Signal analyzing method, signal synthesizing method of complex exponential modulation filter bank, program thereof and recording medium thereof
View Patent ↗Provided is a complex exponential modulation filter bank which can reduce quantity of arithmetic operation, and can realize low electric power consumption or speeding-up. This complex exponential modulation filter bank has a step of calculating a first intermediate signal from an input signal, a step of calculating a second intermediate signal from the first intermediate signal, a step of calculating a third intermediate signal from the second intermediate signal with fast Fourier transform, and a step of calculating a complex band output signal from the third intermediate signal.
1. A signal analyzing method of a complex exponential modulation filter bank, which makes K times sampling of an input signal with sampling frequency fs by K times (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and divides it into M complex band signals with equal bandwidth and sampling frequency fsK/M to be outputted, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and an input signal at sampling time n is set to x(n),
a step of calculating a first intermediate signal w 1 (n) from the input signal by (formula 1), where m is an integer;
w
1
(
n
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
x
(
mM
/
K
-
2
Mi
/
K
-
n
)
h
(
2
Mi
+
Kn
)
(
0
≦
n
≦
2
M
/
K
-
1
)
(
formula
1
)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 2) (j is an imaginary unit);
w 2( n )=( w 1( n )− jw 1( n+M/K )) K exp(− jπnK /(2 M ))(0≦ n≦M/K− 1) (formula 2)
a step of calculating a third intermediate signal W 3 (k) from the second intermediate signal by (formula 3) with fast Fourier transform; and
W
3
(
k
)
=
∑
n
=
0
M
/
K
-
1
w
2
(
n
)
exp
(
-
j2
π
knK
/
M
)
(
0
≦
k
≦
M
/
K
-
1
)
(
formula
3
)
a step of calculating a complex band output signal X(k, mM/K) at sampling time mM/K in k-th band (0≦k≦M/K−1) from the third intermediate signal by (formula 4) (* is a conjugate complex number, and A is a phase for signal analysis) when k is an even number (k=2l), and by (formula 5) when k is an odd number (k=2l+1)
X (2 l,mM/K )= W 3*( l )exp( j π(4 l +1) A /(4 M )) (0≦ l≦M/ (2 K )−1) (formula 4)
X (2 l+ 1, mM/K )= W 3( M/K −1− l )exp( jπ (4 l +3) A /(4 M )) (0≦ l≦M /(2 K )−1) (formula 5).
2. A signal analyzing method of a complex exponential modulation filter bank as set forth in claim 1 , wherein the phase for signal analysis is set up in such a manner that there occurs no multiplication in a step of calculating the complex band output signal from the third intermediate signal.
3. A signal analyzing method of a complex exponential modulation filter bank, which makes K times sampling of an input signal with sampling frequency fs by K times (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and divides it into M complex band signals with equal bandwidth and sampling frequency fsK/M to be outputted, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and an input signal at sampling time n is set to x(n),
a step of calculating a first intermediate signal w 1 (n) from the input signal by (formula 6), where m is an integer;
w
1
(
n
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
x
(
mM
/
K
-
2
Mi
/
K
-
n
)
h
(
2
Mi
+
Kn
)
(
0
≦
n
≦
2
M
/
K
-
1
)
(
formula
6
)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 7) (j is an imaginary unit);
w 2( n )=( w 1( n )+ jw 1( n+M/K )) K exp( jπnK/ (2 M )) (0≦ n≦M/K− 1) (formula 7)
a step of calculating a third intermediate signal W 3 (k) from the second intermediate signal by (formula 8) with inverse fast Fourier transform; and
W
3
(
k
)
=
∑
n
=
0
M
/
K
-
1
w
2
(
n
)
exp
(
j2
π
knK
/
M
)
(
0
≦
k
≦
M
/
K
-
1
)
(
formula
8
)
a step of calculating a complex band output signal X(k, mM/K) at sampling time mM/K in k-th band (0≦k≦M/K−1) from the third intermediate signal by (formula 9) (A is a phase for signal analysis) when k is an even number (k=2l), and by (formula 10) (* is a conjugate complex number) when k is an odd number (k=2l+1)
X (2 l,mM/K )= W 3( l )exp( j π(4 l +1) A /(4 M )) (0≦ l≦M/ (2 K )−1) (formula 9)
X (2 l +1 ,mM/K )= W 3*( M/K− 1 −l )exp( jπ (4 l +3) A/ (4 M )) (0≦ l≦M/ (2 K )−1) (formula 10).
