that the equalizer of

Viterbi would be impractical with current techniques.

The difference between curves I and II illustrates the superiority of the Viterbi algorithm which delivers the estimates

according to maximum likelihood.

Curves III and IV show the results obtained by the process according to the invention, respectively in the cases where p = 1 and p = 2. We see the very significant improvement in the results already obtained for the value p = l compared

- 13 -

to pure "zero forcing".

As an indication, the equalization of a GSM signal frame by the pure Viterbi algorithm under the conditions of FIG. 3 would require of the order of 8.45 million floating point operations, or approximately 1.83 Gflops, whereas the implementation of the present invention under the same conditions requires of the order of 19000 floating point operations (, 4.2 Mflops) in the case op = 1, including the search for the roots of R ( Z) and inverse 1 / RS (Z) filtering by the "zero forcing" method. This number is of the order of 129,000 operations (= 28 Mflops) in the case where p = 2, which remains compatible with the power of signal processing processors.

(DSP) currently available.

- 14 -

Claims (8)

R E V E N D I C A T I ON S 1. Method of digital equalization, for estimating discrete symbols (dk) of a signal transmitted from digital samples (yk) of a signal received via a transmission channel represented by an impulse response finite of W + 1 coefficients (ro, rl, ..., rw), W being an integer greater than 1, comprising the following steps: determine the W roots (al, a2, .., Cw) in the complex plane the Z transform (R (Z)) of the impulse response of the channel; - distribute the W roots into a first set of Wp roots (al, ..., W_p) and a second set of p roots (Cw_p + l, ..., aw), p being an integer greater than 0 and more smaller than W, the roots of the second set being closer to the unit circle than those of the first set according to a distance criterion determined in the complex plane; - obtain an intermediate signal (Y ') by applying to the received signal (Y) a first equalization method on the basis of a finite impulse response whose transform in Z (RS (Z)), formed by a polynomial in Z -1 of degree Wp, has for roots the Wp roots of the first set; and - obtaining estimates (dk) of the discrete symbols of the transmitted signal by applying to the intermediate signal a second equalization method on the basis of a finite impulse response whose transform in Z (RI (Z)), formed by a polynomial in Z-1 of degree p, has for roots the p roots of the second set. 2. The method of claim 1, wherein the first equalization method produces the signal. - 15 - intermediate in the form of a vector Y 'of n + p samples (Y' ,, Y'n + p) obtained according to the relation: Y '= (A'H A) -1 Ai H yo Y is a vector formed of n + W samples (yk) of the received signal, and A 'is a matrix of n + W rows and n + p columns having a Toeplitz structure formed from the coefficients (Sq) of said Z-1 polynomial of degree Wp ( Rs (Z)) 3. The method of claim 1 or 2, wherein the second equalization method comprises the implementation of a Viterbi algorithm. 4. Method according to any one of claims 1 to 3, in which the criterion of distance to the unit circle, used to distribute the W roots al, ..., aw of the transform in Z (R (Z)) of the impulse response of the channel between the first and second sets , is expressed by a distance Èq of the form Èq = 1 - q if | q <1, and of the form 8q = 1 - 1 / | qq if q> 1, for 1 <q W. 5. Radio communication receiver, comprising: - conversion means (1,3,4) for producing digital samples (Yk) from a radio signal received via a transmission channel represented by a response finite impulse of W + 1 coefficients (r0, r1, ..., rw), W being an integer greater than 1; - means (6) for measuring the impulse response of the channel; - means for calculating the W roots (al, U2, ..., aw) in the complex plane of the transform in Z (R (Z)) of the measured impulse response; - 16 - - means for distributing the W roots into a first set of Wp roots (al, ..., Wp) and a second set of p roots (, W_p + i, ..., - W), p being an integer plus greater than O and smaller than W, the roots of the second set being closer to the unit circle than those of the first set according to a distance criterion determined in the complex plane; a first equalization stage for obtaining an intermediate signal by applying to the received signal (yk) a first equalization method on the basis of a finite impulse response whose transform in Z (RS (Z)), formed by a polynomial in Z-1 of degree Wp, has for roots the Wp roots of the first set; and a second equalization stage for obtaining estimates (dk) of discrete symbols of a signal transmitted on the channel by applying to the intermediate signal a second equalization method on the basis of a finite impulse response whose transform into Z (RI (Z)), formed by a polynomial in Z-1 of degree p. has for roots the p roots of the second set. 6. Receiver according to claim 5, wherein the first equalization stage is arranged to produce the intermediate signal in the form of a vector Y 'of n + p samples (Y'1,' .. Y'n + p ) obtained according to the relation: Y '= (A'H A') - 'A'H yo Y is a vector formed of n + W samples (yk) of the received signal, and A' is a matrix of n + W lines and n + p columns having a Toeplitz structure formed from the coefficients (Sq) of said Z-1 polynomial of degree Wp (RS (Z)). - 17 - 7. Receiver according to claim 5 or 6, wherein the second equalization stage is arranged for implement a Viterbi algorithm. 8. Receiver according to any one of the claims 5 to 7, wherein the means of distribution of the roots use, to distribute the W roots between the first and second sets, a criterion of distance to the unit circle expressed by a distance q of the form 3q = 1 - | q if cql <1, and of the form 6q = 1 - 1 / | aq if | q> 1, for l1 <qW.