GB2229898A

GB2229898A - "Determining the coefficients of a transversal equalizer"

Info

Publication number: GB2229898A
Application number: GB9005510A
Authority: GB
Inventors: Risto Kari
Original assignee: Nokia Data Systems Oy
Current assignee: ICL Personal Systems Oy
Priority date: 1989-03-13
Filing date: 1990-03-12
Publication date: 1990-10-03
Anticipated expiration: 2010-03-12
Also published as: GB2229898B; FR2644654B1; FI891186A; SE9000854D0; FI891186A0; GB9005510D0; DE4007989A1; SE510915C2; FI82336B; FR2644654A1; SE9000854L; FI82336C; DE4007989B4

Description

DESCIRIPTION OF INVENTION 1 A method of determining the coefficients of a

transversal equalizer The invention relates to a method according to the preamble of claim 1 f or the determination of the initial values of the coefficients of a transversal equalizer in the receiver of a data communication system.

In a synchronous data communication system, the data to be transmitted is in the f orm of a sequence of bits. In the transmitter (such as a modem) the bits are converted into signalling symbols which are then transmitted over the communication channel at a given signalling rate 1/T, where T is the symbol interval. In the receiver (such as a modem), the received signals are detected and again converted into a sequence of data bits. The communication channel degrades the transmitted signal by various sources of interference, including linear distortion (amplitude and delay distortion) and noise.

To alleviate this problem, the system may be provided with an adaptive equalizer, such as a transversal filter with variable tap coefficients and a tap interval T equal to or smaller than (fractionally-spaced equalizer) the symbol interval T' of the signal.

One method of calculating the initial values of the coefficients of the fractionally-spaced transversal equalizer is disclosed in Rapid Training of a Voiceband Data-Modem Receiver Employing an Equalizer with FractionalT Spaced Coefficients, IEEE Transactions on Communications, Vol. COM-35, p. 869-876, Oct. 1987.

In this prior art method, the data transmitted over a communication channel is preceded by a known

2 cyclic sequence of symbols called a training sequence. The transfer function H(k) of the channel is estimated by first calculating the discrete Fourier transform (DFT) R(k) of at least one period of the received training signal and then dividing it with the DFT S(k) of the transmitted training sequence. The transfer function C(k) of the equalizer is obtained from the ratio C(k) = A(k)/H(k), where A(k) is the reference spectrum, that is, the desired corrected transfer function (the common transfer function of the communication channel and the equalizer). The equalizer coefficients are obtained by inverse DFT of C(k).

The f ractionally- spaced equalizer (T<T'), however, has an infinite number of solutions, and a suitable criterion has to be applied to select a suitable solution.

According to the above-mentioned article, the equalizer spectrum is selected so as to minimize the gain of white noise at the input of the equalizer (cf. Eq. 16 of the article). For a high signal-tonoise ratio and a short equalizer, this, however, is not a good approach because in this case most of the residual- error is caused by the aliasing of the equalizer impulse response.

In many applications, such as in polled multipoint networks, it is also preferable to minimize the length of the training sequence. On the contrary, the ability of the equalizer to compensate for the distortion of the communication channel decreases with decreasing length of training sequence.

The object of the invention is to provide a method of determining the coefficients of a transversal equalizer, in which the criterion applied allows for the aliasing of the equalizer impulse re- 1 3 sponse and which enables the initialization of the equalizer to be performed by a shorter training period than previously.

This is achieved by means of a method of the invention, which is characterized by the features disclosed in claim 1.

The invention is based on the idea that a rapidly decaying impulse response of the equalizer minimizes the aliasing of the equalizer impulse response. Thus it could also be said that the amount of the aliasing of the impulse response of the equalizer is minimized by using a reference spectrum A(k) optimized for the transfer function estimated in each particular case.

