Patent Publication Number: US-9838700-B2

Title: Encoding apparatus, decoding apparatus, and method and program for the same

Description:
TECHNICAL FIELD 
     The present invention relates to a technique for encoding or decoding a time-series signal such as a sound signal. 
     BACKGROUND ART 
     As a method for encoding a sound signal with a low-bit rate (for example, about 10 to 20 kbit/s), adaptive coding for an orthogonal transform coefficient in a frequency domain, such as DFT (Discrete Fourier Transform) and MDCT (Modified Discrete Cosine Transform), is known. For example, MEPG USAC (Unified Speech and Audio Coding), which is a standard technique, has a TCX (transform coded excitation) encoding mode, and, in this mode, MDCT coefficients are normalized for each frame and variable-length encoded after being quantized (see, for example, Non-Patent Literature 1). 
       FIG. 1  shows a configuration example of a conventional TCX-based encoding apparatus. The encoding apparatus in  FIG. 1  is provided with a frequency domain transforming portion  11 , a linear prediction analyzing portion  12 , an amplitude spectral envelope sequence generating portion  13 , an envelope normalizing portion  14  and an encoding portion  15 . Each portion in  FIG. 1  will be described below. 
     &lt;Frequency Domain Transforming Portion  11 &gt; 
     A time domain sound signal is inputted to the frequency domain transforming portion  11 . The sound signal is, for example, a voice signal or an acoustic signal. 
     The frequency domain transforming portion  11  transforms the inputted time domain sound signal to an MDCT coefficient sequence X(0), X(1), . . . , X(N−1) at a point N in a frequency domain for each frame with a predetermined time length. Here, N is a positive integer. 
     The transformed MDCT coefficient sequence X(0), X(1), . . . , X(N−1) is outputted to the envelope normalizing portion  14 . 
     &lt;Linear Prediction Analyzing Portion  12 &gt; 
     A time domain sound signal is inputted to the linear prediction analyzing portion  12 . 
     The linear prediction analyzing portion  12  generates linear prediction coefficients α 1 , α 2 , . . . , α p  by performing linear prediction analysis for a sound signal inputted in frames. Further, the linear prediction analyzing portion  12  encodes the generated linear prediction coefficients α 1 , α 2 , . . . , α p  to generate linear prediction coefficient codes. An example of the linear prediction coefficient code is LSP codes, which are codes corresponding to a sequence of quantized values of an LSP (Line Spectrum Pairs) parameter sequence corresponding to the linear prediction coefficients α 1 , α 2 , . . . , α p . Here, p is a positive integer equal to or larger than 2. 
     Further, the linear prediction analyzing portion  12  generates quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  which are linear prediction coefficients corresponding to the generated linear prediction coefficient codes. 
     The generated quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  are outputted to the amplitude spectral envelope sequence generating portion  13 . Further, the generated linear prediction coefficient codes are outputted to a decoding apparatus. 
     For the linear prediction analysis, for example, a method is used in which linear prediction coefficients are obtained by determining autocorrelation for the sound signal inputted in frames and performing a Levinson-Durbin algorithm using the determined autocorrelation. Otherwise, a method may be used in which linear prediction coefficients are obtained by inputting an MDCT coefficient sequence determined by the frequency domain transforming portion  11  to the linear prediction analyzing portion  12  and performing the Levinson-Durbin algorithm for what is obtained by performing inverse Fourier transform of a sequence of square values of coefficients of the MDCT coefficient sequence. 
     &lt;Amplitude Spectral Envelope Sequence Generating Portion  13 &gt; 
     The quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  generated by the linear prediction analyzing portion  12  are inputted to the amplitude spectral envelope sequence generating portion  13 . 
     The amplitude spectral envelope sequence generating portion  13  generates a smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) defined by the following formula (1) using the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p . In the formula (1), exp(●) indicates an exponential function with a Napier&#39;s constant as a base on the assumption that “●” is a real number, and j indicates an imaginary unit. Further, γ is a positive constant equal to or smaller than 1 and is a coefficient which reduces amplitude unevenness of an amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) defined by the following formula (2), in other words, a coefficient which smoothes the amplitude spectral envelope sequence. 
     