4. A signal analyzing method of a complex exponential modulation filter bank as set forth in claim 3 , wherein the phase for signal analysis is set up in such a manner that there occurs no multiplication in a step of calculating the complex band output signal from the third intermediate signal.
5. A signal synthesizing method of a complex exponential modulation filter bank, which synthesizes M complex band input signals with equal bandwidth and sampling frequency fsK/M (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and makes 1/L times sampling of it (L is a divisor of N and a positive integer greater than or equal to 1), and outputs a signal with sampling frequency fsK/L, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and a complex band input signal at sampling time mM/K in k-th band is set to X(k, mM/K), where m is an integer;
a step of calculating a first intermediate signal W 1 (k) from the complex band input signal by (formula 11) (j is an imaginary unit, * is a conjugate complex number, and B is a phase for signal synthesis) of 0≦k≦M/(2L)−1, and by (formula 12) of M/(2L)≦k≦M/L−1;
W 1( k )= X* (2 k,mM/K )exp(− jπkB/M ) (0≦ k≦M/ (2 L )−1) (formula 11)
W 1( k )= X (2 M/L− 1−2 k,mM/K )exp( jπB (1 /L−k/M )) ( M /(2 L )≦ k≦M/L− 1) (formula 12)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 13) with fast Fourier transform;
w
2
(
n
)
=
∑
k
=
0
M
/
L
-
1
W
1
(
k
)
exp
(
-
j2
π
knL
/
M
)
(
0
≦
n
≦
M
/
L
-
1
)
(
formula
13
)
a step of shifting a third intermediate signal w 3 (n) of 0≦n≦2(N−M)/L−1 to w 3 (n+2M/L), and calculating third intermediate signals w 3 (n) and w 3 (n+M/L) of 0≦n≦M/L from the second intermediate signal by (formula 14) (Re(x) is a real part of a complex number x, and Im(x) is an imaginary part of x); and
w 3( n )= Re{w 2( n )(1/ M )exp(− j π(2 Ln+B )/(4 M ))}
w 3( n+M/L )= Im{w 2( n )(1 /M )exp(− jπ (2 Ln+B )/(4 M ))} (0 ≦n≦M/L− 1) (formula 14)
a step of calculating an output signal x(mM/K+nL/K) at sampling time mM/K+nL/K (0≦n≦N/L−1) from the third intermediate signal by (formula 15)
x
(
mM
/
K
+
nL
/
K
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
{
w
3
(
4
Mi
/
L
+
n
)
h
(
2
Mi
+
Ln
)
+
w
3
(
4
Mi
/
L
+
3
M
/
L
+
n
)
h
(
2
Mi
+
M
+
Ln
)
}
.
(
formula
15
)
6. A signal synthesizing method of a complex exponential modulation filter bank as set forth in claim 5 , wherein the phase for signal synthesis is set up in such a manner that there occurs no multiplication in a step of calculating the first intermediate signal from the complex band input signal.
