1. A homopolymer having a repeating unit represented by the following formula (2):
wherein X1, X2, X3 and X4 are independently of one another an atom belonging to group 16 of the periodic table; R1 and R2 are independently of each other a hydrogen atom or a hydrocarbyl group having 1 to 20 carbon atoms, and R1 and R2 may be linked with each other; and A1, A2, A3, A4, A5, A6, A7, A8, A9 and A10 are independently of one another a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group, an aryloxy group, a disubstituent-having amino group, or a hydrocarbylthio group.
2. The homopolymer according to claim 1, wherein the homopolymer is a diisotactic homopolymer having more than 25% of a threodiisotactic triad.
3. The homopolymer according to claim 1, wherein the homopolymer is a diisotactic homopolymer having more than 25% of an erythrodiisotactic triad.
4. The homopolymer according to claim 1, wherein the homopolymer is a disyndiotactic homopolymer having more than 25% of a threodisyndiotactic triad.
5. The homopolymer according to claim 1, wherein the homopolymer is a disyndiotactic homopolymer having more than 25% of an erythrodisyndiotactic triad.
6. A process for producing a homopolymer having a repeating unit represented by the following formula (2), which comprises the step of homopolymerizing a compound represented by the following formula (3):
wherein X1, X2, X3 and X4 are independently of one another an atom belonging to group 16 of the periodic table; R1 and R2 are independently of each other a hydrogen atom or a hydrocarbyl group having 1 to 20 carbon atoms, and R1 and R2 may be linked with each other; and A1, A2, A3, A4, A5, A6, A7, A8, A9 and A10 are independently of one another a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group, an aryloxy group, a disubstituent-having amino group, or a hydrocarbylthio group.
7. The process for producing a homopolymer according to claim 6, wherein the homopolymerization is carried out in the presence of a polymerization catalyst formed by contacting a transition metal compound with an organic aluminum compound andor boron compound.
8. The process for producing a homopolymer according to claim 7, wherein tie transition. metal compound is a compound represented by the following formula (I):
wherein M is a transition metal atom belonging to group 10 of the periodic table; R3 and R4 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group or an aryloxy group; R5 and R6 are independently of each other a hydrocarbyl group having 1 to 30 carbon atoms; and R7 and R8 are independently of each other a hydrogen atom or a hydrocarbyl group having 1 to 20 carbon atoms, and R7 and R8 may be linked with each other.
9. The process for producing a homopolymer according to claim 7, wherein the transition metal compound is a compound represented by the following formula (II):
wherein M is a transition metal atom belonging to group 10 of the periodic table; R3 and R4 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group or an aryloxy group; R7 and R8 independently of each other a hydrogen atom or a hydrocarbyl group having 1 to 20 carbon atoms, and R7 and R8 may be linked with each other; R9 and R10 independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, a substituent-having silyl group an alkoxy group, an aralkyloxy group, a disubstituent-having amino group, or a hyrocarbylthio group; R11 and R12 are independently of each other an aryl group having 7 to 20; and R13 and R14 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group, an aryloxy group, a disubstituent-having amino group, or a hyrocarbylthio group.
10. The process for producing a homopolymer according to claim 7, wherein the transition metal compound is a compound represented by the following formula (III);
wherein M is a transition metal atom belonging to group 10 of the periodic table; R3 and R4 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group or an aryloxy group; R7 and R8 are independently of each other a hydrogen atom or a hydrocarbyl group having 1 to 20 carbon atoms, and R7 and R8 may he linked with each other; R15, R16, R17 and R18 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, a phenyl group or a phenyl group having 1 to 5 halogen atoms, a substituent-having silyl group, an alkoxy group, an aralkyloxy group, an aryloxy group, a disubstituent-having amino group, or a hyrocarbylthio group; and R19 and R20 are independently of each other a hydrogen atom, a halogen atom, an alkyl group, an aralkyl group, an aryl group, a substituent-having silyl group, an alkoxy group, an aralkyloxy group, an aryloxy group, a disubstituent-having amino group, or a hyrocarbylthio group.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A partitioned block frequency domain adaptive filter for filtering an input signal in dependence on a control signal, the adaptive filter comprising a plurality of parallel arranged filter partitions, each filter partition being arranged for modeling a part of an impulse response of the adaptive filter, each filter partition having update means for updating filter coefficients of that filter partition by circular convoluting a signal representative of the input signal and a signal representative of the control signal, the update means comprising constraint means for intermittently constraining the filter coefficients by eliminating circular wrap-around artifacts of the circular convolution, characterized in that the update means further comprise selection means for selecting and removing at least part of the circular wrap-around artifacts, the selection means comprising an approximation of a rectangular constraint window.
2. The partitioned block frequency domain adaptive filter according to claim 1, characterized in that the selection means are arranged for selecting the at least part of the circular wrap-around artifacts when the filter coefficients are not constrained by the constraint means.
3. The partitioned block frequency domain adaptive filter according to claim 1 or 2, characterized in that time domain values of the approximation are larger than or equal to zero.