       
         
           
             
               
                 
                   
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     The generated smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) is outputted to the envelope normalizing portion  14 . 
     &lt;Envelope Normalizing Portion  14 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) generated by the frequency domain transforming portion  11  and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) outputted by the amplitude spectral envelope sequence generating portion  13  are inputted to the envelope normalizing portion  14 . 
     The envelope normalizing portion  14  generates a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by normalizing each coefficient X(k) of the MDCT coefficient sequence by a corresponding value ^Wγ(k) of the smoothed amplitude spectral envelope sequence. That is, X N (k)=X(k)/^Wγ(k) [k=0, 1, . . . , N−1] is satisfied. 
     The generated normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) is outputted to the encoding portion  15 . 
     Here, in order to realize such quantization that auditory distortion is reduced, the envelope normalizing portion  14  normalizes the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) in frames, using the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), which is a sequence of a smoothed amplitude spectral envelope. 
     &lt;Encoding Portion  15 &gt; 
     The normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  14  is inputted to the encoding portion  15 . 
     The encoding portion  15  generates codes corresponding to the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1). 
     The generated codes corresponding to normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) are outputted to the decoding apparatus. 
     The encoding portion  15  divides coefficients of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by a gain (global gain) g, and causes codes obtained by encoding a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), which is a sequence of integer values obtained by quantizing results of the division, to be integer signal codes. In a technique of Non-Patent Literature 1, the encoding portion  15  decides such a gain g that the number of bits of the integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible. Then, the encoding portion  15  generates a gain code corresponding to the decided gain g and an integer signal code corresponding to the decided gain g. 
     The generated gain code and integer signal codes are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1). 
     [Specific Example of Encoding Process Performed by Encoding Portion  15 ] 
     A specific example of the encoding process performed by the encoding portion  15  will be described. 
       FIG. 2  shows configuration example of the specific example of the encoding portion  15 . As shown in  FIG. 2 , the encoding portion  15  is provided with a gain acquiring portion  151 , a quantizing portion  152 , a Rice parameter deciding portion  153 , a Golomb-Rice encoding portion  154 , a gain encoding portion  155 , a judging portion  156  and a gain updating portion  157 . 
     Each portion in  FIG. 2  will be described below. 
     &lt;Gain Acquiring Portion  151 &gt; 
     The gain acquiring portion  151  decides such a global gain g that the number of bits of integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible from an inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and outputs the global gain g. The global gain g obtained by the gain acquiring portion  151  becomes an initial value of a global gain used by the quantizing portion  152 . 
     &lt;Quantizing Portion  152 &gt; 
     The quantizing portion  152  obtains and outputs a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) as a sequence of an integer part of a result of dividing each coefficient of the inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g obtained by the gain acquiring portion  151  or the gain updating portion  157 . 
     Here, a global gain g used when the quantizing portion  152  is executed for the first time is the global gain g obtained by the gain acquiring portion  151 , that is, the initial value of the global gain. Further, a global gain g used when the quantizing portion  152  is executed at and after the second time is the global gain g obtained by the gain updating portion  157 , that is, an updated value of the global gain. 
     &lt;Rice Parameter Deciding Portion  153 &gt; 
     The Rice parameter deciding portion  153  obtains and outputs Rice parameters r by the following formula (3) from the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  152 . 
     
       
         
           
             
               
                 
                   
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     It is assumed that “●” indicates an arbitrary number, and [●] indicates a rounding operation for “●”. 
     &lt;Golomb-Rice Encoding Portion  154 &gt; 
     The Golomb-Rice encoding portion  154  performs Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  152 , using the Rice parameters r obtained by the Rice parameter deciding portion  153 , to obtain integer signal codes, and outputs the integer signal codes and the number of consumed bits C, which is the number of bits of the integer signal codes. 
     &lt;Judging Portion  156 &gt; 
     When the number of times of updating the gain is a predetermined number of times, the judging portion  156  outputs the integer signal codes as well as outputting an instruction signal to encode the global gain g obtained by the gain updating portion  157  to the gain encoding portion  155 , and, when the number of times of updating the gain is smaller than the predetermined number of times, the judging portion  156  outputs the number of consumed bits C measured by the Golomb-Rice encoding portion  154  to the gain updating portion  157 . 
     &lt;Gain Updating Portion  157 &gt; 
     When the number of consumed bits C measured by the Golomb-Rice encoding portion  154  is larger than the number of allocated bits B, the gain updating portion  157  updates the value of the global gain g to a larger value and outputs the value. When the number of consumed bits C is smaller than the number of allocated bits B, the gain updating portion  157  updates the value of the global gain g to a smaller value and outputs the updated value of the global gain g. 
     &lt;Gain Encoding Portion  155 &gt; 
     The gain encoding portion  155  encodes the global gain g obtained by the gain updating portion  157  in accordance with the instruction signal outputted by the judging portion  156  to obtain and output a gain code. 
     The integer signal codes outputted by the judging portion  156  and the gain code outputted by the gain encoding portion  155  are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     As described above, in the conventional TCX-based encoding, an MDCT coefficient sequence is normalized with the use of a smoothed amplitude spectral envelope sequence obtained by smoothing an amplitude spectral envelope, and, after that, the normalized MDCT coefficient sequence is encoded. This encoding method is adopted in the MPEG-4 USAC described above. 
     PRIOR ART LITERATURE 
     Non-Patent Literature 
     
         
         Non-patent literature 1: M. Neuendorf, et al., “MPEG Unified Speech and Audio Coding—The ISO/MPEG Standard for High-Efficiency Audio Coding of all Content Types,” AES 132 nd  Convention, Budapest, Hungary, 2012. 
       
