CN1882938A - Process and device for determining a transforming element for a given transformation function, method and device for transforming a digital signal and computer readable medium - Google Patents

Abstract

According to the process for determining a transform element for a given transformation function, which transformation function comprises a transformation matrix and corresponds to a transformation of a digital signal from the time domain into the frequency domain or vice versa, the transformation matrix is decomposed into a rotation matrix (306) and an auxiliary matrix (307) which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix. Further, the rotation matrix (306) and the auxiliary matrix (307) are each decomposed into a plurality of lifting matrices (308). Further, the transforming element is determined to comprise of a plurality of lifting stages (309) which correspond to the lifting matrices (308). The invention further provides a method for the transformation of a digital signal from the time domain into the frequency domain according to the transforming element determined by the process described above.

Description

Determine the process and the equipment of the conversion element of given transforming function transformation function, digital signal converting method and equipment and computer-readable medium

The cross reference of related application

The application requires the U.S. Provisional Application No.60/507 of submission on September 29th, 2003, the U.S. Provisional Application No.60/507 that on September 29th, 210 and 2003 submitted to, 440 right of priority is incorporated herein by reference in full in this content with each, to be used for all purposes.

In addition, the following application of owning together is submitted to together simultaneously, introduces in full at this:

" Method for Performing a Domain Transformation of a DigitalSignal from the Time Domain into the Frequency Domain and vice Versa ", application attorney docket No.P100442, and

" Method for Transforming a Digital Signal from the Time Domaininto the Frequency Domain and Vice Versa ", application attorney docket No.P100444.

Technical field

The present invention relates to be used for determine the process and the equipment of the conversion element of given transforming function transformation function, be used for digital signal is transformed from the time domain to frequency domain and the method and apparatus from the frequency domain transform to the time domain, and computer-readable medium.

Background technology

The territory conversion, for example discrete cosine transform (DCT) is widely used in current signal Processing industry.In recent years, because its key player in lossless coding is used, the distortion that is called the DCT of integer DCT has attracted many research interest.Term " can't harm " and means that demoder can produce definitely duplicating of source signal according to bitstream encoded.

Described DCT is real-valued conversion.Even described input block only comprises integer, the IOB of described DCT can comprise the non-integer component.For easy, described input block is called as input vector, and IOB is called as output vector.If vector only comprises integer components, then this vector is called as integer vectors.Contrast in DCT, described integer DCT produces the integer output vector according to the integer input vector.For same integer input vector, the integer output vector of integer DCT is similar to the real output vector of DCT very much.Therefore, integer DCT keeps all good characteristics of described DCT when spectrum analysis.

The key property of described integer DCT is a reversibility.Reversibility means and has integral discrete cosine inverse transformation (IDCT) that if make described integer DCT produce output vector y according to input vector x, then described integer ID CT can recover vector x according to vectorial y.Sometimes integer DCT is also referred to as positive-going transition, and integer ID CT is called as reciprocal transformation or inverse transformation.

The conversion that is called integer improvement discrete cosine transform (IntMDCT) is suggested in recent years and is used in the ISO/IEC MPEG-4 audio compression.Described IntMDCT comes from its prototype-improvement discrete cosine transform (MDCT).In [1], Malvar has provided by a plurality of Givens of DCT-IV piece cascade are rotated the method that realizes MDCT effectively.What known is that the Givens rotation can be broken down into three lifting step, is used for integer is mapped as integer, referring to example [2].

Therefore, the realization of IntMDCT depends on effective enforcement of integer DCT-IV.

By utilizing three lifting step to replace each Givens rotation, can directly be converted to integer transform from the prototype of integer transform.Owing to there is the operation that rounds up in each lifting step, the number of times that always rounds up of integer transform is 3 times of Givens number of revolutions of prototype conversion.For discrete trigonometric transforms (for example discrete Fourier transform (DFT) (DFT) or discrete cosine transform (DCT)), the number of times of related Givens rotation is generally Nlog ₂The N magnitude, wherein N is a described size, the amount of the data symbol that comprises in each piece that promptly described digital signal is divided.Therefore, for the conversion of the same clan of the integer transform of direct conversion, the described total degree that always rounds up also is Nlog ₂The N magnitude.Because only approximate its floating-point prototype of described rounding up, integer transform.Described approximate error increases along with the increase of the number of times that rounds up.

Summary of the invention

The present invention solves following problems, promptly determine the conversion element of given transforming function transformation function, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, make the number of times that rounds up that comprises by described conversion element reduce significantly.The present invention also provides a kind of method, is used for according to described definite conversion element, digital signal is transformed from the time domain to frequency domain or from the frequency domain transform to the time domain.

Utilization meets the feature of independent claims, the process and the equipment of the conversion element by being used for determining given transforming function transformation function, be used for digital signal is transformed from the time domain to frequency domain and the method and apparatus from the frequency domain transform to the time domain, and computer-readable medium solves described problem.

According to the present invention, a process that is used for the conversion element of definite given transforming function transformation function is provided, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, wherein said transformation matrix is broken down into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other, each of described rotation matrix and described companion matrix all is broken down into a plurality of lifting matrixes; And described conversion element is confirmed as comprising a plurality of lifting levels corresponding with described lifting matrix.

In addition, according to the present invention, provide a kind of equipment of carrying out said process that is applicable to.

In addition, according to the present invention, providing a kind of is used to use the conversion element that digital signal is transformed from the time domain to frequency domain or the method from the frequency domain transform to the time domain, wherein said conversion element is corresponding to given transforming function transformation function, this transforming function transformation function comprises transformation matrix, and wherein said conversion element determined by a process, and this process comprises described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other; Each of described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes; And determine that described conversion element comprises a plurality of lifting levels corresponding with described lifting matrix; Wherein each lifting level comprises the processing that utilizes the householder transformation and the sub-piece of unit to described digital signal that round up to carry out.

In addition, according to the present invention, provide a kind of equipment that is suitable for carrying out said method.

In addition, according to the present invention, a kind of computer-readable medium is provided, this computer-readable medium has record program thereon, wherein said program is suitable for making computing machine to carry out the process of the conversion element that is used for definite given transforming function transformation function, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, wherein said transformation matrix is broken down into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other; Each of described rotation matrix and described companion matrix all is broken down into a plurality of lifting matrixes; And described conversion element is confirmed as comprising a plurality of lifting levels corresponding with described lifting matrix.

In addition, according to the present invention, a kind of calculation machine computer-readable recording medium is provided, this computer-readable medium has record program thereon, wherein said program is suitable for making the computing machine execution to be used to use the conversion element that digital signal is transformed from the time domain to frequency domain or the method from the frequency domain transform to the time domain, wherein said conversion element is corresponding to given transforming function transformation function, this transforming function transformation function comprises transformation matrix, wherein said conversion element is determined by a process, this process comprises: described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other, each of described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes; And determine that described conversion element comprises a plurality of lifting levels corresponding with described lifting matrix; Wherein each promotes level and comprises and utilize householder transformation and the unit that rounds up is handled the sub-piece of described digital signal.

In a preferred embodiment, the invention provides a kind of process and method that is used to realize integer IV type dct transform.Compare with the method for prior art, the number of times of the operation that rounds up that the method according to this invention needs significantly reduces.The result is, described approximate error is significantly reduced, and under the situation of DCT-IV, it is from common Nlog ₂It is low as 2.5N that the N magnitude reduces to, and wherein N represents the block size of digital signal.The method according to this invention is low and structurally be modular on computation complexity.

Method and apparatus according to the invention can be used for the digital signal of any kind, such as audio frequency, image or vision signal.But as the digital signal of the signal that is digitized signal corresponding to physical measurement, its can by scanning corresponding analog signal at least one characteristic feature (for example, the brightness value of vision signal and chromatic value, from the amplitude of the analoging sound signal of sensor, or the amplitude of analog sensing signal) and produce.Described digital signal comprises a plurality of data symbols.The data symbol of described digital signal is grouped into a plurality of, and wherein each piece has the data symbol of identical predetermined number based on the sampling rate of described corresponding analog signal.

