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Description  |
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FIELD OF THE INVENTION
This invention relates to a source separation system for processing input
signals formed by instantaneous linear mixtures of primary signals that
result from sources and for producing at least one estimated primary
signal, which mixtures cause mixing coefficients to occur, the system
comprising source separation means which have a number of inputs connected
to the input signals and at least one output for producing the estimated
primary signal, the source separation means adaptively determining
separation coefficients which are used for extracting the estimated
primary signals.
The invention likewise relates to the application of such a system to the
reception of electric signals, sound signals, electromagnetic signals. It
may relate, for example, to an aerial, a car radio or to a hands-free
telephone.
BACKGROUND OF THE INVENTION
The technique of separating primary signal sources, which consists of
processing mixtures of primary signals to produce an estimate of each
primary signal, is known. Said technique is applied to primary signals
resulting from independent sources, which signals are only available in
the form of said mixtures. This may relate to convolutional linear
mixtures or instantaneous linear mixtures. They may be generated by
propagation mechanisms of primary signals and/or by superposition
mechanisms for signals that result from various sources or other causes.
Generally, the technique of source separation works "in the blind", that is
to say, the sources are supposed to be unknown, to be independent, to have
unknown mixtures. Therefore, various samples of said mixtures are detected
on the basis of which the use of separation algorithms permits of
restoring one or various estimates of the original primary signals.
Such a technique, applied to the separation of instantaneous linear signal
mixtures is disclosed, for example, in the document entitled "Separation
of Independent Sources from Correlated Inputs" by J. L. Lacoume and P.
Ruiz, IEEE Transactions on Signal Processing, Vol. 40, No. 12 Dec. 1992,
pages 3074 to 3078.
For effecting the source separation, that is to say, obtaining on the
output an estimate of each source that forms the mixture, this document
reveals a method of calculating cumulants, For this purpose, it teaches to
adapt parameters of the source separation system in such a way that
cumulants of output signals which are expressed as a function of cumulants
measured on the input signals resulting from the mixtures are set to zero,
while the cumulants are of a higher order than the second order. By
setting these cumulants to zero, inverse mixing coefficients are
indirectly derived to obtain an inverse transform of the transform
obtained from the application of mixing operations to the primary signals.
The teaching of this document leads to giving a direct structure to the
system of source separation. Moreover, as the input signals are
statistically dependent because they result from mixtures, the equations
that link the cumulants of the output signals to the cumulants of the
input signals are very complex.
Such a technique has turned out to be very complex to implement and does
not yield a simple solution in the case of real signals which generally
result from instantaneous linear mixtures.
SUMMARY OF THE INVENTION
It is an object of the invention to simplify the processing that leads to
distinguishing the sources by producing estimated primary signals.
This object is achieved with a source separation system that comprises
characterization means which have a number of inputs connected to the
input signals for estimating the mixing coefficients based upon these
signals, and have at least one output for producing estimated mixing
coefficients, while the separation means have at least one further input
for receiving the estimated mixing coefficients and comprise means for
transforming the estimated mixing coefficients into separation
coefficients.
Advantageously, an estimation is obtained of the mixing coefficients
themselves and not an estimation of the inverse mixing coefficients. This
property ensures a source separation that is much more precise, while the
estimated primary signals are nearer to the primary signals. In addition,
the source separation means may thus have a recursive structure which is
more compact.
Preferably, the characterization means comprise estimating means for making
estimates of cumulants of a higher order than the second order based upon
the input signals, and calculating means for calculating estimated
coefficients for transforming the estimates of said cumulants into
estimated mixing coefficients.
Advantageously, the processing is simplified by establishing a relation
between the cumulants of the input signals and the cumulants of the source
signals. As the sources are statistically independent, it follows that the
equations are much simpler, thus easier to resolve than in the prior art,
which calls for less considerable processing means.
More particularly, the estimated coefficient calculating means comprise a
neuron network for transforming, in accordance with a predetermined
transform function, the estimates of said cumulants into estimated mixing
coefficients.
An interesting field of application of the invention is that where the
primary signal sources and/or receivers are mobile, which entail variable
transmission conditions.