    
     SUMMARY OF THE INVENTION 
     Problems to be Solved by the Invention 
     Encoding efficiency of an encoding portion  15  is better when envelope unevenness of an inputted coefficient sequence is as small as possible. In a conventional encoding apparatus, however, since an envelope normalizing portion  14  normalizes an MDCT sequence X(0), X(1), . . . , X(N−1) not by an amplitude spectral envelope sequence but by a smoothed amplitude spectral envelope sequence in order to reduce auditory distortion, a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) inputted to the encoding portion  15  has envelope unevenness though the envelope unevenness is not so large as that of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1). That is, in the conventional encoding apparatus, the encoding efficiency of the encoding portion  15  is bad due to envelope unevenness of the smoothed amplitude spectral envelope sequence. 
     An object of the present invention is to provide an encoding apparatus and a decoding apparatus capable of performing more efficient encoding or decoding than before, and a method and a program for the encoding apparatus and the decoding apparatus. 
     Means to Solve the Problems 
     An encoding apparatus according to an aspect of the present invention is an encoding apparatus for encoding a time-series signal in a frequency domain, the encoding apparatus comprising: a spectral envelope estimating portion estimating a spectral envelope, regarding absolute values of a frequency domain sample sequence corresponding to the time-series signal raised to the power of η as a power spectrum, on the assumption that η is a predetermined positive number other than 2; and an encoding portion performing such encoding that changes bit allocation or that bit allocation substantially changes, for each coefficient of the frequency domain sample sequence corresponding to the time-series signal, based on the estimated spectral envelope. 
     A decoding apparatus according to an aspect of the present invention is a decoding apparatus for obtaining a frequency domain sample sequence corresponding to a time-series signal by decoding in a frequency domain, the decoding apparatus comprising: a linear prediction coefficient decoding portion decoding inputted linear prediction coefficient codes to obtain coefficients transformable linear prediction coefficients; an unsmoothed spectral envelope sequence generating portion obtaining an unsmoothed spectral envelope sequence which is a sequence obtained by raising a sequence of an amplitude spectral envelope corresponding to the coefficients transformable to the linear prediction coefficients to the power of 1/η, on the assumption that η is a predetermined positive number other than 2; and a decoding portion obtaining a frequency domain sample sequence corresponding to the time-series signal by decoding inputted integer signal codes in accordance with such bit allocation that changes or substantially changes based on the unsmoothed spectral envelope sequence. 
     Effects of the Invention 
     It is possible to perform more efficient encoding or decoding than before. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram for illustrating an example of a conventional encoding apparatus; 
         FIG. 2  is a block diagram for illustrating an example of a conventional encoding portion; 
         FIG. 3  is a histogram for illustrating a technical background; 
         FIG. 4  is a block diagram for illustrating an example of an encoding apparatus of the present invention; 
         FIG. 5  is a flowchart for illustrating an example of an encoding method of the present invention; 
         FIG. 6  is a block diagram for illustrating an example of an encoding portion of the present invention; 
         FIG. 7  is a block diagram for illustrating an example of the encoding portion of the present invention; 
         FIG. 8  is a flowchart for illustrating an example of a process of the encoding portion of the present invention; 
         FIG. 9  is a block diagram for illustrating an example of a decoding apparatus of the present invention; 
         FIG. 10  is a flowchart for illustrating an example of a decoding method of the present invention; 
         FIG. 11  is a flowchart for illustrating an example of a process of a decoding portion of the present invention; 
         FIG. 12  is a diagram for illustrating a technical background of the present invention; 
         FIG. 13  is a flowchart for illustrating an example of the encoding method of the present invention; 
         FIG. 14  is a block diagram for illustrating an example of the encoding portion of the present invention; 
         FIG. 15  is a block diagram for illustrating an example of the encoding portion of the present invention; 
         FIG. 16  is a block diagram for illustrating an example of the decoding apparatus of the present invention; and 
         FIG. 17  is a flowchart for illustrating an example of the process of the decoding portion of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Technical Background 
     Normalization of an MDCT sequence X(0), X(1), . . . , X(N−1) by a smoothed amplitude spectral envelope whitens the MDCT sequence X(0), X(1), . . . , X(N−1) less than normalization by an amplitude spectral envelope sequence. Specifically, unevenness of a normalized MDCT coefficient sequence X N (0)=X(0)/^Wγ(0), X N (1)=X(1)/^Wγ(1), . . . , X N (N−1)=X(N−1)/^Wγ(N−1), which is obtained by normalizing an MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by a smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), is larger than unevenness of a normalized sequence X(0)/^W(0), X(1)/^W(1), . . . , X(N−1)/^W(N−1), which is obtained by normalizing the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by an amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), by ^W(0)/^Wγ(0), ^W(1)/^Wγ(1), . . . , ^W(N−1)/^Wγ(N−1). Therefore, when it is assumed that the normalized sequence X(0)/^W(0), X(1)/^W(1), . . . , X(N−1)/^W(N−1), which is obtained by normalizing the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by the amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), is such that envelope unevenness has been smoothed to an extent suitable for encoding by an encoding portion  15 , a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) to be inputted to the encoding portion  15  has envelope unevenness indicated by a sequence of ^W(0)/^Wγ(0), ^W(1)/^Wγ(1), . . . , ^W(N−1)/^Wγ(N−1) (hereinafter referred to as a normalized amplitude spectral envelope sequence ^W N (0), ^W N (1), . . . , ^W N (N−1)) that is left. 
       FIG. 3  shows an appearance frequency of a value of each coefficient comprised in the normalized MDCT coefficient sequence when the envelope unevenness ^W(0)/^Wγ(0), ^W(1)/^Wγ(1), . . . , ^W(N−1)/^Wγ(N−1) of the normalized MDCT sequence takes each value. A curve of envelope: 0.2-0.3 indicates a frequency of a value of a normalized MDCT coefficient X N (k) corresponding to such a sample k that envelope unevenness ^W(k)/^Wγ(k) of the normalized MDCT sequence is equal to or larger than 0.2 and below 0.3. A curve of envelope: 0.3-0.4 indicates a frequency of a value of the normalized MDCT coefficient X N (k) corresponding to such a sample k that the envelope unevenness ^W(k)/^Wγ(k) of the normalized MDCT sequence is equal to or larger than 0.3 and below 0.4. A curve of envelope: 0.4-0.5 indicates a frequency of a value of the normalized MDCT coefficient X N (k) corresponding to such a sample k that the envelope unevenness ^W(k)/^Wγ(k) of the normalized MDCT sequence is equal to or larger than 0.4 and below 0.5. 
     It is seen from  FIG. 3  that, though an average of values of coefficients comprised in the normalized MDCT coefficient sequence is almost zero, variance of the values is relevant to envelope values. That is, the larger the envelope unevenness of the normalized MDCT sequence is, the longer the foot of a curve indicating a frequency is. Therefore, it is seen that the envelope unevenness of the normalized MDCT sequence being large is relevant to variance of the values of the normalized MDCT coefficients being large. In order to realize more efficient compression, encoding utilizing this relevance is performed. Specifically, for each coefficient of a frequency domain sample sequence targeted by encoding, such encoding that changes bit assignment or that bit assignment actually changes, based on a spectral envelope, is performed. 
     For this purpose, for example, (i) in a case of performing Golomb-Rice encoding of a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), Rice parameters decided based on a spectral envelope are used. Further, for example, (ii) in a case of performing arithmetic encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), variance parameters decided based on the spectral envelope are used. 
     First, a technical background in the case of (i) will be described. 
     In a conventional encoding apparatus, Rice parameters used for Golomb-Rice encoding are determined, for example, from a formula (4) below which comprises an average of coefficients comprised in a quantized normalized coefficient sequence, and Golomb-Rice encoding is performed for all coefficients comprised in the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), using the same Rice parameters determined from the formula (4). 
     