It also is integer-valued output signal that the method according to this invention can be used for being that integer-valued supplied with digital signal is transformed to.Transform method according to the present invention is reversible.Can original input signal be returned in described output signal conversion by carrying out transform method according to the present invention.This kind reversibility of the conversion of the method according to this invention can be used for the lossless coding that output signal wherein should be equal to original input signal.

This kind integer transform according to signal of the present invention can be used for many application and system, such as mpeg audio, image and video compress, JPEG2000 or analysis of spectrum (being used to analyze infrared, ultraviolet or nuclear-magnetism radiation signal).It can be under not considering such as the situation in the factor of the overflow under the situation of real-valued signal conversion, easily to realize such as the hardware system of point of fixity digital signal processor (DSP).

The method according to this invention utilizes the conversion element that frequency domain is arrived in described digital signal conversion, and this conversion element is that given transforming function transformation function is determined according to process of the present invention.

Preferably, described conversion element comprises a plurality of lifting levels.

Described conversion element can carry out illustration based on the model that promotes ladder.Described lifting ladder model has two side members (piece), and each parts is used for receiving a group of two groups of data symbols.Two or more cascades are set between described two side members promote level.Each promotes level (input end) received signal at one end, and via addition unit in the other end (output terminal) output signal.The unit that rounds up is arranged at output terminal.In the mode that replaces described lifting level is arranged between the described side members, makes adjacent output (or input) end that promotes level be connected to different side members.

It should be noted, though describe the conversion element with the form that promotes the ladder model, the transform path of the described conversion element of illustration only.Yet the present invention should not be limited to described ladder model.

The number of the lifting level of described conversion element is to be defined by the number that promotes matrix, and the number of this lifting matrix is determined by process according to the present invention.

Discrete cosine transform, discrete sine transform, discrete Fourier transform (DFT) or DISCRETE W TRANSFORM are the examples that can be used as according to the transforming function transformation function of transforming function transformation function of the present invention.The result according to process of the present invention who depends on the conversion element that is used for definite each transforming function transformation function, the number of the lifting level of described conversion element can be different.

Description of drawings

Exemplary embodiments of the present invention is described with reference to the accompanying drawings, wherein:

Fig. 1 shows the architecture of audio coder according to an embodiment of the invention;

Fig. 2 shows the architecture of audio decoder according to an embodiment of the invention, and it is corresponding to the audio coder shown in Fig. 1;

Fig. 3 illustration according to the embodiment of process of the present invention, wherein said transforming function transformation function is the DCT-IV transforming function transformation function;

Fig. 4 shows the process flow diagram of the embodiment of the method according to this invention;

Fig. 5 illustration use the embodiment of DCT-IV as the method according to this invention of transforming function transformation function;

Fig. 6 illustration be used for algorithm according to the inverse transformation of the embodiment of the illustrative method of the present invention of Fig. 5;

Fig. 7 shows the architecture of image archiving system according to an embodiment of the invention;

Fig. 8 illustration use the embodiment of DWT-IV as the method according to this invention of transforming function transformation function;

Fig. 9 illustration be used for algorithm according to the inverse transformation of the embodiment of the illustrative method of the present invention of Fig. 8.

Embodiment

Fig. 1 shows the structure of audio coder 100 according to an embodiment of the invention.

Described audio coder 100 comprises based on the conventional perception base layer coder of improving discrete cosine transform (MDCT) and improves the harmless enhanced encoder of discrete cosine transform (IntMDCT) based on integer.

For example, will provide and carry out digitized sound signal 109 by microphone 110 and offer audio coder 100 by A/D converter 111.Described sound signal 109 comprises a plurality of data symbols.Described sound signal 109 is divided into a plurality of, and wherein each piece comprises a plurality of data symbols of digital signal, and by improving discrete cosine transform (MDCT) equipment 101 each piece is carried out conversion.The described MDCT coefficient that is provided by MDCT equipment 101 is quantized by means of sensor model 102 by quantizer 103.Described sensor model is controlled described quantizer 103 according to a kind of like this mode, makes that the audio distortions that is produced by quantization error is low.Encoded by the output of 104 pairs of quantizers 103 of bitstream encoder subsequently, this bitstream encoder 104 produces the output bit flow 112 of the perceptual coding that diminishes.Described bitstream encoder 104 is utilized such as the standard method of Huffman coding or the distance of swimming (Run-Length) coding and is nondestructively compressed its input to produce an output, the mean bit rate of its input that the mean bit rate of this output will be lower than.Described input audio signal 109 also is transported in the IntMDCT equipment 105 that produces the IntMDCT coefficient.The MDCT of quantification coefficient as the output of quantizer 103 is used to predict described IntMDCT coefficient.The described MDCT coefficient that quantized is transported to contrary-quantizer 106, and contrary-quantizer 106 described outputs are transported to the unit 107 that rounds up, described round up the unit with described contrary-quantizer 106 described outputs are rounded to round values, and remaining IntMDCT coefficient carries out entropy coding by the IntMDCT coefficient of 108 couples of remnants of entropy coder, and the IntMDCT coefficient of described remnants is the poor of the output of unit 107 and IntMDCT coefficient that round up.Described entropy coder (being similar to bitstream encoder 104) 108 nondestructively reduces the mean bit rate of its input, and produces the harmless bit stream 113 that strengthens.Described harmless enhancing bit stream 113 and perceptual coding bit stream 112 carry the essential information of accurate reconstruct input audio signal 109 together.

Fig. 2 shows the architecture of the audio decoder 200 that comprises embodiments of the invention, and it is corresponding to the audio coder shown in Fig. 1 100.

Described perceptual coding bit stream 207 is by 201 decodings of bit stream decoding device, the inverse operation of the operation of the bitstream encoder 104 of these bit stream decoding device 201 execution graphs 1.Described decoded bit stream is provided for contrary-quantizer 202.At its output terminal, apply anti-MDCT by improving inverse discrete cosine transform equipment (anti-MDCT) 203.Therefore, obtain the perceptual coding sound signal 209 of reconstruct.Described harmless enhancing bit stream 208 is by entropy decoder 204 decodings, and the inverse operation of the operation of the entropy coder 108 in these entropy decoder 204 execution graphs 1 produces corresponding remaining IntMDCT coefficient.Output by 205 pairs of contrary-quantizers 202 of the equipment of rounding up rounds up, and is added to described remaining IntMDCT coefficient then, produces described IntMDCT coefficient thus.At last, carry out described integer by 206 pairs of described IntMDCT coefficients of the improved inverse discrete cosine transform equipment of described integer and improve inverse discrete cosine transform, to produce the lossless coding sound signal 210 of described reconstruct.

As mentioned above, the core that IntMDCT has been shown in [2] is integer DCT-IV, and this core is played an important role in lossless audio coding, and is used in the illustrative embodiments of the invention of Fig. 1 and 2.

Fig. 3 illustration according to the process flow diagram of the embodiment of process of the present invention, wherein said transforming function transformation function is the DCT-IV transforming function transformation function.

The embodiment of the process of the present invention of the conversion element that is used for definite DCT-IV transforming function transformation function has been described below.Described definite conversion element is used for the scrambler shown in Fig. 1 and implements IntMDCT, and corresponding inverse transformation element is used for the demoder shown in Fig. 2 to implement anti-IntMDCT.For the description that how to utilize DCT-IV enforcement IntMDCT and anti-IntMDCT, referring to [2].

Have the DCT-IV transforming function transformation function quilt of the real list entries x of N point (n) as give a definition (referring to [2]):

$y\left(m\right)=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}x\left(n\right)\cos \left(\frac{(m+1/2)(n+1/2)\mathrm{\π}}{N}\right)m,n=\mathrm{0,1},\·\·\·,N-1---\left(1\right)$

Suppose Be the transformation matrix of DCT-IV, that is,

$\underset{\‾}{C_N^{\mathrm{IV}}}=\sqrt{\frac{2}{N}}{\left[\cos \left(\frac{(m+1/2)(n+1/2)\mathrm{\π}}{N}\right)\right]}_{m,n=\mathrm{0,1},,N-1}---\left(2\right)$

According to the embodiment of process of the present invention, transformation matrix

Be broken down into rotation matrix and another matrix,, equal permutation matrix and the integer diagonal matrix multiplies each other when this matrix and when self multiplying each other.