These and other aspects of the invention will be apparent from and
elucidated with reference to the embodiments described hereinafter.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows a diagram of a source separation system according to the
invention,
FIG. 2 shows a diagram of a characterization unit according to the
invention,
FIG. 3 shows a diagram of a source separation unit according to the
invention comprising a transformation unit for transforming estimated
mixing coefficients into separation coefficients,
FIG. 4 shows a partial diagram of a first illustrative embodiment of the
separation unit in the case of a direct structure, and
FIG. 5 shows a partial diagram of a second illustrative embodiment of the
separation unit in the case of a recursive structure.
DESCRIPTION OF EMBODIMENTS
In FIG. 1 are represented the primary sources 5, S1 to Sn, formed, for
example, by various RF transmitters. For identifying and separating these
transmitters, sensors C1-Cn are put up at various receiving points. Each
of these sensors is, for example, an antenna followed by an amplifier and
a demodulator. The sensors produce signals E.sub.1 (t) to E.sub.n (t)
resulting from mixtures of primary signals X.sub.1 (t) to X.sub.n (t)
which are either produced directly by the sources S1 to Sn, or are used
for modulating signals broadcast by the sources S1 to Sn.
The mixtures which are produced between the primary signals are directly
linked to the propagation of these signals in space. These mixtures may be
rendered completely or approximately linear and instantaneous by utilizing
transmitters and adequate sensors, notably by selecting the types of
modulators and demodulators used, as well as the operating frequencies of
the system, for example, by selecting transmitters that use an amplitude
modulation.
In FIG. 1 the detected signals E.sub.1 (t) to E.sub.n (t) enter a source
separation unit 10 which produces estimated primary signals X.sub.1 (t) to
X.sub.n (t), where the variable t is, for example, time. The mixed input
signals E.sub.1 (t) to E.sub.n (t) are instantaneous linear mixtures,
generally caused by p unknown primary signals X.sub.1 (t) to X.sub.p (t),
so that:
##EQU1##
where 1.ltoreq.i.ltoreq.n and 1.ltoreq.j.ltoreq.p, where i and j are
current variables. In FIG. 1, n=p.
These primary signals are supposed to be statistically independent,
preferably up to at least the fourth order. This preference may be changed
to at least the third order in certain cases.
The coefficients a.sub.ij are mixing coefficients which define the
contributions of the signal X.sub.j (t) to the mixed signal E.sub.i (t).
The mixing coefficients a.sub.ij, which are unknown, are constant values
or slowly variable values.
For the following description, primary normalized signals are defined, so
that:
Y.sub.j (t)=a.sub.jj.X.sub.j (t) for 1.ltoreq.j.ltoreq.p (2)
and normalized mixing coefficients so that:
.alpha..sub.ij =a.sub.ij /a.sub.jj for i.noteq.j. (3)
For i=j one has .alpha..sub.ii =.alpha..sub.jj =1.
After this normalization, equation (1) becomes
##EQU2##
The normalized mixing coefficients a.sub.ij will henceforth be simply
called mixing coefficients.
For producing estimated primary signals X.sub.k (t) on the basis of the
input signals E.sub.i (t), the separation unit 10 in the case where it has
a direct structure, utilizes separation coefficients C.sub.ki. It produces
estimated primary signals by performing the inverse transformation (k,
output running index):
##EQU3##
The separation coefficients C.sub.ki are determined in the separation unit
10 on the basis of estimated mixing coefficients .alpha..sub.ij which form
an estimate of the mixing coefficients .alpha..sub.ij which are unknown.
This estimate is obtained in a characterization unit 15 which produces the
estimated mixing coefficients (FIG. 1) on the basis of input signals
E.sub.i (t).
Analogously, the separation unit (10) may have a recursive structure in
which separation coefficients d.sub.ki are used.
FIG. 2 shows a diagram of the characterization unit 15. It comprises an
estimation unit 16 followed by an estimated coefficient calculation unit
17. The estimation unit 16 receives the set of input signals E.sub.i (t)
and forms an estimate of a set A of characteristic magnitudes of the input
signals E.sub.i (t). These characteristic magnitudes of set A are selected
so that it is firstly possible to calculate them on the basis of the input
signals E.sub.i (t) and secondly to express them from the mixing
coefficients .alpha..sub.ij and from a set B of characteristic magnitudes
which relate to the normalized primary signals Y.sub.j (t), although they
are unknown, while the characteristic magnitudes of set B are of the same
type as the characteristic magnitudes of set A.