In one embodiment of the invention, the reference spectrum is optimized and a suitable equalizer spectrum is selected by estimating the group delay of the communication channel and by selecting the reference spectrum or equalizer spectrum so that the resulting equalizer uses frequencies at which the estimated group delay distortion is at minimum. As a result, the equalizer impulse response is more rapidly decaying than the impulse response to be obtained with a fixed reference spectrum. The aliasing problem is thus considerably alleviated, so that a shorter training period can be used. This, in turn, involves improved performance of the data communication system and a shorter equalizer start-up period.

The invention will now be described in greater detail by means of examples with reference to the attached drawing, which shows a typical telephone channel amplitude spectrum and a group delAy function.

The overall structure and operation of the data communication system and the transversal equalizer 1 4 are obvious to one skilled in the art, see, e.g., the above-mentioned article and U. S. Patent Specif ication 4,152,649. The invention can be applied in equalizers described by the article and the patent specification as well as in other suitable equalizers.

The article also describes the principles of the method used for the determination of the coefficients of the transversal equalizer. To facilitate the understanding of the invention, the principles of the method will, however, be described in outline before the description of the inventive idea.

Assume that the baseband-equivalent impulse response of a data communication system is to be equalized by a transversal equalizer having tap spacing KT/L S T, where T is the symbol interval of the signal, and K and L are small integers. Prior to data transmission the transmitter sends the training signal s(t) = E s(i) 8 (t-iT) 1 (1) where the sequence s(i) is periodic with the period M = KN/L and N is the number of equalizer tap coefficients. The transmitted signal propagates through the channel and is sampled in the receiver at the sampling frequency L/T. The received signal samples are x(n) = E s(i)y(nT/L-T-iT)e j 2 n& n T 1 L+W (n) (2) 1 where T is the sampling phase, w(n) represents the summed complex-valued noise and,,,Sf is an unknown constant frequency offset. Other channel imperfections (phase jitter, amplitude jitter, non-linearities, etc.) are assumed to be negligible or included in the included in the noise term w(n).

At the receiver the incoming signal is monitored continuously f or the presence of a cyclic training signal, an estimate for the carrier frequency of f set being calculated as described in the above-mentioned article. As soon as the presence of the cyclic training signal has been detected, one period r(n), n = 0,1,...,LM-1, is extracted from the received signal and used for the calculation of the equalizer tap coefficients. The sequence r(n) is ob,tained by copying LM samples from the delay line of the equalizer and by removing the phase rotation introduced by the carrier frequency offset. The received samples may also be averaged over several periods to reduce the effect of noise and other channel imperfections at the expense of increased training time.

In order to completely equalize the received cyclic sequence r(n), the equalizer tap coefficients c(i) are chosen to satisfy the equation N=1 E c(i)r[(Ln-Ki),.dLMI = s(n), n=0,1,...,M-1 (3) i=o where modLM denotes modulo LM operation. In the frequency domain, this may be expressed as L-1 E C[(k+iM). dNIR(k+iM) = LS(k), k=0,1,...,M-1 (4) i=o where C(k), R(k) and S(k) represent the discrete Fourier transforms of the impulse responses of the 6 equalizer, the received sequence and the training sequence, respectively. Eq. (4) may also be expressed in terms of a reference spectrum A(k) as C(k,,dN)R(k) = A(k)S(km.am), k=0,1,...,LM-1 (5) This equation is equivalent to Eq. (4), if the reference spectrum A(k) satisfies the Nyqvist criterion.

L-1 E Mk+Mi) = 1, k=0,1,...,M-1 i=o (6) The reference spectrum A(k) refers to a desired spectrum of the equalized channel before sampling at the symbol frequency.

For a T-spaced equalizer the calculation of the equalizer spectrum is straightforward because there is at most one solution for Eq. (4) or (5). The equalizer spectrum is obtained from C(k) = S(k)/R(k), k=0,1,..., N-1 (7) provided that R(k) is different from zero for all values of k.