       
         
           
             
               
                 
                   
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     It is assumed that “●” indicates an arbitrary number, and [●] indicates a rounding operation for “●”. 
     In comparison, in a second embodiment of the present invention, a Rice parameter for each of coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is calculated from each of values of the normalized amplitude spectral envelope sequence ^W N (0), ^W N (1), . . . ^W N (N−1) corresponding to the coefficients, respectively, and a global gain g by the following formula (5). 
     
       
         
           
             
               
                 
                   
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     In the above formula, σ is a square root of predictive residual energy σ 2 . That is, σ is a positive number. That is, the Rice parameter for each of the coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is an output value in a case of inputting a value of the normalized amplitude spectral envelope corresponding to each of the coefficients to a predetermined monotone non-decreasing function. By doing so, it becomes possible to obtain a Rice parameter suitable for each coefficient without newly adding information to indicate a Rice parameter for each coefficient, and it is possible to increase efficiency of Golomb-Rice encoding. 
     A spectral envelope determined in a method different from the conventional method may be used. Specifically, in a first embodiment of the present invention, a Levinson-Durbin algorithm is performed for what is obtained by performing inverse Fourier transform for a sequence of absolute values of an MDCT coefficients; ^β 1 , ^β 2 , . . . , ^β p  obtained by quantizing linear prediction coefficients obtained thereby are used instead of quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p ; and an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , {circumflex over (α)}H(N−1) and a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) are determined from the following formulas (6) and (7), respectively. 
     
