JP3060767B2 - Modified discrete cosine transform and inverse transform method and apparatus - Google Patents

Abstract

PURPOSE:To provide a high-speed calculating method and its device for efficiently performing the calculation of MDCT in the case of compressing audio signals. CONSTITUTION:This device is provided with a first intermediate data generating part 1 for providing first intermediate data x2(n) (O<=n<=M-1) from the combination of specified two points in input sample data x(n) (0<=n<=2M-1) of two M points, a second intermediate data generating part 2 for providing M/2 pieces of second intermediate data z(n) (0<=n<=M/2-1) by dividing the intermediate data x2(n) into the first half and the latter half, an FTT executing part 3 for providing M/2 pieces of Fourier coefficients Z(k) (0<=k<=M/2-1) by performing high-speed Fourier transformation to the second intermediate data z(n) and an extension part 4 for providing the even-numbered spectrum and odd-numbered spectrum of the input data x(n) from the output Z(k) of the FFT executing means.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a high-speed calculation method and apparatus for transform coding used in compression coding of audio signals and the like.

[0002]

2. Description of the Related Art In recent years, there has been proposed an apparatus for compressing the amount of information of a digital audio signal by positively utilizing the auditory characteristics of a person and transmitting or recording / reproducing the information. In these devices, the digital audio signal is divided into several frequency components using a band division filter or using an orthogonal transform. For each divided signal, the allocation of the information amount is determined according to the ubiquity of the audio signal component and in consideration of the auditory characteristics. By doing this,
Efficient distribution of the information amount becomes possible, and as a result, compression of the information amount is achieved.

As an orthogonal transform suitable for audio signal compression, a modified discrete cosine transform (hereinafter abbreviated as MDCT) has been proposed. About MDCT, iTripleE Transactions on ISSP (IE
EE TRANSACTIONS ON ASSP), Vol. 34, No. 5, pages 1153 to 1161 co-authored by John P. Princen and Allen Bernard Bradley, "Based on Time Domain Aliasing Cancellation." Analysis / Synthesis Filter Bank Design ".

[0004] The MDCT forward transform equation is expressed as follows. Here, x (k) is the k-th spectrum of the MDCT, and x (n) is the n-th input sample of the MDCT.

[0005]

(Equation 3)

Accordingly, X (k) can be obtained by matrix multiplication of a vector X (n) composed of input samples and a cosine coefficient matrix. M X (k)
Requires 2M ² multiplications and M × (2M−1) additions.

[0007] The inverse transform equation of MDCT is expressed as follows. Here, X (k) is a spectrum of MDCT, and y (n) is a sample sequence obtained by the conversion.

[0008]

(Equation 4)

This equation has the same form as the forward transform,
To find y (n), again multiply 2M ² times,
2M × (M−1) additions are required.

[0010]

In (Equation 3), if M is 256, for example, the number of multiplications required for MDCT forward transform is 131072, and the number of additions is 130816. Recognize. This number is M
When the number increases, it increases in proportion to the square of M, and the number of operations is too large to be easily executed by a semiconductor circuit in audio equipment, resulting in an increase in the cost of the apparatus and an increase in the power consumption of the apparatus. was there.

[0011]

SUMMARY OF THE INVENTION The present invention has been made in view of such inconveniences, and has the following configuration to efficiently perform MDCT calculations. That is,
2M points of input sample data x (n) (0 ≦ n ≦ 2M−
1), a combination of two specific points is added or subtracted, or the sign is inverted, and the first intermediate data x of the M points is obtained.
If a first intermediate data generating unit for obtaining 2 (n) (0 ≦ n ≦ M−1) and the intermediate data x2 (n) are divided into a first half and a second half, and the first half x2 (n) is a real part, At the same time x
2 (n + M / 2) is inverted to an imaginary part, and e
xp (-jπn / M) to obtain M / 2 second intermediate data z (n) (0 ≦ n ≦ M / 2-1).
And the second intermediate data z (n)
Is subjected to a fast Fourier transform to obtain M / 2 Fourier coefficients Z
(K) an FFT execution unit that obtains (0 ≦ k ≦ M / 2-1);
By obtaining the real part of the multiplication result of the output Z (k) of the FFT execution means and the corresponding rotation element, an even-numbered spectrum of the input data x (n) is obtained.
The conjugate complex number Z ² (M / 2) of the output Z (k) of the FT execution means
−1-k) and a development unit that outputs an odd-numbered spectrum of the input data x (n) by obtaining a real part of a multiplication result of the corresponding rotation element.