For the sake of clarity, N is assumed to be even number.

Described process starts from step 300.

In step 301, the even number index entry of transforming function transformation function separates with the odd number index entry:

$y\left(m\right)=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}x\left(n\right)\cos \left((m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\π}}{N}\right)$

$=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}x\left(2n\right)\cos \left((m+\frac{1}{2})(2n+\frac{1}{2})\frac{\mathrm{\π}}{N}\right)+\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}x(2n+1)\cos \left((m+\frac{1}{2})(2n+1+\frac{1}{2})\frac{\mathrm{\π}}{N}\right)$

$=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_1\left(n\right)\cos \left((m+\frac{1}{2})(n+\frac{1}{4})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)}\right)+\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_2\left(n\right)\cos \left((m+\frac{1}{2})(n+\frac{3}{4})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)}\right)$

Therefore,

$y\left(m\right)=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_1\left(n\right)\cos ((m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)}-\frac{1}{4}(m+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)})---\left(3\right)$

$+\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_2\left(n\right)\cos ((m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)}+\frac{1}{4}(m+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)})$

Wherein has component x ₁(n)=x (2n), $n=\mathrm{0,1},...,\frac{N}{2}-1$ x ₁Be to comprise that all have the vector of the x of even number index (n), and have component x ₂(n)=x (2n+1), $n=\mathrm{0,1},...,\frac{N}{2}-1$ x ₂Be to comprise that all have the vector of the x of odd number index (n).

Use following two simple formulas:

${\mathrm{\α}}_{m,n}=(m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)},m,n=0,...,N-1---\left(4\right)$

${\mathrm{\β}}_m=\frac{1}{4}(m+\frac{1}{2})\frac{\mathrm{\π}}{\left(\frac{N}{2}\right)}=(m+\frac{1}{2})\frac{\mathrm{\π}}{2N},m=0,...,N-1---\left(5\right)$

Utilize equation (4) and (5), equation (3) can be write as:

$y\left(m\right)=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_1\left(n\right)\cos ({\mathrm{\α}}_{m,n}-{\mathrm{\β}}_m)+\sqrt{\frac{2}{N}}\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_2\left(n\right)\cos ({\mathrm{\α}}_{m,n}+{\mathrm{\β}}_m)---\left(6\right)$

In step 302, use the addition theorem of following two cosine:

cos(α _m，n-β _m)＝cosα _m，n·cosβ _m+sinα _m，n·sinβ _m (7)

cos(α _m，n+β _m)＝cosα _m，n·cosβ _m-sinα _m，n·sinβ _m (8)

Utilize equation (7) and (8), equation (6) can be write as

$y\left(m\right)=\sqrt{\frac{2}{N}}\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_1\left(n\right)\cos {\mathrm{\α}}_{m,n}\cos {\mathrm{\β}}_m+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_1\left(n\right)\sin {\mathrm{\α}}_{m,n}\sin {\mathrm{\β}}_m+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_2\left(n\right)\cos {\mathrm{\α}}_{m,n}\cos {\mathrm{\β}}_m-\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}{x}_2\left(n\right)\sin {\mathrm{\α}}_{m,n}\sin {\mathrm{\β}}_m\}$

$=\sqrt{\frac{2}{N}}\left\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\right[x_1\left(n\right)\cos {\mathrm{\β}}_m+x_2\left(n\right)\cos {\mathrm{\β}}_m]\cos {\mathrm{\α}}_{m,n}+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}[x_1\left(n\right)\sin {\mathrm{\β}}_m-x_2\left(n\right)\sin {\mathrm{\β}}_m\left]\sin {\mathrm{\α}}_{m,n}\right\}$

$=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\left\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\frac{\cos {\mathrm{\β}}_m}{\sqrt{2}}\right[x_1\left(n\right)+x_2\left(n\right)]\cos {\mathrm{\α}}_{m,n}+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\frac{\sin {\mathrm{\β}}_m}{\sqrt{2}}[x_1\left(n\right)-x_2\left(n\right)\left]\sin {\mathrm{\α}}_{m,n}\right\}---\left(9\right)$

For simple formula, the vector of two N/2 sizes x ₁' and x ₂' be defined as comprising following component:

${x_1}^{\′}\left(n\right)=\frac{1}{\sqrt{2}}[x_1\left(n\right)+x_2\left(n\right)],n=0,...,\frac{N}{2}-1,---\left(10\right)$

${x_2}^{\′}\left(n\right)=\frac{1}{\sqrt{2}}[x_1\left(n\right)-x_2\left(n\right)],n=0,...,\frac{N}{2}-1---\left(11\right)$

Utilize (10) and (11), equation (9) is reduced to

$y\left(m\right)=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\cos {\mathrm{\β}}_m{x_1}^{\′}\left(n\right)\cos {\mathrm{\α}}_{m,n}+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\sin {\mathrm{\β}}_m{x_2}^{\′}\left(n\right)\sin {\mathrm{\α}}_{m,n}\}---\left(12\right)$

In step 303, vector

$\underset{\‾}{y}=\left[\begin{array}{c}y\left(0\right)\\ y\left(1\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right]$

Be divided into two parts y ₀With y ₁, wherein

$\underset{\‾}{y_0}=\left[\begin{array}{c}y\left(0\right)\\ y\left(1\right)\\ \·\\ \·\\ \·\\ y(\frac{N}{2}-1)\end{array}\right]---\left(13\right)$

$\underset{\‾}{y_1}=\left[\begin{array}{c}y(N-1)\\ y(N-2)\\ \·\\ \·\\ \·\\ y\left(\frac{N}{2}\right)\end{array}\right]---\left(14\right)$

Vector y ₁Comprise according to backward corresponding to from To the index of N-1 yComponent.

Vector y ₁Component y ₁(m) satisfy following equation:

$m=0,...,\frac{N}{2}-1$ Situation under,

y ₁(m)＝y(N-1-m)

$=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\cos {\mathrm{\β}}_{N-1-m}{x_1}^{\′}\left(n\right)\cos {\mathrm{\α}}_{N-1-m,n}+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\Σ}}}\sin {\mathrm{\β}}_{N-1-m,m}{x}_2\left(n\right)\sin {\mathrm{\α}}_{N-1-m,n}\}---\left(15\right)$

Attention for $m=0,...,\frac{N}{2}-1$

$y(m+\frac{N}{2})=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\cos {\mathrm{\&beta;}}_{m+\frac{N}{2}}{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{m+\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2},n}+\underset{n=0}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\sin {\mathrm{\&beta;}}_{m+\frac{N}{2}}{x_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m+\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2},n}\}---\left(16\right)$

In step 304, y ₀With y ₁In each by the DCT-IV matrix

With the DST-IV matrix

Expression, the size of each is

This realizes in the following manner:

For ${\mathrm{\&beta;}}_m=(m+\frac{1}{2})\frac{\mathrm{\&pi;}}{2N},$ Following equation is set up:

${\mathrm{\&beta;}}_{m+\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}=(m+\frac{1}{2}+\frac{N}{2})\frac{\mathrm{\&pi;}}{2N}=(m+\frac{1}{2})\frac{\mathrm{\&pi;}}{2N}+\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}\&CenterDot;\frac{\mathrm{\&pi;}}{2N}={\mathrm{\&beta;}}_m+\frac{\mathrm{\&pi;}}{4}---\left(17\right)$

${\mathrm{\&beta;}}_{N-1-m}=(N-1-m+\frac{1}{2})\frac{\mathrm{\&pi;}}{2N}=N\&CenterDot;\frac{\mathrm{\&pi;}}{2N}-(m+\frac{1}{2})\&CenterDot;\frac{\mathrm{\&pi;}}{2N}=\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&beta;}}_m---\left(18\right)$

In addition,

$\cos {\mathrm{\&beta;}}_{N-1-m}=\cos (\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&beta;}}_m)=\sin {\mathrm{\&beta;}}_m---\left(19\right)$

$\sin {\mathrm{\&beta;}}_{N-1-m}=\sin (\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&beta;}}_m)=\cos {\mathrm{\&beta;}}_m---\left(20\right)$