7. A signal synthesizing method of a complex exponential modulation filter bank, which synthesizes N complex band input signals with equal bandwidth and sampling frequency fsK/N (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and makes 1/L times sampling of it (L is a divisor of M and a positive integer greater than or equal to 1), and outputs a signal with sampling frequency fsK/L, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and a complex band input signal at sampling time mM/K in k-th band is set to X(k, mN/K), where m is an integer;
a step of calculating a first intermediate signal W 1 (k) from the complex band input signal by (formula 16) (j is an imaginary unit, and B is a phase for signal synthesis) of 0≦k≦M/(2L)−1, and by (formula 17) (* is a conjugate complex number) of M/(2L)≦k≦M/L−1;
W 1( k )= X (2 k,mM/K )exp( jπkB/M ) (0 ≦k≦M/ (2 L )−1) (formula 16)
W 1( k )= X *(2 M/L− 1−2 k,mM/K )exp( jπB ( k/M −1/ L )) ( M/ (2 L )≦ k≦M/L− 1) (formula 17)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 18) with inverse fast Fourier transform;
w
2
(
n
)
=
∑
k
=
0
M
/
L
-
1
W
1
(
k
)
exp
(
j2
π
knL
/
M
)
(
0
≦
n
≦
M
/
L
-
1
)
(
formula
18
)
a step of shifting a third intermediate signal w 3 (n) of 0≦n≦2(N−M)/L−1, to w 3 (n+2M/L), and calculating third intermediate signals w 3 (n) and w 3 (n+M/L) of 0≦n≦M/L from the second intermediate signal by (formula 19) (Re(x) is an real part of a complex number x, and Im(x) is an imaginary part of x); and
w 3( n )= Re{w 2( n )(1 /M )exp( jπ (2 Ln+B )/(4 M ))}
w 3( n+M/L )=− Im{w 2( n )(1 /M )exp( jπ (2 Ln+B)/( 4 M ))} (0≦ n≦M/L− 1) (formula 19)
a step of calculating an output signal x(mN/K+nL/K) at sampling time mN/K+nL/K (0≦n≦M/L−1) from the third intermediate signal by (formula 20)
x
(
mM
/
K
+
nL
/
K
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
{
w
3
(
4
Mi
/
L
+
n
)
h
(
2
Mi
+
Ln
)
+
w
3
(
4
Mi
/
L
+
3
M
/
L
+
n
)
h
(
2
Mi
+
M
+
Ln
)
}
.
(
formula
20
)
8. A signal synthesizing method of a complex exponential modulation filter bank as set forth in claim 7 , wherein the phase for signal synthesis is set up in such a manner that there occurs no multiplication in a step of calculating the first intermediate signal from the complex band input signal.
9. A computer readable recording medium in which is recorded a program for having a computer or a digital signal processor execute a signal analyzing method of a complex exponential modulation filter bank, which makes K times sampling of an input signal with sampling frequency fs by K times (K is a divisor of number of bands N and a positive integer greater than or equal to 1), and divides it into M complex band signals with equal bandwidth and sampling frequency fsK/M to be outputted, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and an input signal at sampling time n is set to x(n),
a step of calculating a first intermediate signal w 1 (n) from the input signal by (formula 1), where m is an integer;
w
1
(
n
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
x
(
mM
/
K
-
2
Mi
/
K
-
n
)
h
(
2
Mi
+
Kn
)
(
0
≦
n
≦
2
M
/
K
-
1
)
(
formula
1
)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 2) (j is an imaginary unit);
w 2( n )=( w 1( n )− jw 1( n+M/K )) K exp(− jπnK /(2 M )) (0≦ n≦M/K− 1) (formula 2)
a step of calculating a third intermediate signal W 3 (k) from the second intermediate signal by (formula 3) with fast Fourier transform; and
W
3
(
k
)
=
∑
n
=
0
M
/
K
-
1
w
2
(
n
)
exp
(
-
j2
π
knK
/
M
)
(
0
≦
k
≦
M
/
K
-
1
)
(
formula
3
)
a step of calculating a complex band output signal X(k, mM/K) at sampling time mM/K in k-th band (0≦k≦M/K−1) from the third intermediate signal by (formula 4) (* is a conjugate complex number, and A is a phase for signal analysis) when k is an even number (k=2l), and by (formula 5) when k is an odd number (k=2+l)
X( 2 l,mM/K )= W 3*( l )exp( jπ (4 l+ 1) A /(4 M )) (0≦ l≦M/ (2 K )−1) (formula 4)
X (2 l+ 1 ,mM/K )= W 3( M/K− 1− l )exp( jπ (4 l+ 3) A /(4 M )) (0≦ l≦M/ (2 K )−1) (formula 5).
