CA1194165A - Method of reducing the convergence time of an echo canceller and apparatus for carrying out said method - Google Patents

Abstract

ABSTRACT:

A method of reducing the convergence time of an echo can-celler including a transversal filter having N coefficients comprises at least the following steps:
- transmission of a training data signal D(n) constituted by data trans-mitted at instants nT, T being the data interval, and being periodically reproduced after a duration LT at least equal to NT and having the property:

where d and d* are the value of the data signal D(n) and its complex conjugate value, respectively;
- calculation of the coefficients of the transversal filter carried out after the instant of appearance of the echo signal produced in response to the training signal in accordance with the expression:

Description

~4~5 PHF 82-548 l 03-06-1983 Method of reducing the convergenee time of an eeho canceller and apparatus for earrying out said method.

The invention relates to a method of redueing the eonver-genee time of an echo eaneeller eonneeted in a transceiver arrangement between one-way transmit and reeeive paths coupled to a two-way path and used to cancel an echo signal produeed in the receive path in response to a signal supplied to the transmit path, said echo canceller eomprising a transversal filter having N controllable eoe ffieients for processing a signal derived from the signal supplied to the transmit path and a difference circuit for producing a difference signal between two signals which are formed from the signal in the reeeive path and the output signal of the transversal filter, respectively.
Echo cancellers are used, for example, in data trans-mission modems whose transmit and receive paths, together forming a four-wire aceess, are often eoupled by a eircuit known as hybrid junetion in a manner such that each modem has a two-wire access to the exterior. It is known that, when establishing a connection between two modems by their t~-wire access, an unti~ely signal may ~e produced in the receive path of each mcdem, said signal termed echo signal being a fraction of the signal in the transmit path of the same mcdem and being due to imperfections of the coupling cireuit and/or to signal reflections in the conneetion. An eeho eanceller has for its object to automatieally caneel this echo signal in order to permit simultaneous full-duplex transmission ~etween two modems connected by their two-wire aecess.
In an eeho eanceller, the coefficients of the transversal filter are controlled to minimize the mean square value oE the difference signal or error signal appearing at the output of the difference circuit. When the echo canceller has converged, the coefficients of the transversal filter are substantially equal to samples of the impulse response of the echo path and the transversal filter provides an echo copy signal substantially equal to the echo signal at the sampling instants.
A disadvantage of the known eeho cancellers is that their convergence time is generally long. The coefficients are ~4~

iteratively controlled by using the gradient algorithm, according to which the correction term of a coefficient at each iteration is the product of a weighting coefficient ~ less than 1, the difference signal and a transmitted datum. In the prior art echo cancellers, this 5 iterative mcde of coefficient control is used both during the initializing period of the eoefficients and during the tracking period in the eourse of the data transmission proper; see, for example, in this respeet the articel of Kurt H. Mueller, entitled: "A new Digital Echo Canceller for Tw~Wire Full Duplex Data Transmission", published in IEEE Trans-10 actions, Vol. CC~q-24, No. 9, September 1976, pages 957 to 962. During the follow-up period, in order to avoid that during full-duplex transmission the echo cancelling process will be disturbed by the data signal originating from the remote transmitter and superimposed on the echo signal in the error signal, a weighting coefficient 9~ of l5 very low value is used, which implies small corrections of the coefficients and a very long convergence time amounting to several seeonds, whieh may adversely affect the tracking possibilities of the echo canceller.
For instanee from the aforesaid artiele, it is known 20 to transmit a particular se~uence known as maximum length sequence during the initializing period in order to produce an echo signal and to correct iteratively the eoefficients of the transversal filter by using a weighting coefficient o( = 1/N, N characterizing the maximum delay produced by the transversal filter, said fixed coefficient leading 25 to the fastest eonvergence possible. It is also known to improve t~is iterative correction process in order to reduce to some extent the eonvergenee time during the initializing period by using a weighting eoeffieient o~ which is variable in the course of said period, taking serval deereasing values. At the beginning of the initializing period, 30 the eeho signal is relatively high with respect to the noise which ineludes the signal from the remote transmitter and a weighting eoeffieient o( of high value may be used, whieh permits a fairly important eorreetion of the coefficients, whereas towards the end of said period, where the echo signal tends tcwards zero, a small 35 weighting eoeffieient ~( has to be used, whieh provides only a small correetion of the eoefficients of the transversal filter.
It will be obvious, however, that even after this improve~ent an iterative method of coefficient eontrol during the PHF 82-5~8 3 03-06-1983 initializingperiodcannot lead to a very fast convergence of the echo canceller since the coefficients tend asymptotically towards their optimum values and towards the end of the initializing period sub-stantially the same conditions prevail as in the tracking period with coefficient corrections which are necessarily very small.
The present invention provides a quite different method of reducing the convergence time of an echo canceller during the initializing period by utilizing the fact that in a modem provided with an echo canceller it is possible to locally transmit a particular data sequence selected for producing an echo signal that can be rapidly cancelled without successive corrections of the coefficients of the transversal filter.
According to the invention a method of reducing the convergence time of an echo canceller comprises at least the following steps:
- transmission of a training data signal D~n) constituted by data transmitted at instants nT, T being the data interval, and being periodically reproduced after a duration LT equal to at least NT and satisfying the property:

> d(n) . d~ l(n-i)mcdulo L~ = O for i ~ O and n = 0 1 ~ i ~ N-1 where d and dx are the value of a datum of the data signal D(n) and its complex conjugate value, respectively;
- calculation of the coefficients of the transversal filter carried out after the instant of appearance of the echo signal produced in response to said training signal in accordance with the expression:

C CO L ~2 ~ e(n) . D (n) n = 0 where CO and C are the vectors of the N coefficients of the trans-versal filter at the beginning and at the end of the period of calcu-lating the coefficients, respectively, e(n) is the difference signal, D~(n) is the vector of the complex conjugate values of the N data stored in the transversal filter, 1~9416~

~2 is a constant term representative of the power of each of the transmitted data.
If the echo signal to be cancelled does not include a d.c. component, which occurs, for example, in a modem where the echo is produced by a modulated carrier signal and where cancellation is effected on the received signal prior to its demodulation, it is also possible to use as a training signal maximum length sequences having the property:

~ d(n) . d~ ~(n-i) modulo L ~ = -1 for i ~ 0 and n = 0 1 ~ i ~ N-1 The method according to the invention thus permits of obtaining the coefficients of the transversal filter for cancelling the echo signal after a calculation interval LT, which may be equal to NT and which immediately follows the appearance of the echo signal produced in response to the training signal. It is advantageous for the coefficients ~O of the transversal filter to be made equal to zero at the beginning of this calculation interval.
The method according to the invention may be used in a duplex process for simultaneouslY initializing the coefficients of the transversal filters of the echo cancellers of two transceiver arrangements interconnected by a two-way transmission path. In this case the training signals D(n) and G(n) transmitted in each arrangement have one or the other pro~erty mentioned above and must have, in addition, the property:

d~ L(n-i) modulo L~ . g ~n-i') modulo L~ = 0 (or = -1) n = 0 for and l' such that 0 ~ i ~ N-1 and S i' ~ N-1, where g is the value of a datum of the data signal G(n).
The method according to the invention can be used in a canceller for echos subjected to frequency off-set comprising, apart from a transversal filter havingNcontrollable coefficients, a phase-shifting circuit connected between the output of the transversal filter and an input of the difference circuit and receiving a simulated phase from a phase generator for compensating the phase of the echo signal ;5 in the difference signal. In this case the method according to the invention comprises at least the following steps:
- transmission of said training signal D(n) during two time intervals [P1~LP2~ each having said duration LT, in the course of which the phase of the echo signal has the values ~(P1) and ~(P2), each being substantially constant, the space between said two intervals being choser. so that ~(P2) - ~(P1) has an appreciable magnitude;
during the time interval ~P1~, calculation of the coe fficients of the transversal filter;
- during the time interval LP2~ :
. maintaining the coefficients of the transversal filter at their calculated values, . maintaining the simulated pahse applied to said phase-shifting circuit at zero, . calculating a quantity s(P2) formed by accumulating products of two factors, one derived from the output signal of the phase-shifting circuit and the other from the difference signal (or from the received signal);
and - at the end of the time interval ~P2~ a processing operation to derive from the quantity s(p2) the phase difference term ~(P2) ~ ~(P1) and a processing operation to derive from said phase difference term a phase variation term ~`~ .T formed by using the expression:

~ (P2) _ ~(P1) t - t where ~ represents the angular frequency variation corresponding to the frequency off-set, t2 ~ t1 is the average time difference between the time intervals LP2i and [P11 ~ the two terms thus formed being used to initialize said phase generator circuit.
The invention will now be described by way of example with reference to the accompanying drawings, in which:
Fig. 1 shows the structure of an echo canceller arranged 3s in a modem to which the method of the invention can be applied;
Fig. 2 shows an em~odiment of the circuit for calculating coefficients in accordance with the invention;

~1~4~

Fig. 3 shows a signal diagram intended to explain the operation of the circuit for calculating the coefficients;
Figs. 4 and 5 show two further emkodiments of the circuit for calculating the coefficients;
5Fig. 6 shows the general structure of a canceller for echos subjected to frequency off-set;
Fig. 7 represents as a function of time the phase of an echo signal with frequency off-set to illustrate the method of the invention applied to a canceller for echos subjected to frequency lo off-set;
Fig. 8 shows the apparatus for carrying out the method of the invention in a canceller for echos subjected to frequency off-set.
Figs. 9 and 10 show two emkodiments of the processing lS circuit supplying a phase difference and a phase variation for initia-lizing a canceller for echos subjected to frequency off-set.
The block diagram of a modem provided with an echo canceller is shown in Fig. 1. The mcdem comprises a transmit path 1 including a modulator 2 receiving data from a terminal (not shown) and a receive path 3 including a receiver 4 supplying data to this terminal.
The output of the modulator 2 is connected to a transmit access of a coupling circuit 5 and a receive access of this coupling circuit is connected, by way of a given number of elements to be described here-inafter, tothe input of the receiver 4. This coupling circuit 5 per-mits of coupling the transmit and receive paths of the modem to atwo-way path 6 for full duplex connections with a distant mcdem which is coupled in the same manne7with the transmission path 6.
When the modulator 2 transmits in the transmit path 1 a signal modulated by the data to be transmitted towards the remote mcdem, an untimely echo signal may be produced in the receive path 3 due to imperfections of the coupling circuit 5 or to reflections in the transmission path 6, which signal may disturb in the receiver 4 the demodulation of the received signal originating from the remote modem. The echo canceller incorporated in the modem of Fig. 1 has to 3s eliminate the echo signal in the receive path 3. Let it be assumed that the echo signal is of the linear type, that is to say, in the echo path to which the output singal of the modulator is applied there only occur operations of linear character which do not cause the phase 1194~