For ${\mathrm{\&alpha;}}_{m,n}=(m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\&pi;}}{\left(\frac{N}{2}\right)},$ Following equation is set up:

${\mathrm{\&alpha;}}_{N-1-m,n}=(N-1-m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\&pi;}}{\left(\frac{N}{2}\right)}=(N-m-\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\&pi;}}{\left(\frac{N}{2}\right)}$

$=N(n+\frac{1}{2})\frac{\mathrm{\&pi;}}{\left(\frac{N}{2}\right)}-(m+\frac{1}{2})(n+\frac{1}{2})\frac{\mathrm{\&pi;}}{\left(\frac{N}{2}\right)}=2\mathrm{\&pi;}(n+\frac{1}{2})-{\mathrm{\&alpha;}}_{m,n}=2\mathrm{\&pi;n}+\mathrm{\&pi;}-{\mathrm{\&alpha;}}_{m,n}---\left(21\right)$

In addition,

cosα _N-1-m，n＝cos(2πn+π-α _m，n)＝cos(π-α _m，n)＝-cosα _m，n (22)

sinα _N-1-m，n＝sin(2πn+π-α _m，n)＝sin(π-α _m，n)＝sinα _m，n (23)

Utilize equation (19), (20), (22) and (23), equation (15) can be written as

$y(N-1-m)=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\cos {\mathrm{\&beta;}}_{N-1-m}{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{N-1-m,n}+\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\sin {\mathrm{\&beta;}}_{N-1-\mathrm{<figure-callout\; id="2"\; label="m\; x"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">m</figure-callout>}}{x_{}}^{}{{}_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{N-1-m,n}\}$

$=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\sin {\mathrm{\&beta;}}_m{x_1}^{\&prime;}\left(n\right)(-\cos {\mathrm{\&alpha;}}_{m,n})+\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\cos {\mathrm{\&beta;}}_m{x_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m,n}\}$

$=-\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\sin {\mathrm{\&beta;}}_m{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{m,n}+\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\cos {\mathrm{\&beta;}}_m{x_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m,n}$

$=-\sin {\mathrm{\&beta;}}_m\left[\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{m,n}\right]+\cos {\mathrm{\&beta;}}_m\left[\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}{{\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">x</figure-callout>}}_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m,n}\right]$

$m=0,...,\frac{N}{2}-1---\left(24\right)$

Because another equation (12) draws

$y\left(m\right)=\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\{\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\cos {\mathrm{\&beta;}}_m{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{m,n}+\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}\sin {\mathrm{\&beta;}}_m{x_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m,n}\}$

$=\cos {\mathrm{\&beta;}}_m\left[\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}{x_1}^{\&prime;}\left(n\right)\cos {\mathrm{\&alpha;}}_{m,n}\right]+\sin {\mathrm{\&beta;}}_m\left[\sqrt{\frac{2}{\left(\frac{N}{2}\right)}}\underset{n}{\overset{\frac{N}{2}-1}{\mathrm{\&Sigma;}}}{{\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">x</figure-callout>}}_2}^{\&prime;}\left(n\right)\sin {\mathrm{\&alpha;}}_{m,n}\right]---\left(25\right)$

Can form according to (24) and (25) y ₀With y ₁Expression formula:

${\underset{\&OverBar;}{y}}_0=\underset{\&OverBar;}{\mathrm{diag}\left[\cos {\mathrm{\&beta;}}_m\right]}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\&CenterDot;\underset{\&OverBar;}{{x_1}^{\&prime;}}+\underset{\&OverBar;}{\mathrm{diag}\left[\sin {\mathrm{\&beta;}}_m\right]}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\&CenterDot;\underset{\&OverBar;}{{x_2}^{\&prime;}},m=0,...,\frac{N}{2}-1---\left(26\right)$

${\underset{\&OverBar;}{y}}_0=\underset{\&OverBar;}{\mathrm{diag}[-\sin {\mathrm{\&beta;}}_m]}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\&CenterDot;\underset{\&OverBar;}{{x_1}^{\&prime;}}+\underset{\&OverBar;}{\mathrm{diag}\left[\cos {\mathrm{\&beta;}}_m\right]}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\&CenterDot;\underset{\&OverBar;}{{x_2}^{\&prime;}},m=0,...,\frac{N}{2}-1---\left(27\right)$

Diag[a wherein _m] expression N/2 * N/2 diagonal matrix, the m behavior a of this matrix _m, Be the transformation matrix of DCT-IV conversion, and

It is the transformation matrix of IV type discrete sine transform (DST-IV).

Two equatioies (26) and (27) can be expressed as single equation in the following manner:

Below, use Matrix

With

As can be seen

$\left[\begin{array}{c}y\left(0\right)\\ y\left(2\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y(N-1)\end{array}\right]=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{c}{y}_0\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y_0(\frac{N}{2}-1)\\ y_1\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y_1(\frac{N}{2}-1)\end{array}\right]---\left(31\right)$

It can simply be

$\underset{\&OverBar;}{y}=\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="I\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\frac{N}{}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{c}\underset{\&OverBar;}{y_0}\\ \underset{\&OverBar;}{y_1}\end{array}\right]---\left(32\right)$

It can also be seen that, adopt N * N matrix, R _PrMay be defined as

$\underset{\&OverBar;}{R_{\mathrm{pr}}}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \underset{\&OverBar;}{I_{N/2}}\\ \underset{\&OverBar;}{I_{N/2}}& \underset{\&OverBar;}{-I_{N/2}}\end{array}\right]---\left(33\right)$

Following equation is set up:

$\left[\begin{array}{c}{x_1}^{\&prime;}\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {x_1}^{\&prime;}(\frac{N}{2}-1)\\ {x_2}^{\&prime;}\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {x_2}^{\&prime;}(\frac{N}{2}-1)\end{array}\right]=\underset{\&OverBar;}{R_{\mathrm{pr}}}\left[\begin{array}{c}{x}_1\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_1(\frac{N}{2}-1)\\ x_2\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_2(\frac{N}{2}-1)\end{array}\right]---\left(34\right)$

Equation (34) can simply be

$\left[\begin{array}{c}\underset{\&OverBar;}{{x_1}^{\&prime;}}\\ \underset{\&OverBar;}{{x_2}^{\&prime;}}\end{array}\right]=\underset{\&OverBar;}{R_{\mathrm{pr}}}\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]---\left(35\right)$

In addition, suppose P _EoBe the strange matrix of idol, that is, permutation matrix, it is by will be corresponding to the component of even index and component branch corresponding to the strange index described vector of rearrangement that comes xComponent, wherein

$\underset{\&OverBar;}{x}=\left[\begin{array}{c}x\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-1)\end{array}\right]$

Make following formula set up

$\left[\begin{array}{c}{x}_1\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_1(\frac{N}{2}-1)\\ x_2\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_2(\frac{N}{2}-1)\end{array}\right]=\underset{\&OverBar;}{P_{\mathrm{eo}}}\left[\begin{array}{c}x\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-1)\end{array}\right]---\left(36\right)$

Or be abbreviated as

$\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]=\underset{\&OverBar;}{P_{\mathrm{eo}}}\underset{\&OverBar;}{x}---\left(37\right)$

By equation (28) and equation (31), (34) and (36) are merged, can draw

Utilize above-mentioned simple formula and following simple formula

With

Equation (38) can be abbreviated as

$\underset{\&OverBar;}{y}=\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="I\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\frac{N}{}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{C}& \underset{\&OverBar;}{S}\\ -\underset{\&OverBar;}{S}& \underset{\&OverBar;}{C}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\end{array}\right]\underset{\&OverBar;}{R_{\mathrm{pr}}}\underset{\&OverBar;}{P_{\mathrm{eo}}}\underset{\&OverBar;}{x}---\left(41\right)$

In step 305, utilize Represent This utilizes following equation to finish.