The spectrum for a fractionally-spaced equalizer is more difficult to calculate because Eq. (4) yields only M equations and the number of the variables C(k) is N > M. It appears from Eq. (5) that there is an infinite number of solutions for a fractionally-spaced equalizer unless there are L spectral nulls at intervals of 1/T Hz in the spectrum of the received signal. A suitable criterion has to be applied to select an appropriate one out of the infinite number of solutions given by Eq. (4).

Ideally, a reference spectrum A(k) should be 7 found which gives the optimum N-length equalizer for the channel impulse response. This is generally not possible since it requires knowledge about the channel transfer function at all frequencies and the reference spectrum for the optimum length-N equalizer does not necessarily satisfy the Nyqvist criterion. As a result, some suboptimum approach must be used.

Depending on the reference spectrum, the resulting equalizer is a more or less aliased version of the "imaginary" infinite-length equalizer. However, the infinite-length equalizer will not be calculated at any stage. In theory, an infinite-length equalizer could be calculated by increasing the period of the training sequence infinitely.

In the method described in the above-mentioned article the reference spectrum is selected so that the noise gain is minimized. With a high signal-tonoise ratio and/or a short equalizer, however, this is not a good approach.

According to the invention the aliasing of the equalizer impulse response of the equalizer is minimized by the criterion N-1 J = E f(k)IC(k) 1 2 k=0 (8) where f(k) is a weight function that puts more weight on frequencies difficult in terms of aliasing and less weight an frequencies easy in that respect.

The weight function f(k) is selected so that the resulting infinitelength equalizer impulse response becomes as rapidly decaying as possible. In other words, variation in the spectrum of the group delay of the equalizer is minimized at frequencies 1 8 where the channel amplitude function is sign:Lf icant. At the same time abrupt changes in the amplitude of the equalizer spectrum are avoided, because they also result in slowly decaying equalizer impulse response.

If the reference spectrum is assumed to have a linear phase, the equalizer group delay function becomes necessarily a mirror image of the channel group delay. In one preferred embodiment of the method of the invention, the equalizer amplitude is therefore minimized at frequencies where the channel group delay dif fers much from the average while keeping the equalizer amplitude spectrum as smooth as possible.

This is achieved if the weight function is selected to be, for instance, F(k) = J-c(k)--ca,. J' + F.

(9) where z(k) is the estimated group delay of the channel at frequency k and -c... is the average group delay and n k 1. Smoothness of the equalizer spectrum is guaranteed by adding a positive constant F, to the squared group delay difference. As a result of this weight function, the equalizer uses mainly frequencies. at which the group delay is close to the average, and amplifies less frequencies at which the group delay differs much from the average. The weight function F(k) may also be some other function of the group delay difference.

For instance, it may be seen from the figure, which shows the amplitude spectrum D and the group delay E of a typical severely distorted telephone channel, that the frequencies A and B overlap when the output signal of the equalizer is sampled at intervals of 1/T. It is thereby preferable to modify the equalizer spectrum so that it uses mainly the 9 1 frequency B since the channel group delay deviates much more from the average at the frequency A.

For the weight function F(k) described above, the channel group delay function has to be determined. Given a channel transfer function H(f) in complex exponential form as H(f) = IH(f) jel 9 If) where 9(f) is the continuous phase response, the group delay function of the channel is T(f) = - I- dg(f) 2ndf In the method of the invention, estimation of the channel group delay consists of calculating the phase of the channel transfer fun-tion H(k) = R(k)/S(k) and computing from it the group delay function by numerical derivation.

As the phase function of H(k) is calculated using a standard inverse tangent routine, all the resulting values range between -n and n. In order to obtain samples of the continuous phase function, it is necessary to add an appropriate integer multiple of 2n to the samples of the principal value. The right multiple of 2n can be determined from the samples of the principal value, if the samples are sufficiently close together for the discontinuities to be detected.

For a very short equalizer and a severely distorted channel, the channel group delay variation may be much larger than the time span of the equalizer. This means that the phase difference between two ad- 1 jacent frequencies in H(k) may be larger than n, making it impossible to determine which multiple of 2n should be added to the principal value of the phase.