or the frequency of the signal to vary.
The echo canceller of Fig. 1 uses a ccmplex data signal corresponding to -the data applied to the modulator. Assuming, for example, the modem uses phase modulation or phase and amplitude modu-lation for the transmission of the data, the complex data signal usedin the echo canceller can be obtained from a modulator 2 built up as follows. This m~dulator comprises an encoding circl~it 7 receiving the data to be transmitted and supplying a pair of signals representative of the amplitudes A(n) and the phase changes ~ (n) to be assigned to the carrier as a function of the data at instants nT determined by a generator 8 of frequency 1/T, 1/T being the modulation rate and n being an integer varying from -QO to +oo . In order to take account of the phase variation ~ ~ of the (unmodulated) carrier during each mcdulation interval T, an adding circuit 9 is used to supply at each instant nT the sum V;(n) + i~- , which is rep~esentative of the absolute phase ~J(n) of the modulated carrier to be transmitted.
The t~ signals A(n) and ~ (n) are applied to a circuit 10 which forms the real component A(n) cos ~ (n) and the imaginary component A(n)sin ~ (n) of the complex signal D(n). In the modulator 2 these t~ components are applied to bandpass filters 11 and 12, the output signals of which are added in an adding circuit 13 to form the analog modulated carrier signal which is directly applied to the transmit access of the coupling circuit 5.
The co~olex signal D(n) thus formed in the modulator

2 is also used in the echo canceller. It should be noted here that in the various diagrams of the present patent application the two-line connections convey both the real and imaginary components of a complex singal, but in most cases reference will only ke made to complex signals conveyed by these connections. Likewise, in processor circuits processing complex number signals, the operations carried cut in practice on the real and imaginary components of these complex number signals will, in general, not be explicitly dealt with.
The echo canceller included in the modem of Fig. 1 comprises a transversal filter 15 receiving the complex signal D(n) sampled at the instants nT and assumed, for example, to be of the analogue type. me transversal filter 15 is provided with a coefficient control circuit 16. me complex output signal of the filter 15 is applied to the (-)input of a difference circuit 17. To the (+)output ~41~5 of this difference circuit 17 are applied sanples of an analog signal, complex in general, which are formed starting from the signal appearing at the receive access of the coupling circuit 5. In order to form these samples a circuit 18 is used which supplies a complex signal, whose real part is the signal from the coupling circuit 5 and whose imaginary part is the same signal subjected to a 90 phase shift. The complex signal supplied by the circuit 18 is applied to a sample-and-hold circuit 19, in which sampling is carried out at a frequency fe supplied by the generator 8. This frequency fe is a multi-ple of the modulation frequency 1/T and has a value such that theShannon theorem is satisfied with regard to the echo signal, that is to say, with regard to the signal sup~lied by the modulator 2.
However, in order to simplify the explanations the following description will consider only those of the samples supplied by the circuit 19 which are produced at the frequency 1/T at the same instants nT as the samples of the data signal D(n), it being understcod that the other sequences of samples at the frequency 1/T have to be treated in the same manner.
The complex difference signal supplied by the difference circuit 17 is applied to the control circuit 16 in which it is used to control the coefficients of the transversal filter 15. When these coefficients are suitably controlled, the echo copy signal supplied by the transversal filter 15 is substantially equal to the complex version of the echo signal appearing at the receive access of the coupling cirucit 5 so that the echo signal is practically cancelled in the difference signal appearing at the output of the circuit 17.
Of this complex difference signal, thus freed from the echo signal, only the real component is used in the receiver 4 of the modem after having been previously filtered in a low-pass filter 20.
For an accurate description of the operation of such an echo canceller and an expL~nation of how the method according to the invention can be applied thereto, it is useful to indicate the calculations to be carried out with the aid of a vectorial notation.
For example, when at an instant nT the transversal filter 15 stores the N preceding samples of the data signal D(n) applied to its input, these N samples can be represented by the vector D(n). The N coefficients of the filter 15 at an instant nT can also be represen-ted by a vector C(n3.

With this notation the echo copy signal ~ (n) supplied by the transversal filter results from the operation:

~(n) = D(n) C(n) (1) D(n) being the transpose of vector D(n).
There may also ke defined a vector k having N components which are samples of the impulse response of the echo path to which the data signal D(n) is applied. It can then be written that the complex echo signal ~(n) appearing at the (+)input of the difference lQ circuit 17 results from the operation:
~(n) = D(n) . k (2) It will now ke assumed that the remote modem does not transmit any data signal that can be superimposed on the echo signal ~ (n). Then lS the error signal e(n) appearing at the output of the difference circuit 17 can be written:
e(n) = ~ (n) - c (n) = D(n).Lk - C(n)¦
This error signal e(n) is cancelled when C(n) = k.
In order to obtain this result in practice, which means that the echo signal is cancelled, the prior art echo cancellers use the criterion consisting in minimizing the mean square value of the error signal e(n) and kecoming manifest, while using the gradient algori-thm, by iterative control of the coefficients of the transversal filter 15 in accordance with the recursion formula C(n+1) = C(n) + ~ .e(n) . D~(n) (3) in which ~ is a coefficient less than 1, and D (n) is a vector whose components are complex conjugate to those of the vector D(n).
By using the fonlulae (1) and (2) the recursion formula

(3) may also be written:
.__ ~ = _ ._ C(n+1) = C(n) + X .A(n) . k - C(n) (4) 3s in which A(n) is matrix of the order N such that:

A(n) = D(n) D~(n) (5)

4~iS

If the components of vector D(n) and D (n) are indicated by d(n-i) and d (n-j) with integers i and ~ between O and N-1, the comFonents of the matrix A(n) can ke written:
aij(n) = d(n-i) d~tn-i) for i ~ j (6) aii(n) = ¦d(n-i)¦

In the known echo cancellers, the control of the coefficients of the transversal filter is carried out in the same manner, that is to say, by iteration in accordance with the recursion formula (3) or the e~uivalent formula (4), both during the training period intended to initialize the coefficients as well as during the tracking period in the course of the data transmission. Therefore, during the tracking period a weighting coefficient C~ is used which is small with respect lS to 1 in order to prevent the coefficients of the transversal filter from being disturbed by the data signal originating from the remote modem during the full duplex transmission. This results in small corrections of the coefficients and a slow convergence of the echo canceller. At the beginning of the training period it is possible to operate with a greater weighting coefficient ~ , but towards the end of this period it is necessary to operate with a weightin~ coefficient ~ of the same order of magnitude as that used during the tracking period. Finally, with an iterative control of the coefficients the convergence of the echo canceller remains slow during the training period.
The present invention provides another method of initiali-zing the coefficients allowing a training period of reduced duration.
This method is based on a combination of several steps which will now be explained.
If a first step, the coefficients of the transversal filter, instead of being modified at each sampling instant nT, are mcdified at all "L" sampling instants with a modification term which is the average of the L modifications calculated in accordance with the classical formulae (3) or (4). This results in that the algorithm 3s for modifying the coefficients can be written in accordance with a formula derived from formula (3):
(p+l)L-1 _~
C l(p+1)L ¦ = C rPL~ + L > e(n) . D (n) (7) n=pL

.,, ~199~165 This algorithm for modifyinq the coefficients may also be written according to a formula derived from formula (4):
1' _ ~ ~._ . = -~ ~ ~
_ (p+1)Ll = C~pL] + L B(pL) k - C(pL)¦ (8) in which B(pL) is a matrix of the order N deriving from matrix A(n) defined by the formula (5):
(p+1)L-1 B(pL) = > A(n) (9) n=pL

It can then be easily derived from formula (6) giving the ccmponents of the matrix A(n) that the components of the matrix -B(pL) are:
(p+1)L-1 bii = ~ d(n-i) . d~(n-j) for i ~ j and (10) . n-pL i,j = 0, 1, ... , N-1 bii = > ¦d(n-i)¦
n=pL

A further step in the method according to the invention consists in that for producing the echo signal and for cancelling it, a periodic data signal D(n) is used with a period LT such that L ~ N, where NT is the total delay produced by the transversal filter. This means that the components bij and bii f the matrix B(pL) occur periodically with a period defined by L; they are independent of the variable p, if p ~ 1. These components may, therefore, be written:

~bij = ~ dv~n-i~ . d~n~ for i ~ j and n=0 i,j = 0, 1, ... , N-1 (11)~ L-1 ii ~ I dtn-i~ 12 n=0 where ~n-i~ = (n-i) modulo L
~n-j~-= (n-j) modulo L

"

11~41t~5 It can be shcwn that owing to the periodicity of the data signal D(n) the components bii are independent of the index i and have, in accordance with the second fonnula (10), the value bo such that:

~ ,~
bo = > ¦d(n)¦ (12) n=0 This value bo of the component bii can be written in the form:

bo = L ~2 where ~ 2 is the power of each dabum of the signal D(n).
It can otherwise be shown that the component bij only depends on the index difference (i-j) and that bij = b~ji. The c~nponents bij with a positive difference (i-j) have values bi such that:

bi = d(n) . d~ ~n-i~¦ (13) n=0 The components bji with a positive difference(j-i) have values bj such that bj = b~i.
Finally, taking into account these properties of the components of the matrix B(pL) provided by the periodicity of the dat~ signal D(n), this matrix can be written:
bo b1 b2 ~ bN_1 .
b~1 bo b1 ~ ~