$\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}=\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}\&CenterDot;\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="D\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">D</figure-callout>}}_{\frac{N}{}}}---\left(42\right)$

Wherein

It is the diagonal matrix on the following N/2 rank that provide

Utilize equation (42), equation (41) can be write as

$\underset{\&OverBar;}{y}=\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="I\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\frac{N}{}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{C}& \underset{\&OverBar;}{S}\\ -\underset{\&OverBar;}{S}& \underset{\&OverBar;}{C}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{I_{\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\frac{N}{}}^{\frac{}{2}}{D}_{\frac{N}{}}{\frac{}{2}}_{}}\end{array}\right]\underset{\&OverBar;}{R_{\mathrm{pr}}}\underset{\&OverBar;}{P_{\mathrm{eo}}}\underset{\&OverBar;}{x}---\left(44\right)$

In step 306, calculate N * N rotation matrix R _Po, this rotation matrix comprises three matrixes of beginning in the equation (44):

$\underset{\&OverBar;}{R_{\mathrm{po}}}=\left[\begin{array}{cc}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="I\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\frac{N}{}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{C}& \underset{\&OverBar;}{S}\\ -\underset{\&OverBar;}{S}& \underset{\&OverBar;}{C}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{I_{\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}}& \\ & \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="J\; N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">J</figure-callout>}}_{\frac{N}{}}}\end{array}\right]---\left(45\right)$

Carry out three multiplications of matrices in the equation (45), obtain

In step 307, calculate companion matrix, when described companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other.

Described companion matrix comprises the 4th and the 5th matrix of equation (44):

$\underset{\&OverBar;}{T}=\left[\begin{array}{cc}\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}& \\ & \underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}\end{array}\right]\underset{\&OverBar;}{R_{\mathrm{pr}}}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}& \underset{\&OverBar;}{C_{N/2}^{\mathrm{VI}}}\\ \underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}& -\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}\end{array}\right]---\left(47\right)$

Note

Be T and self multiply each other and equal permutation matrix and the integer diagonal matrix multiplies each other.

Utilize equation (46) and (47), equation (44) can be reduced to

$\underset{\&OverBar;}{C_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_{\mathrm{po}}}\underset{\&OverBar;}{T}\underset{\&OverBar;}{P_{\mathrm{eo}}}---\left(49\right)$

Therefore, described transformation matrix

Be broken down into rotation matrix R _Po, companion matrix TAnd the strange matrix of idol P _Eo, wherein work as companion matrix TWhen multiplying each other, equal permutation matrix and multiply by the integer diagonal matrix with self.

In step 308, R _PoWith TEach all be factorized as the product that promotes matrix.According to

   

Following T is carried out factorization:

$\underset{\&OverBar;}{T}=\underset{\&OverBar;}{T_1{T}_2{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">T</figure-callout>}}_2}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{K_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\left[\begin{array}{cc}-\underset{\&OverBar;}{D_{N/2}}& \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">K</figure-callout>}}_2}\\ & \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\begin{array}{c}\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{K_3}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(50\right)\end{array}$

Wherein

$\underset{\&OverBar;}{K_1}=-(\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}+\sqrt{2}\underset{\&OverBar;}{I_{N/2}})\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}---\left(51\right)$

$\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">K</figure-callout>}}_2}=\frac{C_{N/2}^{\mathrm{IV}}}{\sqrt{2}}---\left(52\right)$

$\underset{\&OverBar;}{K_3}=\sqrt{2}\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}+\underset{\&OverBar;}{I_{N/2}}---\left(53\right)$

And come factorization according to following equation R _Po:

$\underset{\&OverBar;}{R_{\mathrm{po}}}=\underset{\&OverBar;}{R_1{R}_2{R}_3}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">H</figure-callout>}}_2}\\ & \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\begin{array}{c}\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_3}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(54\right)\end{array}$

Wherein

With

Equation (49) can further be written as

$\underset{\&OverBar;}{C_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_1{R}_2{R}_3{T}_1{T}_2{T}_3{P}_{\mathrm{eo}}}---\left(57\right)$

In step 309, will promote the matrix merger together as much as possible.Matrix S is defined as the product of R3 and T1, promptly

$\underset{\&OverBar;}{S}=\underset{\&OverBar;}{R_3{T}_1}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_3}+\underset{\&OverBar;}{K_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(58\right)$

Because SAlso be to promote matrix, so this merger is possible.

According to (57) and (58), the formula of the last factorization of the DCT-IV matrix of acquisition is

$\underset{\&OverBar;}{C_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_1{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\underset{\&OverBar;}{S}\underset{\&OverBar;}{T_2{T}_3{P}_{\mathrm{eo}}}---\left(59\right)$

Equation (59) expression comprises that according to the conversion element of integer DCT-IV of the present invention conversion five promote level.

Because last factorization formula is determined, so at step S310, described process stops.

Fig. 4 shows the process flow diagram 400 of the embodiment of the method according to this invention, and this method uses five to promote level, and first promotes level 401, second promotes level the 402, the 3rd and promote level the 403, the 4th and promote level the 404 and the 5th and promote level 405.This method is preferred in the anti-InrMDCT equipment 206 of the IntMDCT equipment 105 of Fig. 1 and Fig. 2, to realize IntMDCT and anti-IntMDCT respectively.In Fig. 4, x ₁With x ₂Be respectively first and second of digital signal, z ₁, z ₂With z ₃Be M signal, y ₁With y ₂Be respectively and first and second of described digital signal corresponding output signals.

Fig. 5 shows the process flow diagram of the embodiment of the method according to this invention, and wherein said transforming function transformation function is the DCT-IV transforming function transformation function.The conversion element of Shi Yonging is corresponding to equation (59) in this embodiment, that is, it is the conversion element that the embodiment by illustrative process among Fig. 3 determines.

Described conversion element comprises that five promote level, these five five lifting matrixes that promote level corresponding to equation (59).

In addition, described conversion element comprises and described permutation matrix P _EoCorresponding data recombination (shuffling) level.

In Fig. 5, first input that promotes level is two pieces of digital signal x ₁With x ₂, z ₁, z ₂With z ₃Be M signal, y ₁With y ₂Be respectively and first and second of described digital signal corresponding output signals.

The input of described conversion element xPromote two input blocks of level with first of described conversion element x ₁With x ₂Satisfy equation

$\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]=\underset{\&OverBar;}{P_{\mathrm{eo}}}\underset{\&OverBar;}{x}---\left(60\right)$

(consistent) with equation (37).

Below, explain that first promotes level 501, it is and promotes matrix T ₃Corresponding lifting level.Suppose $\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]$ Be the output vector of the first lifting level when not being rounded up to round values, promptly

$\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]=\underset{\&OverBar;}{T_3}\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]---\left(61\right)$

Use is provided by equation (50) T ₃Definition, equation (61) can be rewritten as

$\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]=\left[\begin{array}{cc}{I}_{\frac{N}{2}}& \\ \underset{\&OverBar;}{K_3}& \underset{\&OverBar;}{I_{\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}}\end{array}\right]\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]---\left(62\right)$

Owing to, be included in integer-valued rounding up at this embodiment that the reversible algorithm that is used for integer DCT-IV is provided.Therefore,, promote in the first step 506 of level 501 first according to equation (62), x ₁With K ₃Multiply each other.In step 507, the result of this multiplication is rounded to round values.In step 508, be added to subsequently through the value after rounding up x ₂Therefore, M signal z ₁Satisfy equation:

z ₁＝_ K ₃· x ₁_+ x ₂

Wherein, _ * _ expression operation that rounds up.

The matrix that corresponds respectively to owing to illustrative conversion element among Fig. 5 T ₂, S, R ₂With R ₁All the other four promote level 502,503,504 and 505, have with first and promote the identical structure of level 501, so the descriptions thereof are omitted.Only it should be noted, promote in the addition step 509 of level 502 second, according to T ₂Definition, x ₁With- D _N/2Multiply each other.

The lifting level of the conversion element of inverse transformation is described with reference to Fig. 6 below.

Fig. 6 illustration the lifting level of conversion element of inverse transformation of illustrative conversion among Fig. 5.

In Fig. 6, first input that promotes level is two pieces of digital signal y ₁With y ₂, z ₁, z ₂With z ₃Be M signal, x ₁With x ₂Be respectively and first and second of described digital signal corresponding output signals.