10. A computer readable recording medium in which is recorded a program for having a computer or a digital signal processor execute a signal analyzing method of a complex exponential modulation filter bank, which makes K times sampling of an input signal with sampling frequency fs by K times (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and divides it into M complex band signals with equal bandwidth and sampling frequency fsK/M to be outputted, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and an input signal at sampling time n is set to x(n),
a step of calculating a first intermediate signal w 1 (n) from the input signal by (formula 6), where m is an integer;
w
1
(
n
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
x
(
mM
/
K
-
2
Mi
/
K
-
n
)
h
(
2
Mi
+
Kn
)
(
0
≦
n
≦
2
M
/
K
-
1
)
(
formula
6
)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 7) (j is an imaginary unit);
w 2( n )=( w 1( n )+ jw 1( n+M/K )) K exp( jπnK /(2 M )) (0 ≦n≦M/K− 1) (formula 7)
a step of calculating a third intermediate signal W 3 (k) from the second intermediate signal by (formula 8) with inverse fast Fourier transform; and
W
3
(
k
)
=
∑
n
=
0
M
/
K
-
1
w
2
(
n
)
exp
(
j2
π
knK
/
M
)
(
0
≦
k
≦
M
/
K
-
1
)
(
formula
8
)
a step of calculating a complex band output signal X(k, mM/K) at sampling time mM/K in k-th band (0≦k≦M/K−1) from the third intermediate signal by (formula 9) (A is a phase for signal analysis) when k is an even number (k=2l), and by (formula 10) (* is a conjugate complex number) when k is an odd number (k=2l+1)
X (2 l,mM/K )= W 3( l )exp( jπ (4 l +1) A /(4 M )) (0 ≦l≦M /(2 K )−1) (formula 9)
X (2 l +1 ,mM/K )= W 3*( M/K− 1− l )exp( jπ (4 l+ 3) A /(4 M )) (0 ≦l≦M /(2 K )−1) (formula 10).
11. A computer readable recording medium in which a program for having a computer or a digital signal processor execute a signal synthesizing method of a complex exponential modulation filter bank, which synthesizes M complex band input signals with equal bandwidth and sampling frequency fsK/M (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and makes 1/L times sampling of it (L is a divisor of M and a positive integer greater than or equal to 1), and outputs a signal with sampling frequency fsK/L, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and a complex band input signal at sampling time mM/K in k-th band is set to X(k, mM/K), where m is an integer;
a step of calculating a first intermediate signal W 1 (k) from the complex band input signal by (formula 11) (j is an imaginary unit, * is a conjugate complex number, and B is a phase for signal synthesis) of 0≦k≦M/(2L)−1, and by (formula 12) of M/(2L)≦k≦M/L−1;
W 1( k )= X *(2 k,mM/K )exp(− jπkB/M ) (0 ≦k≦M /(2 L )−1) (formula 11)
W 1( k )= X (2 M/L− 1−2 k,mM/K )exp( jπ (1 /L−k/M )) ( M /(2 L )≦ k≦M/L− 1) (formula 12)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 13) with fast Fourier transform;
w
2
(
n
)
=
∑
k
=
0
M
/
L
-
1
W
1
(
k
)
exp
(
-
j2
π
knL
/
M
)
(
0
≦
n
≦
M
/
L
-
1
)
(
formula
13
)
a step of shifting a third intermediate signal w 3 (n) of 0≦n≦2(N−M)/L−1 to w 3 (n+2M/L), and calculating third intermediate signals w 3 (n) and w 3 (n+M/L) of ≦n≦M/L from the second intermediate signal by (formula 14) (Re(x) is a real part of a complex number x, and Im(x) is an imaginary part of x); and
w 3( n )= Re{w 2( n )(1 /M )exp(− jπ (2 Ln+B)/( 4 M ))}
w 3( n+M/L )= Im{w 2( n )(1 /M )exp(− jπ (2 Ln+B )/(4 M ))} (0 ≦n≦M/L− 1) (formula 14)
a step of calculating an output signal x(mM/K+nL/K) at sampling time mM/K+nL/K (0≦n≦M/L−1) from the third intermediate signal by (formula 15)
x
(
mM
/
K
+
nL
/
K
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
{
w
3
(
4
Mi
/
L
+
n
)
h
(
2
Mi
+
Ln
)
+
w
3
(
4
Mi
/
L
+
3
M
/
L
+
n
)
h
(
2
Mi
+
M
+
Ln
)
}
.