       
         
           
             
               
                 
                   
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                       5 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         H 
                         ^ 
                       
                       γ 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         
                            
                           
                             1 
                             + 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 p 
                               
                               ⁢ 
                               
                                 
                                   
                                     β 
                                     ^ 
                                   
                                   n 
                                 
                                 ⁢ 
                                 
                                   γ 
                                   n 
                                 
                                 ⁢ 
                                 
                                   exp 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         kn 
                                         / 
                                         N 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ^ 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         
                            
                           
                             1 
                             + 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 p 
                               
                               ⁢ 
                               
                                 
                                   
                                     β 
                                     ^ 
                                   
                                   n 
                                 
                                 ⁢ 
                                 
                                   exp 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         kn 
                                         / 
                                         N 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     By dividing each coefficient of the determined unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) by a corresponding coefficient of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), a normalized amplitude spectral envelope sequence ^H N (0)=^H(0)/^Hγ(0), ^H N (1)=^H(1)/^Hγ(1), . . . , ^H N (N−1)=^H(N−1)/^Hγ(N−1) is obtained. Rice parameters are calculated from the normalized amplitude spectral envelope sequence and the global gain g by the following formula (8). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       6 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     max 
                     ( 
                     
                       
                         [ 
                         
                           
                             log 
                             2 
                           
                           ( 
                           
                             
                               
                                 
                                   ( 
                                   
                                     ln 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                               g 
                             
                             ⁢ 
                             
                               
                                 
                                   H 
                                   ^ 
                                 
                                 N 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ] 
                       
                       , 
                       0 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In the formula (8) also, the Rice parameter for each of the coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is the output value in the case of inputting a value of the normalized amplitude spectral envelope corresponding to each of the coefficients to a predetermined monotone non-decreasing function. 
     The above technique is based on a minimization problem based on a code length at the time of performing Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1). Derivation of the above technique will be described below. 
     A code length when Golomb-Rice encoding is performed for each quantized normalized coefficient X Q (k) by a Rice parameter r(k) is represented by the following formula (9) when influence of a rounding error is ignored. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       7 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           r 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         + 
                         
                           
                              
                             
                               
                                 X 
                                 Q 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                           
                           
                             2 
                             
                               r 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     It is assumed that positive and negative signs are separately encoded. Consideration will be made on determining a Rice parameter sequence r(0), r(1), . . . , r(N−1) based on linear prediction coefficients which have been already quantized and encoded, in order to reduce the code length. The above formula (9) can be rewritten as below by performing formula transformation. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       8 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         L 
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 
                                   2 
                                   
                                     r 
                                     ⁡ 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       X 
                                       Q 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                    
                                 
                                 
                                   2 
                                   
                                     r 
                                     ⁡ 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 e 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                        
                                       
                                         
                                           X 
                                           Q 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                        
                                     
                                     
                                       
                                         ( 
                                         
                                           
                                             log 
                                             2 
                                           
                                           ⁢ 
                                           e 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         2 
                                         
                                           r 
                                           ⁡ 
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   - 
                                   
                                     ln 
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           
                                             X 
                                             Q 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                          
                                       
                                       
                                         
                                           ( 
                                           
                                             
                                               log 
                                               2 
                                             
                                             ⁢ 
                                             e 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           2 
                                           
                                             r 
                                             ⁡ 
                                             
                                               ( 
                                               k 
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             N 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     log 
                                     2 
                                   
                                   ⁢ 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 + 
                                 
                                   
                                     log 
                                     2 
                                   
                                   ⁢ 
                                   e 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                
                               
                                 
                                   X 
                                   Q 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 e 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   D 
                                   IS 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             log 
                                             2 
                                           
                                           ⁢ 
                                           e 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         2 
                                         
                                           r 
                                           ⁡ 
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                       
                                     
                                     | 
                                     
                                        
                                       
                                         
                                           X 
                                           Q 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     It is assumed that ln indicates a logarithm with a Napier&#39;s constant as a base, C indicates a constant for the Rice parameters, and D IS (X|Y) indicates an Itakura Saito distance of X from Y 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       9 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       D 
                       IS 
                     
                     ⁡ 
                     
                       ( 
                       
                         X 
                         | 
                         Y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Y 
                       X 
                     
                     - 
                     
                       ln 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Y 
                         X 
                       
                     
                     - 
                     1 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     That is, a minimization problem of a code length L for the Rice parameter sequence comes down to a minimization problem of a sum total of Itakura Saito distances between (log 2  e)2 r(k)  and X Q (k). Here, though it is possible to make an optimization problem for determining linear prediction coefficients to minimize a code length if one of correspondence relationships between the Rice parameter sequence r(0), r(1), . . . , r(N−1) and linear prediction coefficients β 1 , β 2 , . . . , β p  and between the Rice parameter sequence r(0), r(1), . . . , r(N−1) and the predictive residual energy σ 2  is decided, association will be made as shown below here in order to use a conventional faster method. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       10 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       log 
                       2 
                     
                     ( 
                     
                       
                         1 
                         
                           
                             ( 
                             
                               
                                 log 
                                 2 
                               
                               ⁢ 
                               e 
                             
                             ) 
                           