The illustrative first lifting level, 501 contraries among illustrative last lifting level 605 and Fig. 5 among Fig. 6.Therefore, in the end promote in the first step 606 of level 605, x ₁With K ₃Multiply each other.In step 607, the result of this multiplication is rounded to round values.Subsequently in step 608, from z ₁In deduct through the value after rounding up.Therefore, signal x ₂Satisfy equation:

x ₂＝ z ₁-_ K ₃· x ₁_ (64)

Wherein, _ * _ expression operation that rounds up.

Because all the other four of illustrative conversion element is respectively the contrary lifting level 601,602,603 and 604 that promotes level 505,504,503 and 502 among Fig. 6, have and the last grades 605 identical structures that promote, so the descriptions thereof are omitted.Only it should be noted, the 4th promote the addition step 609 of level 604 after, the result of addition step 609 with- D _N/2Multiply by generation mutually x ₁

As can be seen, the lifting level among Fig. 6 605,604,603,602 and 601 is respectively lifting level 501 to 505 contrary among Fig. 5.Because and matrix P _EoThe also reversible and described inverse transformation element of displacement of corresponding input signal comprises corresponding data recombination level, so the method that is provided is reversible.Therefore, if in Fig. 1 and Fig. 2, use in illustrative audio coder 100 and the audio decoder 200, just obtained being used for the method and apparatus of lossless audio coding.

Provide the analysis of the number of times that rounds up that uses in this embodiment in the ending of instructions of the present invention.

Fig. 7 shows the architecture of image archiving system according to an embodiment of the invention.

In Fig. 7, image source 701, for example camera provides analog picture signal.By A/D converter 702 this picture signal is handled, so that the corresponding digital picture signal to be provided.Carry out lossless coding by 703 pairs of these data image signals of lossless image scrambler, it comprises the conversion from the time domain to the frequency domain.In this embodiment, time domain is corresponding to the coordinate space of described image.Picture signal behind the described lossless coding is stored in the memory device 704, for example hard disk or DVD.When the described image of needs, picture signal from described memory device 704 behind the described lossless coding of taking-up, and provide it to lossless image demoder 705, picture signal behind 705 pairs of lossless codings of this lossless image demoder is decoded, and the described original image signal of reconstruct and loss of data can not occur.

For example, be the Error Graph of semiconductor wafer and must be stored being used under the situation with post analysis that this of picture signal kind of harmless filing is important at described image.

Below, the another embodiment that is used for digital signal is transformed from the time domain to frequency domain and the method from the frequency domain transform to the time domain according to the present invention is described.This embodiment preferably uses in the lossless image scrambler 703 of illustrative image archiving system and the lossless image demoder 705 in Fig. 7.

Fig. 8 illustration the embodiment of the method according to this invention, it uses DWT-IV as transforming function transformation function.

The DWT-IV of the real list entries x of N point (n) is as giving a definition:

$y\left(m\right)=\sqrt{\frac{2}{N}}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}x\left(n\right)\sin (\frac{\mathrm{\&pi;}}{4}+\frac{(m-1/2)(n+1/2)2\mathrm{\&pi;}}{N})m,n=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,N-1---\left(65\right)$

The transformation matrix of DWT-IV Following providing

$\underset{\&OverBar;}{W_N^{\mathrm{IV}}}=\sqrt{\frac{2}{N}}{\left[\sin (\frac{\mathrm{\&pi;}}{4}+\frac{(m+1/2)(n+1/2)2\mathrm{\&pi;}}{N})\right]}_{m,n=\mathrm{0,1},,N-1}---\left(66\right)$

According to process of the present invention, described DWT-IV matrix is factorized as following form:

$\underset{\&OverBar;}{W_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_N}\underset{\&OverBar;}{T}\underset{\&OverBar;}{P_N}---\left(67\right)$

R _NBe rotation matrix as the N * N that gives a definition:

$\underset{\&OverBar;}{R_N}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \underset{\&OverBar;}{J_{N/2}}\\ -\underset{\&OverBar;}{J_{N/2}}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(68\right)$

I _N/2It is the unit matrix (consistent) on N/2 rank with equation (29). J _N/2It is the anti-unit matrix (consistent) on N/2 rank with equation (30).

P _NIt is the permutation matrix of N * N

$\underset{\&OverBar;}{P_N}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ & \underset{\&OverBar;}{J_{N/2}}\end{array}\right]---\left(69\right)$

TBe matrix as the N * N that gives a definition:

$\underset{\&OverBar;}{T}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}& -\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}\\ \underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}& \underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}{D}_{N/2}}\end{array}\right]---\left(70\right)$

Wherein Be the DCT-IV matrix on N/2 rank,

$\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}=\sqrt{\frac{2}{(N/2)}}[\cos \left(\frac{(m+1/2)(n+1/2)\mathrm{\&pi;}}{(N/2)}\right){]}_{m,n=\mathrm{0,1},,N/2-1}---\left(71\right)$

D _N/2It is the diagonal matrix on the following N/2 rank that provide

According to process of the present invention, R _NWith TCan further be factorized as and be promoted the long-pending of matrix:

$\underset{\&OverBar;}{T}=\underset{\&OverBar;}{T_1{T}_2{T}_3}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{K_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\left[\begin{array}{cc}\underset{\&OverBar;}{D_{N/2}}& \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">K</figure-callout>}}_2}\\ & \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{K_3}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(73\right)$

Wherein, $\underset{\&OverBar;}{K_1}=(\sqrt{2}\underset{\&OverBar;}{I_{N/2}}-\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}\underset{\&OverBar;}{D_{N/2}})\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}},\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">K</figure-callout>}}_2}=\frac{-\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}}{\sqrt{2}},\underset{\&OverBar;}{K_3}=\sqrt{2}\underset{\&OverBar;}{C_{N/2}^{\mathrm{IV}}}\underset{\&OverBar;}{D_{N/2}}-\underset{\&OverBar;}{I_{N/2}}$

With

$\underset{\&OverBar;}{R_N}=\underset{\&OverBar;}{R_1}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\underset{\&OverBar;}{R_3}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">H</figure-callout>}}_2}\\ & \underset{\&OverBar;}{I_{N/2}}\end{array}\right]\&CenterDot;\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_3}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(74\right)$

Wherein

With

Therefore, equation (67) can be written as following form

$\underset{\&OverBar;}{W_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_1}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\underset{\&OverBar;}{R_3}\underset{\&OverBar;}{T_1}\underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">T</figure-callout>}}_2}\underset{\&OverBar;}{T_3}\underset{\&OverBar;}{P_N}---\left(77\right)$

According to process of the present invention, merger lifting step as much as possible.

In this embodiment, promote matrix R ₃With T ₁Can be merged to promoting matrix S:

$\underset{\&OverBar;}{S}=\underset{\&OverBar;}{R_3}\underset{\&OverBar;}{T_1}=\left[\begin{array}{cc}\underset{\&OverBar;}{I_{N/2}}& \\ \underset{\&OverBar;}{H_3}+\underset{\&OverBar;}{K_1}& \underset{\&OverBar;}{I_{N/2}}\end{array}\right]---\left(78\right)$

According to (77) and (78), can obtain the following factorization formula of DWT-IV matrix:

$\underset{\&OverBar;}{W_N^{\mathrm{IV}}}=\underset{\&OverBar;}{R_1{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">R</figure-callout>}}_2}\underset{\&OverBar;}{S}\underset{\&OverBar;}{T_2{T}_3}\underset{\&OverBar;}{P_N}---\left(79\right)$

Equation (79) expression comprises that according to the conversion element of integer DWT-IV of the present invention conversion five promote level.

In addition, described conversion element comprises and described permutation matrix P _NCorresponding data recombination level.Described data recombination level is rearranged the component order in each input block.According to P _N, rearrange input data vector in the following manner: the first half parts of vector remain unchanged, and the second half parts of vector turn upside down, and promptly

$\underset{\&OverBar;}{P_N}\underset{\&OverBar;}{x}=\underset{\&OverBar;}{P_N}\left[\begin{array}{c}{x}_1\\ x_2\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_{N/2}\\ x_{N/2+1}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_{N-1}\\ x_N\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_{N/2}\\ x_N\\ x_{N-1}\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x_{N/2+1}\end{array}\right]---\left(80\right)$

In Fig. 8, first input that promotes level is two pieces of digital signal x ₁With x ₂, z ₁, z ₂With z ₃Be M signal, y ₁With y ₂Be respectively and first and second of described digital signal corresponding output signals.

