Method of and apparatus for identifying unknown system using adaptive filter

A method of and an apparatus for identifying an unknown system using an adaptive filter wherein the step size is subsequently supplied to the adaptive filter so that the identification error may be decreased. Value of the step size is calculated by using a gradient of power of error signal according to the LMS or LIM (learning identification method) algorithm, and limited by the previous value of step size. Further, when a sudden change in the error signal is detected, the step size can be re-setted or the limitation to the step size can be released.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to a method of and an apparatus for identifying an 
unknown system using an adaptive filter. 
2. Description of the Related Art 
Identification of an unknown system using an adaptive filter proceeds such 
that the same signal is inputted both to an unknown system to be 
identified and to an adaptive filter, and coefficients of the adaptive 
filter are updated using the identification error obtained by subtracting 
an output of the adaptive filter from the output of the unknown system. 
The identification error will hereinafter be referred to as an error 
signal. This identification of an unknown system using an adaptive filter 
can be applied, for example, to an echo canceler for removing echoes which 
may be produced at a two-line to four-line converting portion on a 
transmission line, an equalizer for removing intersymbol interference 
between codes which may take place in transmission lines, a noise canceler 
for removing noise which may leak into a sound inputting microphone, a 
howling canceler for removing howling which may be caused by acoustic 
coupling from a loudspeaker to a microphone, and various other apparatus. 
An exemplary common document regarding an adaptive filter is B. Widrow et 
al., "Adaptive Signal Processing", (Prentice-Hall, N.J., 1985) 
(hereinafter referred to as reference 1.) Here, operation of an adaptive 
filter is described taking a noise canceler as an example. 
A noise canceler generally operates such that, using an adaptive filter 
having a transfer function approximated to an impulse response of a path 
from a noise source to a main input terminal, a noise replica 
corresponding to a noise component mixed into the main input terminal is 
produced to suppress the noise which is admitted into the main input 
terminal and gives a disturbance to a signal. In this instance, 
coefficients at taps of the adaptive filter are successively modified by 
taking the correlation between a difference signal obtained by subtraction 
of the noise replica from a mixed signal in which noise and an original 
signal are mixed and a reference noise available at a reference input 
terminal. As an exemplary algorithm for this coefficient modification of 
an adapive filter, that is, for convergence of a noise canceler, the LMS 
algorithm disclosed in reference 1 above and the learning identification 
method (LIM) disclosed in J. Nagumo et al., IEEE Transactions on Automatic 
Control, Vol. AC-12, No. 3, pp. 282-287 (1967) (hereinafter referred to as 
reference 2) are known. 
FIG. 1 is a block diagram showing the construction of an example of a 
conventional noise canceler. Referring to FIG. 1, a mixed signal in which 
a signal detected at main input terminal 1 and noise are mixed is supplied 
to subtracter 4. A reference noise signal detected at reference input 
terminal 2 is supplied to adaptive filter 3. A noise replica generated by 
adaptive filter 3 is subtracted from the mixed signal at subtracter 4. 
Consequently, the noise components in the mixed signal are cancelled and a 
signal wherein the noise is removed is supplied to output terminal 6. 
Output of subtracter 4 is supplied also to multiplier 5, at which it is 
multiplied by 2.alpha.. Output of multiplier 5 is used for updating of the 
coefficients of adaptive filter 3. Here, .alpha. is a constant and called 
a step size. Now, if a signal is represented by s.sub.k (k is an index 
representative of time), reference noise by n.sub.k, noise to be erased by 
v.sub.k, and additional noise that signal s.sub.k undergoes by 
.delta..sub.k, then signal w.sub.k to be supplied from input terminal 1 to 
subtracter 4 is given by: 
EQU w.sub.k =s.sub.k +v.sub.k +.delta..sub.k ( 1) 
The object of the noise canceler is to produce replica u.sub.k of noise 
component v.sub.k defined by equation (1) to cancel noise. By producing 
noise replica u.sub.k adaptively using a closed loop circuit consisting of 
adaptive filter 3, subtracter 4 and multiplier 5 shown in FIG. 1, 
difference signal d.sub.k given by the following equation can be obtained 
as an output signal of subtracter 4: 
EQU d.sub.k =s.sub.k +v.sub.k -u.sub.k ( 2) 
Since it is considered that generally .delta..sub.k is small enough 
compared with s.sub.k, .delta..sub.k is ignored here. (v.sub.k -u.sub.k) 
in equation (2) above is called residual noise, and from the point of view 
of system identification, this residual noise is equal to an error signal. 
If the LMS algorithm is assumed, then m-th coefficient c.sub.m,k+1 of 
adaptive filter 3 is updated in accordance with the following equation: 
EQU c.sub.m,k+1 =c.sub.m,k +2.alpha..multidot.d.sub.k .multidot.n.sub.m,k( 3) 
If equation (3) is represented in a matrix form for all N coefficients, 
then 
EQU c.sub.k+1 =c.sub.k +2.alpha..multidot.d.sub.k .multidot.n.sub.k( 4) 
where c.sub.k and n.sub.k are given by the following equations, 
respectively: 
EQU c.sub.k =[c.sub.0,k c.sub.1,k . . . c.sub.N-1,k ].sup.T ( 5) 
EQU n.sub.k =[n.sub.k n.sub.k-1 . . . n.sub.k-N+1 ].sup.T ( 6) 
where [.].sup.T represents transposition of the matrix. On the other hand, 
according to LIM, updating of coefficients are performed in accordance 
with following equation (7) in place of equation (4): 
EQU c.sub.k+1 =c.sub.k +(2.mu./N.delta..sub.n.sup.2).multidot.d.sub.k 
.multidot.n.sub.k ( 7) 
where .mu. is a step size in LIM, and .sigma..sub.n.sup.2 is an average 
power inputted to adaptive filter 3. Parameter .sigma..sub.n.sup.2 is used 
so that the value of step size .mu. may increase in inverse proportion to 
the average power in order to assure stabilized convergence. Various 
methods are available to calculate .sigma..sub.n.sup.2, and 
.sigma..sub.n.sup.2 can be calculated, for example, in accordance with 
following equation (8): 
##EQU1## 
Step sizes .alpha. and .mu. in equations (4) and (7) define the rate of 
convergence of the adaptive filter and the level of residual noise after 
convergence. In the case of LMS, as the value of .alpha. increases, the 
convergence occurs more rapidly, but the residual noise level increases. 
On the contrary, in order to attain a sufficiently low residual noise 
level, .alpha. having a corresponding low value must necessarily be 
adopted, which results in deterioration in convergence rate. This 
similarly applies to step size .mu. in LIM. 
In order to satisfy the contradictory requirements for the step sizes and 
the residual noise, an algorithm wherein the step size is variable has 
been proposed and is disclosed in Proceeding of International Conference 
on Acoustics, Speech and Signal Processing, pp. 1385-1388, 1990 
(hereinafter referred to as reference 3.) The algorithm will be 
hereinafter referred to as SGA-GAS (Stochastic Gradient Adaptive Filters 
with Gradient Adaptive Step Size). 
SGA-GAS uses .alpha..sub.k in place of step size .alpha. of the LMS 
algorithm of equation (4). Parameter .alpha..sub.k is defined by equation 
(9) below as a value which increases in proportion to the negative 
gradient of power d.sub.k.sup.2 of difference signal .alpha..sub.k : 
##EQU2## 
wherein .rho. is a positive constant and normally a very small value is 
used therefor. Equation (9) can be rewritten, using noise n.sub.k, as: 
EQU .alpha..sub.k =.alpha..sub.k-1 +.rho.d.sub.k d.sub.k-1 n.sub.k-1.sup.T 
n.sub.k ( 10) 
Further, .alpha..sub.k must satisfy the following condition: 
##EQU3## 
where {.}.sup.T is transposition of the matrix, tr{.} a trace of the 
matrix, and R an autocorrelation matrix of noise n.sub.k. 
FIG. 2 is a block diagram showing an example of construction of a noise 
canceler which employs the SGA-GAS algorithm. While the step size is fixed 
in the noise canceler shown in FIG. 1, it is variable in the noise 
canceler shown in FIG. 2. In particular, step size .alpha..sub.k 
represented by equation (10) is calculated by step size controller 8 and 
then limited by limiter 9, whereafter it is supplied to adaptive filter 3 
via multiplier 5. An example of construction of step size controller 8 is 
shown in a block diagram of FIG. 3. 
Referring to FIG. 3, step size controller 8 has two input terminals 90 and 
93 and single output terminal 100 and includes correlation calculating 
circuit 94, three multipliers 92, 96 and 97, two delay elements 91 and 99 
and adder 98. Difference signal d.sub.k is supplied to first input 
terminal 90 while reference noise n.sub.k is supplied to second input 
terminal 93. Step size .alpha..sub.k which is an output of step size 
controller 8 is supplied from output terminal 100 to limiter 9 of FIG. 2. 
Signal d.sub.k supplied to first input terminal 90 is delayed by one 
sample period by first delay element 91 to make d.sub.k-1 and is supplied 
to first multiplier 92. Signal d.sub.k is also supplied to first 
multiplier 92, and d.sub.k .multidot.d.sub.k-1 which is an output of first 
multiplier 92 is transmitted to second multiplier 96: 
Reference noise n.sub.k supplied to second input terminal 93 is transmitted 
to correlation calculating circuit 94. Correlation C.sub.k, which is 
defined by the following equation, is calculated by correlation 
calculating circuit 94 and transmitted to second multiplier 96. 