                           ⁢ 
                           g 
                           ⁢ 
                           
                             
                               
                                 H 
                                 ^ 
                               
                               γ 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       · 
                       
                         
                           
                             σ 
                             2 
                           
                           / 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                             ) 
                           
                         
                         
                           
                              
                             
                               1 
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   p 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     β 
                                     n 
                                   
                                   ⁢ 
                                   
                                     exp 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           - 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           kn 
                                           / 
                                           N 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     When influence of quantization is ignored, each quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) can be represented as X Q (k)=X(k)/(g^Hγ(k)) with the use of the MDCT sequence X(0), X(1), . . . , X(N−1), the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) and the global gain g. Therefore, terms depending on the Rice parameters of the formula (10) can be represented as Itakura Saito distances between the absolute values of an MDCT coefficient sequence and an all-pole spectral envelope by the formula (11). 
     
       
         
           
             
               [ 
               
                 Formula 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 11 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     log 
                     2 
                   
                   ⁢ 
                   e 
                 
                 ) 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     D 
                     IS 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             log 
                             2 
                           
                           ⁢ 
                           e 
                         
                         ) 
                       
                       ⁢ 
                       
                         2 
                         
                           r 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             X 
                             Q 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       log 
                       2 
                     
                     ⁢ 
                     e 
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       D 
                       IS 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             1 
                             
                               g 
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   γ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                           · 
                           
                             
                               σ 
                               2 
                             
                             / 
                             
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 ) 
                               
                               
                                 
                                    
                                   
                                     1 
                                     + 
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           1 
                                         
                                         p 
                                       
                                       ⁢ 
                                       
                                         
                                           β 
                                           n 
                                         
                                         ⁢ 
                                         
                                           exp 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 - 
                                                 j 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               2 
                                               ⁢ 
                                               π 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 kn 
                                                 / 
                                                 N 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                         | 
                         
                           
                              
                             
                               X 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                           
                           
                             g 
                             ⁢ 
                             
                               
                                 
                                   H 
                                   ^ 
                                 
                                 γ 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   ( 
                   
                     
                       log 
                       2 
                     
                     ⁢ 
                     e 
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       D 
                       IS 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               σ 
                               2 
                             
                             
                               g 
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   γ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             1 
                             
                               
                                  
                                 
                                   1 
                                   + 
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       p 
                                     
                                     ⁢ 
                                     
                                       
                                         β 
                                         n 
                                       
                                       ⁢ 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                             ⁢ 
                                             π 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               kn 
                                               / 
                                               N 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                         | 
                         
                            
                           
                             X 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Conventional linear prediction analysis, that is, analysis in which the Levinson-Durbin algorithm is applied to what is obtained by performing inverse Fourier transform for a power spectrum is known as an operation of determining a linear prediction coefficient minimizing an Itakura Saito distance between the power spectrum and an all-pole spectral envelope. Therefore, as for the code length minimization problem described above, an optimal solution can be determined by applying the Levinson-Durbin algorithm to an amplitude spectrum, that is, what is obtained by performing inverse Fourier transform for absolute values of an MDCT coefficient sequence, similarly to the conventional method. 
     Next, a technical background in the case of (ii) will be described. 
     Since there are various probability distributions to which encoding targets belong, there is a possibility that, when optimal bit assignment on the assumption of an encoding target belonging to certain probability distribution (for example, Laplace distribution) is performed for an encoding target belonging to probability distribution departing from the assumption, compression efficiency decreases. 
     Therefore, as probability distribution to which an encoding target belongs, generalized Gaussian distribution represented by the following formula, which is distribution capable of expressing various probability distributions, will be used. 
     
       
         
           
             
               [ 
               
                 Formula 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 12 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     f 
                     GG 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         X 
                         | 
                         ϕ 
                       
                       , 
                       η 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       A 
                       ⁡ 
                       
                         ( 
                         η 
                         ) 
                       
                     
                     ϕ 
                   
                   ⁢ 
                   
                     exp 
                     ⁡ 
                     
                       ( 
                       
                         - 
                         
                           
                              
                             
                               
                                 B 
                                 ⁡ 
                                 
                                   ( 
                                   η 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 X 
                                 ϕ 
                               
                             
                              
                           
                           η 
                         
                       
                       ) 
                     
                   
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                 
                   A 
                   ⁡ 
                   
                     ( 
                     η 
                     ) 
                   
                 
                 = 
                 
                   
                     η 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       B 
                       ⁡ 
                       
                         ( 
                         η 
                         ) 
                       
                     
                   
                   
                     2 
                     ⁢ 
                     
                       Γ 
                       ⁡ 
                       
                         ( 
                         
                           1 
                           / 
                           η 
                         
                         ) 
                       
                     
                   
                 
               
               , 
               
                 
                   B 
                   ⁡ 
                   
                     ( 
                     η 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       Γ 
                       ⁡ 
                       