(
formula
15
)
12. A computer readable recording medium in which a program for having a computer or a digital signal processor execute a signal synthesizing method of a complex exponential modulation filter bank, which synthesizes M complex band input signals with equal bandwidth and sampling frequency fsK/M (K is a divisor of number of bands M and a positive integer greater than or equal to 1), and makes of it 1/L times sampling of it (L is a divisor of M and a positive integer greater than or equal to 1), and outputs a signal with sampling frequency fsK/L, comprising:
assuming that a filter coefficient of a linear phase non-recursive type prototype filter is set to h(n) (N is a filter order, 0≦n≦N), and values of first and last filter coefficients are set to zero, and a complex band input signal at sampling time mM/K in k-th band is set to X(k, mM/K), where m is an integer;
a step of calculating a first intermediate signal W 1 (k) from the complex band input signal by (formula 16) (j is an imaginary unit, and B is a phase for signal synthesis) of 0≦k≦M/(2L)−1, and by (formula 17) (* is a conjugate complex number) of M/(2L)≦k≦M/L−1:
W 1( k )= X (2 k,mM/K )exp( jπkB/M ) (0 ≦k≦M /(2 L )−1) (formula 16)
W 1( k )= X *(2 M/L− 1−2 k,mM/K )exp( jπB ( k/M− 1 /L )) ( M /(2 L )≦ k≦M/L− 1) (formula 17)
a step of calculating a second intermediate signal w 2 (n) from the first intermediate signal by (formula 18) with inverse fast Fourier transform;
w
2
(
n
)
=
∑
k
=
0
M
/
L
-
1
W
1
(
k
)
exp
(
j
2
π
knL
/
M
)
(
0
≦
n
≦
M
/
L
-
1
)
(
formula
18
)
a step of shifting a third intermediate signal w 3 (n) of 0≦n≦2(N−M)/L−1, to w 3 (n+2M/L), and calculating third intermediate signals w 3 (n) and w 3 (n+M/L) of 0≦n≦M/L from the second intermediate signal by (formula 19) (Re(x) is an real part of a complex number x, and Im(x) is an imaginary part of x); and
w 3( n )= Re{w 2( n )(1 /M )exp( jπ (2 Ln+B )/(4 M ))}
w 3( n+M/L )=− Im{w 2( n )(1 /M )exp( jπ (2 Ln+B )/(4 M ))} (0 ≦n≦M/L− 1) (formula 19)
a step of calculating an output signal x(mM/K+nL/K) at sampling time mM/K+nL/K (0≦n≦M/L−1) from the third intermediate signal by (formula 20)
x
(
mM
/
K
+
nL
/
K
)
=
∑
i
=
0
N
/
(
2
M
)
-
1
(
-
1
)
i
{
w
3
(
4
Mi
/
L
+
n
)
h
(
2
Mi
+
Ln
)
+
w
3
(
4
Mi
/
L
+
3
M
/
L
+
n
)
h
(
2
Mi
+
M
+
Ln
)
}
.
(
formula
20
)