EQU C.sub.k =n.sub.k-1.sup.T n.sub.k ( 12) 
Correlation C.sub.k is multiplied by d.sub.k .multidot.d.sub.k-1 by second 
multiplier 96 and is further multiplied by .rho. by third multiplier 97, 
whereafter it is transmitted as .rho..multidot.d.sub.k .multidot.d.sub.k-1 
.multidot.n.sub.k-1.sup.T .multidot.n.sub.k to adder 98. At adder 98, the 
signal from third multiplier 97 is added to the signal which is an output 
of second delay element 99 and was obtained one sample period earlier, and 
the sum signal is transmitted to output terminal 100. Accordingly, signal 
.alpha..sub.k transmitted to output terminal 100 is given by 
.alpha..sub.k-1 +.rho.d.sub.k d.sub.k-1 n.sub.k-1.sup.T n.sub.k, which 
coincides with equation (10) above. 
FIG. 4 is a block diagram showing construction of correlation calculating 
circuit 94. Correlation calculating circuit 94 includes N-input adder 123, 
N delay elements 121.sub.1 to 121.sub.N and N multipliers 122.sub.1 to 
122.sub.N. Delay elements 121.sub.1 to 121.sub.N are connected in series 
and form a tapped delay line, to which multipliers 122.sub.1 to 122.sub.N 
are connected respectively. Input terminal 120 of correlation calculating 
circuit 94 corresponds to second input terminal 93 in FIG. 3 and receives 
n.sub.k thereat. Signal n.sub.k is supplied to the delay line with taps 
and first multiplier 122.sub.1. For 2.ltoreq.i.ltoreq.N, outputs of 
(i-1)-th and i-th delay elements 121.sub.i-1 and 121.sub.i are supplied to 
i-th multiplier 122.sub.i. Accordingly, (n.sub.k, n.sub.k-1), (n.sub.k-1, 
n.sub.k-2), . . . , (n.sub.k-N+1, n.sub.k-N) are inputted to multipliers 
122.sub.1, 122.sub.2, . . . , 122.sub.N, respectively, and outputs of the 
multipliers are given by n.sub.k n.sub.k-1, n.sub.k-1 n.sub.k-2, . . . , 
n.sub.k-N+1 n.sub.k-N. 
Outputs of multipliers 122.sub.1 to 122.sub.N are all supplied to adder 
123, and a value given by 
##EQU4## 
is outputted from adder 123. Output n.sub.k-1.sup.T .multidot.n.sub.k is 
transmitted as correlation C.sub.k to output terminal 124 of correlation 
calculating circuit 94. 
Referring now to FIG. 5, limiter 9 includes 2-input minimum value circuit 
22 and 2-input maximum value circuit 21. Step size .alpha..sub.k is 
supplied to minimum value circuit 22 from step size controller 8 by way of 
input terminal 23. Th.sub.H, which is a threshold value for a maximum 
value, is supplied to the other input terminal of minimum value circuit 
22, and a smaller value of .alpha..sub.k and Th.sub.H is supplied as a 
minimum value from minimum value circuit 22 to maximum value circuit 21. 
Th.sub.L, which is a threshold value for a minimum value, is supplied to 
the other input terminal of maximum value circuit 21, and a greater value 
of the output of minimum value circuit 22 and Th.sub.L is supplied as a 
maximum value from maximum value circuit 21 to output terminal 20. In 
other words, step size .alpha..sub.k supplied to input terminal 23 is 
limited, at upper and lower limits thereof, with minimum value Th.sub.L 
and maximum value Th.sub.H, respectively, so that it is outputted as 
.alpha..sub.k given by the following equation: 
##EQU5## 
Here, if Th.sub.L =0 and Th.sub.H =2/(3.multidot.tr{R}), then a value 
equivalent to a value obtained by execution of equation (11) is outputted 
from limiter 9. 
Maximum value circuit 21 and minimum value circuit 22 can each be realized 
by a combination of selector 31 and comparator 32 as shown in FIG. 6. 
Minimum value circuit 21 will be described first. The two input terminals 
of minimum value circuit 22 in FIG. 5 correspond to two input terminals 33 
and 34, respectively, of FIG. 6. Signals supplied to input terminals 33 
and 34 are transmitted simultaneously to both selector 31 and comparator 
32. Comparator 32 compares the two input signals with each other and 
generates a controlling signal such that a smaller signal is selected by 
selector 31. The controlling signal is transmitted to selector 31, and the 
signal from input terminal 33 or 34 selected by selector 31 is transmitted 
as a minimum value to output terminal 30. On the other hand, in the case 
of maximum value circuit 21, comparator 32 generates a controlling signal 
such that selector 31 selects the greater signal of the two input signals 
supplied thereto. Selector 31 then supplies the signal selected in 
accordance with the controlling signal as a maximum value to output 
terminal 30. 
Identification of an unknown system based upon the SGA-GAS algorithm has 
been described so far by way of an example of a noise canceler. Here, in 
an ideal case wherein d.sub.k =v.sub.k -u.sub.k is met with difference 
signal d.sub.k, the negative gradient of power d.sub.k.sup.2 of difference 
signal d.sub.k coincides with a system identification error, and 
consequently, d.sub.k.sup.2 can be used for step size control. However, in 
an actual noise canceler, d.sub.k is given by d.sub.k =s.sub.k +v.sub.k 
-u.sub.k as represented in equation (2) and accordingly is influenced by 
signal s.sub.k. Therefore, a correct gradient of d.sub.k.sup.2 cannot be 
obtained, and the step size cannot be controlled correctly. In some other 
apparatus described above other than a noise canceler, even when s.sub.k 
=0, when additional noise .delta..sub.k in equation (1) cannot be ignored, 
difference signal d.sub.k is given by 
EQU d.sub.k =v.sub.k -u.sub.k +.delta..sub.k ( 15) 
and consequently, .delta..sub.k makes an obstruction to error signal 
v.sub.k -u.sub.k similar to s.sub.k. The presence of parameter 
.delta..sub.k or s.sub.k will result in elongation in converging time or 
an increase in final misadjustment after convergence. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a method of identifying 
an unknown system using an adaptive filter which is robust even with an 
obstruction signal to an error signal v.sub.k -u.sub.k and short in 
converging time and can minimize the final misadjustment after 
convergence. 
It is another object of the present invention to provide an apparatus for 
identifying an unknown system using an adaptive filter which is robust 
even with an obstruction signal to an error signal v.sub.k -u.sub.k and 
short in converging time and can minimize the final misadjustment after 
convergence. 
The first object of the present invention is achieved by a method of 
identifying an unknown system using an adaptive filter, which delays an 
input signal successively by one sample period to form a plurality of 
samples, calculates products between the samples and corresponding samples 
of a plurality of multiplicands provided correspondingly to the plurality 
of samples, and outputs the sum total of the products, wherein the unknown 
system is identified by adding to each of the multiplicands a product 
among a difference signal obtained by subtraction of an output of said 
adaptive filter from an output of the unknown system, the sample 
corresponding to the multiplicand and a parameter as an updated amount for 
unit updating to update the multiplicands so that the difference signal 
may be decreased, characterized in that a value which increases in 
proportion to a gradient of the difference signal with respect to the 
parameter is added to the parameter to obtain a sum and the parameter is 
modified using a limited sum obtained by applying a limit to the sum, and 
a threshold value for applying the limit is determined using a previous 
limited sum or sums. 
The first object of the present invention can also be achieved by a method 
of identifying an unknown system using an adaptive filter, which delays an 
input signal successively by one sample period to form a plurality of 
samples, calculates products between the samples and corresponding samples 
of a plurality of multiplicands provided correspondingly to the plurality 
of samples, and outputs the sum total of the products, wherein the unknown 
system is identified by adding to each of the multiplicands a product 
among a difference signal obtained by subtraction of an output of said 
adaptive filter from an output of the unknown system, the sample 
corresponding to the multiplicand and a parameter as an updated amount for 
unit updating to update the multiplicands so that the difference signal 
may be decreased, characterized in that a value which increases in 
proportion to a gradient of the difference signal with respect to the 
parameter is added to the parameter to obtain a sum and a limit is applied 
to the sum to obtain a limited sum, and when it is detected that there is 
a sudden change in the difference signal, the sum is regarded as the 
limited sum only for a period of time of a number of clocks equal to a 
predetermined first constant, and then the parameter is modified using the 
limited sum, and a threshold value for applying the limit is determined 
using a previous limited sum or sums. 
Further, the first object of the present invention can also be achieved by 
a method of identifying an unknown system using an adaptive filter, which 
delays an input signal successively by one sample period to form a 
plurality of samples, calculates products between the samples and 
corresponding samples of a plurality of multiplicands provided 
correspondingly to the plurality of samples, and outputs the sum total of 
the products, wherein the unknown system is identified by adding to each 
of the multiplicands a product among a difference signal obtained by 
subtraction of an output of said adaptive filter from an output of the 
unknown system, the sample corresponding to the multiplicand and a 
parameter as an updated amount for unit updating to update the 
multiplicands so that the difference signal may be decreased, 
characterized in that a value which increases in proportion to a gradient 
of the difference signal with respect to the parameter is added to the 
parameter to obtain a sum and a limit is applied to the sum to obtain a 
limited sum, and then, an average value of these limited sums is 
calculated and is determined as the parameter for calculation of the 
updated amount. 