                         ( 
                         
                           3 
                           / 
                           η 
                         
                         ) 
                       
                     
                     
                       Γ 
                       ⁡ 
                       
                         ( 
                         
                           1 
                           / 
                           η 
                         
                         ) 
                       
                     
                   
                 
               
               , 
               
                 
                   Γ 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                   ⁢ 
                   
                     
                       e 
                       
                         - 
                         t 
                       
                     
                     ⁢ 
                     
                       t 
                       
                         
                             
                         
                         ⁢ 
                         
                           x 
                           - 
                           1 
                         
                       
                     
                     ⁢ 
                     dt 
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     By changing a shape parameter (&gt;0), the generalized Gaussian distribution can express various distributions, for example, Laplace distribution at the time of η=1 and Gaussian distribution at the time of η=2 as shown in  FIG. 12 . Here, η is a predetermined number larger than 0, and η may be a predetermined number larger than 0 other than 2. Specifically, η may be a predetermined positive number smaller than 2. The value of η may be decided in advance or may be selected or variable for each frame, which is a predetermined time section. Further, φ in the above formula is a value corresponding to variance of distribution. Information about unevenness of a spectral envelope is incorporated with this value as a variance parameter. That is, variance parameters φ(0), φ(1), . . . , φ(N−1) are generated from a spectral envelope; for the quantized normalized coefficient X Q (k) at each frequency k, such an arithmetic code that becomes optimal when being in accordance with f GG (X|φ(k),η) is configured; and encoding is performed with the arithmetic code based on the configuration. 
     It is assumed below that one shape parameter η has been decided. 
     In a third embodiment of the present invention, distribution information to be used in addition to the information of the predictive residual energy σ 2  and the global gain g is further adopted, and a variance parameter for each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is calculated, for example, by the following formula (A1). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       13 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ϕ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       η 
                       
                         1 
                         / 
                         η 
                       
                     
                     ⁢ 
                     
                       B 
                       ⁡ 
                       
                         ( 
                         η 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           H 
                           ^ 
                         
                         N 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         σ 
                         
                           2 
                           / 
                           η 
                         
                       
                       g 
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     In the above formula, σ is a square root of σ 2 . 
     Specifically, the Levinson-Durbin algorithm is performed for what is obtained by performing inverse Fourier transform for a sequence of values obtained by raising absolute values of MDCT coefficients to the power of η; and, using ^β 1 , ^β 2 , . . . , ^β p , which are obtained by quantizing linear prediction coefficients obtained thereby, instead of the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p , the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) are determined from the following formulas (A2) and (A3), respectively. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       14 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       H 
                       ^ 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           1 
                           
                             
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       n 
                                     
                                     ⁢ 
                                     
                                       exp 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             kn 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                     
                       1 
                       / 
                       η 
                     
                   
                 
               
               
                 
                   ( 
                   A2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         H 
                         ^ 
                       
                       γ 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           1 
                           
                             
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       n 
                                     
                                     ⁢ 
                                     
                                       γ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       exp 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             kn 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                     
                       1 
                       / 
                       η 
                     
                   
                 
               
               
                 
                   ( 
                   A3 
                   ) 
                 
               
             
           
         
       
     
     By dividing each of coefficients of the determined unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) by a corresponding each of coefficients of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), a normalized amplitude spectral envelope sequence ^H N (0)=^H(0)/^Hγ(0), ^H N (1)=^H(1)/^Hγ(1), . . . , ^H N (N−1)=^H(N−1)/^Hγ(N−1) is obtained. From the normalized amplitude spectral envelope sequence and the global gain g, variance parameters are calculated with the above formula (A1). 
     Here, σ 2/η /g in the formula (A1) is a value closely related with entropy, and fluctuation of the value for each frame is small when a bit rate is fixed. Therefore, it is possible to use a predetermined fixed value as σ 2/η /g. In the case of using a fixed value as described above, it is not necessary to newly add information for the method of the present invention. 
     The above technique is based on a minimization problem based on a code length at the time of performing arithmetic encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1). Derivation of the above technique will be described below. 
     When it is assumed that quantization has been performed sufficiently in detail, a code length at the time of encoding each quantized normalized coefficient X Q (k) with an arithmetic code using generalized Gaussian distribution of the shape parameter η by a corresponding variance parameter φ(k) is in proportion to a length represented by the following formula (A4). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       15 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         - 
                         
                           log 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           f 
                           GG 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   X 
                                   Q 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               | 
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             , 
                             η 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   A4 
                   ) 
                 
               
             
           
         
       
     
     Consideration will be made on determining the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) based on linear prediction coefficients which have been already quantized and encoded, in order to reduce the code length. The above formula (A4) can be rewritten as below by performing formula transformation. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       16 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         L 
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 log 
                                 2 
                               
                               ⁢ 
                               e 
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               [ 
                               
                                 
                                   
                                      
                                     