In order to perform the inverse transform calculation of MDCT,
It is configured as follows. That is, M input spectra
Even data of the data X (k) (0 ≦ k ≦ M−1)
Data into M pieces of first intermediate data U (k) (0 ≦ k ≦ M−
1) placed in the first half (0 ≦ k ≦ M / 2-1), and
After inverting the sign of the data and further reversing the order,
1 second half of the intermediate data U (k) (M / 2 ≦ k ≦ M−1)
A first intermediate data generating unit arranged in the
The data U (k) is divided into the first half and the second half, and the first half U (k)
Is the real part and the second half is the negative sign of U (k + M / 2)
Is inverted to an imaginary part, and exp (−jπk / M) is further changed to
Multiplied to obtain M / 2 pieces of second intermediate data Z (k) (0
≦ k ≦ M / 2-1), a second intermediate data generating unit,
A fast Fourier transform is applied to the second intermediate data Z (k).
To obtain M / 2 Fourier coefficients z (n) (0 ≦ n ≦ M / 2
-1) FFT execution unit that obtains -1), and output z of FFT execution means
(N) and the corresponding predetermined rotation element exp (-j
π (2n + ／) / 2M)
Thus, the even-numbered third intermediate data y1 (n)
Value and the output z (M / 2-1-1) of the FFT execution means.
complex conjugate z of n) ^Two(M / 2-1-n) and its counterpart
The corresponding predetermined rotation element exp (−jπ · (2n + 3 /
2) / 2M), the real part of the result of the multiplication with
To obtain an odd-numbered value of the intermediate data y1 (n)
An intermediate data generator and the third intermediate data twice
Developing unit that uses and arranges in a predetermined order or reverses the sign
It is equipped with.

[0013]

According to the structure of the present invention, the vector x (n) and the vector X
Instead of performing matrix multiplication of (k) and the cosine coefficient matrix, the calculation procedure is configured so that the algorithm of the fast Fourier transform can be used by cleverly utilizing the symmetry of the cosine function. Since the number of provided samples is also halved, the calculation procedure can be made much more efficient.

[0014]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS First, a high-speed calculation method for MDCT forward transform will be described. The definition formula of MDCT can be expressed by the following formula. Here, x (n) is the input sample and X (k) is the k-th spectrum.

[0015]

(Equation 5)

FIG. 1 shows a flow of a high-speed MDCT forward transform calculation method according to the present invention. In step 1 of FIG. 1, x2 (n) is obtained from x (n) according to the following equation.

[0017]

(Equation 6)

This will be described. First, in order to utilize the cosine symmetry, the order of x (n) and the sign are converted, and x1 (n) is defined as in the following equation.

[0019]

(Equation 7)

By rewriting (Equation 3) using x1 (n), the following equation is obtained.

[0021]

(Equation 8)

Here, the operation of (Expression 7) can be represented as shown in FIG. In FIG. 3, the sequence x (n) is divided into four parts. A portion is a portion of x (n) where 0 ≦ n ≦ M / 2-1, B is a portion of M / 2 ≦ n ≦ M−1, and C is a portion of M ≦
where n ≦ 3M / 2−1, and D is 3M / 2 ≦ n ≦ 2.
M-1. In (Equation 7), the portion of ABC in FIG. 3 is shifted backward by M / 2 samples. Also, x
The D portion of (n) is inverted at the negative sign and placed at the beginning of x (n). By doing so, M / 2 in the cosine term of (Equation 5) is eliminated. Next, by rewriting (Equation 8) by the symmetry of the cosine function,

[0023]

(Equation 9)

Here,

[0025]

(Equation 10)

In summary,

[0027]