In the following description, the selected characteristic magnitudes will
be cumulants calculated on the basis of the mixed signals E.sub.i (t).
The relations linking set A and set B via mixing coefficients form an
intrinsic property of the described system. According to the invention,
the estimation unit 16 determines an estimate of set A, which estimate is
sufficient for estimated mixing coefficients .alpha..sub.ij to be derived
therefrom in the estimated coefficient calculation unit 17.
According to a preferred embodiment, the characteristic magnitudes which
form set A are autocumulants associated to each mixed signal E.sub.i (t)
which has previously been centered, as well as cross-cumulants associated
to each pair of mixed signals E.sub.f (t) and E.sub.g (t) previously
centered (f and g being integral indices, 1.ltoreq.f.ltoreq.n,
1.ltoreq.g.ltoreq.n). Analogously, the characteristic magnitudes which
form set B are autocumulants associated to each normalized primary signal
Y.sub.i (t) that has previously been centred. The definition of these
magnitudes is well known per se to a person skilled in the art.
According to the invention, one considers the cumulants (autocumulants and
cross-cumulants) of an order higher than the second order. The cumulants
which can be calculated in the simplest way are those that have the lowest
order. Nevertheless, the third-order cumulants must be sufficiently large
for the calculation to be limited to only these cumulants. This condition
occurs only for a limited number of types of sources. Preferably,
according to the invention, fourth-order cumulants are calculated which
allows of coverage of a larger spectrum of experimentally found signals. A
person of ordinary skill in the art may choose cumulants of a different
order, without departing from the scope of the invention.
For, clarity's sake, a repetition of the definitions of the cumulants will
be given hereinafter, while only the cumulants necessary for the discussed
example will be confined to, that is to say, the autocumulants and the
fourth-order cross-cumulants, while two signals will be confined to, and
without causing a time shift to occur between the two signals of interest.
In the general case, let us consider two signals U(t) and V(t) which vary
as a function of a variable t, for example, time. Centred signals are
defined by:
u(t)=U(t)-E{U(t)} (6)
v(t)=V(t)-E {V(t)} (7)
In the equations described above, the notation E{.} represents the
mathematical expectation of the expression in parentheses.
The five fourth-order cumulants associated with these centered signals are
(while abbreviating by u and v, the signals u(t) and v(t), respectively):
cum4(u)=cum(u,u,u,u)=E{u.sup.4 }-3›E{u.sup.2 }!.sup.2 (8)
cum31(u,v)=cum(u,u,u,v)=E{u.sup.3 v}-3E{u.sup.2 }E{uv} (9)
cum22(u,v)=cum(u,u,v,v)=E{u.sup.2 v.sup.2 }-E{u.sup.2 }E{v.sup.2
}-2›E{uv}!.sup.2 (10)
cum31(v,u)=cum(u,v,v,v)=E{v.sup.3 u}-3E{v.sup.2 }E{uv} (11)
cum4(v)=cum(v,v,v,v)=E{v.sup.4 }-3›E{v.sup.2 }!.sup.2 (12)
In practice, the exact cumulants are not calculated, but only estimated
cumulants. Various techniques for calculating cumulants are known to a
person skilled in the art. By way of example, only two methods will be
provided hereinafter.
A first, simple method for calculating cumulants consists of executing the
following operations:
1. In one time window, take M samples for each signal U(t) and V(t).
2. Based upon these samples, make estimates of the mathematical
expectations of the signals U(t) and V(t) by calculating two estimators
<U> and <V>. Each estimator is, for example, the arithmetic mean of the
signal of interest calculated over the M samples taken. These estimators
<U> and <V> are real numbers.
3. Estimators <u(t) > and <v(t) > which relate to the centred signals u(t)
and v(t) are derived from the equations 6 and 7 and estimators <U> and
<V>. These first estimators contain M samples which are, for example,
calculated in the following manner. The respective estimator <u(t)> or
<v(t)> of the centred signal u(t) or v(t) is calculated by subtracting
from each sample of the initial signal U(t), V(t) respectively, the
estimator <U> of the mathematical expectation E{U(t)}, estimator <V> of
the mathematical expectation E{V(t)}, respectively. ›equations 6 and 7
applied to the estimators!.