The first object of the present invention can also be achieved by a method 
of identifying an unknown system using an adaptive filter, which delays an 
input signal successively by one sample period to form a plurality of 
samples, calculates products between the samples and corresponding samples 
of a plurality of multiplicands provided correspondingly to the plurality 
of samples, and outputs the sum total of the products, wherein the unknown 
system is identified by adding to each of the multiplicands a normalized 
value of a product among a difference signal obtained by subtraction of an 
output of said adaptive filter from an output of the unknown system, the 
sample corresponding to the multiplicand and a parameter as an updated 
amount for unit updating to update the multiplicands so that the 
difference signal may be decreased, said normalized value being obtained 
by normalization of the product with a value of input power to said 
adaptive filter, characterized in that a value which increases in 
proportion to a gradient of the difference signal normalized with the 
value of input power to said filter with respect to the parameter is added 
to the parameter to obtain a sum, and the parameter is modified using the 
sum. 
Furthermore, the first object of the present invention can also be achieved 
by a method of identifying an unknown system using an adaptive filter, 
which delays an input signal successively by one sample period to form a 
plurality of samples, calculates products between the samples and 
corresponding samples of a plurality of multiplicands provided 
correspondingly to the plurality of samples, and outputs the sum total of 
the products, wherein the unknown system is identified by adding to each 
of the multiplicands a normalized value of a product among a difference 
signal obtained by subtraction of an output of said adaptive filter from 
an output of the unknown system, the sample corresponding to the 
multiplicand and a parameter as an updated amount for unit updating to 
update the multiplicands so that the difference signal may be decreased, 
said normalized value being obtained by normalization of the product with 
a value of input power to said adaptive filter, characterized in that a 
value which increases in proportion to a gradient of the difference signal 
normalized with the value of input power to said filter with respect to 
the parameter is added to the parameter to obtain a sum and the parameter 
is modified using a limited sum obtained by applying a limit to the sum, 
and a threshold value for applying the limit is calculated using a 
previous limited sum or sums. 
The second object of the present invention is achieved by an identifying 
apparatus for identifying a characteristic of an unknown system using an 
adaptive filter, characterized in that it comprises: a subtracter for 
subtracting an output of said adaptive filter from an output signal of the 
unknown system to obtain a difference signal; a step size controller for 
receiving the difference signal and an input signal to said adaptive 
filter and successively calculating a step size for use for the updating 
of coefficients of said adaptive filter; a limiter for receiving an output 
of said step size controller and limiting the received output; a first 
delay element for feeding back an output of said limiter to said limiter 
and said step size controller; and a first multiplier for multiplying the 
output of said limiter by the difference signal; and in that an output of 
said first multiplier is used as the step size for the updating of 
coefficients of said adaptive filter. 
The second object of the present invention can be also achieved by an 
identifying apparatus for identifying a characteristic of an unknown 
system using an adaptive filter, characterized in that it comprises: a 
subtracter for subtracting an output of said adaptive filter from an 
output signal of the unknown system to obtain a difference signal; a first 
delay element; an error change detecting circuit for receiving the 
difference signal and detecting a sudden change of the received difference 
signal; a counter for receiving an output of said error change detecting 
circuit and counting clocks; a first selector for receiving an output of 
said error change detecting circuit and an output of said first delay 
element and selectively outputting one of the two received outputs in 
response to an output of said counter; a step size controller for 
receiving the difference signal and an input signal to said adaptive 
filter and successively calculating a step size for use for the updating 
of coefficients of said adaptive filter; a limiter for receiving an output 
of said step size controller and limiting the received output; a second 
selector for receiving an output of said limiter and the output of said 
step size controller and selectively outputting one of the two received 
outputs in response to an output of said first selector; a second delay 
element for feeding back an output of said second selector to said 
limiter; and a first multiplier for multiplying the output of said second 
selector by the difference signal; and in that said first delay element 
delays the output of said first selector and feeds back the delayed output 
to an input of said first selector; and an output of said first multiplier 
is used as a step size for the updating of coefficients of said adaptive 
filter. 
Further, the second object of the present invention can be also achieved by 
an identifying apparatus for identifying a characteristic of an unknown 
system using an adaptive filter, characterized in that it comprises: a 
subtracter for subtracting an output of said adaptive filter from an 
output signal of the unknown system to obtain a difference signal; a step 
size controller for receiving the difference signal and an input signal to 
said adaptive filter and successively calculating a step size for use for 
the updating of coefficients of said adaptive filter; a limiter for 
receiving an output of said step size controller and limiting the received 
output; and an averaging circuit for receiving an output of said limiter 
and calculating an average value from the received output; and in that an 
output of said averaging circuit is used as a step size for the updating 
of coefficients of said adaptive filter. 
Furthermore, the second object of the present invention can be also 
achieved by an identifying apparatus for identifying a characteristic of 
an unknown system using an adaptive filter, characterized in that it 
comprises: a subtracter for subtracting an output of said adaptive filter 
from an output signal of the unknown system to obtain a difference signal; 
a correlation calculating circuit for receiving an input signal to said 
adaptive filter and calculating and outputting a power value and a 
correlation of the input signal; a first delay element for receiving the 
power value and delaying the received power value by one sample period; a 
second delay element; a step size controller for receiving the difference 
signal, the correlation value, an output of said first delay element and 
an output of said second delay element and successively calculating a step 
size for use for the updating of coefficients of said adaptive filter; a 
limiter for receiving an output of said step size controller and limiting 
the received output; a first multiplier for multiplying an output of said 
limiter by the difference signal; and a first normalizing circuit for 
normalizing an output of said first multiplier with the power value; and 
in that said second delay element delays the output of said limiter by one 
sample period and feeds back the thus delayed output to said step size 
controller; and an output of said first normalizing circuit is used as a 
step size for updating of coefficients of said adaptive filter. 
The above and other objects, features and advantages of the present 
invention will be apparent from the following description referring to the 
accompanying drawings which illustrate an example of a preferred 
embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
An apparatus for identifying an unknown system using an adaptive filter 
according to a first preferred embodiment of the present invention is 
shown in FIGS. 7 to 14. Referring first to FIG. 7, the identifying 
apparatus is generally constructed such that it removes, from a mixed 
signal s.sub.k +v.sub.k of a signal s.sub.k and noise v.sub.k, which is 
inputted to main input terminal 1, the noise and outputs a resultant 
signal from output terminal 6. The identifying apparatus includes adaptive 
filter 3, subtracter 4, multiplier 5, step size controller 18, limiter 17 
and delay element 10. The apparatus is different from the conventional 
apparatus shown in FIG. 2 in construction of the limiter, step size 
controller and further in that output of limiter 17 is fed back to limiter 
17 and step size controller 18 by way of delay element 10. The function 
blocks in FIG. 7 which are denoted by the same reference numerals to those 
of FIG. 2 have the same functions. Limited step size .alpha..sub.k is 
supplied from limiter 17 to delay element 10. 
FIG. 8 shows a first example of construction of step size controller 18, 
and function block in FIG. 8 which are denoted by same reference numerals 
as those of FIG. 3 have same functions. Comparing with above-mentioned 
step size controller 8 (FIG. 3), the delay element for delaying the output 
of adder 98 and feeding back the delayed output to adder 98 is not 
provided in step size controller 18. Instead of this delay element, input 
terminal 89 which is connected to adder 98 and transmits output of outer 
delay element 10 (FIG. 7) to adder 98 is provided. 
FIG. 9 shows a first example of construction of limiter 17. Referring to 
FIG. 9, limiter 17 is composed of a pair of multipliers 25 and 26, 3-input 
maximum value circuit 27 and 3-input minimum value circuit 28. Output of 
delay element 10 (FIG. 7), i.e., .alpha..sub.k-1, is transmitted to 
multipliers 25 and 26 by way of input terminal 24. At multipliers 25 and 
26, .alpha..sub.k-1 is multiplied by coefficients a and b and is 
outputted as a.alpha..sub.k-1 and b.alpha..sub.k-1, respectively. 
a.alpha..sub.k-1 and b.alpha..sub.k-1 are supplied to maximum value 
circuit 27 and minimum value circuit 28, respectively. Step size 
.alpha..sub.k is supplied from step size controller 18 (FIG. 7) to minimum 
value circuit 28 by way of input terminal 23. Th.sub.H, which is a 
threshold value for a maximum value, is supplied to the remaining input 
terminal of minimum value circuit 28. The three inputs, i.e., 
.alpha..sub.k, Th.sub.H and b.alpha..sub.k-1, are compared with one 
another, and the minimum input is transmitted as an output of minimum 
value circuit 28 to maximum value circuit 27. Supplied to the other two 
input terminals of maximum value circuit 27 are Th.sub.L which is a 
threshold value for a minimum value and output a.alpha..sub.k-1 of second 
multiplier 25, and the maximum input of the three inputs is supplied to 
output terminal 20. In other words, step size .alpha..sub.k supplied to 
input terminal 23 is limited at the lower limit thereof with Th.sub.L and 
a.alpha..sub.k-1 and at the upper limit thereof with Th.sub.H and 
b.alpha..sub.k-1 and is outputted as .alpha..sub.k given by the 
following equation: 
##EQU6## 
Amplitude distributions of s.sub.k and v.sub.k -u.sub.k are generally 
independent of each other. Accordingly, 
EQU E[.vertline.s.sub.k .vertline.].multidot.E[.vertline.v.sub.k -u.sub.k 
.vertline.]=0 (17) 
stands. Here, E[.multidot.] is the mathematical expectation of 
[.multidot.]. By using .alpha..sub.k in place of .alpha..sub.k, even if 
s.sub.k makes an obstacle to v.sub.k -u.sub.k, an instantaneous influence 
of s.sub.k upon v.sub.k -u.sub.k can be suppressed by means of limiter 17 
to obtain a stabilized step size. While stabilization of the step size is 
achieved in FIG. 9 by limitation for both a maximum value and a minimum 
value, even limitation for only one maximum value or one minimum value is 
effective. 