B(pL) = b 2 _ b 1 bo ~ ~ ~ b2 _b~N_1 ~ b~ - b~ ` b A further step of the method according to the invention consists in using a data signal D(n) such that the values bi of the c~nponents bij of the matrix B(pL) as provided by the formula (13) are ~ero for i ~ 0 and i = 1, 2, ..., N-1. This property of the PHF 82~5~8 13 03-06-1983 signal D(n) can thus be written:

d(n) . d~ [~n-i~] = for i ~ 0 and i = 1, 2, N-1 n=0 (14) This property (14) means that the autocorrelation function of the data signal D(n) between the data and the complex conjugate data is zero when their shift measured m~dulo LT is T, 2T, ..~ 1)T. When this shift is zero or a multiple of LT, the auto-correlation function assumes a maximum value corresponding with the value bo = L ~ for each element of the diagonal of the matrix B(pL).
Of course, if the da-ta signal D(n) is chosen so that its autocorrelation function is zero for any shift differing from zero and the multiples of ~T, the required property (14) is respected strictly, since L is chosen to be L > N.
With a periodic data signal satisfying condition (14), all components of the matrix B(pL) are æero with the exception of those of the diagonal, which have the value bo = L ~2. Therefore, this matrix B(pL) is equal to L ~ . IIN where ~ is the identity matrix.
The formula (8) giving the algorithm for modifying the coefficients ean then be written:
C l(p+1)L3 = C(pL~ + ~ o-2~k - C(pL)~
or C [p+1)L~ = C(pL) ~ r2~ + ~ ~ 2k'- (15) If the weighting coefficient ~ is chosen such that ~ = 1/~
the formula (15) becomes ~ .~
C ~p+1~L1 = k (16) Therefore, irrespective of the value of the coefficients C(pL) of the transversal filter at an instant pLT, the optimum value of _ ,_ these coefficients C l(p+1)L~ at an instant (p+1)LT can be obtained, which value is equal to the samples k of the impulse response of the echo path and permits of cancelling the echo signal by carrying out the calculation given by the formula (7) under the conditions PE~ 82-548 14 03-06-1983 indicated akove in detail. Since the remote m~dem does not transmit any data signal, the operations to ke performed in the local modem may be summarized as follcws:
- Transmission of a periodic training data signal D(n) satisfying the property (14); since the pexiod LT of this data signal is such that L ~ N, the transversal filter providing a delay NT cannot store more than one period of the data signal, - Calculation of the coefficients of the transversal filtex according to formula (7) by choosing a weighting coefficient ~ such that -~= 1/ ~ 2, whexe a~2 corresponds to the constant power of each datum of the signal D(n). As formula (7) shows, the coefficients C l(p+1)L~ are obtained by a single mDdification of the coefficients CLPL7 this modification resulting from the calculation, during a pexiod LT, of the sum of the products e(n) . D (n) calculated at each instant r.T of the period.
As is shown int he foregoing, the resultant coefficier.ts C L(p+1)L~ have, after this unique m~dification, the optimum values k, which permits the cancelling the echo signal.
Since the method according to the invention is in fact used for training the ech~ cancellex, it is easier to write the formula (7) for the period of the duration LT of the simulated signal such that p = 0. With CC = 1/~r2, the formula (7) then kecomes:

C(L~ = C(0) + 1 2 ~ e(n) . D~(n) (17) 2s L ~

In order to avoid noise on the coefficients due to non-infinite precision of the calculation, it is preferred to make the coefficients C(0) at the beginning of the calculation period equal to zexo so that in this practical case the coefficients C(L) obtained at the end of this period are f~rmed in accordance with the eYpression:

C(L) = 1 2 ~ e(n) . D (n) (18) ~ n=0 In the case concerned, in which the coefficients C(0) are zero, the output signal _ (n) of the transversal filter remains zero during -the ~7hole calculation period so that during this period 1:199~ 5 e(n) = (n). ~herefore, -the coefficients of the filter may also be calculated in accordance with the expressicno C(L) = 1 2 ~ ~(n) D (n) (19) s L ~ ~ -In the method according to the invention described above, the calculation of the coefficients of the transversal filter is carried out in a single period of duration LT of the data signal used for the training O the echo canceller. As a matter of course, in order to obtain the opt~lmcoefficients from said calculation, it is necessary for the echo signal produced ir. response to the training signal D(n) to be present at the input of the difference circuit of the echo canceller at the instant of calculation. In order to obtain the optimum coefficients, the training signal has to be first transmitted during the time required for the excitation of the echo path, which is in fact equal to the delay NT provided by the transversal filter, when the latter is dimensioned most correctly, and then during the calculation period LT of the coefficients. By the method according to the invention the convergence time of the echo canceller, as from the instant of starting the trans-mission of the training sequence ean, therefore, be reduced to (L+N)T and to the minimum value 2~ if L = N. In the latter ease, in order to cancel, for example, an echo of 20 msec, the required time is 40 msec, as compared with the convergence time in the order of a second 2s obtained by the prior art echo cancellers.
It ean be noted that in a general case in which the data signal D(n) is complex, the relation (14) defining the property of this signal to be satisfied is expressed by two relations to be simultaneously satisfied and concerning, respectively, the real part and the imaginary part of the first term of the relation (14). With respect to the choice of the training signal satifsying these relations, it is preferred to use the data values which are normally transmitted by the mcdem and which have values such as 0, +1, +], +\~, +~i2, etc.
A training signal satisfying the relation (14) and comprising 16 elements in one period may be formed with the aid of the sequence:
1, 1, 1, 1, 1, j, -1, -j, 1, -1, 1, -1, 1, -j, -1, j.
With a training data signal D(n) satisfying the property defined by relation (14) cancellation of the echo signal is obtained ~194~65 in the manner described without any limitation regarding the spectrum of this signal. In particular, such a training signal is suitable for cancelling an echo signal which may comprise or not comprise a d.c. component. The case of an echo comprising a d.c. component occurs, for example, in an echo canceller used in a baseband transmission modem or in the case in which the echo cancellation is carried out after the dem~dulation of a signal transmitted by carrier modulation. The case of an echo without d.c. component occurs, for example, in the system represented in Fig. 1, where the echo is produced in response to a carrier modulated signal and cancelation is performed directly on the received signal prior to its demcdulation.
In a case as shown in Fig. 1, in which the echo signal does not comprise a d.c. component, a variant of the invention uses a periodic training signal D(n), which instead of satisfying condition (14), satisfies the condition:

d(n) . d~ ~ n-i~ ~ = -1 for i ~ 0 and (20) n=0 i = 1, 2, ... , N-1 This condition (20) can, in particular, be satisfied with the aid of data se~lences kncwn as maximum length sequences, whose autocorrelation function, apart from a constant factor, has the value -1 for any shift differing from zero and multiples of the period LT of the sequence and has the value L for a shift zero or a multiple of LT.
If a trainlng signal D(n) satisfying this condition (20) is used, the matrix B(pL) has its components bo on the diagonal still equal to LG- , whereas all other components are equal to -1.
3Q It can be shcwn that in this case, when formula (8) for calculating the coefficients is applied with ~ = 1/ ~2 and C(pL) = 0 (coefficient initialized at zero), the result is:
~_ ` 1 C l(p-~1)Ll ~-- k ~ 2 d In this expression d is a vector, all N components of which are equal to ko + k1 + + kN 1~ that is to say, to the sum of the components of the vector k. This vector d thus represents the d.c. component of 11~41f~i5 the impulse response of the echo path. The echo signal can only be ~ ._ . ~
cancelled when C ~p+1) L3 = k.
Therefore, in order for a training signal satisfying condition (20) to permit cancellation of the echo, this echo must not

5 comprise a d.c. component.
It will generally ke assumed hereinafter that the training signal satisfies condition (14), which is convenient for all cases, it keing understood that it may also satisfy condition (20) when the echo signal does not comprise a d.c. component.
Fig. 2 shows a possible configuration for the circuit calculating the coefficients of the transversal filter 15 during he training period of an echo canceller using the method according to the invention, while the signal diagrams 3a to 3c of Fig. 3 are intended to explain the operation thereof.
lS Fig. 2 shows a generator 25 supplying a complex training signal D(n) of a period LT having the autocorrelation properties as expalined akove. D~ring the training period the generator 25 is connected on the one hand to the transmit path 1 of the modem through the assembly of the circuits 11, 12, 13 supplying the modulated carrier 20 signal and, on the other hand, to the input of a memory 26 arranged to supply in parallel at N outputs the N last samples of the signal applied to its input for a duration NT. This memory 26 may be the one which is normally used in the transversal filter 15 of the echo canceller. In the diagram 3a of Fig. 3 the double-crosshatched zone 25 represents the time during which, in the configuration of Fig. 2, the training signal D(n) is supplied to the transmit path 1 of the modem and to the memory 26, starting from an initial instant ti.
There will be considered the general case in which the period Lrr of the training signal exceeds the storage time NT of the ~r~ory 26, which 30 time N'r is at least equal to the excitation time of the echo path.
From the time to = ti + NT i-t is ensured that the echo signal ~ (n~
is present in the receive path 3 of the modem. The calculation circuit of Fig. 2 permits of calculating the N coefficients of the transversal filter 15 during the period LT of the training signal extending from 35 the instant to to the instant tL = to + LT. This coefficient calculating circuit comprises N identical circuits which are connected to the N
outputs of the Ir~ory 26 to receive at a given instant nT the respective samples d(n), d(n-1), ... d(n - N+1) of the -training signal and which 1~416S