4. The same calculation method is applied for the estimators of all the
mathematical expectations which occur in the equations 8 to 12. For
example, the estimator of E{u.sup.4 } will be obtained by calculating the
arithmetic mean of ›<u(t)>!.sup.4 of M samples. Similar calculations hold
for the other mathematical expectations to be calculated.
5. An estimate for each cumulant defined by the equations 8 to 12 is
derived therefrom by replacing the mathematical expectations by their
respective estimates.
A greater precision can be obtained in the estimate of the cumulants by
applying a second method which consists of subdividing the window that
contains the M samples into various sub-windows. The first method is
applied for each sub-window to produce partial cumulants. Then, the
cumulants are calculated for the set of M samples (equations 8 to 12) by
calculating the arithmetic means of the partial cumulants.
Other calculation methods for calculating the cumulants can be used.
For a proper understanding of the invention, let us consider, by way of
example, the simple case of the separation of 2 sources while 2 sensors
are used. In this example there will only be considered the fourth-order
autocumulants and cross-cumulants. Two signals E.sub.1 (t) and E.sub.2 (t)
are detected which, after centering, become the signals e.sub.1 (t) and
e.sub.2 (t). Similarly, y.sub.1 (t) and Y.sub.2 (t) are the centering
signals corresponding to the normalized signals Y.sub.1 (t) and Y.sub.2
(t) assumed to be independent.
By applying that which has previously been discussed, the following 5
equations are obtained:
cum4(e1)=cum4(y1)+.alpha..sup.4.sub.12 cum4(y2) (13)
cum31(e1,e2)=.alpha..sub.21 cum4(y1)+.alpha..sup.3.sub.12 cum4(y2)(14)
cum22(e1,e2)=.alpha..sup.2.sub.21 cum4(y1)+.alpha..sup.2.sub.12
cum4(y2)(15)
cum31(e2,e1)=.alpha..sup.3.sub.21 cum4(y1)+.alpha..sub.12 cum4(y2)(16)
cum4(e2)=.alpha..sup.4.sub.21 cum4(y1)+cum4(y2) (17)
These equations contain:
i) the elements of already defined set A, that is to say, the elements that
characterize the signals produced by the sensors, that is: cum4(e1),
cum31(e1,e2), cum22(e1,e2), cum31(e2,e1), cum4(e2).
ii) the elements of already defined set B, that is to say, the elements
characterizing the normalized primary signals, that is: cum4(y1), cum4(y2)
which are unknown.
iii) two unknown mixing coefficients: .alpha..sub.12, .alpha..sub.21. While
disposing of 5 equations, it is thus theoretically possible to determine
the unknown elements.
In the case of more than two sources, with a number of sensors equal to the
number of sources, the cumulants cum4 (equations 13 and 17) and the
cumulants cum31 (equations 14 and 16) are, for example, generalized
respectively, so that:
##EQU4##
with 1.ltoreq.j.ltoreq.n, j.noteq.r and j.noteq.s, and
1.ltoreq.r.ltoreq.n, 1.ltoreq.s.ltoreq.n, and r.noteq.s,r,s being current
integer values.
The aim of this calculation (the case of two sources and two sensors) is to
determine the mixing coefficients .alpha..sub.12, .alpha..sub.21 in
reality, the calculation is made to determine solely estimates of the
mixing coefficients .alpha..sub.12, .alpha..sub.21 in the form of
estimated coefficients .alpha..sub.12, .alpha..sub.21.
The calculation of the estimated coefficients .alpha..sub.12,
.alpha..sub.21 is made in the characterization unit 15 (represented in
FIG. 2). Firstly, the estimation unit (16) determines the estimates of the
cumulants of the mixed signals, after which estimated coefficient
calculation unit (17) produces the estimated coefficients .alpha..sub.12,
.alpha..sub.21.
The estimation unit (16) calculates the cumulants cum4(e1) to cum4(e2) in
accordance with equations 8 to 12. The estimation unit DSP can be formed
by a calculator, a microprocessor or a digital signal processing unit.
When all the previous cumulants have been calculated, they are processed
by the coefficient calculation unit 17 which calculates the mixing
coefficients .alpha..sub.12 and .alpha..sub.21 on the basis of the
equations 13 to 17. In reality, the resolution of these equations does not
give the real coefficients .alpha..sub.12 and .alpha..sub.21, but
approximate values which form the estimated coefficients .alpha..sub.12
and .alpha..sub.21, because these are the estimated cumulants and not the
exact cumulants that are used for resolving the equations 13 to 17.