If the maximum value and the minimum value of the step size are limited 
using a previous value or values of the step size as described above with 
reference to FIG. 9, then some defect may take place in the case that a 
characteristic of the unknown system to be identified presents a sudden 
change so that the error signal is increased suddenly. In this instance, 
the step size must necessarily be increased suddenly so that the adaptive 
filter may follow up the change of the characteristic of the unknown 
system. However, when the value of the Step size is limited by a previous 
value or values thereof, it can present only a moderate change; 
accordingly, the speed of following up a sudden change of the system is 
slow. An example of step size controller 18 which can cope with this 
defect is shown in FIG. 10. 
Function blocks in FIG. 10 which are denoted by the same reference numerals 
as those of FIG. 8 have the same functions. Step size controller 18 shown 
in FIG. 10 is different from that of FIG. 8 in that it includes sudden 
change detecting circuit 42 for error to which error signal d.sub.k is 
inputted, and selector 43 is interposed between adder 98 and input 
terminal 100. An output of adder 98 is inputted to an input terminal of 
selector 43, and .alpha..sub.0 is inputted to the other input terminal of 
selector 43. Parameter .alpha..sub.0 is used for re-setting of the step 
size. Selector 43 selects .alpha..sub.0 or an output of adder 98 in 
response to a controlling signal from sudden change detecting circuit 42 
and transmits the same as an output signal, i.e., as a step size, to 
output terminal 100. Accordingly, when selector 43 selects a signal from 
adder 98 and transmits it to output terminal 100, step size controller 18 
operates quite similarly to the conventional controller shown in FIG. 8. 
Thus, the step size is calculated adding a variation to a previous value. 
On the other hand, when selector 43 supplies .alpha..sub.0 to output 
terminal 100, the step size is equal to .alpha..sub.0 irrespective of a 
previous value, which corresponds to re-setting of the step size. Sudden 
change detecting circuit 42 receives as an input thereto an error signal 
supplied to input terminal 90 and detects a sudden change of the error 
signal. When a sudden change is detected, sudden change detecting circuit 
42 produces "1", but it produces "0" in all other cases. Selector 43 
selects .alpha..sub.0 when "1" is supplied thereto from sudden change 
detecting circuit 42, but selects an output signal of adder 98 when "0" is 
supplied thereto. 
Subsequently, an example of construction of sudden change detecting circuit 
42 will be described with reference to FIG. 11. The sudden change 
detecting circuit is provided to detect a sudden change of an error signal 
inputted to input terminal 50 thereof, and is composed of two multipliers 
51 and 56, selector 52, counter 53, 2-input maximum value circuit 54, 
delay element 55 and comparator 57. The error signal supplied to input 
terminal 50 is squared by multiplier 51 and transmitted to selector 52 and 
comparator 57. Constant "0" is supplied to the other input terminal of 
selector 52. Selector 52 is controlled by an output of counter 53 and 
transmits either the output of multiplier 51 or "0" to an input terminal 
of maximum value circuit 54. The output of maximum value circuit 54 is fed 
back to the other input terminal of maximum value circuit 54 by way of 
delay element 55. Counter 53 continues its counting up to predetermined 
integral number N.sub.c and forwards to selector 52, during counting up, a 
controlling signal in accordance with which selector 52 selects a signal 
supplied from multiplier 51. After completion of counting up to N.sub.c, 
counter 53 forwards to selector 52 another controlling signal in 
accordance with which selector 52 selects "0". Since an output of maximum 
value circuit 54 which was obtained one sample interval prior to the 
present output is fed back to maximum value circuit 54, the maximum value 
of values of the first to N.sub.c -th samples is detected by and held in 
the feedback loop. A sample here is the output of multiplier 51 to the 
second power, i.e., an error signal to the second power. When a sample 
following the N.sub.c -th sample is inputted, selector 52 selects "0" and 
transmits it to maximum value circuit 54; accordingly, the output of 
maximum value circuit 54 is a maximum value of the signals supplied from 
delay element 55, i.e., the maximum value of values of the first to 
N.sub.c -th samples. This signal is transmitted to multiplier 56. At 
multiplier 56, the maximum value supplied from maximum value circuit 54 is 
multiplied by constant e.sub.th, and the result of the multiplication is 
transmitted to comparator 57. Meanwhile, the squared value of the error 
signal, which is the output of multiplier 51, is supplied to the other 
input terminal of comparator 57. Comparator 57 outputs "1" when the 
squared value of the error signal is higher in magnitude than the output 
of maximum value circuit 54, but outputs "0" in all other cases, and 
transmits it to output terminal 58. Selector 43 (FIG. 10) is controlled by 
the signal supplied to output terminal 58, and .alpha..sub.0 or 
.alpha..sub.k is outputted as the step size from selector 43. If the error 
signal is represented by e.sub.k, inputs to comparator 57 are given by 
e.sub.k.sup.2 and e.sub.th .multidot.max{e.sub.k.sup.2 
.vertline.1.ltoreq.k.ltoreq.N.sub.c }. Accordingly, step size 
.alpha..sub.k supplied to output terminal 100 in FIG. 10 is represented in 
the following equation: 
##EQU7## 
Step size .alpha..sub.k obtained from equation (18) is transmitted as an 
output of step size controller 18 to limiter 17. 
In the sudden change detecting circuit for error, it is possible to utilize 
an absolute value of an error signal in place of the squared value of an 
error signal. FIG. 12 shows an example of an error fluctuation detecting 
circuit which utilizes the absolute value of an error signal. The sudden 
change detecting circuit differs from the one shown in FIG. 11 only in 
that absolute value calculating circuit 59 is provided in place of 
multiplier 51 connected to input terminal 50. Absolute value calculating 
circuit 59 calculates the absolute value of an error signal inputted to 
input terminal 50 and transmits it to selector 52 and comparator 57. 
Accordingly, when the sudden change detecting circuit shown in FIG. 12 is 
employed, step size .alpha..sub.k supplied to output terminal 100 shown in 
FIG. 10 is represented in the following equation: 
##EQU8## 
Step size .alpha..sub.k obtained from equation (19) is transmitted as an 
output of step size controller 18 to limiter 17. 
While the first preferred embodiment of the present invention is described 
so far, various forms of limiters can be used for limiter 17 in the 
present embodiment. FIG. 13 is a block diagram showing a second example of 
construction of limiter 17. 
Referring to FIG. 13, the limiter shown includes 3-input maximum value 
circuit 27, 3-input minimum value circuit 28 and a pair of multipliers 200 
and 201. It is to be noted that function blocks in FIG. 13 which are 
denoted by the same reference numerals to those of FIG. 9 have the same 
functions. 
The output of delay element 10 (FIG. 7), i.e., .alpha..sub.k-1, is 
transmitted to multipliers 200 and 201 by way of input terminal 24. 
.alpha..sub.k-1 is multiplied by positive constant .eta. at multiplier 
201 and is then multiplied further by .alpha..sub.k-1 by the other 
multiplier 200, so that it is supplied as .eta..alpha..sub.k-1 .sup.2 to 
maximum value circuit 27 and minimum value circuit 28. Step size 
.alpha..sub.k is supplied from step size controller 18 (FIG. 7) to minimum 
value circuit 28 by way of input terminal 33. The threshold value for the 
maximum value, i.e., Th.sub.H, is supplied to the remaining input terminal 
of minimum value circuit 28. The three inputs, i.e., .alpha..sub.k, 
Th.sub.H and .eta..alpha..sub.k-1 .sup.2, are compared with one another, 
and the minimum of the three inputs is transmitted as the output of 
minimum value circuit 28 to maximum value circuit 27. .eta..alpha..sub.k-1 
.sup.2 and Th.sub.L, which is the threshold value for the minimum value, 
are supplied to the other two input terminals of maximum value circuit 27. 
The maximum of the three inputs is outputted as the maximum value from 
maximum value circuit 27 and is supplied to output terminal 20. In other 
words, step size .alpha..sub.k supplied to input terminal 23 is limited 
for the minimum value thereof by threshold value Th.sub.L and 
.eta..alpha..sub.k-1 .sup.2 and for the maximum value thereof by threshold 
value Th.sub.H and .eta..alpha..sub.k-1 .sup.2, so that it is outputted as 
.alpha..sub.k given by the following equation: 
##EQU9## 
Amplitude distributions of s.sub.k and v.sub.k -u.sub.k are generally 
independent of each other. Accordingly, 
EQU E[.vertline.s.sub.k .vertline.].multidot.E[.vertline.v.sub.k -u.sub.k 
.vertline.]=0 (21) 
stands. By using .alpha..sub.k in place of .alpha..sub.k, even if s.sub.k 
makes an obstacle to v.sub.k -u.sub.k, an instantaneous influence of 
s.sub.k upon v.sub.k -u.sub.k can be suppressed by means of limiter 17 to 
obtain a stabilized step size. 