4. The partitioned block frequency domain adaptive filter according to any one of the claims 1 to 3, characterized in that the approximation in time domain has substantially high slopes near the positions which correspond to the positions of the transitions in a rectangular constraint window.
5. The partitioned block frequency domain adaptive filter according to any one of claims 1 to 4, characterized in that frequency domain values of the approximation each comprise a real value and complex conjugate imaginary values, whereby at least part of the imaginary values form a row of numbers, the numbers being obtainable from one another by multiplication.
6. The partitioned block frequency domain adaptive filter according to claim 5, characterized in that the frequency domain values of the approximation are defined as:
30
(
G
_
2
N
)
i
=
{
2
N
a
for
i
=
0
0
for
0
<
i
<
2
N
,
i
even
–
2
N
a
2
(
m
–
(
i
–
1
2
)
–
m
–
(
2
N
–
1
–
i
2
)
)
j
for
0
i
<
2
N
,
i
odd
with i being an index number, m being a multiplication factor, a being a mean value.
7. The partitioned block frequency domain adaptive filter according to claim 6, characterized in that m is substantially equal to 2.166.
8. An acoustic echo canceller comprising a partitioned block frequency domain adaptive filter for filtering an input signal in dependence on a control signal, the adaptive filter comprising a plurality of parallel arranged filter partitions, each filter partition being arranged for modeling a part of an impulse response of the adaptive filter, each filter partition having update means for updating filter coefficients of that filter partition by circular convoluting a signal representative of the input signal and a signal representative of the control signal, the update means comprising constraint means for intermittently constraining the filter coefficients by eliminating circular wrap-around artifacts of the circular convolution, characterized in that the update means further comprise selection means for selecting and removing at least part of the circular wrap-around artifacts, the selection means comprising an approximation of a rectangular constraint window.
9. The acoustic echo canceller according to claim 8, characterized in that the selection means are arranged for selecting the at least part of the circular wrap-around artifacts when the filter coefficients are not constrained by the constraint means.
10. The acoustic echo canceller according to claim 8 or 9, characterized in that time domain values of the approximation are larger than or equal to zero.
11. The acoustic echo canceller according to any one of the claims 8 to 10, characterized in that the approximation in time domain has substantially high slopes near the positions which correspond to the positions of the transitions in a rectangular constraint window.
12. The acoustic echo canceller according to any one of claims 8 to 11, characterized in that frequency domain values of the approximation each comprise a real value and complex conjugate imaginary values, whereby at least part of the imaginary values form a row of numbers, the numbers being obtainable from one another by multiplication.
13. The acoustic echo canceller according to claim 12, characterized in that the frequency domain values of the approximation are defined as:
31
(
G
_
2
N
)
i
=
{
2
N
a
for
i
=
0
0
for
0
<
i
<
2
N
,
i
even
–
2
N
–
a
2
(
m
–
(
i
–
1
2
)
–
m
–
(
2
N
–
1
–
i
2
)
)
j
for
0
i
<
2
N
,
i
odd
with i being an index number, m being a multiplication factor, a being a mean value.
14. The acoustic echo canceller according to claim 13, characterized in that m is substantially equal to 2.166.
15. A method of adaptively filtering an input signal in dependence on a control signal, the method comprising the steps of:
partitioning the input signal into partitions,
for each partition updating filter coefficients by circular convoluting a signal representative of the input signal and a signal representative of the control signal,
for each partition intermittently constraining the filter coefficients by eliminating circular wrap-around artifacts of the circular convolution, characterized in that the method further comprises the step of selecting and removing at least part of the circular wrap-around artifacts by means of an approximation of a rectangular constraint window.
16. The method according to claim 15, characterized in that the selection of the at least part of the circular wrap-around artifacts is performed only when the filter coefficients are not constrained by the constraint means.
17. The method according to claim 15 or 16, characterized in that time domain values of the approximation are larger than or equal to zero.
18. The method according to any one of the claims 15 to 17, characterized in that the approximation in time domain has substantially high slopes near the positions which correspond to the positions of the transitions in a rectangular constraint window.
19. The method according to any one of claims 15 to 18, characterized in that frequency domain values of the approximation each comprise a real value andor an imaginary value andor a conjugate imaginary value, whereby at least part of the imaginary values form a row of numbers, the numbers being obtainable from one another by multiplication.
20. The method according to claim 19, characterized in that the frequency domain
32
(
G
_
2
N
)
i
=
{
2
N
a
for
i
=
0
0
for
0
<
i
<
2
N
,
i
even
–
2
N
–
a
2
(
m
–
(
i
–
1
2
)
–
m
–
(
2
N
–
1
–
i
2
)
)
j
for
0
i
<
2
N
,
i
odd
values of the approximation are defined as:
with i being an index number, m being a multiplication factor, a being a mean value.
21. The partitioned block frequency domain adaptive filter according to claim 20, characterized in that m is substantially equal to 2.166.