[Equation 11]

Can be expressed as Where Re [x] is x
Means the real part of Here, by summing (Formula 7) and (Formula 10), (Formula 6) is derived as shown in FIG. In FIG. 4, x1 (2n) and x1 (2n)
M-1-2n), and x2
(N) = x1 (2n) -x1 (2M-1-2n) is shown. For example, if n is in the range of 0 ≦ n ≦ M / 4-1, x1 (2n) = − x (3M / 2 + 2n),
Since x1 (2M-1-2n) = x (3M / 2-1-2n), x2 (n) = x1 (2n) -x1 (2M-1
−2n) = − x (3M / 2 + 2n) −x (3M / 2-1)
-2n).

Next, the term following the cumulative sum Σ of (Equation 11) is expressed as X
Assuming 2 (k),

[0030]

(Equation 12)

The sum range n = 0 to M-1 is defined as n = 0 to M
/ 2-1 and n = M / 2 to M-1 and rewritten,
Furthermore, replacing k with 2k,

[0032]

(Equation 13)

Here,

[0034]

[Equation 14]

In other words, X2 (2k) is

[0036]

(Equation 15)

Can be expressed as follows. Where DFT [x] is the discrete Fourier transform of x. Therefore, step 2 in FIG.
Then, from x2 (n) obtained in step 1, the complex number z
(N). Then, in the subsequent step 3, the discrete Fourier transform of z (n) is executed using the algorithm of the fast Fourier transform to obtain Z (k).

X2 (M-1-k) = X2 ^* (k)
X2 (2k + 1) is obtained from the following equation.
However, * represents a complex conjugate.

[0039]

(Equation 16)

Here, by rewriting (Equation 11) using (Equation 15) and (Equation 16),

[0041]

[Equation 17]

[0042]

(Equation 18)

In step 4 of FIG. 1, X
(K) is obtained from Z (k). To summarize the above,
The MDCT can be efficiently calculated in the following steps, as shown in FIG.

[0044]

[Equation 19]

FIG. 6 is a block diagram of an apparatus for executing the MDCT forward conversion step shown in FIG. In FIG. 6, the first intermediate data generating unit 1 receives input data x (n)
Then, x2 (n) as shown in (Equation 6) is calculated and output. The second intermediate data generator 2 calculates (Equation 14) from x2 (n) output from the first intermediate data generator 1 and calculates z
(N) is output. The FFT execution unit 3 executes a fast Fourier transform on z (n) output from the second intermediate data generation unit 2 and outputs the result as Z (k). Deployment part 4
Calculates X (k) from the output Z (k) of the FFT execution unit 3 according to (Equation 17) and (Equation 18). This deployment part 4
Is the MDCT of the input x (n).

Next, the inverse transform of MDCT will be described. The inverse transform can be calculated by a procedure similar to the above-described forward transform. The inverse transform of MDCT is represented by the following equation.
Where X (k) is the spectrum of MDCT and y
(N) is a sample sequence obtained by the conversion.

[0047]

(Equation 20)

To take advantage of the symmetry of the cosine function,
y1 (n) is defined as follows.

[0049]

(Equation 21)

From equation (16), y1 (n) can be expressed as follows.

[0051]

(Equation 22)

Here, U (k) is defined as follows.

[0053]

(Equation 23)

By rewriting (Equation 22) using (Equation 23),

[0055]

(Equation 24)

(Formula 24) has the same form as (Formula 11) except that 2 / M is not multiplied. Therefore, (Equation 2)
The calculation of 4) can use the same calculation method as the MDCT forward conversion using (Equation 11). At this time (Equation 2)
0 ≦ n ≦ of 2M Y1 (n) defined in 0)
Although only M of M−1 are obtained, there is a relationship of y1 (n) = − y (2M−1−n) as derived from (Equation 22). Therefore, as shown in FIG. Y of ≤n≤2M-1
1 (n) is immediately obtained from y1 (n) satisfying 0 ≦ n ≦ M−1. That is, A, B, C, and D at 0 ≦ n ≦ M−1 of y1 (n) in FIG. 5 are copied to −D, −C, − after negative sign inversion to M ≦ n ≦ 2M−1. B, -A. Therefore, considering (Equation 21), after all, y (n)
Is obtained from y1 (n) as follows. That is,

[0057]

(Equation 25)

To summarize the above, MDCT
Can be obtained by the following steps as shown in FIG.