The coefficient calculation unit 17 may also be formed by a calculator, a
microprocessor or a digital signal processing device which applies an
equation resolution method.
For resolving these equations, it is possible to use non-linear equation
resolution methods which are known to a person of ordinary skill in the
art. They may be Gauss-Newton, Levenberg-Marquardt, Powell-Fletcher
algorithms or conventional gradient slope methods.
Another approach consists of reducing the number of equations by combining
the equations 13 to 17. Thus, the following equations are obtained:
.alpha..sub.21.sup.3 cum4(e.sub.1)+.alpha..sub.12
cum4(e.sub.2)-.alpha..sub.21.sup.3 .alpha..sub.12
cum31(e.sub.1,e.sub.2)-cum31(e.sub.2,e.sub.1)=0 (20)
.alpha..sub.21 cum4(e.sub.1)+.alpha..sub.12.sup.3
cum4(e.sub.2)-cum31(e.sub.1,e.sub.2)-.alpha..sub.12.sup.3 .alpha..sub.21
cum31(e.sub.2,e.sub.1)=0 (21)
The coefficient calculation unit 17 is in that case programmed for
resolving the equations 20 and 21, for example, via one of the known
methods indicated previously.
It is possible to combine these two equations to retain only equations
containing a single mixing coefficient .alpha..sub.12 or .alpha..sub.21
which are easier to resolve.
Another approach can consist of not resolving exactly the system of
equations 13 to 17 as has just been observed, but to confine oneself to
making an approximation of the function that links the calculated cumulant
estimates to the estimated mixing coefficients .alpha..sub.ij which are to
be determined. In effect, the equations 13 to 17 may coarsely be described
in the form of:
.alpha..sub.ij =F(A). (22)
Equation 22 has only for its aim to link the unknown coefficients
.alpha..sub.ij to set A, without considering set B which, although
likewise unknown, need not be calculated.
If G(.) is a function that forms an approximation of F(.), and if G(.) is
simpler to use experimentally, one may restrict oneself to calculating:
.alpha..sub.ij =G(A). (23)
This function F(.) is characteristic of the number of primary signals and
sensors which are necessary for utilizing the application of interest.
This forms an advantage, because the system itself is independent of the
other parameters of the application. It will easily be understood that the
coefficient calculation unit 17 may first undergo in a first phase a
learning process to resolve the function F(.). In a second phase, the data
which are measured experimentally can be processed in accordance with the
learned method. It is known that an approximate value of the function F(.)
can be obtained via a function G(.) by utilizing a multilayer neuron
network or a multilayer perceptron or a tree-like neuron network.
The use of a neuron network is particularly interesting, because it does
not require each time for each batch of cumulants the resolution of the
equations 13 to 17 in the separation system itself. It is sufficient that
the neuron network learns beforehand to resolve the function G(.), so that
it can subsequently be used for each batch of cumulants, while the source
separation system thus becomes assigned to the type of function G(.)
learned. To adapt the source separation system to a number of sensors or
to a different number of primary signals to be separated, it is sufficient
to have the neuron network again learn the new function G(.) that relates
to this new number of sensors and/or to this new number of primary
signals. To generate the array of examples necessary for the learning
process, arbitrary autocumulant values are chosen for the primary signals
as well as arbitrary values for the normalized mixing coefficients
.alpha..sub.ij (terms on the right in the equations 13 to 17). With the
aid of these equations, the autocumulants and the cross-cumulants of the
mixed signals are calculated (term on the left in said equations). Then,
these cumulants of the mixed signals are introduced on the inputs of the
neuron network and there is imposed that its outputs produce the
normalized mixing coefficients a.sub.ij selected previously. These
produced coefficients .alpha..sub.ij form the results to be obtained on
the output of the neuron network. The learning of the neuron network will
thus consist of determining the synaptic coefficients of the neuron
network to approximate the input/output signal correspondence of the
neuron network in conformity with the precalculated results of the
equations 13 to 17. This provides the advantage of not having to resolve
the complex equations 13 to 17.
The estimated mixing coefficients a.sub.ij thus calculated are applied to
source separation unit 10 to demix the input signals E.sub.i (t) and
produce estimated primary signals X.sub.k (t).