Subsequently, a third example of construction of limiter 17 will be 
described with reference to FIG. 14. The limiter is generally constructed 
such that third multiplier 202 for squaring .alpha..sub.k-1 inputted to 
input terminal 24 is additionally provided to the construction of the 
limiter shown in FIG. 9 (first example of construction.) The output of 
delay element 10 (FIG. 7), i.e., .alpha..sub.k-1, is transmitted to 
multiplier 202 by way of input terminal 24. .alpha..sub.k-1 is squared by 
multiplier 202 and is then multiplied by positive constants a and b at 
first and second multipliers 25 and 26, respectively. Outputs 
a.alpha..sub.k-1 .sup.2 and b.alpha..sub.k-1 .sup.2 thus obtained are 
supplied to maximum value circuit 27 and minimum value circuit 28, 
respectively. Thereafter, the limiter operates in a similar manner as in 
the first example of construction shown in FIG. 9. In particular, step 
size .alpha..sub.k supplied to input terminal 23 is limited for the 
minimum value thereof by threshold value Th.sub.L and a.alpha..sub.k-1 
.sup.2 and for the maximum value thereof by threshold value Th.sub.H and 
b.alpha..sub.k-1 .sup.2, so that it is outputted as .alpha..sub.k given 
by the following expression from output terminal 20: 
##EQU10## 
While stabilization of the step size is achieved in the third example of 
construction by limitation of both the maximum value and the minimum 
value, even limitation of only one of the maximum value and the minimum 
value is effective. 
Also in the second and third examples of construction of the limiter 
described above with reference to FIGS. 13 and 14, the maximum value and 
the minimum value of the step size are limited using a previous value of 
the step size, and some defect many take place in the case that a 
characteristic of the unknown system to be identified presents a sudden 
change so that the error signal is increased suddenly. In this instance, 
the step size must necessarily be increased suddenly so that the adaptive 
filter may follow up the change of the characteristic of the unknown 
system. However, when the value of the step size is limited by a previous 
value thereof, it can present only a moderate change; accordingly, the 
speed of following up a sudden change of the system is slow. Therefore, it 
is very effective to employ a step size controller which includes sudden 
change detecting circuit 42 described above with reference to FIG. 10. 
The first preferred embodiment of the present invention has been described 
so far. In the embodiment, when a step size which is used for the updating 
of coefficients is calculated using a gradient of power of an error 
signal, a limit which relies upon the earlier step size is provided for a 
variation of the step size obtained so that the step size may be prevented 
from being made excessively different from the correct value due to noise 
obstruction or other factors. Further, the power of the identification 
error signal is monitored so that, when it is detected that a 
characteristic of the unknown system of an object for identification has 
presented a sudden change, the step size is reset in order to achieve both 
quick convergence and low identification error. 
Subsequently, a second preferred embodiment of the present invention will 
be described. FIG. 15 is a block diagram showing construction of an 
apparatus for identifying an unknown system using an adaptive filter in 
the present embodiment. 
The present apparatus is different from the apparatus of the first 
embodiment described above in the manner of re-setting the step size. In 
particular, while the step size is first re-set and then limitation of the 
step size is performed by means of a limiter in the first embodiment, in 
the present embodiment, re-setting is performed for the step size limited 
by means of a limiter. 
The present apparatus is generally constructed such that sudden change 
detecting circuit 48 for error and selector 41 are added to the apparatus 
of the first embodiment shown in FIG. 7. Sudden change detecting circuit 
48 is provided independently of step size controller 18, and receives an 
error signal as an input thereto and outputs "1" when the error signal 
presents a sudden change, but outputs "0" when the error signal presents 
no significant change. Sudden change detecting circuit 42 in the first 
embodiment described hereinabove with reference to FIG. 11 or 12 can be 
employed as is as sudden change detecting circuit 48. 
Selector 41 is provided on the output side of limiter 17 and receives an 
output of limiter 17 and step size .alpha..sub.0 for re-setting. An output 
of selector 41 is supplied to multiplier 5 and delay element 11. An output 
of delay element 11 is fed back to limiter 17 and step size controller 18 
similarly to delay element 10 (FIG. 7) in the first embodiment. Selector 
41 selectively outputs .alpha..sub.0 when "1" is supplied thereto as a 
controlling signal from sudden change detecting circuit 48, but outputs 
.alpha..sub.k which is an output of limiter 17 when "0" is supplied 
thereto. 
Further, any of the limiters described above in connection with the first 
embodiment (FIGS. 9, 13 and 14) can be used as limiter 17. A step size 
controller which does not include a sudden change detecting circuit 
therein, such as, for example, one described hereinabove with reference to 
FIG. 8 can be employed as step size controller 18. Accordingly, in the 
present apparatus, when a sudden change is detected from an error signal 
by sudden change detecting circuit 48, re-set step size .alpha..sub.0 is 
transmitted to multiplier 5 by means of selector 41. 
The second embodiment of the present invention has been described so far. 
In the present embodiment, when a step size which is used for the updating 
of coefficients is calculated using a gradient of power of an error 
signal, a limit which relies upon the squared value of a step size in 
advance is provided for a variation of the step size obtained so that the 
step size may be prevented from being made excessively different from the 
correct value due to noise obstruction or other factors. Further, the 
power of the identification error signal is monitored so that, when it is 
detected that a characteristic of the unknown system of an object for 
identification has presented a sudden change, the step size is reset in 
order to achieve both quick convergence and low identification error. 
Subsequently, a third preferred embodiment of the present invention will be 
described. FIG. 16 shows construction of an apparatus for identifying an 
unknown system using an adaptive filter in the present embodiment. The 
present apparatus is generally constructed such that counter 44, selector 
45 and delay element 46 are further added to the apparatus of the second 
embodiment shown in FIG. 15. Selector 41 is provided on the output side of 
limiter 17, and output .alpha..sub.k of step size controller 8 is directly 
inputted to selector 41 in place of step size .alpha..sub.0 for 
re-setting. Function blocks in FIG. 16 which are denoted by the same 
reference numerals to those of FIG. 15 have the same functions. 
Output of sudden change detecting circuit 48 for error is connected to 
counter 44 and an input terminal of selector 45 newly added. An output of 
selector 45 is fed back to the other input terminal of selector 45 by way 
of delay element 46. Counter 44 starts counting after the output of sudden 
change detecting circuit 48 changes from "0" to "1", and controls selector 
45 such that selector 45 selectively outputs the output of sudden change 
detecting circuit 48 when the count value of counter 44 is equal to 1, but 
selectively outputs a signal supplied from delay element 46 while the 
count value of counter 44 counts from "2" to a threshold value k.sub.th 
supplied from the outside. Accordingly, selector 45 continues to output 
"1" for a period of time of k.sub.th clocks at counter 44 after the output 
of sudden change detecting circuit 48 changes from "0" to "1". The output 
of selector 45 controls selector 41. Selector 41 selectively outputs a 
signal from limiter 17 when "0" is supplied thereto as a controlling 
signal, but selectively outputs a signal from step size controller 18 when 
" 1" is supplied thereto as the controlling signal. Accordingly, step size 
.alpha..sub.k, which does not undergo any limitation, is supplied to 
multiplier 5 for a period of time of k.sub.th clocks after a sudden change 
is detected from the error signal, but limited step size .alpha..sub.k 
obtained from limiter 17 is supplied to multiplier 5 in all other cases. 
The limiter described above with reference to FIG. 14 can be used as is as 
limiter 17. The sudden change detecting circuit for error described 
hereinabove with reference to FIG. 11 can be used as it is as sudden 
change detecting circuit 48. 
The third embodiment of the present invention has been described so far. In 
the present embodiment, when a step size which is used for the updating of 
coefficients is calculated using a gradient of power of an error signal, a 
limit which relies upon a step size in advance is provided for a variation 
of the step size obtained so that the step size may be prevented from 
being made excessively different from a correct value due to noise 
obstruction or other factors. Further, the power of an identification 
error signal is monitored so that, when it is detected that a 
characteristic of the unknown system of an object for identification has 
presented a sudden change, the limitation to the step size is cancelled 
for a fixed period of time in order to achieve both quick convergence and 
low identification error. 
Next, a fourth preferred embodiment of the present invention will be 
described. FIG. 17 is a block diagram showing construction of an apparatus 
for identifying an unknown system using an adaptive filter in the present 
embodiment. Function blocks in FIG. 17 which are denoted by the same 
reference numerals to those of FIG. 2 have the same functions. The present 
apparatus is different from the conventional apparatus shown in FIG. 2 in 
that averaging circuit 47 is interposed between limiter 9 and multiplier 
5. Accordingly, a step size obtained from limiter 9 is supplied to 
multiplier 5 after it is averaged by averaging circuit 47. 