First of the input x of described conversion element and described conversion element promotes two input blocks of level x ₁With x ₂Satisfy equation

$\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]=\underset{\&OverBar;}{P_N}\underset{\&OverBar;}{x}---\left(81\right)$

Below, explain that first promotes level 801, it is and promotes matrix T ₃Corresponding lifting level.Suppose $\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]$ Be the output vector of the first lifting level when not being rounded up to round values, promptly

$\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]=\underset{\&OverBar;}{T_3}\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]---\left(82\right)$

Use has equation (73) to provide T ₃Definition, equation (82) can be rewritten as

$\left[\begin{array}{c}\underset{\&OverBar;}{v_1}\\ \underset{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}\end{array}\right]=\left[\begin{array}{cc}{I}_{\frac{N}{2}}& \\ \underset{\&OverBar;}{K_3}& \underset{\&OverBar;}{I_{\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}}}\end{array}\right]\left[\begin{array}{c}\underset{\&OverBar;}{x_1}\\ \underset{\&OverBar;}{x_2}\end{array}\right]---\left(83\right)$

Because the reversible algorithm that is used for integer DCT-IV that provides in this embodiment is included in integer-valued rounding up.Therefore,, promote in the first step 806 of level 801 first according to equation (83), x ₁With K ₃Multiply each other.In step 807, the result of this multiplication is rounded to round values.In step 808, be added to subsequently through the value after rounding up x ₂Therefore, M signal z ₁Satisfy equation:

z ₁＝_ K ₃· x ₁_+ x ₂ (84)

Wherein, _ * _ expression operation that rounds up.

Because all the other four of illustrative conversion element lifting levels 802,803,804 and 805 correspond respectively to matrix among Fig. 8 T ₂, S, R ₂With R ₁, it has with first and promotes the identical structure of level 801, so the descriptions thereof are omitted.Only it should be noted, promote in the addition step 809 of level 802 second, according to T ₂Definition, x ₁With D _N/2Multiply each other.

The lifting level of the conversion element of inverse transformation is described with reference to Fig. 9 below.

Fig. 9 illustration the lifting level of conversion element of inverse transformation of illustrative conversion among Fig. 8.

In Fig. 9, first input that promotes level is two pieces of digital signal y ₁With y ₂, z ₁, z ₂With z ₃Be M signal, x ₁With x ₂Be respectively and first and second of described digital signal corresponding output signals.

The illustrative first lifting level, 801 contraries among illustrative last lifting level 905 and Fig. 8 among Fig. 9.Therefore, in the end promote in the first step 906 of level 905, x ₁With K ₃Multiply each other.In step 907, the result of this multiplication is rounded to round values.Subsequently in step 908, from z ₁In deduct through the value after rounding up.Therefore, signal x ₂Satisfy equation:

x ₂＝z ₁-_K ₃·x ₁_ (85)

Wherein, _ * _ expression operation that rounds up.

Because all the other four of illustrative conversion element is respectively the contrary lifting level 901,902,903 and 904 that promotes level 805,804,803 and 802 among Fig. 9, have and the last grades 905 identical structures that promote, so the descriptions thereof are omitted.Only it should be noted, the 4th promote the addition step 909 of level 904 after, the result of addition step 909 with D _N/2Multiply by generation mutually x ₁

As can be seen, the lifting level among Fig. 9 905,904,903,902 and 901 is respectively lifting level 801 to 805 contrary among Fig. 8.Because and matrix P _NThe also reversible and described inverse transformation element of displacement of corresponding input signal comprises corresponding data recombination level, so the method that is provided is reversible.Therefore, if in Fig. 7, use in illustrative lossless image scrambler 703 and the lossless image demoder 705, lossless image Methods for Coding and device have so just been obtained being used for.

Though in the embodiment of described explanation, method according to DCT-IV of the present invention is used to audio coding, and be used to picture coding according to the method for DWT-IV of the present invention, but the method according to DCT-IV of the present invention can be used for picture coding equally, and can be used for audio coding equally according to the method for DWT-IV of the present invention, and these two kinds of methods may be used to the coding of other digital signals, such as vision signal.

Consider equation (63) and (64), as can be seen, carry during upgrading the existence at each and round up for N/2 time.Therefore, considering equation (59), as can be seen, is five times of N/2 according to the number of times that always rounds up of the conversion element of DCT-IV algorithm of the present invention, 2.5N just, and it significantly is lower than the Nlog according to prior art ₂N.

Consider equation (59) once more, as can be seen, when N is when being worth greatly, N=1024 for example, main calculated amount be used in corresponding to Four N/2 point DCT-IV subroutines of multiplication on.Because can be according to equation (47) and (49), use two and half length DCT-IV add the rotation of going forward and calculate floating-point DCT-IV with the back rotation, so the algorithm complex of the integer DCT-IV of described proposition can be estimated as the twice of the algorithm complex of floating-point DCT-IV roughly.

Can obtain similar conclusion for integer DWT-IV transforming function transformation function.

Below, the another embodiment that uses discrete Fourier transform (DFT) is described.

Transformation matrix as the transformation matrix of N rank normalization FFT F _NFollowing providing:

${\underset{\&OverBar;}{F}}_N=\sqrt{\frac{1}{N}}\left[\exp \left(\frac{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mn}}{N}\right)\right]---\left(86\right)$

m，n＝0，1，…，N-1

Wherein transform size N is an even number.

Follow base-2 (radix-2) decimation in time fft algorithm, can decompose in the following manner F _N:

${\underset{\&OverBar;}{F}}_N=\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}& {\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}\\ {\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}& -{\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}\end{array}\right]\left[\begin{array}{cc}& {\underset{\&OverBar;}{I}}_{N/2}\\ \underset{\&OverBar;}{W}& \end{array}\right]\left[\begin{array}{cc}& {\underset{\&OverBar;}{F}}_{N/2}\\ {\underset{\&OverBar;}{F}}_{N/2}& \end{array}\right]{\underset{\&OverBar;}{P}}_{\mathrm{eo}}---\left(87\right)$

Wherein, as mentioned above, P _EoBe the strange matrix of idol, that is, permutation matrix, it is by will be corresponding to the component of even index and component branch corresponding to the strange index rearrangement vector that comes xComponent, wherein

$\underset{\&OverBar;}{x}=\left[\begin{array}{c}x\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-1)\end{array}\right]$

Make

$\left[\begin{array}{c}x\left(0\right)\\ x\left(2\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-2)\\ x\left(1\right)\\ x\left(3\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-1)\end{array}\right]=\underset{\&OverBar;}{P_{\mathrm{eo}}}\left[\begin{array}{c}x\left(0\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ x(N-1)\end{array}\right]---\left(88\right)$

Suppose F _N/2Be that exponent number is the transformation matrix of the normalization FFT of N/2.

Suppose WIt is the following diagonal matrix that provides

W wherein _N=e ^{-j2 π/N}

As above, I _N/2The unit matrix of expression exponent number N/2.

In equation (87), the first left matrix is the strange matrix of idol P _Eo, the component in its input vector of only resequencing.

In equation (87), the second left matrix can be three by factorization in the following manner and promote matrixes:

$\left[\begin{array}{cc}& {\underset{\&OverBar;}{F}}_{N/2}\\ {\underset{\&OverBar;}{F}}_{N/2}& \end{array}\right]=\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}& \\ -\underset{\&OverBar;}{Q}{\underset{\&OverBar;}{F}}_{N/2}& {\underset{\&OverBar;}{I}}_{N/2}\end{array}\right]\left[\begin{array}{cc}-\underset{\&OverBar;}{Q}& {\underset{\&OverBar;}{F}}_{N/2}\\ & {\underset{\&OverBar;}{I}}_{N/2}\end{array}\right]\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}& \\ {\underset{\&OverBar;}{F}}_{N/2}& {\underset{\&OverBar;}{I}}_{N/2}\end{array}\right]---\left(89\right)$

Wherein Q is the permutation matrix that the following exponent number that provides is N/2

$\underset{\&OverBar;}{Q}=\left[\begin{array}{cc}1& 0\\ 0& {\underset{\&OverBar;}{J}}_{N/2-1}\end{array}\right]$

And J _N/2-lMatrix is drawn in the counterclaim that is the N/2-1 rank.