[0059]

(Equation 26)

[0060]

[Equation 27]

[0061]

[Equation 28]

[0062]

(Equation 29)

[0063]

[Equation 30]

FIG. 7 is a block diagram of an apparatus for executing the inverse transform step of the MDCT shown in FIG. In FIG. 7, the first intermediate data generating unit 5 receives input data X (k)
Then, U (k) expressed by (Equation 23) is calculated and output. The second intermediate data generator 6 calculates (Equation 27) from U (k) output from the first intermediate data generator 5 and calculates Z
(K) is output. The FFT execution unit 7 executes fast Fourier transform on Z (k) output from the second intermediate data generation unit 6 and outputs the result as z (n). The third intermediate data generation unit 8 executes (Equation 29) from the output z (n) of the FFT execution unit 7 to execute the third intermediate data y1
(N) is obtained and output. The expansion unit 9 rearranges the output y1 (n) of the third intermediate data generation unit 8 according to (Equation 30) or inverts the sign to obtain y (n) and outputs it.
The output of the developing unit 9 is used as the inverse MDCT transform of the input data X (k).

As described above, the forward and inverse transforms of the MDCT can be efficiently calculated using the present invention.

[0066]

For example, in (Equation 5), if M = 256, the number of multiplications required in the forward transform of MDCT is
512 × 256 = 131072 times, the number of additions is 511 ×
256 = 130816 times, which requires a huge amount of calculations. However, according to the high-speed calculation method of the present invention, the number of operations required for each step in the forward transform of the MDCT is as follows.

(Number of calculations in step 1) Addition M = 25
Six times (the number of operations in step 2) Since multiplication is performed between complex numbers, M multiplications and M / 2 additions are required to obtain the real part and the imaginary part. Therefore, the addition M = 256 times, the multiplication 2M = 512 times (the number of operations in step 3), and the FFT of M / 2 points is (M /
2) · log ₂ M butterfly operations are required. Therefore, when M = 256, 1024 butterfly operations are required. Since one butterfly operation includes two complex additions and one complex multiplication, if this is converted into a real number operation, the number of additions is six and the number of multiplications is four. 6144 times multiplication 1024 × 4 = 4096 times (the number of operations in step 4) (2 / M) A0 (k) to (2
/ M) A3 (k) can be calculated in advance and prepared as a table, so that for one X (k), two multiplications and one addition
Times are needed. Therefore, when adding these 256 times, multiplying 512 times, and calculating these, the amount of calculation required as the MDCT forward transform becomes 6912 times of addition and 5120 times of multiplication.
The number of operations for both multiplication and addition is reduced to about 25 times. Similarly, in the inverse transform of the MDCT, the number of calculations is greatly reduced.

As described above, according to the present invention, the calculation of the MDCT can be performed very efficiently.

[Brief description of the drawings]

FIG. 1 is a flowchart of a MDCT forward transform calculation procedure according to an embodiment of the present invention.

FIG. 2 is a flowchart of an MDCT inverse transform calculation procedure according to an embodiment of the present invention.

FIG. 3 is an explanatory diagram of (Equation 7).

FIG. 4 is an explanatory diagram of (Equation 6).

FIG. 5 is an explanatory diagram of (Equation 30).

FIG. 6 is a block diagram of an MDCT forward conversion apparatus according to an embodiment of the present invention.