The step size obtained from limiter 9 is obstructed by noise s.sub.k since 
noise s.sub.k is included in output s.sub.k +v.sub.k -u.sub.k of 
subtracter 4, and is varied from the original step size by an amount which 
relies on noise amplitude. Noise amplitude s.sub.k is a stochastic 
process, and the amplitude distribution thereof has no correlation with 
the amplitude distribution of error signal v.sub.k -u.sub.k. In other 
words, since the equation 
EQU E[s.sub.k (v.sub.k -u.sub.k)]=0 (23) 
stands and since 
EQU E[s.sub.k ]=0 (24) 
also stands, averaging of the output of limiter 9 will average a variation 
of the step size caused by the influence of noise. The original step size 
can be obtained approximately at an output of averaging circuit 47, and 
the bad influence of step size variation caused by noise can be 
suppressed. 
FIG. 18 shows an example of construction of averaging circuit 47. The 
averaging circuit is composed of two coefficient multipliers 75 and 79, 
adder 76 and delay element 78. Step size .alpha..sub.k is supplied from 
limiter 9 to input terminal 74. Step size .alpha..sub.k is multiplied by 
.beta. into .beta..alpha..sub.k by coefficient multiplier 75 and 
transmitted to adder 76. An output of coefficient multiplier 75 and an 
output of the other coefficient multiplier 79 are added at adder 76 to 
obtain averaged step size .alpha..sub.k, which is transmitted to output 
terminal 77. Meanwhile, the output of adder 76 is fed back to adder 76 by 
way of delay element 78 and the other coefficient multiplier 79. Output 
.alpha..sub.k-1 of delay element 78 is multiplied by (1-.beta.) at 
multiplier 79 into (1-.beta.).alpha..sub.k-1, which makes a feedback 
signal to adder 76. Accordingly, output signal .alpha..sub.k, of averaging 
circuit 47 is given, using input signal .alpha..sub.k, by 
EQU .alpha..sub.k =.beta..alpha..sub.k +(1-.beta.).alpha..sub.k-1 (25) 
The convergence characteristic and the final error of the present apparatus 
rely upon parameter .beta.. When .beta. is great, the convergence is slow, 
but the final error is small. When .beta. is small, the convergence is 
fast, but the final error is great. In order to solve the problem, 
parameter .beta. may be made variable using the output of limiter 9. 
The fourth embodiment of the present invention has been described so far. 
In the present embodiment, when a step size which is used for the updating 
of coefficients is calculated using a gradient of power of an error 
signal, the step size obtained is averaged so that it may be prevented 
from being made excessively different from the correct value due to noise 
obstruction or other factors. 
Subsequently, a fifth preferred embodiment of the present invention will be 
described. The present apparatus is different from the apparatus of the 
fourth embodiment described above in that parameter .beta. which is 
provided to an averaging circuit is varied in response to an averaged step 
size which is an output of the averaging circuit and in that an sudden 
change detecting circuit for error is provided to detect a sudden change 
or fluctuation of the unknown system to change over the step size to 
.beta..sub.0. 
FIG. 19 shows construction of an apparatus for identifying an unknown 
system using an adaptive filter in the present embodiment. The present 
apparatus is different from the apparatus shown in FIG. 17 in that it 
additionally includes sudden change detecting circuit 48, counter 44, two 
selectors 45 and 61 and multiplier 62. Further, averaging circuit 60 is 
connected to receive variable parameter .beta.. Function blocks in FIG. 19 
which are denoted by the same reference numerals as those of FIG. 16 or 17 
have the same functions. 
Output of averaging circuit 60 is transmitted to multiplier 5 and newly 
added multiplier 62. Latter multiplier 62 multiplies an output signal of 
averaging circuit 60 by positive constant .delta. and transmits the 
product to selector 61. Another positive constant .beta..sub.0 is 
supplied to the other input terminal of selector 61. Selector 61 thus 
selectively outputs the output of multiplier 62 or .beta..sub.0 as a 
parameter to averaging circuit 60 in response to a controlling signal from 
other selector 45. When the output of multiplier 62 is supplied to 
averaging circuit 60, the value obtained by multiplication of the averaged 
step size by positive constant .delta. is fed back to averaging circuit 
60. 
FIG. 20 shows an example of the detailed construction of averaging circuit 
60. Averaging circuit 60 is a modification of averaging circuit 47 shown 
in FIG. 18 in that multipliers 301 and 302 replace coefficient multipliers 
75 and 79, respectively, for determining a time constant for averaging. 
Common parameter .beta..sub.i is supplied to multipliers 301 and 302 by 
way of parameter input terminal 300. Multiplier 301 multiplies a signal 
from input terminal 74 by .beta..sub.i and transmits the product to adder 
76. Multiplier 302 multiplies the output of delay element 78 by 
(1-.beta..sub.i) and transmits the product to adder 76. 
If the step size is averaged using a circuit having a time constant which 
varies in response to a step size in advance, then some defect takes place 
in such a case wherein a characteristic of an unknown system to be 
identified presents a sudden change so that an error signal is suddenly 
increased. In this instance, step size must necessarily be increased 
suddenly so that the adaptive filter may follow up the change of the 
characteristic of the unknown system. However, when the value of step size 
is averaged with a great time constant, it can present only a moderate 
change; accordingly, the speed of following up sudden changes of the 
system is slow. In the present embodiment, sudden change detecting circuit 
48 is provided so that, when a sudden change of an error signal is 
detected, the time constant for averaging is decreased. 
The sudden change detecting circuit for error described hereinabove with 
reference to FIG. 11 can be used as sudden change detecting circuit 48. 
Referring back to FIG. 19, an output of sudden change detecting circuit 48 
is supplied to counter 44 and selector 45. A feedback signal from delay 
element 46 is also supplied to selector 45. Counter 44 starts its counting 
up after the output of sudden change detecting circuit 48 changes from "0" 
to "1", and controls selector 45 such that selector 45 selectively outputs 
an output of sudden change detecting circuit 48 when the count value of 
counter 44 is equal to 1, but selectively outputs a signal supplied from 
delay element 46 while the count value of counter 44 counts from 2 to 
threshold value k.sub.th supplied from the outside. Accordingly, selector 
45 continues to output "1" for a period of time of k.sub.th clocks after 
the output of sudden change detecting circuit 48 changes from "0" to "1". 
The output of selector 45 controls selector 61. Selector 61 selectively 
outputs a signal from multiplier 62 when "0" is supplied thereto as a 
controlling signal, but selectively outputs positive constant .beta..sub.0 
when "1" is supplied thereto as the controlling signal. Accordingly, Be is 
supplied as a parameter to averaging circuit 60 for a period of time of 
k.sub.th clocks after a sudden change is detected from an error signal; 
consequently, the time constant for step size averaging at averaging 
circuit 60 is decreased. On the other hand, at any time other than for a 
period of time of k.sub.th clocks after a sudden change is detected from 
an error signal, the step size averaged by averaging circuit 60 and 
multiplied by .delta. is supplied as a parameter to averaging circuit 60. 
In this instance, the time constant of averaging circuit 60 increases as 
step size decreases. Accordingly, the time constant is maximum in the 
converged condition. 
The fifth embodiment of the present invention has been described so far. In 
the present embodiment, when the step size which is used for the updating 
of coefficients is calculated using a gradient of power of an error 
signal, the step size obtained is averaged so that the step size may be 
prevented from being made excessively different from a correct value due 
to noise obstruction or other factors. Further, power of the 
identification error signal is monitored, and when it is detected that a 
characteristic of the unknown system for an object of identification has 
changed suddenly, the time constant for averaging is held decreased for a 
fixed period of time, thereby achieving both quick convergence and low 
identification error. 
In the first to fifth embodiments described above, adaptive filter 3 is 
controlled using parameter .alpha..sub.k (.alpha..sub.k), which is a 
variable value of step size .alpha., based on the LMS algorithm. As is 
apparent from equations (4) and (7) given above, the difference between 
the LMS algorithm and the LIM algorithm resides in whether .alpha. or .mu. 
which is obtained by division of .alpha. by an average power 
.sigma..sub.n.sup.2 inputted to the filter is used as the step size. 
Accordingly, the methods of varying the step size described so far can be 
applied as is to the LIM algorithm. 
Next, a sixth preferred embodiment of the present invention will be 
described. FIG. 21 is a block diagram showing construction of an apparatus 
for identifying an unknown system using an adaptive filter according to 
the present embodiment. Function blocks in FIG. 21 which are denoted by 
the same reference numerals as those of FIG. 2 have the same functions. 
The significant difference of the present apparatus from the apparatus 
shown in FIG. 2 resides in that an output of limiter 17 is fed back to 
step size controller 108 by way of delay element 101 and normalizing 
circuit 114 is interposed between multiplier 5 and adaptive filter 3. 
Further, construction of step size controller 108 is different from that 
of step size controller 8 shown in FIG. 2 and correlation calculating 
circuit 107 is externally provided for step size controller 108. 
Correlation calculating circuit 107 has a pair of outputs, one of which is 
connected directly to step size controller 108. The other output of 
correlation calculating circuit 107 is connected to normalizing circuit 
114 and also to step size controller 108 by way of delay element 102. 
Normalizing circuit 114 is provided to normalize the output of multiplier 
5 with input power to adaptive filter 3. 