supply the N respective coefficients C(0), C(1), ..., C(N-1) at the end of the calculation period.
The circuit calculating the coefficient C(0), for example, comprises a circuit 28 forming the comple~ conjugate value d (n) of the sample d(n) of the training signal. This complex conjugate value is multiplied in a multiplier 29 by the echo signal ~ (n), it being supposed that formula (19) is used. Each resultant product (n).d~(n) is applied to a multiplier 30, in which it is multiplied by the constant factor 1/L ~2. The output signal of the multiplier 30 lo is applied to a gate 31 which may be blocked hy a signal AC stopping the calculation, as will be e~plained hereinafter. The output of the gate 31 is connected to an accumulator formed by a memory 33 and an adder 32. The contents of memory 33 can be reset to zero by a signal RAZ and supplies the coefficient C(0) of the transversal filter at the end of the calculation period. The other coefficient calculating circuits use in the same manner the echo signal ~ (n), the constant factor 1/L ~ and the control signals AC and RAZ.
A coefficient calculating circuit, such as the circuit supplying the coefficient C(0), operates as follows: The control signals AC and RAZ have the form shown in the diagrams 3b and 3c of Fig. 3. The calculation stop signal AC is low until the instant tL
and up to this instant it permits the application of the output signal of the multiplier 30 through the gate 31 to the accumula-tor 33, 32.
Ho~ever, the effective calculation of the coefficients only starts at the instant to~ at which a pulse appears in the signal RAZ which resets memory 33 of the accumulator to zero. Starting from this instant to~ the accumulator forms the sum of the terms ~(n).d~(n).1/(L~
supplied by the multiplier 30 and at the instant tL = to + LT this sum formed in the accumulator constitutes the coefficient C(0).
At the instant tL the calculation stop signal AC becomes high and blocks gate 31 so that after the instant tL the calculated coefficient C(0) remains available at the output of the accumulator memory in order to be used in the transversal filter 15. The other coefficients C(1) to C(N-1) of the transversal filter are calculated in the same manner during the period of time from to to tL and after the instant tL they are simultaneously available for use in the transversal filter.
After the training period, which has thus permitted to initialize the coefficients of the transversal filter of the echo ~L194~65 canceller, the generator 25 of the training sequence is disconnected and the m~dem is put in the configuration of Fig. 1 to transmit the useful data. During the transmission of the data the coefficients of the transversal filter 15 of the echo canceller can be controlled by known processes generally consisting in iteratively modifying the coefficients in accordance with the recursion formula (3) mentioned above. It will readily be seen that the circuits described with reference to Fig. 2, which permit of calculating the coefficients during the training period, also permits of controlling the coefficients during data transmission, if the signal AC permanently remains lcw in order to keep gate 31 conducting and if the contents of the accumulator are not reset to zero by the signal RAZ, while the constant factor ~ may be adjusted to a value differing from the value 1/L ~2 used during the training period.
Fig. 4 shows an other possible configuration of the circuit calculating the coefficients of the transversal filter 15 during the training period. Instead of supplying these coefficients simultaneously the circuit of Fig. 4 supplies the coefficients successive-ly.
Fig. 4 shows the generator 25 supplying the training signal D(nj and being connected, during the training period, on the one hand to the transmit path 1 of the modem, as shor~n in Fig. 2, and on the other hand through an interrupter 35 to the input of a memory 36 arranged to supply in parallel at L outputs the L last samples of the signal appl ed to its input. In the case that L = N, memory 36 may be completely constituted by the one arranged for N samples, with which the transversal filter 15 is normally provided. The inter-rupter 35, flrst being assumed to be closed, is opened by a control signal S1 at an instant to be defined hereinafter, in a manner such that at the L outp~lts of memory 36 the samples d(n), d(n-1), ... d(n-L + 1) of the training signal remain available.
The coefficient calculating circuit proper has the form of a transversal filter 38, whose input 39 receives the echo signal ~(n) and whose output 40 supplies the signal y(n), which is identifiable to the N coefficients in series of the transversal filter 15 of the echo canceller during a well-defined time interval NT. The samples of the echo signal ~ (n) are applied to an input of a memory 41 arranged to supply in parallel at L outputs the L last samples of the signal 1:194165 applied to its input. At the instant at which the interrupter 35 is open, the samples appearing at the L outputs of memory 41 are ~ (n), ~(n-1), ... (n-L + 1). These samples are respectively applied to an input of multipliers 42(0), 42(1), ..., 42(L-1), the other input of which receives the complex conjugate values d~(n), d~(n-1), ..., d~(n - L+l) of the samples of the training signal available at the L outputs of memory 36. These complex conjugate values constitute the coefficients of the transversal filter 38. They are respectively formed with the aid of circuits 43(0), 43(1), .... 43(L-1). The samples o at the outputs of the multipliers 42(0), 42(1), .... 42(L-1) are summed with the aid of the chain of adders 44(1) ... 44(L-1). At the output of this chain the output signal y(n) of the transversal filter 38 is obtained. This signal y(n) is applied to a multiplier 46 to be multiplied by the constant factor 1/(L ~2). The product y(n) .1/(L 2) lS is then applied to an interrupter 45, which under the control of a signal S2 provides for a time window of duration NT, during which N
successive samples are obtained which constitute the N coefficients C(n) of the transversal filter 15 of the echo canceller.
The timing conditions which permit of obtaining this result may be specified with the aid of the diagrams 3d and 3e of Fig. 3, respectively representing the control signals S1 and S2 controlling the closure of the interrupters 35 and 45. From the initial instant ti to the instant tL the interrupter 35 is closed by the control signal S1 and the interrupter 45 is opened by the control signal 2s S2. At the instant to~ the echo signal begins appearing at the input 39 of memory 41 and during the period LT of the training signal from to to tL~ memory 41 stores samples of the echo signal, whereas me ry 36 stores samples of the training signal. At the instant tL~
the interrupter 35 opens under the control of signal S1 and L
samples d(n) to d(n-L + 1) remain stored in memory 36. At the instant tL~ the interrupter 45 closes under the control of signal S2. Just after the instant tL~ the filter 38 calculates a first sample y(n) such that L-1 35 y(n) = > = ~(n-m) . d~(n-m) m=0 This first sample y(n) multiplied by~the factor 1/(L ~2) constitutes the first coefficient C(0) of the transversal filter 15 of the ~1~4165 echo canceller which is transferred bytlle interrupter 45. This circuit remains open for a period of duration NT from the instant tL to the instant tM. For this entire period represented by a single-crosshatched zone in diagram 3a the transversal filter 38 uses the s same coefficients d~(n) to d~(n-L + 1) and after each instant tL + iT
(0 ~ N-1) the filter 38 calculates a sample y(n+i) such that y(n+i) = ~ ~(n+i-m~ . d~(n-m) m=0 The samples y(n) to y(n + N-1) supplied by filter 38 and multiplied by the constant factor 1/(Lor2) are transferred by the interrupter 45 until the instant tM to form the N coefficients 'l5 C(0) to C(N-1) of the transversal filter 15 of the echo canceller.
It will be noted~that with the configuration of Fig. 4 the time tM
required to obtain these N coefficients in series exceeds by NT the time tL required in the configuration of Fig. 2 to obtain the N
coefficients in parallel.
In the configuration of the coefficient calculating circuit of Fig. 4, the transversal filter 38 may be constructed in a different way, for example, in accordance with the diagram shown in Fig. 5 which is kncwn ~ se. In Fig. S the elements having the same function as those of Fig. 4 are designated by the same reference 2s numerals. From Fig. 5 it is apparent that each sample ~(n) of the echo signal is directly applied to L multipliers 50(0) to 50(L-1), which otherwise receive the complex conjugate values of the samples d(n) to d(n-L + 1) of the training signal from circuits 43(0) to 43(L-1).
As is indicated in E'ig. 4, these samples are formed at the outputs 30 of memory 36. The filter 38 also comprises L memory registers 51(0) to 51(L-1) each of which provides a delay T with respect to their input samples. The ~utput samples of the registers 51(0) to 51(L-2) are applied to an input of L-1 adders 52(1) to 52(L-1). The output of multiplier 50(0) is connected to the input of register 51(0).
35 The output of multipliers 50(1) to 50(L-1) is connected to an input of the respective adders 52(1) to 52(L-1). The output of adder 52(L-1) is connected to the output 40 of the transversal filter 38 ,. through the register 51(L-1). The output signal y(n) of this transversal i5 filter is processed int he same manner as in Fig. 4, that is to say, it is multiplied hy the constant factor 1/(L ~2) with the aid of circuit 46, after which it is applied to interrupter 45, the closure of which is controlled by signal S2.
With a filter 38 constituted in the way described, the memory registers 51(0) to 51(L-2) have to be reset to zero by a pulse of the signal RAZ indicated in diagram 3c and producelat the instant to~ During the time interval LT from the instant to to the instant tL~ partial results appear at the output 40 of transversal filter 38 and they are not used since the interrupter 45 is then open.
During the time interval NT from the instant tL to the instant tM~
complete results of the calculation appear at the output 40 of filter 38 and at the output of the interrupter 45, which is then closed~the N coefficients C(0) to C(N-1) of the transversal filter 15 of the echo lS canceller can then be obtained.
So far it has been assumed that the method according to the invention and the corresponding calculation circuits are used in a local mcdem in the ahsence of any singal from the remote modem, that is to say under the conditions actually imposed by CCITT for the training period of the echo canceller. However, the variants of the method according to the invention using similar circuits also permit of reducing the convergence time of an echo canceller when a data signal is received which orginates from the remote modem.
There will first be considered the case in which the remote mcdem transmits an arbitrary data signal, whilst the local modem tries to recover a lost echo cancellation. This situation may occur, for example, when the coefficient control circuit of the echo canceller has not correctly operated during a full duplex transmission and when it is not possible to stop the transmission of data by the distant mcdem. In this case, the signal (n) applied to the difference cirucit 17 of the echo canceller has not only the value of the echo signal D(n).k as indicated in formule (2), but it has the value:
~(n) = D(n) . k ~ b(n) ~21) In this expression for ~(n), b(n) is an additive noise term provided by the data signal originating from the distant modem.
When using the method according to the invention, carrying out the calculation in accordance with formula (19) for the coeffi.cients C(L) of the transversal filter 15 of the echo canceller, instead of the coefficients C(L) = k permitting precise cancellation of the echo signal, the following coefficients will be obtained with 5 a signal ~(n) according to formula (21):
1 L-1 -~P
C(L) = k + L 2 ~ b(n) . D (n) (22) ~ n=0 The noise b(n) in the signal (n) ~eccmes manifest as lo a noice ~ C on the coefficients of the transversal filter 15 of the echo canceller, which noise ~C is equal to the second term of the expression for the coefficients in formula (22). When G~b2 represents the square error of the noise b(n), the mean square error of the noise ~ C on the coefficients can ~e written:

1 I ~ L2 2 L Crb . N = N
Therefore, the noise C of the coefficients determines at the output of the difference circuit 17 a residual echo signal ê(n), the square error of which is:

E ~ e(n)¦2~ = E ~¦~C12~ Cr2 N ~2 It may be inferred that the ratio R of the power of the residual echo e(n) to the power of the noise b(n) produced by the remote modem is:
R = E rle!n)l ~ = N (23) Gb This formula (23) permits of fixing the duration LT
of the training signal to be taken into account in calculating the coefficients of the transversal filter 15 in order to obtain a given ratio R. In order to obtain, for example, a residual echo e(n) whose power exceeds by 20 dB the power of the noise b(n), R = L = 100 or 3s L = 100 N has to be chosen. The duration LT may be the effective period of the training signal, which may be very long, for example when L = 100 N. In order to avoid the necessity of forming a traininq signal having a very long period, there may also be used a trair.ing ~1~4~S
PHF 82-548 2~ 03-06-1983 signal of shorter period, for example, of a duration L'T = NT, taking into account for the calculation of the coefficients a time interval of a duration LT which is a multiple of L'T and corresponds to the ratio chosen.
In the method according to the invention as descrlbed a~ove for th.e case in which the distant m~dem transmits an arbitary data signal, the coefficients of the transversal filter of the echo canceller are calculated by integration of correction terms of the coefficients for a duration LT which is sufficiently long for the residual echo to attaiu~ a desirable small value. In a variant of the invention used for the same case, a training signal is used which has a period LT equal to NT and also has the autocorrelation properties defined ab~ve. However, the coefficients of the transversal filter of the echo cancelle.r are m~difi.ed in several successive steps, during each of whichthecalculated corrections are integrated for a duration LT = NT before being effectively applied to the coefficients. The modi-fications of the coefficients after each step p, are performed in accor-dance with the recursion formula (7) with ~c = 1/~- , the coefficients being initialized at zero value at the beginning of the first step p = 0, which means using the formula (18) for calculating the coefficients during this first step. The various variants of the invention applied in the case in which the remote m~dem transmits an arbitrary data signal, provide convergence times of the echo canceller of the same order of magnitude.
The method according to the invention applied to a training period preceding the transmission of the data permits, by using a particular training signal, of obtaining a rapid convergence of the echo canceller of the local modem provided the distant modem does not transmit any signal; starting from an initial instant ti, at which the transmission of the training signal begins, the convergence time may be reduced to the value 2~. In order to cause the echo cancellers of two modems of a duplex connection to converge, th.e method so far described has to be successively applied to these echo can-cellers, which doubles at least the convergence time for the whole connection and requires a particular procedure that cannot always be carried into effect.
A variant of the invention permits of avoiding these difficulties by allowing duplex convergence of the t~ echo cancellers 11941~iS

of a connection by means of other properties of the training signals transmitted by both modems.
As in the foregoing D(n) designates the vector of N
components characteristic of the samples of the training data signal D(n) transmitted by the local mcdem and k designates the vector of N
components characteristic of the samples of the impulse response of the echo path for the local rr~odem. For the remote mc~em G(n) designates the vector of N cornponents characterizing the samples of a training signal G(n). Each datum of the signal G(n) at an instant nT has the 0 value g(n). On the other hand, the samples of the impulse response of the path between the distant modem and the local modem are charac-terized by the vector h of N components.
Under these conditions the signal ~ (n) appearing at the (+)input of the difference cirucit 17 of the echo canceller is written:
.(n) = D(n) . k + G(n) . h It is now assumed that the signals D(n) and G(n) only have the properties already described a~ove, that is to say, they are periodical with a period LT and have the autocorrelation properties of the kind defined by fon~ula (14). When the coefficients of the transversal filter of the local echo canceller are calculated by integration over a period LT according to formula (19), the value for these coefficients is found to be:

~~ [ n~0 ~ (24) The coefficients C(L) calculated in this manner differ from the desired values k by a quantity depending on the matrix F such that F = = D (n) . G(n) n=0 The matrix F is a square matrix of the order N, the components of ~hich are:

~9416~

ii = ~ d~(n-i).g(n-j) for i ~ j and n=0 i,j = 0, 1, ... , N-1 fii ~__ d~(n-i).g(n-i) n=0 Since the signals D(n) and G(n) have the same period LT, the components fii and fii can ke written:

ij ~ d ~n-i~ .g ~n-j}
n=0 (25) fii ~ d~`~n-i~ g ln-n=0 In formula (25) the braces 4 .~ mean that the quantities between them are taken modulo L.
It is now assumed that the training signals D(n) and G(n) of the same period have, in addition to their autocorrelation properties, the property of being completel~ decorrelated relative to one another. This additional property beccmes manifest by the fact that: L-1 25 > = d~ ~n-i~ .g.~n-j~ = 0 for i,j = 0, 1, .................. ...,N-1 (26) n=0 This propery (26) can be obtained, for example, when G(n) is a delayed version of D(n) or the time-inverted version of D(n).
When the training signals D(n) and G(n) have the proper-ty (26), all components fii and fii of the matrix F are zero so that according to formula (24) the coefficients C(L) of he local echo canceller assume the desired values k of the impulse response of the echo path of the local modem. I~en the t~ echo cancellers of a connection simultaneously operate in -this manner with two training signals D(n) and G(n) completely decorrelated relative to one another at the point of the signal received in each modem, the coefficients of the t~ echo cancellers are simultaneously obtained after a period 4~

LT of integrating the correetions of the coeffieients.
If for the t~o mcdems at the ends of a conneetion the eeho signal does not eomprise a d.e. eomponent, periodie training signals D(n) and G(n) ean be used whieh satisfy the autocorrelation property (20) and, in addition, the extra property:

d~ tn-i~ . g ~n-j~ = 1 for i,j = 0, 1,...,N-1 n=0 So far it has been eonsidered that the sampling frequeney fe in the reeeive path of the mcdem is equal to the modulation frequency 1/T, that is to say, the sampling frequeney of the data in the training signal such as D(n). Now, normally the sampling l~ frequeney fe is a multiple of the frequeney 1/T. An echo canceller using the method aecording to the present invention with a sampling frequency fe such that fe = q.1/T and reeeiving the data signal D(n) sampled at the frequency 1/T has to be arranged for caleulating q sets of coeffieients of the transversal filter. A first step consists in eausing the eeho eaneeller to operate in time-sharing to ealculate these q sets of eoeffieients during a single integration period LT.
A seeond step requiring less rapid eireuits eonsists in suecessively calculating the ~ sets of coeffieients. The ealeulation of eaeh set requires an integration period LT so that the total convergence time of the eeho eaneeller becomes qLT.
The method according to the invention can be carried out with a training sequence D(n) having the partieular properties defined above by using the sign-algorithm, provided, however, the coeffieients of the transversal filter be modified in several sueeessive steps, during eaeh of whieh the caleulated correetions are integrated for a duration LT before being effeetive applied to the eoefficients. me modifications of the coefficients after each step p are then performed in accordance with the following reeursion formula, derived from formula (7):
(p+1)L-1 C ~(p+1)L1 = C(pL~ + 2 /\ sign re(n) ~ (n) n=pL
.

PHF. 82-548 28 At the beginning of the first step such that p = O, the coefficients are initialized at zero value. It will readily be seen that with the sign-algorithm the ccefficients are not obtained with sufficient pre-cision. The procedure is, therefore, repeated during a certain number of steps until the coefficients are obtained with the desired precision.
The use of the sign-algorithm in the method according to the invention thus raises the convergence time of the echo canceller, but it permits an important simplification of the calculating circuits.
So far there has been described the application of the method according to the invention to a linear echo canceller, that is to say, designed to canoe l an echo signal only resulting from linear operations in the echo path. Elowever, there may also be non-linear echoes, in particular, in certain carrier current transmission systems which produ oe an echo signal affected by a carrier frequency off-set with respect to the carrier frequency of the transmitted signal. A
linear echo canoeller is poorly adapted to correct such an echo signal, the phase of which is~variable with the frequency off-set. There are known echo canoe llers designed to cancel echo signals having frequency off-set, for example,~ as descrihed in our Canadian Patent Application 373,390 which was filed on March 19, 1981. These echo cancellers generally CQmprise a circuit w~ich produces, starting fron the received signal, a simulated phase of the echo signal, which simulated phase serves for phaæ correction, in the suitable direction, of the received echo signal or the echo copy signal produced by the transversal filter.
Fig. 6 shows by way of example the general possible structure of a canoeller for an echo havi~g frequency off-set. The elements having ~he same functions as in Fig. 1 are designated in the same manner.
Fig. 6 shows the differen oe circuit 17 receiving at its (+) input the echo signal f(n) affected by frequency off-set and at its (-) input the echo copy signal ~f(n). me echo signal f(n), by using the vectorial notation, can be written:
~f(n) ~ D(n) ~ k . exp j 0 (n) (27) where 0(n) represents the phase of the echo signal.
In order to obtain the echo copy signal ~f(n) a transversal filter 15 is used, the coefficients of which are determined in the device 16, and a phase-shifting circuit 60, which ~194~65 ,~ ~
n~difies by ~ (n) the phase of the output signal signal ~d(n) of the filter 15, where ~(n) is the simulated pahse of the echo signal supplied by a phase generator 61, which phase.m.odification is perfonred by forlTing the product d(n) . exp j~(n). The coefficients 5 are determined in circuit 16 starting from the difference signal e(n) or the received echo signal ~f(n). The simulated phase 0(n) is determined in the generator 61 starting from the signal e(n) or ~f(n), and from the signal ~: f(n) supplied by the phase-shifting circuit 60 so that it is equal to the phase ~(n) of the echo signal.
The method according to the invention can be applied to an echo canceller of the kind set forth, the echo copy signal ~f(n) then being obtained in two steps, one providing the coefficients of the transversal filter 15, the other providing the simulated pahse of the echo signal.
This two-step method will be described with reference to Fig. 7 illustrating the variation as a function of time of the phase ~(n) of an echo signal with frequency off-seti this phase variation may be considered to be lineæ. The first step consists in calculating in circuit 16 the coefficients of the transversal filter 20 15 by carrying out an integration of the corrections in accordance with the method described above during the period LT of the training signal D(n). As is shown in Fig. 7, this first step is performed in the time interval [P17 centered around the instant t1. Since the coefficients are initialized at zero at the beginning of the interval, 25 the calculation of the coefficients are performed in accordance with the expression (19), which may ke more briefly written for the interval rPl l ~ n C( P1) = j ~ f(n) . D~g(n) ) . 1 2 For the small frecraency off-set found in practice (of the order of 0.1 Hz) it may be rightfully assumed that during the whole interval rP11 the phase ,(Z~(n) has the constant value ,0(P1) 35 which is the mean value of the phase in the interval rp1~. Therefore, the calculation of the coefficients by integration during the interval rp1J provides co fficient values:
C(p1) ~ k . exp j ~(P1) (28) PHF 82-548 1~165 03~06-1983 This means that at the end of the first step the coefficients C(p1) of the transversal filter have the values which permit of compensating the echo signal having the phase ~(P1). During the remainder of the training period the coefficients are fixed at this value calculated 5 in the first step.
In the second step, an integration of the products ~ f~ (n) . ~f(n) or products f~(n) . e(n) is carried out during a time interval ~P2~ equal to the period LT of the training signal, centered around the instant t2, as is shcwn in Fig. 7. This integration o provides a term s(p2)~ which can be represented by the expression:

~P2~ f ( ) f( ) ~ L ~ (29) or S(P2) = ( / _ f (n) e( ) )- L ~2 where the terms ~f(n) and f ~(n) represent the received echo signal and the complex conjugate value of the echo copy signal f(n), respec-tive]y.
During the whole interval rp2~ the simulated phase ~(n) applied to the phase-shifting circuit 60 is set to zero so that the signal ~f(n) is equal. to the signal ~d(n) supplied by the transversal filter and therefore, it be written that:
~f(n) = D(n) C(p1) or, taking account of the calculation of C(p1) in the first step:
~f(n) ~ D~n) k exp j ~(P1) By using this expression for f(n) and the expression (27) for ~`f(n) the calculation of s(p2) in accordance with expression (29) yields:

~194~

S(P2) = ( ~ ]~ , D~(n) D(n~ k exp j L~(n) ~ 0(P1)l ~ Lqr2 = ¦k¦ exp i L0(P2) ~ 0(P1)~ (/ D~(n) D(n)j L ~2 ~2~

Since the training signal D(n) has the autocorrelation property (14), the final result is:
S (P2) ~ Ikl ~P iL~(P2) ~(P1)~

where ~(P2) is the supposedly constant value of the phase 0(n) dur.ing the time interval ~P2~-Since the interval ~P2l is chosen to be sufficiently remote from the time interval ~p;~ the phase shift 0(P2) ~ ~(P1) has an appreciable magnitude which can be determined after the calculation of s(p2). From formula (30) is can be derived:
0(P2) - 0(P1) = argument ~s(p2j~ = arc tan.llem~s~(p-P)2~ (31) where ~eLs(p2)~ and Ilm ~P2)¦ represent the real part and the imaginary part of s(p2)~ respectively.
After the calculation of the phase shift ~(P2) - 0(P1)~
there can finally be calculated a phase variation term ~ ~T = 2rr ~f T
proportional to the frequency off-set ~f by using the formula:
~ T = ~(P2) - 0(P1) (32) ~P2~ rP1l where rP2l ~ rP11 is the time interval t2 ~ t1~ measured in numbers of periods T~between the in-tervals ~P2~ and rp1~ -Since at the end of the time interval ~p1l the coefficients of the transversal filter 15 are fixed so as to correctan echo of mean phase 0(P1) in this interval, it is possible to correct after the end of the time interval rP2l an echo signal having ~19416S
PHF. 82-548 32 a frequency off-set ~ f, by producing a simulated phase 0(n) initial-ized just after the interval rp2] to the value 0(P2) - 0(Pl) calcu-lated in acoordance with expression (31), and subsequently varying with the slope ~ ~.T calculated in accordance with expression (32).
It will now be described how the method according to the invention can be carried into effect in a canceller for an echo having frequency off-set. After a training time period, the method pen~its of initializing various~parameters of the echo canceller.
During this whole time period a training signal is transmitted, the properties of which are defined above. The first step, which permits of initializing the coefficients of the transversal filter 15 by an integration calculation of the corrections of the coefficients during a time interval [Pl~ can be exactly OE ried out as described with reference to Figs. 2, 4 or 5. In order to explain how the seoond step can ke carried out, which serves to initialize parameters concerning the simulated phase of the echo signal, it is useful to consider again the n~nner in which this simulated phase is controlled in a canceller for an echo haying frequency off-set~ For example, in the aforesaid Canadian Paten-t Application No. 373,390, there is described an echo can oeller; the simulated phase of which is controlled in accordance with the recursion formula~
~(n+l) = ~(n) + ~ IIm [~ f(n) . ~ f ( )]
In this recursion formula, ~ is a coefficient of small value with respect to 1, which coefficient determines the magnitude of the correction to be applied to the phase 0(n~ to obtain the phase 0tn+1) at the next iteration. In this control system the phase cor-rections are performed at each sampling period.
Fig. 8 shows primarily the simulated phase generator 61, which effectuates the recursion formula (33) in the echo canceller of Fig. 6. This generator, illustrated in Fig. 8 of the aforesaid Canadian Patent Application, ccmprises a circuit 62 forming the term ~ f~(n), a multiplier 63 forming the product ~ f~(n) . ~f(n) and a circuit 64 only retaining the ima~i~ary part of this product. This imaginary part, which constitutes the phase shift J ~ between the real phase and the simulated phase of the echo signal, is multiplied by the ooefficient ~3 in a multiplier 65. I~e output of circuit 64 ~9416S
PHF. 82-548 33 forms the terms of the pha æ correction ~ IIm L~f~n~ ~f~(n)3 which are accumulated in an aecumulator formed by an adder 66 and a memory 67. m is accumulator supplies the simulated phase 0(n~
to the phase-shifting eircuit 60. In the generator 61, multiplier 65, multiplying by the coeffieient ~ , plays the part of a loop filter of the first order in the phase control-loop and, as stated in the aforesaid Canadian Patent Applieation 373,390, it is ad~antageous to use a loop filter of the second order whieh is formed by the circuits included in a block 68 indicated by broken lines. This filter 68 is formed by a multiplier 69, which multiplies the phase shift ~ 0 by a ooefficient ~less than 1. The product ~0 is applied to an accumu-lator for~ed by an adder 70 and a memory 71. The output signal of the aeeumulator 70, 71 is added by an adder 72 to the output signal of multiplier 65. The output signal o the adder 72 is applied to the aceumNlator 66, 67 which supplies the si~hlated phase 0(n).
Before continuing the operation of this control eireuit of the sim~lated phase, tw3 parameters:have to be initialized on the one:hand the contents of memory 67 of the aeeumulator 66, 67 and on the other hand the contents of memory 71 of the aecumulator 70, 71.
This initialization takes place at the end of the second step by transferring in me~ory 67 the.ealeulated phase differenoe 0(P2) -0(Pl), which is the initial phase attributed to be simulated phase 0(n) and by transferring in memory 71 the quantity a~J.T, which permits of obtaining the initial slope of the simulated phase.
In order to obtain these tw~ initial quantities first the quantity s(p2) is ealculated in aecordanee with the method of the invention. Thereto a eireuit 73 is used which realizes formula (29). m is eireuit 73, whieh is arranged in a oonventional m~nner to form an aeeumulator, reeeives the produets ~f(n) . ~f~(n) formed at the output of multiplier 63 and aee~mulates these products during the time interval ~P2] after having been set to zero at the beginning of this interval. The oomplex quantity s(p2) thus formed at the end of the in.terval [P2~ is applied to a eireui-t 74 whieh fo.rms the phase differeno~ 0(Pl) ~ 0(Pl) by using formula (31). Finally the differen oe 0(P2) ~ 0(Pl) is applied to a eireuit 75, to whieh is also applied the ti~e differenoe ~P2~ ~ [Pl~ to form the phase variation a w .T
in aeeordance with formula (32). The ealeulated quantities 0(P2)-0(Pl) and ~ W~T are simultaneously transferred to the memories 67 and 71.

~9~65 PHF. 82.548 34 As is shcwn by the example described above for con-trolling the simulated phase 0(n) during the tracking period or for calculating the quantity s(p2) during the initializing period, use is made of the same quantity ~f~(n) . ~f(n) which is representative of the phase difference between the echo signal and the echo copy signal. In other embodiments of cancellers for echos having a variable phase as disclosed in the previously mentioned Canadian Patent Applica-tion 373,390 a quantity representativ,e of this phase differenoe may he calculated in another way, which phase differenoe is at any rate necessary for a circuit for iteratively controlling the simulated phase. It will be clear that in these em~odiments it is preferred to use the quantity representative of the phase difference already calcu-lated to form the quantity s(p2).
Instead of using formula (31) to obtain the phase difference 0(P2) ~ 0(Pl) there may be used more easily implementable algorithms. One of these algorithms consists in updating an auxiliary variable INT at a very high clock rate in accordance with the recur-sion formula:

INT(m+l) = INT(m) + ~ L~Sm s(p2) - INT(m) . r(p2)~ (34) In this formwla Itm S(P2) is the imaginary part of the quantity s(p2) already defined, yis an,integration constant and r(p2) is a quantity calculated by either of the following expressions:
r(P2) = ( ~ ~f~n~ . ~f~n ~ L G2 r(P2) = ~ f(n) . ~f (n)~ . L ~
~LP2~ J
It can be shown that the quantity r(p2) is substantially equal to the absolute value ¦k¦ of the quantity S(P2) see formula (30).
In this case the recursion formLla (34) can be written:

INT(m+l) = INT(m) Ll ~ ~.~k¦2~ + ~ ¦k¦2 sin ~0(P2) ~ 0(P

PHF 82-548 ~ 1 9~16S

It is inferred therefrom that after a suffieient number of iterations and provided ~.lki2 is less than 1, the auxiliary variable INT practically assumes the value INT(~',~ ) = sinL~(P2) ~(P1j1 The phase difference ~(P2) - ~(P1) can then be readily obtained by reading a look-up table for the function arc sin, using INT as address.
Fig. 9 shows how the algorithm (34) can be implemented to caleulate ~(P2) - 0(P1) starting from the signals ~f(n) and ~f(n).
The quantity s(p2) is formed as shown in Fig. 8 and is available at the output of circuit 73. On the other hand a multiplier 76 supplies the products c f(n) . ~'f~(n) appearing in formula (35). These products are accumulated during the time interval ~P2~ in a circuit 77 which thus supplies the quantity r(p2). In order to update the auxiliary variable INT according to the reeursion formula (34), an accumulator is used which is formed by an adder 78 and a memory 79 which contains the auxiliary variable. The adder 78 has an input connected to the output of memory 79, a second input reeeiving the product ~. IIm s(p2) formed with the aid of circuits 80 and 94, and a third input receiving the product - ~ T(m).r(p2) formed with the aid of cireuits 82, 81, 95 connected as indicated in the Fig. 9. The aceumu-lation in the aecumulator 78, 79 is performed at a clock frequency h whieh is very high as eompared with the mcdulation rate 1/T so that in praetiee very soon after the time interval rp2~ the quantity INT(~o ) is obtained at the output of the aecumulat,or. This quantity serves as an address to read in a look-up table 83 for thefunetion are sin the phase difference ~(P2) - ~(P1)~ which is used as described with reference to Fig. 8.
The phase difference 0(P2) - ~(P1) can be obtained even more simply by using another algorithm which consists in producing an auxiliary variable INT according to the recursion formula INT(m+1) = INT(m) - ~. IIm Ls (~, . exp j.INT(m)~ (36) Taking into aceount the value of s(p2) given by formula (30) it ean be s'nown that the recursion formula (36) ean be ~ritten:

1:1941f~S

INT(m+1) = INT(m) _ ~ 2 sin [INT(m) - ~(P2)-0(P1))~
An approximate solution of this e~uatlon is:
INT(m+1) = INT(m) ~ kl 3 ~ ~ Ikl2~(P2~ - ~(P1~

It is irferred therefrom that after a sufficient number of iteration the auxiliary variable INr substantially assumes the value:
INT( ~ ) ~(P2) 0(P1) Fig. 10 shows how the algorithm (36) can be used to obtain the phase differnece 0(P2) ~ ~(P1)- The ~uantity s(p2) formed as is indicated in Fig. 8 is applied to a circuit 85 which derives therefrom the complex conjugate value s~(p2). The auxiliary variable INT is contained in a memory 86 of an accumulator formed by this m~mory and an adder 87. The output of memory 86 is connected t~a circuit 88 which forms the quantity exp.j.INT(m). This quantity is multiplied by s~(p2) with the aid of a multiplier 89.
A circuit 90 extracts from the resultant product the imaginary part which is multiplied by the coefficient ~ with the aid of a multiplier 96, the sign of the latter product being changed by means of a circuit 91. The output of circuit 91 is connected to an input of adder 87.
The accumulation in the accumulator 86, 87 is carried out at a very high clock rate h and very soon after the time interval rp2~ the desired phase difference ~(P2) - ~(P1) is obtained at the output of the accumulator.
The phase variation ~ ~'.T also required for initializing the echo canceller can be obtained by modifying the algorithm (36) in the following manner:
INT(m+1) = INT(m) - IIm ~ (p2).exp i ~ EPIN~r(mEP~-~ (37) It will now be easily seen that after a sufficient number of iterations the auxiliary variable INT assumes the value:

~P2~ lP1~

PHF 82-5~8 37 03-06-1983 or:
rNT ( cO ) = ~ r.

