JPH06232824A - Correcting discrete cosine transformation, inverse transforming method and its device - 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 transform coding high-speed calculation method and apparatus used in compression coding of audio signals and the like.

[0002]

2. Description of the Related Art In recent years, there has been proposed a device for compressing the information amount of a digital audio signal by positively utilizing human auditory characteristics and transmitting or recording / reproducing. In these devices, a digital audio signal is divided into several frequency components by using a band division filter or an orthogonal transformation. For each of the divided signals, the allocation of the information amount is determined according to the ubiquity of the components of the audio signal and considering the auditory characteristics. By doing this,
The information amount can be efficiently distributed, and as a result, the compression of the information amount is achieved.

A modified discrete cosine transform (hereinafter abbreviated as MDCT) has been proposed as an orthogonal transform suitable for compressing audio signals. For MDCT, I Triple E Transactions on EYSP (IE
EE TRANSACTIONS ON ASSP) Vol. 34, No. 5, pp. 1153 to 1161, by John P. Princen and Allen Bernard Bradley, "Based on Aliasing Cancellation in the Time Domain" Detailed analysis / synthesis filter bank design ”.

The forward transform formula of MDCT is expressed as follows. Here, x (k) is the kth spectrum of MDCT, and x (n) is the nth input sample of MDCT.

[0005]

[Equation 3]

Therefore, X (k) can be obtained by matrix multiplication of the vector X (n) consisting of input samples and the cosine coefficient matrix. M X (k)
2M ² multiplications and M × (2M−1) additions are required to obtain

The inverse transform formula of MDCT is expressed as follows. However, X (k) is the spectrum of MDCT and y (n) is the sample sequence obtained by the conversion.

[0008]

[Equation 4]

This equation has the same form as the forward transform, and is 2M
To obtain the number of y (n), 2M ² multiplications are required and
2M × (M−1) additions are required.

[0010]

In Equation 3, if M is 256, for example, the number of multiplications required for MDCT forward conversion is 131072, and the number of additions is 130816, which requires enormous calculation. Recognize. This number is M
Is increased in proportion to the square of M, the number of calculations is too large to be easily executed by a semiconductor circuit in an audio device, resulting in an increase in device cost and an increase in device power consumption. was there.

[0011]

SUMMARY OF THE INVENTION The present invention has been made in view of such inconvenience, and has the following configuration to efficiently perform MDCT calculation. That is,
Input sample data x (n) at 2M points (0 ≦ n ≦ 2M−
The first intermediate data x at M points is added or subtracted or the negative sign is inverted for the specific combination of two points in 1).
If the first intermediate data generator that obtains 2 (n) (0 ≦ n ≦ M−1) and the intermediate data x2 (n) are divided into the first half and the second half, and the first half x2 (n) is the real part. At the same time x in the latter half
Invert the negative sign of 2 (n + M / 2) to form the imaginary part, and then e
The second to obtain M / 2 second intermediate data z (n) (0 ≦ n ≦ M / 2−1) by multiplying xp (−jπn / M)
Intermediate data generating section of the second intermediate data z (n)
Fast Fourier transform to 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 predetermined rotation element, an even-numbered spectrum of the input data x (n) is obtained, and F
Conjugate complex number Z ² (M / 2 of output Z (k) of FT execution means
-1-k) and the expansion part that outputs the odd-numbered spectrum of the input data x (n) by obtaining the real part of the multiplication result of the corresponding predetermined rotation element.

In addition, in order to perform the inverse transform calculation of MDCT,
It is configured as follows. That is, M input spectra
Data X (k) (0 ≦ k ≦ M−1) of even-numbered data
Data of M first intermediate data U (k) (0 ≦ k ≦ M−
Arranged in the first half of (1) (0 ≦ k ≦ M / 2−1) and odd-numbered
After inverting the data in the negative sign and inverting the order,
Second half of the intermediate data U (k) of 1 (M / 2 ≦ k ≦ M−1)
A first intermediate data generating section arranged in
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 at the same time the negative sign of U (k + M / 2) in the latter half
Is inverted to the imaginary part, and exp (-jπk / M)
Multiplying by M / 2 second intermediate data Z (k) (0
A second intermediate data generating unit for obtaining ≦ k ≦ M / 2−1),
Fast Fourier transform is applied to the second intermediate data Z (k).
Then, M / 2 Fourier coefficients z (n) (0 ≦ n ≦ M / 2
-1) and an output z of the FFT execution means
(N) and a predetermined rotation element exp (-j corresponding to it)
π (2n + 1/2) / 2M) to find the real part of the multiplication result
By doing so, even-numbered third intermediate data y1 (n)
Value, and the output z (M / 2-1-) of the FFT execution means.
n) conjugate complex number z ²(M / 2-1-n) and it
Corresponding predetermined rotation element exp (-jπ · (2n + 3 /
2) / 2M) to obtain the real part of the result of multiplication
3rd intermediate data y1 (n) to obtain the odd-numbered value of the third
The intermediate data generator and the third intermediate data are provided twice
Expanding part that uses and arranges in a predetermined order, or reverses the negative sign
And are equipped with.