First, construction of correlation calculating circuit 107 will be 
described with reference to FIG. 22. Correlation calculating circuit 107 
is composed of N delay elements 131.sub.1 to 131.sub.N, 2N multipliers 
132.sub.1 to 132.sub.N and 135.sub.1 to 135.sub.N, first N-input adder 133 
and second N-input adder 136. Delay elements 131.sub.1 to 131.sub.N are 
connected in series and form a delay line with taps. Reference signal 
n.sub.k is supplied to first delay element 131.sub.1 by way of input 
terminal 130. Multipliers 132.sub.1 to 132.sub.N calculate products of 
inputs to and outputs of corresponding delay elements 131.sub.1 to 
131.sub.N, respectively, and output the products to first adder 133. In 
other words, the product of input n.sub.k-i+1 and output n.sub.k-i of i-th 
(1.ltoreq.i.ltoreq.N) delay element 131, is calculated by i-th multiplier 
132.sub.i. First adder 133 calculates the sum total of outputs of 
multipliers 132.sub.1 to 132.sub.N and outputs the sum total by way of 
output terminal 134. As a result, the value given by 
##EQU11## 
is outputted from first adder 133, and this n.sub.k-1.sup.T 
.multidot.n.sub.k is outputted as correlation C.sub.k from output terminal 
134. 
Multipliers 135.sub.1 to 135.sub.N calculate squares of inputs to 
corresponding delay elements 131.sub.1 to 131.sub.N, respectively, and 
output the results to second adder 136. For example, at i-th multiplier 
135.sub.i, square n.sub.k-i+1.sup.2 of input n.sub.k-i+1 to i-th delay 
element 131.sub.i is calculated. Second adder 136 calculates the sum total 
of outputs of multipliers 135.sub.1 to 135.sub.N and outputs the sum total 
to the outside by way of output terminal 137. Accordingly, the value given 
by 
##EQU12## 
is outputted from output terminal 137. Here, P.sub.k is an input power to 
adaptive filter 3. Correlation C.sub.k is supplied to step size controller 
108, and filter input power P.sub.k is supplied to delay element 102 and 
normalizing circuit 114. 
Construction of step size controller 108 will be described with reference 
to FIG. 23. Step size controller 108 is composed of delay element 141, 
three multipliers 142, 143 and 145, normalizing circuit 147 and adder 148. 
Normalizing circuit 147 can be constituted from a dividing circuit. 
Difference signal d.sub.k inputted to input terminal 140 is delayed by one 
sample period by delay element 141 to make d.sub.k-1, which is supplied to 
multiplier 142. Signal d.sub.k is also supplied to multiplier 142, and 
d.sub.k d.sub.k-1 which is an output of multiplier 142 is supplied to 
second multiplier 143. Correlation C.sub.k (=n.sub.k-1.sup.T n.sub.k) is 
also supplied to second multiplier 143 from correlation calculating 
circuit 107 by way of input terminal 144. An output of multiplier 143 is 
multiplied by .rho. by multiplier 145 and supplied to normalizing circuit 
147. An output of delay element 102 (FIG. 21), i.e., filter input power 
P.sub.k-1, is supplied to normalizing circuit 147 by way of input terminal 
146. Normalizing circuit 147 normalizes output .rho..multidot.d.sub.k 
.multidot.d.sub.k-1 .multidot.n.sub.k-1.sup.T .multidot.n.sub.k of 
multiplier 145 with filter input power P.sub.k-1 and outputs the result to 
adder 148. The result of normalization is .rho..multidot.d.sub.k 
.multidot.d.sub.k-1 n.sub.k-1.sup.T .multidot.n.sub.k /P.sub.k-1. 
An output of delay circuit 101 is also supplied to adder 148 by way of 
input terminal 146. Since an input to delay circuit 101 is an output of 
limiter 17, the output of delay circuit 101 makes output .alpha..sub.k-1 
of limiter 17 one sample period earlier. An output of adder 148 is 
connected to output terminal 150 of step size controller 108. Accordingly, 
the value given by 
EQU .alpha..sub.k =.alpha..sub.k-1 +.rho..multidot.d.sub.k .multidot.d.sub.k-1 
.multidot.n.sub.k-1.sup.T .multidot.n.sub.k /P.sub.k-1 (28) 
is obtained at output terminal 150. Equation (28) has a normalized form of 
the second term of the right side of equation (10) with P.sub.k-1. By 
normalization, stabilized control of step size can be achieved even with 
an unsteady signal. 
The limiter of conventional construction described above with reference to 
FIG. 5 can be used as is as limiter 17. Output .alpha..sub.k of limiter 17 
is transmitted to delay element 101 and multiplier 5. At multiplier 5, 
multiplication of the output of limiter 17 and signal d.sub.k is 
performed, and product .alpha..sub.k .multidot.d.sub.k thereof is supplied 
to normalizing circuit 114. Also filter input power P.sub.k is supplied to 
normalizing circuit 114 from correlation calculating circuit 107. 
Normalizing circuit 107 normalizes .alpha..sub.k .multidot.d.sub.k with 
P.sub.k and transmits thus normalized value .alpha..sub.k 
.multidot.d.sub.k /P.sub.k to adaptive filter 3. At adaptive filter 3, 
updating of coefficients is performed using this value .alpha..sub.k 
.multidot.d.sub.k /P.sub.k. Accordingly, the coefficient updating equation 
at adaptive filter 3 is given by 
EQU c.sub.k =c.sub.k-1 +.alpha..sub.k .multidot.d.sub.k .multidot.n.sub.k-1 
/P.sub.k (29) 
Considering that P.sub.k is defined by equation (27), equation (29) is 
equivalent to that of the LIM algorithm given by equation (7), except that 
step size is controlled adaptively. In particular, in the present 
embodiment, quick and stabilized convergence can be realized for an 
unsteady signal compared with the conventional SGA-GAS algorithm. The 
relation of convergence by the present embodiment to that by the 
conventional SGA-GAS algorithm is contrasted to the relation of the LIM 
algorithm to the LMS algorithm. 
The sixth embodiment of the present invention is described so far, and in 
the present embodiment, when the step size to be used for the updating of 
coefficients is calculated using a gradient of power of an error signal, 
the step size is normalized with a filter input power so that stabilized 
quick convergence is realized for a nonstationary signal. 
Subsequently, a seventh preferred embodiment of the present invention will 
be described. FIG. 24 shows construction of an apparatus for identifying 
an unknown system using an adaptive filter according to the present 
embodiment. The present apparatus is different from the apparatus of the 
sixth embodiment described above in construction of limiter 17 and also in 
that the output of delay element 101 is also fed back to limiter 17. 
The limiter in the first embodiment described above with reference to FIG. 
9 can be used as is as limiter 17 in the present embodiment. In this 
instance, .alpha..sub.k is supplied from step size controller 108 to input 
terminal 23 of FIG. 9. Limited step size .alpha..sub.k-1 obtained at 
output terminal 20 is fed back to input terminal 24 by way of delay 
element 101. Accordingly, as described above in connection with the first 
embodiment, step size 60 supplied to input terminal 23 of limiter 17 is 
defined for the lower limit thereof by threshold value Th.sub.L and 
a.alpha..sub.k-1 and for the upper limit thereof by threshold value 
Th.sub.H and b.alpha..sub.k-1 so that the following limited step size 
.alpha..sub.k given by the following equation is obtained: 
##EQU13## 
where max{A, B} and min{A, B} represent the maximum value and the minimum 
value of A and B, respectively. Limited step size .alpha..sub.k is 
transmitted as an output of limiter 17 to multiplier 5 and delay element 
101. Since the output of multiplier 5 is applied to adaptive filter 3 by 
way of normalizing circuit 114, calculation B similar to that of the sixth 
embodiment described above will lead to the following coefficient updating 
equation at adaptive filter 3: 
EQU c.sub.k =c.sub.k-1 +(.alpha..sub.k /P.sub.k).multidot.d.sub.k 
.multidot.n.sub.k-1 (31) 
Amplitude distributions of s.sub.k and v.sub.k -u.sub.k are generally 
independent of each other. Accordingly, 
EQU E[.vertline.s.sub.k .vertline.].multidot.E[.vertline.v.sub.k -u.sub.k 
.vertline.]=0 (32) 
stands. By using .alpha..sub.k in place of .alpha..sub.k, even if s.sub.k 
causes an obstacle to v.sub.k -u.sub.k, the instantaneous influence of 
s.sub.k upon v.sub.k -u.sub.k can be suppressed by means of limiter 17 to 
obtain a stabilized step size. While stabilization of a step size is 
achieved by limitation of both the maximum value and the minimum value in 
the limiter, it is also effective to provide a limitation only for either 
the maximum value or the minimum value. Further, it is also possible to 
set a limiting value for an increase of step size and a limiting value for 
a decrease of step size equal to each other. An example of a limiter which 
limits the step size with an equal limiting value whether step size is 
increasing or decreasing is shown in FIG. 25. 
In the limiter shown in FIG. 25, a feedback signal supplied from delay 
element 101 to input terminal 24 is multiplied by a at multiplier 25 and 
is transmitted as a.alpha..sub.k-1 to maximum value circuit 27 and 
minimum value circuit 28. Thereafter, the identifying apparatus which 
employs the limiter shown in FIG. 25 operates in the same manner as the 
identifying apparatus which employs the limiter shown in FIG. 9 except 
that the maximum step size value is limited using min{Th.sub.H, 
a.alpha..sub.k-1 } in place of min{Th.sub.H, b.alpha..sub.k-1 } in 
equation (30). 