FIG. 7 is a block diagram of an MDCT inverse transform device according to an embodiment of the present invention.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 1st intermediate data generation part 2 2nd intermediate data generation part 3 FFT execution part 4 Development part 5 1st intermediate data generation part 6 2nd intermediate data generation part 7 FFT execution part 8 3rd intermediate data generation Part 9 Development part

────────────────────────────────────────────────── ─── Continued on the front page (72) Inventor Tetsushi Kasahara 1006 Kadoma Kadoma, Osaka Prefecture Matsushita Electric Industrial Co., Ltd. (58) Field surveyed (Int. Cl. ⁷ , DB name) H04B 14/00- 14/06 G10L 11/00 G10L 21/04 H03M 7/30

Claims (4)

(57) [Claims] 1. An input sample data x (n) of 2M points
(0 ≦ n ≦ 2M-1 ) from the first intermediate data x M points
2 (n) (0 ≦ n ≦ M−1), and the first intermediate data generator includes 0 ≦ n ≦ M /
In 4-1, as x2 (n), -x (3M / 2 +
2n) -x (3M / 2-1-2n) is calculated and substituted. When M / 4 ≦ n ≦ 3M / 4-1, x2 (n) is set.
X (2n-M / 2) -x (3M / 2-1-2n)
When 3M / 4 ≦ n ≦ M−1, x2 (n) is calculated as x
Calculate (2n-M / 2) -x (7M / 2-1-2n)
Assignment Te, and the intermediate data x2 (n) is divided into first and second halves, the first half of x2 (n) of the second half at the same time when the real part x2 (n +
M / 2) is inverted to an imaginary part, and exp (−
jπn / M) to obtain M / 2 second intermediate data z (n) (0 ≦ n ≦ M / 2-1), and the second intermediate data The data z (n) is subjected to Fourier transform to obtain M / 2 Fourier coefficients Z (k) (0 ≦ k ≦ M / 2−2).
(1) obtaining an FFT execution unit, and obtaining a real part of a multiplication result of an output Z (k) of the FFT execution means and a corresponding rotation element exp (−jπ · (2k + ／) / 2M) And input data x
(N) is obtained, and the conjugate complex number Z * (M / 2-2-1) of the output Z (k) of the FFT execution means is obtained.
k) and the real part of the result of multiplication of the predetermined rotation element exp (−jπ · (2k + 3/2) / 2M) corresponding to the input data x
A modified cosine transform device comprising: (n) a developing unit that outputs an odd-numbered spectrum. 2. M input spectrum data X (k)
(0 ≦ k ≦ M−1), the even-numbered data is converted into M first data.
The first half (0 ≦ k) of the intermediate data U (k) (0 ≦ k ≦ M−1)
k ≦ M / 2-1), the odd-numbered data is negatively inverted, and the order is further inverted.
The first intermediate data generation unit arranged in the second half (M / 2 ≦ k ≦ M−1) of (k), and the first intermediate data U (k) are divided into the first half and the second half, and the first half U (k) k) is converted to a real part, and at the same time, the negative sign of U (k + M / 2) is inverted to an imaginary part, and further multiplied by exp (−jπk / M) to obtain M /
Two second intermediate data Z (k) (0 ≦ k ≦ M / 2−
1), and a Fourier transform of the second intermediate data Z (k) to obtain M / 2 Fourier coefficients z (n) (0 ≦ n ≦ M / 2-1). FFT to get)
An execution unit, an output z (n) of the FFT execution unit, and a predetermined rotation element exp (−jπ (2n + ／)) corresponding to the output z (n)
/ 2M) to obtain the real part of the result of multiplication with the third intermediate data y1 (n).
The conjugate complex number z of the output z (M / 2-1-n) of the T execution means
The third intermediate data y1 is obtained by calculating the real part of the result of multiplication of * (M / 2-l-n) and the corresponding rotation element exp (-jπ · (2n + 3/2) / 2M).
A third intermediate data generating unit for obtaining an odd-numbered value of (n); and a developing unit for arranging the third intermediate data twice in a predetermined order or inverting the sign of the third intermediate data. Inverse converter for modified cosine transform. 3. A modified discrete cosine transform method, characterized in that in a calculation of a modified discrete cosine transform (abbreviated as MDCT), the following steps are performed to perform the transform. (Equation 1) 4. An inverse transform method of the modified discrete cosine transform, wherein the inverse transform is performed by performing the following steps in the inverse transform calculation of the modified discrete cosine transform (abbreviated as MDCT). (Equation 2)