In equation (87), the third left matrix is to oppose the angular moment battle array, and it is only with half component in the input vector and the complex multiplication on unit circle.

This is interpreted as the rotation on the complex plane.

Suppose x=x _r+ jx _rBe the such one-component in the input vector.

In addition, suppose ${{\underset{\&OverBar;}{W}}^k}_N=e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A20048003407600392.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}/N}=\cos (2\mathrm{k\&pi;}/N)-j\sin (2\mathrm{k\&pi;}/N)=c_k-j{s}_k$ Be plural number, promptly WComponent, input vector with equation (87) in the right first matrix and second matrix multiple after, when described input vector with WWhen multiplying each other, W _N ^kMultiply each other with x.

The result is $y=y_r+j{y}_i=W_N^{k}x=(c_k{x}_i+s_k{x}_i)+j(c_k{x}_i-s_k{x}_i),$ This equals x inhour rotation 2k π/N radian on complex plane.This kind rotation can be three lifting step by following factorization:

$\left[\begin{array}{c}{y}_r\\ y_i\end{array}\right]=\left[\begin{array}{cc}{c}_k& s_k\\ -s_k& c_k\end{array}\right]\left[\begin{array}{c}{x}_r\\ x_i\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \frac{c-1}{s}& 1\end{array}\right]\left[\begin{array}{cc}1& s\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ \frac{c-1}{s}& 0\end{array}\right]\left[\begin{array}{c}{x}_i\\ x_i\end{array}\right]---\left(90\right)$

In equation (87), fourth right matrix quilt factorization in the following manner is three lifting matrixes:

$\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}& {\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}\\ {\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}& -{\underset{\&OverBar;}{I}}_{N/2}/\sqrt{2}\end{array}\right]=---\left(91\right)$

$\left[\begin{array}{cc}(\sqrt{2}-1){\underset{\&OverBar;}{I}}_{N/2}& {\underset{\&OverBar;}{I}}_{N/2}\\ {\underset{\&OverBar;}{I}}_{N/2}& \end{array}\right]\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}& -1/\sqrt{2}{\underset{\&OverBar;}{I}}_{N/2}\\ & {\underset{\&OverBar;}{I}}_{N/2}\end{array}\right]\left[\begin{array}{cc}{\underset{\&OverBar;}{I}}_{N/2}& \\ (\sqrt{2}-1){\underset{\&OverBar;}{I}}_{N/2}& {\underset{\&OverBar;}{I}}_{N/2}\end{array}\right]$

Base-2 decimation in frequency fft algorithms only are the transposition of the base-2 decimation in time fft algorithm in the equation (87).

Therefore, said process also can be used for the FFT matrix of factorization decimation in frequency mode F _N

In promoting matrix, use the factorization of the right-hand side in the equation (87),, determine the conversion element by producing the lifting level that promotes matrix corresponding to each.

Because in the above to how to describe in detail, and in this embodiment, the content of this respect is similar to foregoing according to promoting matrix generation lifting level.Therefore, omitted explanation here.

Introduce following document here by reference in this manual:

H.S.Malvar，“Signal?Processing?With?Lapped?Transforms”ArtechHouse，1992；

R.Geiger，T.Sporer，J.Koller，K.Brandenburg，“Audio?Coding?basedon?Integer?Transforms”AES?111 ^th?Convention，New?York，USA，Sept.2001.

Claims (11)

1, process that is used for determining the conversion element of given transforming function transformation function, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, wherein

-described transformation matrix is broken down into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equals permutation matrix and the integer diagonal matrix multiplies each other;

Each of-described rotation matrix and described companion matrix all is broken down into a plurality of lifting matrixes;

-described conversion element is confirmed as comprising a plurality of lifting levels corresponding with described lifting matrix.

2, process as claimed in claim 1, wherein, described transforming function transformation function is DCT-I transforming function transformation function, DCT-IV transforming function transformation function, DST-I transforming function transformation function, DST-IV transforming function transformation function, DFT-I transforming function transformation function, DFT-IV transforming function transformation function, DWT-I transforming function transformation function or DWT-IV transforming function transformation function.

3, process as claimed in claim 1 or 2, wherein, each in the described lifting matrix is the block-tridiagonal matrix that has two reversible INTEGER MATRICES on a diagonal angle.

4, process as claimed in claim 3, wherein, each described reversible INTEGER MATRICES that promotes in the matrix is unit matrix or negative unit matrix.

5, as the described process of any one claim in the claim 1 to 4, wherein, described conversion element comprises that five promote level.

6, as the described process of any one claim in the claim 1 to 5, wherein, sound signal or vision signal are used as described digital signal.

7, equipment that is used for determining the conversion element of given transforming function transformation function, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, described equipment comprises:

-the first resolving cell is used for described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equals permutation matrix and the integer diagonal matrix multiplies each other;

-the second resolving cell, each that is used for described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes;

-determining unit is used for described conversion element is defined as comprising a plurality of lifting levels corresponding with described lifting matrix.

8, a kind ofly be used to use the conversion element that digital signal is transformed from the time domain to frequency domain or the method from the frequency domain transform to the time domain, wherein:

Described conversion element is corresponding to given transforming function transformation function, and this transforming function transformation function comprises transformation matrix, and wherein said conversion element determines that by a process this process comprises

-described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other;

-each of described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes;

-determine that described conversion element comprises a plurality of lifting levels corresponding with described lifting matrix;

-each lifting level comprises utilizes the householder transformation and the unit that rounds up that the sub-piece of described digital signal is handled.

9, a kind of being used for transforms from the time domain to frequency domain or the equipment from the frequency domain transform to the time domain with digital signal, comprises converter unit, and this converter unit utilizes the described digital signal of conversion unit conversion usually, wherein

-described conversion element is corresponding to given transforming function transformation function, and this transforming function transformation function comprises transformation matrix, and wherein said conversion element determines that by a process this process comprises

-described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other,

-each of described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes;

-determine that described conversion element comprises a plurality of lifting levels corresponding with described lifting matrix;

-comprise householder transformation unit that the sub-piece of described digital signal is handled and the unit that rounds up that the sub-piece of described digital signal is handled for each this equipment of lifting level.

10, a kind of computer-readable medium, this computer-readable medium has record program thereon, wherein said program is suitable for making computing machine to carry out the process of the conversion element that is used for definite given transforming function transformation function, this transforming function transformation function comprise transformation matrix and corresponding to digital signal from the time domain to the frequency domain or from the conversion of frequency domain to time domain, wherein

-described transformation matrix is broken down into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equals permutation matrix and the integer diagonal matrix multiplies each other;

Each of-described rotation matrix and described companion matrix all is broken down into a plurality of lifting matrixes;

-described conversion element is confirmed as comprising a plurality of lifting levels corresponding with described lifting matrix.

11, a kind of calculation machine computer-readable recording medium, this computer-readable medium has record program thereon, and wherein said program is suitable for making the computing machine execution to be used to use the conversion element that digital signal is transformed from the time domain to frequency domain or the method from the frequency domain transform to the time domain, wherein

-described conversion element is corresponding to given transforming function transformation function, and this transforming function transformation function comprises transformation matrix, and wherein said conversion element determines that by a process this process comprises

-described transformation matrix is decomposed into rotation matrix and companion matrix, when this companion matrix and when self multiplying each other, equal permutation matrix and the integer diagonal matrix multiplies each other,

-each of described rotation matrix and described companion matrix is decomposed into a plurality of lifting matrixes;

-determine that described conversion element comprises a plurality of lifting levels corresponding with described lifting matrix;

-each lifting level comprises utilizes the householder transformation and the unit that rounds up that the sub-piece of described digital signal is handled.

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