Approximation method for series expansion of input function and system using the same
An approximation method and system for series expansion of functions include the steps and functions, respectively, of: expanding an input function in Taylor series up to an (N−1)-th term; expanding the input function in Taylor series up to the N-th term; multiplying the expanded result up to the (N−1)-th term by a predetermined weight α; combining the expanded result up to the (N−1)-th term, multiplied by α, and the expanded result up to the N-th term; and dividing the combined result by (α+1).
1. An approximation system for a series expansion of an input function with a finite number of terms N to minimize an approximation error, said system including an operational processing unit, said operational processing unit comprising:
means for expanding the input function in Taylor series up to an (N−1)-th term to obtain a first expansion result;
means for expanding the input function in Taylor series up to an N-th term to obtain a second expansion result;
means for multiplying the first expansion result by a predetermined weight α to obtain a multiplication result;
means for combining the multiplication result and the second expansion result to obtain a combined result; and
means for dividing the combined result by (α+1);
whereby to minimize the approximation error.
2. The system of claim 1 , wherein α is greater than 0 and no greater than 1.
3. The system of claim 1 , wherein α obtained for a corresponding respective N is selected so as to minimize a maximum approximation error.
4. The system of claim 1 , wherein the operational processing unit further comprises:
means for using the approximation to obtain a phase rotation value for a first input sample of k=1, wherein sin(2πê) and cos(2πê) are series-expanded to minimize the approximation error;
means for using a phase rotation value for a previous input sample including k=1 to obtain a phase rotation value for a subsequent input sample; and
means for compensating the phase rotation values for all input samples.
5. The system of claim 4 , wherein an estimated carrier frequency effect ê is represented by
ɛ
^
=
1
2
π
arctan
{
∑
i
=
1
L
Im
(
y
(
-
i
)
y
*
(
L
-
i
)
)
∑
i
=
1
L
Re
(
y
(
-
i
)
y
*
(
L
-
i
)
)
}
,
where Re and Im represent a real part and an imaginary part, respectively, of a complex number, y(i) represents an i-th received sample, L is a fast fourier transformation (FFT) size, and {circumflex over (ε)} is an estimated and normalized carrier frequency offset of Δ{circumflex over (ƒ)}T.
6. The system of claim 5 , wherein the phase rotation value for a k-th sample is calculated by:
For
k
=
1
,
cos
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
2
n
(
2
n
)
!
sin
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
(
2
n
+
1
)
(
2
n
+
1
)
!
For
k
≥
2
,
cos
(
k
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
-
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
sin
(
k
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
+
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
.
7. An approximation system for a series expansion of an input function with a finite number of terms N to minimize an approximation error, said system including an operational processing unit, said operational processing unit comprising:
means for expanding the input function in Taylor series up to an (N−1)-th term to obtain an expansion result;
means for multiplying an N-th term of the expansion result by a predetermined weight value to obtain a multiplication result; and
means for combining the expansion result and the multiplication result to obtain an approximation function ƒ for the series expansion function;
whereby to minimize the approximation error.
8. The system of claim 7 , wherein the predetermined weight value is
(
-
1
)
N
(
α
+
1
)
.
9. The system of claim 8 , wherein α obtained for corresponding respective N is selected to minimize a maximum approximation error.
10. The system of claim 9 , wherein α is obtained by:
(a) selecting a minimum input in a given input x area;
(b) calculating the approximation function ƒ for the input function with the finite number of terms N
(c) obtaining and storing an error E N,x by subtracting approximation function ƒ from a nominal function value of the input x;
(d) determining whether the input x has reached a maximum value in the given input x area, adding a predetermined increment ξ to x when x has not yet reached the maximum value, and repeating steps (b), (c) and (d);
(e) selecting a maximum error value among all the stored errors E N,x for all inputs when x has reached a maximum value; and
(f) searching α to minimize the maximum error value, and storing α as the weight value for a corresponding N.
11. The system of claim 7 , said operational processing unit further comprising:
means for using the approximation to obtain a phase rotation value for a first input sample of k=1, wherein sin(2πê) and cos(2ê) are series-expanded to minimize the approximation error;
means for using a phase rotation value for a previous input sample including k=1 to obtain a phase rotation value for a subsequent input sample; and
means for compensating the phase rotation values for all input samples.
12. The system of claim 11 , wherein an estimated carrier frequency effect ê is represented by
ɛ
^
=
1
2
π
arctan
{
∑
i
=
1
L
Im
(
y
(
-
i
)
y
*
(
L
-
i
)
)
∑
i
=
1
L
Re
(
y
(
-
i
)
y
*
(
L
-
i
)
)
}
,
where Re and Im represent a real part and an imaginary part, respectively, of a complex number, y(i) represents an i-th received sample, L is a fast fourier transformation (FFT) size, and {circumflex over (ε)} is an estimated and normalized carrier frequency offset of Δ{circumflex over (ƒ)}T.
13. The system of claim 12 , wherein the phase rotation value for a k-th sample is calculated by:
For
k
=
1
,
cos
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
2
n
(
2
n
)
!
sin
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
(
2
n
+
1
)
(
2
n
+
1
)
!
For
k
≥
2
,
cos
(
k
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
-
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
sin
(
k
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
+
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
.
14. An orthogonal frequency division multiplexing (OFDM) system for compensating a carrier frequency offset, said system comprising:
an estimator for estimating the carrier frequency offset ê by using a series expansion of a function arctan(x);
a first phase rotation calculator for using the estimated carrier frequency offset to obtain a phase rotation value for a first input sample of k=1, wherein sin(2πê) and cos(2πê) are series-expanded to minimize an approximation error;
a second phase rotation calculator for using a phase rotation value for a previous input sample including k=1 to obtain a phase rotation value for a subsequent input sample; and
a compensator for compensating the phase rotation values for all input samples, thereby compensating the carrier frequency offset.
15. The system of claim 14 , wherein the estimated carrier frequency offset ê is represented by
ɛ
^
=
1
2
π
arctan
{
∑
i
=
1
L
Im
(
y
(
-
i
)
y
*
(
L
-
i
)
)
∑
i
=
1
L
Re
(
y
(
-
i
)
y
*
(
L
-
i
)
)
}
,
where Re and Im represent a real part and an imaginary part, respectively, of a complex number, y(i) represents an i-th received sample, L is a fast fourier transformation (FFT) size, and ê is an estimated and normalized carrier frequency offset of Δ{circumflex over (ƒ)}T.
16. The system of claim 15 , wherein the phase rotation value for a k-th sample is calculated by:
For
k
=
1
,
cos
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
2
n
(
2
n
)
!
sin
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
(
2
n
+
1
)
(
2
n
+
1
)
!
For
k
≥
2
,
cos
(
k
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
-
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
sin
(
k
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
+
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
.
17. The system of claim 14 , wherein the phase rotation value for a k-th sample is calculated by:
For
k
=
1
,
cos
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
2
n
(
2
n
)
!
sin
(
Δ
ω
^
T
s
)
=
∑
n
=
0
N
(
-
1
)
n
Δ
ω
^
T
s
(
2
n
+
1
)
(
2
n
+
1
)
!
For
k
≥
2
,
cos
(
k
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
-
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
sin
(
k
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
+
Δ
ω
^
T
s
)
=
sin
(
(
k
-
1
)
Δ
ω
^
T
s
)
cos
(
Δ
ω
^
T
s
)
+
cos
(
(
k
-
1
)
Δ
ω
^
T
s
)
sin
(
Δ
ω
^
T
s
)
.