Further, the limiter in the first embodiment described above with reference 
to FIG. 14 can be used as limiter 17 in the present embodiment. In case 
the limiter shown in FIG. 14 is used, considering that step size is 
normally smaller than 1, limiting values max{Th.sub.L, 
.eta..alpha..sub.k-1 .sup.2 } and min{Th.sub.H, .theta..alpha..sub.k-1 
.sup.2 } for increase and decrease of step size increase in proportion to 
the squared value of limited step sizes .alpha..sub.k-1 one sample period 
earlier. Accordingly, the smaller the step size, the stronger the 
limitation. As the influence of s.sub.k increases, that is, as step size 
decreases, it becomes more stable. 
Further, the limiter in the first embodiment described above with reference 
to FIG. 13 can be used as limiter 17 in the present embodiment. In the 
limiter 17 shown in FIG. 13, as in the limiter shown in FIG. 25, the 
limiting value for increase of the step size and the limiting value for 
decrease of the step size are set equal to each other. In this instance, 
limited step size .alpha..sub.k is given by the following equation: 
##EQU14## 
The seventh embodiment of the present invention is described so far. In the 
present embodiment, when the step size to be used for the updating of 
coefficients is calculated using a gradient of power of an error signal, 
step size is normalized with a filter input power so that stabilized quick 
convergence is realized for an unsteady signal. Further, since a limit 
which depends upon the earlier step size is provided for the variation of 
a step size thus obtained, step size can be prevented from being made 
excessively different from a correct value due to noise obstruction or 
other factors. 
Subsequently, an eighth preferred embodiment of the present invention will 
be described. FIG. 26 is a block diagram showing construction of an 
apparatus for identifying an unknown system using an adaptive filter 
according to the present embodiment. 
If the maximum value and the minimum value for step size are limited using 
a previous value or values of the step size as in the seventh embodiment 
described above, then in the case that a characteristic of an unknown 
system to be identified presents a sudden change so that an error signal 
is increased suddenly, the identifying apparatus cannot cope with such a 
sudden change. In this instance, step size must necessarily be increased 
suddenly so that the adaptive filter may follow up the change of the 
characteristic of the unknown system. However, when the step size value is 
limited by a previous value or values thereof, it can present only a 
moderate change; accordingly, the speed of following up a sudden change of 
the system is slow. Thus, in the present embodiment, a sudden change in an 
unknown system to be identified is detected by means of a sudden change 
detecting circuit for error and re-setting of a step size is performed in 
accordance with the detected result. 
The apparatus shown in FIG. 26 is generally constructed such that sudden 
change detecting circuit 111 for error and selector 112 are added to the 
construction of the apparatus shown in FIG. 24. Sudden change detecting 
circuit 111 receives error signal d.sub.k as an input thereto and outputs 
a controlling signal to selector 112. Selector 112 is interposed between 
limiter 17 and multiplier 5. Limited step size .alpha..sub.k, which is an 
output of limiter 17, is inputted to an input terminal of selector 112, 
and predetermined step size .alpha..sub.0 is inputted to the other input 
terminal of selector 112. Selector 112 selectively transmits a signal from 
limiter 17 to multiplier 5 when the controlling signal from sudden change 
detecting circuit 111 is "0", but selectively transmits predetermined step 
size .alpha..sub.0 to multiplier 5 when the controlling signal is "1". 
Sudden change detecting circuit 42 described above in connection with the 
first embodiment can be used as is as sudden change detecting circuit 111 
of the present embodiment. In particular, any of sudden change detecting 
circuits 42 shown in FIGS. 11 and 12 can be used as sudden change 
detecting circuit 111. 
In the sudden change detecting circuit for error shown in FIG. 11, a sudden 
change in error signal d.sub.k is detected based on squared value 
d.sub.k.sup.2 of error signal d.sub.k. In the sudden change detecting 
circuit shown in FIG. 12, a sudden change in error signal d.sub.k is 
detected based on absolute value .vertline.d.sub.k .vertline. of error 
signal d.sub.k. Whichever two sudden change detecting circuits for error 
is used, when a sudden change in error signal d.sub.k is detected, a 
controlling signal of "1" is outputted, and in any other case, another 
controlling signal of "0" is outputted, from sudden change detecting 
circuit 111. When a sudden change in error signal d.sub.k is detected, 
predetermined step size .alpha..sub.0 is outputted from selector 112 to 
multiplier 5. But when no significant change of error signal d.sub.k is 
detected, limited step size .alpha..sub.k from limiter 17 is supplied to 
multiplier 5. That predetermined step size .alpha..sub.0 is inputted in 
place of limited step size .alpha..sub.k to multiplier 5 means that step 
size re-setting has been performed. Since predetermined step size 
.alpha..sub.k does not rely on a previous step size, even if error signal 
d.sub.k presents a sudden change, the identifying apparatus can cope with 
the sudden change. 
The eighth embodiment of the present invention is described so far, and in 
the present embodiment, when step size to be used for the updating of 
coefficients is calculated using a gradient of power of an error signal, 
step size is normalized with a filter input power so that stabilized quick 
convergence is realized for an unsteady signal. Further, since a limit or 
limits which depend on previous value of step size are provided for 
variation of the step size thus obtained, the step size can be prevented 
from being made excessively different from the correct value due to noise 
obstruction or other factors. Still further, an identification error 
signal is monitored, and when it is detected that a characteristic of the 
unknown system for an object of identification has changed suddenly, the 
step size is re-set to achieve both quick convergence and small 
identification error. 
Subsequently, a ninth preferred embodiment of the present invention will be 
described. FIG. 27 is a block diagram showing construction of an apparatus 
for identifying an unknown system using an adaptive filter according to 
the present embodiment. While, in the eighth embodiment described above, 
re-setting of a step size is performed when a sudden change in error 
signal d.sub.k is detected, in the present embodiment, the step size 
limitation is cancelled for a fixed period of time. The apparatus of the 
present embodiment is different from the apparatus of the eighth 
embodiment described above in that holding circuit 103 is interposed 
between sudden change detecting circuit 111 and selector 112 and output 
.alpha..sub.k of step size controller 108 is inputted to selector 112 in 
place of predetermined step size .alpha..sub.0. 
The sudden change detecting circuit for error described above with 
reference to FIG. 11 or 12 can be used as is as sudden change detecting 
circuit 111, as in the eighth embodiment described above. When a sudden 
change of error signal d.sub.k is detected by sudden change detecting 
circuit 111, holding circuit 103 outputs "1" to selector 112 for a fixed 
period of time beginning with the moment of detection of the change. 
An example of the construction of holding circuit 103 is shown in FIG. 28. 
Holding circuit 103 is composed of selector 152, counter 153 and delay 
element 154. A signal from sudden change detecting circuit 111 is supplied 
to input terminal 151 and is transmitted to selector 152 and counter 153. 
An output of selector 152 is supplied to output terminal 155 and is fed 
back simultaneously to selector 152 by way of delay element 154. Counter 
153 resets its counting when "1" is supplied to input terminal 151. When 
counter 153 has a value between "0" and K.sub.th inclusive, it controls 
selector 152 so as to select a feedback signal from delay element 154 and 
transmit the same to output terminal 155 so that selector 152 may normally 
select a signal supplied thereto from input terminal 151. Accordingly, 
when a sudden change of error signal d.sub.k is detected by sudden change 
detecting circuit 111, the value "1" is inputted to input terminal 151 of 
holding circuit 103; consequently, value "1" will continue to be supplied 
to output terminal 155 for a K.sub.th +1 sample period after the change is 
detected. 
Selector 112 selectively transmits output .alpha..sub.k of limiter 17 to 
multiplier 5 when "0" is outputted from holding circuit 103, but 
selectively transmits output .alpha..sub.k of step size controller 108 to 
multiplier 5 when "1" is outputted from holding circuit 103. Consequently, 
when a sudden change in error signal d.sub.k is detected, the limitation 
of the step size is cancelled for a period of time defined by parameter 
K.sub.th at holding circuit 103. The identifying apparatus can cope with a 
sudden change in error signal d.sub.k. 
The ninth embodiment of the present invention is described so far. In the 
present embodiment, when the step size to be used for the updating of 
coefficients is calculated using a gradient of power of an error signal, 
step size is normalized with a filter input power so that stabilized quick 
convergence is realized for an unsteady signal. Further, since a limit or 
limits which depend on a previous value of step size are provided for 
variation of the step size thus obtained, step size can be prevented from 
being made excessively different from the correct value due to noise 
obstruction or other factors. Still further, an identification error 
signal is monitored, and when it is detected that a characteristic of the 
unknown system for an object of identification has changed suddenly, the 
step size limitation is cancelled for a fixed period of time; 
consequently, both quick convergence and low identification error can be 
achieved. 
In the sixth to ninth embodiments described above, the normalized value of 
step size .alpha..sub.k (.alpha..sub.k) with filter input power P.sub.k-1 
(P.sub.k) is inputted to adaptive filter 3. The difference between the LMS 
algorithm and the LIM algorithm resides in whether .alpha. is used or a 
value (.mu.) obtained by division of .alpha. by a filter average input 
power is used as the step size. The sixth to ninth embodiments can each be 
considered to be equivalent to the LIM algorithm in which step size .mu. 
is made variable. 
It is to be understood that variations and modifications of the method of 
and the apparatus for identifying an unknown system using an adaptive 
filter disclosed herein will be evident to those skilled in the art. It is 
intended that all such modifications and variations be included within the 
scope of the appended claims.