The algorithm (36) can ke implemented as is shcwn in 5 Fig. 10, by means of a circuit 92 ~indicated by broken lines) ~ich multiplies the quantity INT(m) applied to circuit 88 by the coefficient LP2~ ~ ~P1~
This results in that the contents of the accumulator 86, 87 rapidly assume the value of the phase variation ~ .T. Finally the algorithms (36) and (3,) can be implemented in the same manner in accordance with the scheme of Fig. 10. In a first time segment, the coefficient applied to multiplier 92 is fixed at the value 1 and accumulator 86, 87 supplies the phase difference 0(P2) - ~(P1) which is transferred to lS memory 67 of Fig. 8. In a second time segment, this coefficient is e at the value rP2l ~-CP-~ and the variation ~L~T becOmes available in memory 86, which may be the same as memory 71 of Fig. 8.
In some echo cancellers, it is known to directly process the real received signal without using circuits such as circuit 18 of Fig. 1 to form a complex signal. In these echo cancellers, the difference circuit 17 of the echo canceller has applied to it on the one hand the sampled received signal and on the other hand the real part of the complex signal ~ (n) supplied by the transversal filter 15. The latter always receives a complex data signal and always operates with complex coefficients supplied by a control circuit such as circuit 16 of Fig. 1.
A method according to the invention also permits of reducing the convergence time of an echo canceller of this type. First the case of a linear echo canceller will be examined. The echo signal received in the receive path of the mcdem and directly used in the echo canceller under consideration may be regarded as being the real part of the complex echo signal which would be formed by circuit 18 in the case of the echo canceller of Fig. 1. After sampling, this received echo signal is ~R(n) and can, therefore, be ~ritten:
~R =TRe L~ (n)~ = ~(n) + ~(n) apart from the factor 1/2.
This results in that a data signal D(n) transmitted S
P~ 82-548 38 03-06-1983 during a first time interval provides an echo siynal R1(n) of the form:
3~ x R1(n) = D(n) k + D (n) . k (38) If in accordance with the ordinary method of the invention this signal D(n) is the periodic training signal having the autocorrelation properties (14) and if the coefficients are calculated by integration of the corrections of the coefficients during an interval rq1~, in which the echo produced in response to the transmitted siqnal is received, the calculated values of these coefficients can be written in a simplified manner, apart from a constant factor:

C1(q) = / ~R1(n) D (n) L--ql~

or, by using formula (38):
C1(q) = k + E(n) . k (39) where E(n) is a square matrix of the order N such that:

E(n) = ~ D~(n) . D (n) n=0 In order to obtain, at the end of a single integration interval ~q1~, the samples k of the impulse response of the echo path for the coeffi-cients C1(q) of the transversal filter, the matrix E(n) should be zero, which can be obtained by a training signal D(n) which has to satisfy not only the condition (14), i.e.:

\ d(n) . d~(n-i) = 0 for l ~ 0 and n=0 i = 1, 2, ... , N-1 but also the condition:

~ d~(n) . d~(n-i) = O for i ~ O and n=0 i = 1, 2, ... , N-1 1194~6~
P~ 82-548 39 03-06-1983 In order to avoiud this new condition to be imposed on the training signal D(n) a different training signal D+(n) is transmitted in accordance with an advantageous variant of the method of the invention, during a second time interval when the echo provided by the first signal D(n) has disappeared. miS signal D+(n) may be D+(n) = j D(n)-This transmitted signal results in an echo signal R2(n) such that R2 = D+(n) . k + D ~). k~
If on the basis of the signal D+(n) the coefficients of the trans-versal filter are calculated during a time interval Lq2~, this calculation can be written:
n C2(q) = / ~R2(n) D+~(n Lq2~1 Taking into account the above expression for ~R2(n) it is derived that:
~ ~ = ~E
C2(q) = k + F(n) . k (40) where F(n) is a square matrix of the order N such that:

F(n) = > D~(n) . D+~(n) n=0 Since D+(n) = j D(n), F(n) = -E(n).
Thus, according to the formulae (39) and (40), when forming the sum of the coefficients C1(q)~calculated by integration r and C2 during the time intervals rq1~ and rq2~, the desired c oe fficients are obtained, apart from a constant factor, i.e.:
C(q) = C1(q) + C2(q) = k It will now be shcwn h~ the me-thod according to the invention can be applied to canceller for echos having frequency off-set, in ~,Jhich the received signal is also processed without forming 11~416~i PHF. 82-548 40 a corresponding complex signal. An echo canceller of this kind is shcwn by way of ex~l~le in Fig. 11 of the aforesaid Canadian Patent Application 373,390. m is echo canceller has the general structure shown in Fig. 6 of the present patent application, but only real sig-nals are used for the control of circuit 16 controlling the coefficientof the transversal filter 15 and for the control of the control circuit 61 for the simulated phase of the echo signal.
For initializing the parameters of such an echo canoeller a method is used which is similar to that described earlier in the present patent application for canceller for echos having frequency off-set processing a complex echo signal. This method thus comprises two steps, one serving to initialize the coefficients of the transversal filter and the other serving to initialize the simulated phase of the echo signal and the slope of the variation of this phase. However, for carrying out each of these steps in an echo canceller using only the real part of the echo signal it is necessary to first transmit the training signal D(n) and then the signal D+(n~ = jD(n) as stated above.
A signal D(n~ transmitted during either of these tw~
steps generates a real echo signal R1(n) of the form:
Rl (n) = D~n) ~ k . exp j 0(n) + D~(n) . k~ . exp -j0(n) where 0(n) is the phase o the echo signal supposed to be constant during the step ooncerned. If the signal D+(n) = jD(n) is transmitted, an echo signal &R2(n) is obtained having the form:
~R2(n) = j D~n) . k . exp j0(n) + j D~(n) . k~ . exp -j0(n) r~herefore, instead of calculating the coefficients Cl(p) in a single time interval rPl~ with the signals D(n) and (n) (see Fig. 7), the first step calculates also coefficients C2(p) with uhe signals jD(n) and ~2(n) in a second ir.terval ~p;3 of the same duration and sufficiently close for the phase of the echo signal to have substantiaIly the value 0(Pl) during these tw~ time intervals.
In the same manner as described ahove it can be shcwn that the sum of the cDefficients thus calculated has the value mentioned below, apart frcm a ccefficient 1/2:

1~41fi~

C(p) = C1(p) + C2(p) = k exp j ~(P1) During the remainder of the training period the coefficients of the transversal filter 15 are fixed at this value, which permits of compensating the echo signal having the phase 0(P1).
A possible method of perforning the second step is to approach the method already described for a complex echo signal and based on the calculation of a quantity s(p2) according to formula (29).
In this method a quantity s1(p2) is calculated in a time interval ~P2 with the signals D(n) and ~R1(n) such that:

S1(P2) = / / f (n) ~ R1(n) )-~CP2~
lS Then, a quantity s2(p2) is calculated with the signals D+(n) = jD(n) and ~ R2(n) in a time interval rp'2~ of the same duration as rp2~ and sufficiently close for the phase of the echo signal to have the same value ~(P2) such that:

S2(P2) = ~ > ~f (n) ER2(n) ~ 1o_2 In the same manner as described in the foregoing it can be shown that by forming the s~m of the two quantities s1(p2) and s2(P2) the quantity s(p2) is obtained as:
s(P2) = S1(P2) + s2(P2) ~ ¦kl exp i L0 (P2) - ~ (Pl ~]

The quantity s(p2) thus calculated has exactly the same value as that of the formula (30). By the same calculations or the same algorithms as described above ~he initial phase difference ~(P2) ~ ~(P1) and the initial phase variation ~ ~J .T to be transferred to the memories of the simualted phase generators of the echo canceller can be derived therefrom.
A further possible method of performing the second step consists in calculating the quantities s'1(p2) and s'2(P2) rather than the quantities 51(P2) and s2(P2) such that:

~941~iS
PHF. 82-548 42 l(P2) Il(n) ~Rl( [P2~

2(P2) I2(n) ~R2(n) [P2 with I(n) = ~ m ~ f It can be shown that forming the sum of the tWD quantities s'l(p2) and Sl2(p2) the quantity sl(P2) = Sll(p2) + sl(P2) = 4~ ¦kl sin L~(p2) ~ 0(Pl)]

is obtained. From the quantity sl(p2) can be derived the phase differ-enoe 0(P2) ~ 0(Pl) and the phase variation a ~J.T.
In the latter methcd for the second step, the multi-plications required for the calculation of the quantities s'l(p2) and s'2(p2) only relate to two real signals, which is simpler than the multiplication requir~d in the other method for calculating the quan-tities sl(p2) and s2(p2) which relate to a real signal and a complex signal. It should be noted that the latter method is particularly suitable for initializing the echo canceller of Fig. 11 of the afore-said Canadian Patent Application 373,390, where the simulated phase is controlled by corrections proportional to the products of the same real signals. Finally, in the foregoing formulae, which Fermit of calculating quantities Sl(p2)~ s2(p2), s 1(P2)~ 2(P2 real signals ~Rl(n) and ~R2(n) can be replaoed by the corresponding differen oe signals el(n) and e2(n).