[0013]

According to the configuration of the present invention, the vector x (n) and the vector X are calculated in the calculation of the forward transform and the inverse transform of MDCT.
Instead of performing the matrix multiplication of (k) and the cosine coefficient matrix, the calculation procedure is configured so that the fast Fourier transform algorithm can be used by skillfully utilizing the symmetry of the cosine function, and further the fast Fourier transform is performed. Since the number of samples provided is halved, the calculation procedure can be greatly streamlined.

[0014]

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. Where x (n) is the input sample and X (k) is the kth spectrum.

[0015]

[Equation 5]

FIG. 1 shows the flow of a high-speed calculation method of MDCT forward transform according to the present invention. In step 1 of FIG. 1, x2 (n) is calculated 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 negative sign are converted, and x1 (n) is defined as in the following equation.

[0019]

[Equation 7]

Rewriting (Equation 3) using this x1 (n) gives the following equation.

[0021]

[Equation 8]

The operation of (Equation 7) can be expressed as shown in FIG. In FIG. 3, the sequence x (n) is divided into four parts. The part A is 0 ≦ n ≦ M / 2−1 of x (n), the part B is M / 2 ≦ n ≦ M−1, and the part C is M ≦.
n ≦ 3M / 2−1, and D is 3M / 2 ≦ n ≦ 2
It is the part of M-1. In (Equation 7), the portion ABC in FIG. 3 is shifted backward by M / 2 samples. Also, x
The sign D of (n) is inverted by 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, rewriting (Equation 8) according to the symmetry of the cosine function,

[0023]

[Equation 9]

Here,

[0025]

[Equation 10]

When summarized as

[0027]

[Equation 11]

It can be expressed as However, Re [x] is x
Means the real part of. Here, by combining (Equation 7) and (Equation 10), (Equation 6) is derived as shown in FIG. In FIG. 4, x1 (2n) and x1 (2n) for each section of n.
M-1-2n) is shown, and for each section, x2
(N) = x1 (2n) -x1 (2M-1-2n) is shown. For example, when 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 X.
2 (k)

[0030]

[Equation 12]

The range of cumulative sum n = 0 to M-1, n = 0 to M
/ 2-1 and n = M / 2 to M-1 and rewrite,
Replace k with 2k,

[0032]

[Equation 13]

Here,

[0034]

[Equation 14]

Putting this, X2 (2k) is

[0036]

[Equation 15]

Can be expressed as 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
Seeking (n). Then, in the subsequent step 3, the fast Fourier transform algorithm is used to execute the discrete Fourier transform of z (n) to obtain Z (k).

Further, X2 (M-1-k) = X2 ^* (k)
From the relationship of, X2 (2k + 1) is calculated by 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]

Then, 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]

[Formula 19]

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

Next, the inverse transform of MDCT will be described. The inverse transformation can also be calculated by a procedure similar to the forward transformation described above. The inverse transform of MDCT is expressed by the following equation.
However, 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,
Define y1 (n) as follows.

[0049]

[Equation 21]

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

[0051]

[Equation 22]

Here, U (k) is defined by the following equation.

[0053]

[Equation 23]

Rewriting (Equation 22) using (Equation 23),

[0055]

[Equation 24]

(Equation 24) has the same form as (Equation 11) except that it is not multiplied by 2 / M. Therefore, (Equation 2
The calculation of 4) can use the same calculation method as the forward transform of MDCT using (Equation 11). At this time (Equation 2
0) of 2M Y1 (n) defined in 0)
Only M of M−1 are obtained, but since there is a relation of y1 (n) = − y (2M−1−n) as derived from (Equation 22), as shown in FIG. ≤n≤2M-1 y
1 (n) is immediately obtained from y1 (n) with 0 ≦ n ≦ M−1. That is, A, B, C and D in 0 ≦ n ≦ M−1 of y1 (n) in FIG. 5 are copied to M ≦ n ≦ 2M−1 after negative sign inversion and −D, −C, −. It is calculated as B, -A. Therefore, considering (Equation 21), y (n)
Is calculated from y1 (n) as follows. That is,

[0057]

[Equation 25]

Summarizing what has been described above, MDCT
The inverse transformation of can be obtained in the next step, 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 performing the steps of inverse transform of MDCT shown in FIG. In FIG. 7, the first intermediate data generator 5 receives the input data X (k)
From, the U (k) expressed in (Equation 23) is calculated and output. The second intermediate data generator 6 calculates (Equation 27) from U (k) output by the first intermediate data generator 5, and Z
Output (k). 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 generate y1 which is the third intermediate data.
(N) is calculated 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 negative sign to obtain y (n) and outputs it.
The output of the expansion unit 9 is the MDCT inverse transform of the input data X (k).

