Apparatus and method for low complexity combinatorial coding of signals
The invention utilizes low complexity estimates of complex functions to perform combinatorial coding of signal vectors. The invention disregards the accuracy of such functions as long as certain sufficient properties are maintained. The invention in turn may reduce computational complexity of certain coding and decoding operations by two orders of magnitude or more for a given signal vector input.
1. A method for combinatorial coding and decoding, the method comprising the steps of:
receiving a vector comprising speech, audio, image, or video;
receiving a value n based on a number of positions in the vector;
receiving a value d based on a number of occupied positions within the vector;
generating a logarithmic approximation of a combinatorial function based on n and d;
generating a value F′(n,d) based on the logarithmic approximation of the combinatorial function of n and d, such that
F
′
(
n
,
d
)
>
n
!
d
!
(
n
-
d
)
!
wherein generating the value comprising generating a value Q′(d), wherein Q′(d) is an approximation of a sum of logarithm of numbers from 1 to d and comprising
generating a value F′(n,d)=R′(k), where R′(k) is an approximation of the function a k , and where a is a logarithm base;
generating an integer component k i and a fractional component k f , based on a value k such that k=k i +k f;
generating a value K f , wherein K f is based on a low resolution Talyor series expansion of 2 k f , and
producing the value R′(k) based on K f ; and
using the value F′(n,d) to code or decode the vector to produce coded or decoded speech, audio, image, or video.
2. The method of claim 1 wherein Q′(d) is an approximation of the function
Q
(
d
)
=
∑
j
=
1
d
log
a
(
j
)
,
where a is a logarithm base.
3. The method of claim 1 wherein the step of generating the value based on the logarithmic approximation of the combinatorial function of n and d further comprises the step of generating a value Q′(d), is generated by a summation wherein each quantity in the summation is less than logarithm of numbers from 1 to d.
4. The method of claim 1 wherein the step of producing the value R′(k) based on K f further comprises generating:
R ′( k )=└2 k i −l └2 l K f ┘┘.
5. The method of claim 3 wherein the summation is generated as
Q
′
(
d
)
=
∑
j
=
2
d
2
-
l
(
j
)
⌊
2
l
(
j
)
log
2
(
j
)
-
1
⌋
,
where l(j) is a shift factor.
6. The method of claim 1 wherein the approximation of the function a k is obtained using a Taylor series expansion method.
7. An apparatus comprising:
vector generator circuitry receiving a vector comprising speech, audio, image, or video;
combinatorial function generator circuitry receiving a value n based on a number of positions in the vector, receiving a value d based on a number of occupied positions within the vector, generating a logarithmic approximation of a combinatorial function based on n and d, and generating a value F′(n,d) based on the logarithmic approximation of the combinatorial function of n and d, such that
F
′
(
n
,
d
)
>
n
!
d
!
(
n
-
d
)
!
and
F
′
(
n
,
d
)
>
F
′
(
n
-
1
,
d
)
+
F
′
(
n
-
1
,
d
-
1
)
,
wherein generating the value comprising generating a value Q′(d), wherein Q′(d) is an approximation of a sum of logarithm of numbers from 1 to d and generating a value F′(n,d)=R′(k), where R′(k) is an approximation of the function a k , and where a is a logarithm base, generating an integer component k i and a fractional component k f , based on a value k such that k=k i +k f , generating a value K f , where K f is based on a low resolution Talyor series expansion of 2 k f , and producing the value R′(k) based on K f ; and
coder/decoder circuitry using the value F′(n,d) to code or decode the vector to produce coded or decoded speech, audio, image, or video.
8. The apparatus of claim 7 wherein Q′(d) is an approximation of the function
Q
(
d
)
=
∑
j
=
1
d
log
a
(
j
)
,
where a is a logarithm base.
9. The apparatus of claim 7 wherein the combinatorial function generator circuitry generates the value based on Q′(d), wherein
Q
′
(
d
)
=
∑
j
=
2
d
2
-
l
(
j
)
⌊
2
l
(
j
)
log
2
(
j
)
-
1
⌋
,
where l(j) is a shift factor.
10. The apparatus of claim 7 wherein F′(n,d)=R′(k), where R′(k) is an approximation of the function a k , and where a is a logarithm base.