Claims (19)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS

1. A method of reducing the convergence time of an echo canceller connected in a transceiver arrangement between one-way transmit and receive paths coupled to a two-way path and used to cancel an echo signal occurring in the receive path in response to a signal supplied to the transmit path, said echo canceller comprising a transversal filter having N controllable coefficients for processing a signal derived from the signal supplied to the transmit path and a difference circuit for producing a difference signal between two signals which are formed from the signal in the receive path and the output signal of the trans-versal filter, respectively, said method being characterized in that it comprises at least the following steps:
- transmission of a training data signal D(n) constituted by data transmitted at instants nT, T being the data interval, and being periodically reproduced after a duration LT at least equal to NT and having the property:
where d and d* are the value of a datum of the data signal D(n) and its complex conjugate value, respectively;
- calculation of the coefficients of the transversal filter carried out after the instant of appearance of the echo signal produced in response to said training signal in accordance with the expression:
where ?O and ? are the vectors of the N coefficients of the transversal filter at the beginning and at the end of the period of calculating the coefficients, respectively, e(n) is the difference signal, ?x(n) is the vector of the complex conjugate values of the N data stored in the transversal filter, and .sigma.2 is a constant term representative of the power of each of the transmitted data.

2. A method of reducing the convergence time of an echo canceller connected in a transceiver arrangement between one-way trans-mit and receive paths coupled to a two-way path and used to cancel an echo signal occurring in the receive path in response to a signal supplied to the transmit path and not comprising a d.c. component, said echo canceller comprising a transversal filter having N control-lable coefficients for processing a signal derived from the signal supplied to the transmit path and a difference circuit for producing a difference signal between two signals which are formed from the signal in the receive path and the output signal of the transversal filter, respectively, said method being characterized in that it comprises at least the following steps:
- transmission of a training data signal D(n) constituted by data trans-mitted at instants nT, T being the data interval, and being periodically reproduced after a duration LT at least equal to NT and having the property:
where d and d* are the value of a datum of the data signal D(n) and its complex conjugate value, respectively;
- calculation of the coefficients of the transversal filter carried out after the instant of appearance of the echo signal produced in response to said training signal in accordance with the expression:
where ?O and ? are the vectors of the N coefficients of the transversal filter at the beginning and at the end of the period of calculating the coefficients, respectively, e(n) is the difference signal, ?*(n) is the vector of the complex conjugate values of the N data stored in the transversal filter, and .sigma.2 is a constant term representative of the power of each of the trans-mitted data.

3. A method as claimed in Claim 1 or 2, characterized in that the coefficients of the adaptive filter are maintained at zero during the period of calculating said coefficients, whereby in said expression for calculation of the coefficients, ?O is equal to zero and the difference signal e(n) is equal to the received signal .epsilon.(n).

4. A method as claimed in Claim 1 or 2 and used in an echo canceller in which the difference signal e(n) is a complex signal resulting from the difference between two real signals, one derived from the received signal and the other from the output signal of the transversal filter, characterized in that the coefficient vector ? of the transversal filter is calculated while transmitting a single training signal D(n).

5. A method as claimed in Claim 1 or 2 and used in an echo canceller in which the difference signal e(n) is a real signal resulting from the difference between two real signals, one derived from the received signal and the other from the output signal of the transversal filter, characterized in that the coefficient vector ? of the transversal filter is obtained by forming the sum of coefficients calculated while transmitting a first training signal D(n) and of coefficients calculated while transmitting a second training signal D+(n) = jD(n).

6. A method as claimed in Claim 1 or 2 and used in a local transceiver arrangement receiving, apart from the echo signal produced in response to the training data signal D(n) - or jD(n) -, a signal produced in response to the transmission of a training data signal G(n) - or jG(n) - by a remote transceiver arrangement, said signal G(n) having the same property as the signal D(n), characterized in that the signals D(n) and G(n) have, in addition, the property:
for i and i' such that 0 ? 1 ? N-1 and 0 ? i' ? N-1, where g is the value of a datum of the data signal G(n).

7. A method as claimed in Claim 1 or 2 and used in a local transceiver arrangement receiving, apart from a training data signal, a noise signal produced in response to the transmission of an arbitrary data signal by the distant transceiver arrangement, charac-terized in that the coefficients are calculated during a period LT chosen so that L/N is substantially equal to R, where R is the desired ratio between the power of the residual echo signal and the power of the noise signal.

8. A method as claimed in Claim 2 and used in a canceller for echos subjected to frequency off-set comprising, apart from a transversal filter having N controllable coefficients, a phase-shifting circuit connected between the output of the transversal filter and an input of the difference circuit and receiving a simulated phase from a phase generator for compensating the phase of the echo signal in the difference signal, said method being characterized in that it comprises at least the following steps:
- transmission of said training signal D(n) during two time intervals [p1], [p2] each having said duration LT, in the course of which the phase of the echo signal has the values ? (p1), ? (p2), each being substantially constant, the space between said two intervals being chosen so that ? (p2) - ? (p1) has an appreciable magnitude;
- during the time interval [p1] , calculation of the coefficients of the transversal filter;
- during the time interval [p2] :
. maintaining the coefficients of the transversal filter at their calculated values.
. maintaining the simulated phase applied to said phase-shifting circuit at zero, . calculating a quantity s(p2) formed by accumulating products of two factors, one derived from the output signal of the phase-shifting cir-cuit and the other from the difference signal (or the received signal);
and - at the end of the time interval [p2] a processing operation to derive from the quantity s(p2) the phase difference term ? (p2) - ? (p1) and a processing operation to derive from said phase difference term a phase variation term .DELTA.w.T formed by using the expression:
where .DELTA.W represents the angular frequency variation corresponding to the frequency off-set, t2 - t1 is the average time difference between the time intervals [p2] and [p1], the two terms thus formed being used to initialize said phase generator circuit.

9. A method as claimed in Claim 8 and used in an echo can-celler in which the difference signal e(n) is a complex signal result-ing from the difference between two complex signals, one derived from the received signal and the other from the output signal of the phase-shifting circuit, characterized in that the coefficient vector ? of the transversal filter and said quantity s(p2) are calculated while transmitting a single training signal D(n).

10. A method as claimed in Claim 8 and used in an echo can-celler in which the difference signal e(n) is a real signal resulting from the difference between two real signals, one derived from the received signal and the other from the output signal of the phase-shifting circuit, characterized in that the coefficient vector C of the transversal filter is obtained by forming the sum of coefficients calculated while transmitting a first training signal D(n) and of coefficients calculated while transmitting a second training signal D+(n) = jD(n), and in that said quantity s(p2) is obtained by forming the sum of a quantity s1(p2) calculated while transmitting the first training signal D(n) and of a quantity s2(p2) calculated while trans-mitting the second training signal D+(n).

11. A method as claimed in Claim 8, characterized in that the calculation of a quantity s(p2), s1(p2) or s2(p2) is carried out in accordance with either of the two expressions:
where is the conjugate value of the signal ?f (n) supplied by the phase-shifting circuit, and .epsilon.f(n) is a complex or a real signal derived from the received signal.

12. A method as claimed in Claim 10 characterized in that the calculation of a quantity s1(p2) or s2(p2) is carried out in accordance with either of the two expressions:
where .epsilon.? (n) is the imaginary part of the signal supplied by the phase-shifting circuit.

13. A method as claimed in Claim 11 characterized in that the processing operation at the end of the time interval [p2] to form the phase difference ? (p2) - ? (p1) consists in calculating this phase difference in accordance with the expression:
where TRe [s(p2)] and IIm [s(p2]
respectively are the real part and the imaginary part of the quantity s(p2).

14. A method as claimed in Claim 11 characterized in that during the time interval [p2] the quantity r(p2) is calculated, in addition, in accordance with either of the two expressions:
and the processing operation at the end of the time interval [p2] to form the phase difference ?(p2) - ?(p1) consists in:
- updating an auxiliary variable INT in accordance with the recursion formula:
INT (m+1) = INT(m) + ?,IIm [s(p2)] - ?.INT(m).r(p2) at a high rate so that the auxiliary variable practically attains its final value INT (oo) within a relatively short time, - calculating the phase difference ? (p2) - ? (p1) in accordance with the expression:
? (p2) - ? )p1) = arc sin [INT(oo)].

15. A method as claimed in Claim 11 characterized in that the processing operation at the end of the time interval [p2] to form the phase difference ? (p2) - ? (p1) consists in updating an auxiliary variable INTo in accordance with the recursion formula:
INTo(m+1) = INTo(m) - ?.II m [s* (p2).exp j.INTo(m)]
at a high rate so that the auxiliary variable practically attains its final value within a relatively short time, said final value being equal to the phase difference ? (p2) - ? (p1).

16. A method as claimed in Claim 15 characterized in that the phase variation .DELTA.W.T is obtained by updating said auxiliary variable INTo in accordance with the recursion formula:
at a high rate so that the auxiliary variable practically attains its final value within a relatively short time, said final value being equal to the phase variation .DELTA.W.T

17. An apparatus for carrying out a method as claimed in Claim 1 or 2 characterized in that for calculating the coefficients of the transversal filter of the echo canceller said apparatus com-prises:
- calculation means to form at any instant nT the products of the N
complex conjugate values of the N data stored in the transversalfilter and the difference signal for the received signal).
- calculation means for weighting these products by the weighting coefficient 1/(L.sigma.2), - accumulators for accumulating the weighted products from the beginning to the termination of the time interval of a duration LT, the desired coefficients of the adaptive filter being simultaneously obtained in said accumulators at the end of said time interval.

18. An apparatus for carrying out the method claimed in Claim 1 or 2 characterized in that for calculating the N coefficients of the transversal filter of the echo canceller said apparatus com-prises a further transversal filter having L coefficients and receiving as an input signal the difference signal (or the received signal), utilizing as coefficients the complex conjugate values of the L data stored during the calculation time in a memory and supplying an output signal which is weighted by the weighting coefficient 1/(L.sigma.2) and transmitted through a time window of duration NT so that the desired N coefficients of the transversal filter of the echo canceller are formed in series.

19. An apparatus for carrying out the method claimed in Claim 8 characterized in that for the calculation of a quantity s(p2) said apparatus comprises:
- calculation means to form at any instant nT the product of a signal derived from the output signal of the phase-shifting circuit and a signal derived from the difference signal for of the received signal), - an accumulator for accumulating said products from the beginning to the end of a time interval [p2] of a duration LT, - a circuit processing the quantity s(p2) to derive therefrom the phase difference ?2 - ?1 and the phase variation .DELTA.W.T.