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

[0066]

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

(Calculation count in step 1) Addition M = 25
6 times (the number of operations in step 2) Since it is a multiplication of complex numbers, M times of multiplication and M / 2 times of addition are required to obtain each of the real number part and the imaginary number part. Therefore, addition M = 256 times multiplication 2M = 512 times (the number of calculations in step 3) The FFT of M / 2 points is (M /
2) ・ Log ₂ M butterfly calculations are required. Therefore, M = 256 requires 1024 butterfly calculations. Since one butterfly operation includes two complex additions and one complex multiplication, this is 6 additions and four multiplications when converted to a real number operation, so the total is 1024 × 6 = additions. 6144 times Multiplication 1024 × 4 = 4096 times (number of calculations in step 4) (2 / M) A0 (k) to (2
/ M) A3 (k) can be calculated in advance and prepared as a table, so for one X (k), two multiplications and one addition
Needs times. Therefore, when these are calculated as 256 additions and 512 multiplications, the amount of calculation required for the forward transform of MDCT is 6912 additions and 5120 multiplications.
The number of operations for both multiplication and addition is reduced to about 1/25. Similarly, in the inverse transform of MDCT, the number of calculations is greatly reduced.

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

[Brief description of drawings]

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

FIG. 2 is a flow chart of an inverse transform calculation procedure of MDCT according to the 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 transform device 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]

 1 1st intermediate data generation part 2 2nd intermediate data generation part 3 FFT execution part 4 expansion part 5 1st intermediate data generation part 6 2nd intermediate data generation part 7 FFT execution part 8 3rd intermediate data generation Department 9 Development Department

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Satoshi Kasahara 1006 Kadoma, Kadoma City, Osaka Prefecture Matsushita Electric Industrial Co., Ltd.

Claims (4)

[Claims] 1. Input sample data x (n) at 2M points
The first intermediate data x2 (n) (0 ≦ n ≦ M−) at M points is added or subtracted or the sign is inverted for a specific combination of two points (0 ≦ n ≦ 2M−1). 1) which obtains 1) and the above intermediate data x2 (n) is divided into the first half and the second half, and the first half x2 (n) is used as the real part, and at the same time, the second half x2 (n + M / 2) is negative Signal is inverted to the imaginary part, and then multiplied by exp (-jπn / M),
M / 2 second intermediate data z (n) (0 ≦ n ≦ M / 2
−1) and a second intermediate data generator z, and fast Fourier transform is performed on the second intermediate data z (n) to obtain M / 2 Fourier coefficients Z (k) (0 ≦ k ≦ M / 2 −1) to obtain the real part of the multiplication result of the FFT execution unit, the output Z (k) of the FFT execution means, and the predetermined rotation element corresponding to the output Z (k). By obtaining a spectrum and obtaining the real part of the multiplication result of the conjugate complex number Z ^* (M / 2−1−k) of the output Z (k) of the FFT execution means and the corresponding predetermined rotation element, A modified cosine transform device, comprising: an expansion unit that outputs an odd-numbered spectrum of input data x (n). 2. M pieces of input spectrum data X (k)
The even-numbered data of (0 ≦ k ≦ M−1) is converted into M first data.
Intermediate data U (k) (0 ≦ k ≦ M−1) of the first half (0 ≦
k ≦ M / 2-1), the odd-numbered data is negative-inverted, the order is further inverted, and then the first intermediate data U
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) is the real part, and at the same time, the negative sign of U (k + M / 2) in the latter half is inverted to form the imaginary part, and then exp (-jπk / M) is multiplied to obtain
Two pieces of second intermediate data Z (k) (0 ≦ k ≦ M / 2−
1) and the second intermediate data Z (k) are subjected to fast Fourier transform to obtain M / 2 Fourier coefficients z (n) (0 ≦ n ≦ M / 2− F) to get 1)
The output z (n) of the FT execution unit and the FFT execution unit and a predetermined rotation element exp (-jπ (2n + 1 /
2) / 2M) to obtain the real part of the multiplication result to obtain an even-numbered value of the third intermediate data y1 (n), and
A conjugate complex number z ^* (M / 2-1-n) of the output z (M / 2-1-n) of the FFT execution means and a predetermined rotation element exp (-jπ · (2n + 3/2) / 2M corresponding thereto. ) And the real part of the multiplication result, the third intermediate data y
A third intermediate data generating unit for obtaining an odd-numbered value of 1 (n); and a developing unit for arranging the third intermediate data twice in a predetermined order or inverting the negative sign. An inverse transform device for modified cosine transform. 3. A modified discrete cosine transform method characterized by performing the following steps in a modified discrete cosine transform (abbreviated as MDCT) calculation. [Equation 1] 4. An inverse transform method of modified discrete cosine transform (Abbreviated as MDCT) in which inverse transform is performed by executing the following steps. [Equation 2]