In the method of the invention this problem is solved by assuming that the channel group delay cannot have abrupt changes between two adjacent frequency points, so that the correct multiple of 2n for the phase function at a predetermined frequency point can be selected by observing the difference between the group delay value calculated at this particular point and a group delay value calculated at one or more preceding frequency points and by comparing it to the general known group delay characteristics of the communication channel. With the telephone channel, for instance, is it known that the group delay function should be a generally parabolic function (see the figure).

The calculation of the group delay function is preferably started from the DC frequency or some other frequency at which the group delay is assumed to be small, then proceeding separately towards the two band edges.

The examples presented above are only intended to illustrate the invention. In its details, the method of the invention may vary within the scope of the attached claims.

11

Claims

Claims:

1. A method of determining the initial values of the coefficients of a fractional-type transversal equalizer in a data communication system comprising a communication channel, the method comprising the steps of a) transmitting a predetermined periodic data sequence with the discrete Fourier transform S(k) through the communication channel; b) calculating the discrete Fourier transform R(k) of one period of the periodic sequence passed through the communication channel; c) determining the ratio C(k) = A(k)S(k)/R(k), where A(k) is the reference spectrum; and d) determining the values of the coef f icients of the equalizer by calculating the inverse discrete Fourier transform of the ratio C(k), c h a r a c t e r i z e d in that the reference spectrum A(k) is chosen in step c) so as to give the equalizer an impulse response as rapidly decaying as possible.

2. A method according to claim 1, c h a r a c t e r i z e d in that the reference spectrum A(k) is chosen by means of the estimated group delay of the communication channel by decreasing the amplitude of the equalizer at frequencies where the group delay is difficult to the equalizer, and that no rapid changes occur in the amplitude of the equalizer spectrum.

3. A method according to claim 1 or 2, c h a ra c t e r i z e d in that the reference spectrum A(k) is chosen so that the equalizer uses mainly frequencies at which the estimated group delay of the communication channel is close to the average of the group delay, and amplifies less frequencies at which 12 the group delay differs considerably from the average.

4. A method according to claim 2 or 3, c h a ra c t e r i z e d in that the determination of the group delay of the communication channel comprises calculating the phase of the estimated transfer function of the communication channel and calculating the group delay from the phase by numerical derivation.

5. A method according to claim 4, c h a r a c t e r i z e d In that the determination of the group delay of the communication channel comprises the steps of a) calculating the phase of the transfer function of the communication channel using an inverse tangent routine giving phase values ranging between -n and n; b) forming a continuous phase function from the calculated phase values by adding an appropriate integer multiple of 2n to each one of the initial phases values and calculating the group delay values of the communication channel by numerical derivation from the values of the continuous phase function, the right multiple of 2n being chosen so that the resulting group delay value together with the group delay values calculated at preceding frequency points best follow the assumed group delay function of the communication channel.

6. A method according to claim 5, c h a r a c t e r i z e d in that the right multiple of 2n is chosen so that the resulting group delay value differs least from the group delay value calculated at the preceding frequency point.

7. A method according to claim or 6, c h a r a c t e r i z e d in that the calculation of the group delay is started from the frequency at which 1 13 the group delay of the communication channel is assumed to be at smallest.

8. A method according to claim 7, c h a r a c t e r i z e d in that the right multiple of 2n is chosen so that the resulting group delay value is a larger value differing least from the group delay value calculated at the preceding frequency point.

9. A method according to claim 1 and substantially as hereinbefore described.

10. Any novel feature or combination of features described herein.

I>ublished 1990 at The Patent Office. State House. 6671 High Holborn. London WC1R4TP. Further copies maybe obtLMedfrorn The Patent =ice Sales Branch, St M&ry Cray. Orpington. Kent BR5 5RD- Printed by Multiplex techniques ltd. St Mary CrAY, Kent, Con. 1187