The source separation effected by the source separation unit 10 may be
written in the form of:
##EQU5##
for the direct structure, and
##EQU6##
for a recursive structure.
The coefficients C.sub.ki, d.sub.ki are separation coefficients which are
derived from the estimated mixing coefficients .alpha..sub.ij which enter
the separation unit 10.
The transformation of the estimated mixing coefficients .alpha..sub.ij into
separation coefficients C.sub.ki, d.sub.ki depends on the direct or
recursive structure of the separation unit 10.
If the separation unit 10 has a recursive structure, one may be led to
performing either of the following transformations:
d.sub.ki =.alpha..sub.ki or
d.sub.ki =.alpha..sub.k,.sigma.(i) /.alpha..sub.i,.sigma.(i)(26)
The first of these transformations corresponds to the case where the
estimated primary signal corresponding to the k.sup.th order primary
signal is found back on the k.sup.th output of the separation unit.
But cases may occur where stability problems do not allow this
correspondence to be established. It may thus be necessary to provide that
the estimated primary signal corresponding to the k.sup.th order primary
signal is produced on another output of the separation unit. Appropriate
separation coefficients d.sub.ki must thus be determined which make it
possible to perform this permutation. This is the aim of the second
transformation which uses a permutation law .sigma.(i) for the indices.
In the case of two sources, with two estimated coefficients .alpha..sub.12
and .alpha..sub.21, the separation coefficients d.sub.ki are selected such
that:
if .alpha..sub.12..alpha..sub.21 .ltoreq.1, one has d.sub.12 .alpha..sub.12
; d.sub.21 =.alpha..sub.21. (27)
for the first transformation, and
if .alpha..sub.12..alpha..sub.21 <1, one has d.sub.12 =1/.alpha..sub.21 ;
d.sub.21 =1/.alpha..sub.12 (28)
for the second transformation.
When the source separation unit 10 has a direct structure, a matrix
inversion is to be made so that:
##EQU7##
with C.sub.kk =1.
The transformations of the equations 26 to 28 in the case of the recursive
structure or the matrix inversion of the equation 29 in the case of the
direct structure are realized in the transformation unit 11 DSP, for
example, a calculator, a microprocessor or a digital signal processing
unit. In this case (FIG. 3) the source separation unit 10 comprises the
transformation unit 11 which transforms the estimated mixing coefficients
.alpha..sub.ij into separation coefficients C.sub.ki, d.sub.ki, and a
separation sub-unit 12 which receives on a number of inputs the mixed
signals E.sub.i (t) and which produces at least one estimated primary
signal X.sub.k (t). The separation coefficients C.sub.ki, d.sub.ki arrive
on another number of inputs.
FIG. 4 represents part of a separation sub-unit 12.sub.k which has a direct
structure. It comprises a number of inputs I.sub.1 to I.sub.n which
receive each a mixed signal E.sub.1 (t) to E.sub.n (t). Each of these
inputs is connected to a multiplier means 13.sub.1, to 13.sub.n for said
signal via the separation coefficient C.sub.k1 to C.sub.kn assigned to the
input. The outputs of all the multiplier means 13.sub.1 to 13.sub.n are
connected to a summator 125 for adding together all the signals and
producing the estimated primary signal X.sub.k (t). The separation
sub-unit 12 (FIG. 3) contains as many partial sub-units 12.sub.k as there
are estimated primary signals X.sub.k (t).
FIG. 5 represents a separation sub-unit 12 which has a recursive structure
for an example intended to produce two estimated primary signals X.sub.1
(t) and X.sub.2 (t) based on two mixed signals E.sub.1 (t) and E.sub.2
(t). This structure comprises a first summator 112 which has an input
connected to the signal E.sub.1 (t) and an output which produces the
estimated primary signal X.sub.1 (t). A second summator 212 has an input
connected to the signal E.sub.2 (t) and an output which produces the
estimated primary signal X.sub.2 (t). Another input of the first summator
112 is connected to the output of the second summator 212 via a multiplier
means 111 which weights the output signal of the second summator with a
coefficient -d.sub.12. Similarly, another input of the second summator 212
is connected to the output of a first summator 112 via another multiplier
means 211 which weights the output signal of the first summator with a
coefficient -d.sub.21. The summators 112, 212 and the multiplier means
111, 211 may form part of a calculator, a microprocessor or a digital
signal processing unit correctly programmed for performing the described
functions.
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