                                       
                                         
                                           B 
                                           ⁡ 
                                           
                                             ( 
                                             η 
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             X 
                                             Q 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                       
                                       
                                         ϕ 
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                   η 
                                 
                                 + 
                                 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     ⁡ 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     A 
                                     ⁡ 
                                     
                                       ( 
                                       η 
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 log 
                                 2 
                               
                               ⁢ 
                               e 
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               [ 
                               
                                 
                                   
                                     1 
                                     η 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                              
                                             
                                               
                                                 X 
                                                 Q 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 k 
                                                 ) 
                                               
                                             
                                              
                                           
                                           η 
                                         
                                         
                                           
                                             
                                               ϕ 
                                               η 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               k 
                                               ) 
                                             
                                           
                                           / 
                                           
                                             ( 
                                             
                                               η 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   B 
                                                   η 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   η 
                                                   ) 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         ln 
                                         ⁢ 
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   X 
                                                   Q 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   k 
                                                   ) 
                                                 
                                               
                                                
                                             
                                             η 
                                           
                                           
                                             
                                               
                                                 ϕ 
                                                 η 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 k 
                                                 ) 
                                               
                                             
                                             / 
                                             
                                               ( 
                                               
                                                 η 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   
                                                     B 
                                                     η 
                                                   
                                                   ⁡ 
                                                   
                                                     ( 
                                                     η 
                                                     ) 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               1 
                               η 
                             
                             ⁢ 
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             η 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 B 
                                 η 
                               
                               ⁡ 
                               
                                 ( 
                                 η 
                                 ) 
                               
                             
                           
                           + 
                           
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   X 
                                   Q 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                             
                           
                           + 
                           
                             1 
                             η 
                           
                           - 
                           
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               ⁡ 
                               
                                 ( 
                                 η 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               
                                 η 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ln 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   D 
                                   IS 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ϕ 
                                           η 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                       
                                         η 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             B 
                                             η 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             η 
                                             ) 
                                           
                                         
                                       
                                     
                                     | 
                                     
                                       
                                          
                                         
                                           
                                             X 
                                             Q 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                          
                                       
                                       η 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   A5 
                   ) 
                 
               
             
           
         
       
     
     It is assumed that ln indicates a logarithm with a Napier&#39;s constant as a base, C indicates a constant for the variance parameters, and D IS (X|Y) indicates an Itakura Saito distance of X from Y 
     
       
         
           
             
               [ 
               
                 Formula 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 17 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 D 
                 IS 
               
               ⁡ 
               
                 ( 
                 
                   X 
                   | 
                   Y 
                 
                 ) 
               
             
             = 
             
               
                 Y 
                 X 
               
               - 
               
                 ln 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   Y 
                   X 
                 
               
               - 
               1 
             
           
         
       
     
     That is, a minimization problem of a code length L for the variance parameter sequence comes down to a minimization problem of a sum total of Itakura Saito distances between φ η (k)/(ηB η (η)) and |X Q (k)| η . Here, though it is possible to make an optimization problem for determining linear prediction coefficients to minimize a code length if one of correspondence relationships between the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) and the linear prediction coefficients β 1 , β 2 , . . . , β p  and between the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) and the predictive residual energy σ 2  is decided, association will be made as shown below in order to use the conventional faster method. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       18 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ϕ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         g 
                         ⁢ 
                         
                           
                             
                               H 
                               ^ 
                             
                             γ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     · 
                     
                       
                         ( 
                         
                           
                             η 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 B 
                                 η 
                               
                               ⁡ 
                               
                                 ( 
                                 η 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 σ 
                                 2 
                               
                               / 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                                 ) 
                               
                             
                           
                           
                             
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       β 
                                       n 
                                     
                                     ⁢ 
                                     
                                       exp 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             kn 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                         ) 
                       
                       
                         1 
                         / 
                         η 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A6 
                   ) 
                 
               
             
           
         
       
     
     When influence of quantization is ignored, each quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) can be represented as X Q (k)=X(k)/(g^Hγ(k)) with the use of the MDCT sequence X(0), X(1), . . . , X(N−1), the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) and the global gain g. Therefore, terms depending on the variance parameters of the formula (A5) are represented as Itakura Saito distances between absolute values of an MDCT coefficient sequence and an all-pole spectral envelope by the formula (A6). 
     
       
         
           
             
               [ 
               
                 Formula 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 19 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 1 
                 
                   η 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ln 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     D 
                     IS 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             ϕ 
                             n 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         
                           η 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               B 
                               η 
                             
                             ⁡ 
                             
                               ( 
                               η 
                               ) 
                             
                           
                         
                       
                       | 
                       
                         
                            
                           
                             
                               X 
                               Q 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                         
                         η 
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 
                   1 
                   
                     η 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ln 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       D 
                       IS 
                     
                     ( 
                     
                       
                         
                           
                             ( 
                             
                               1 
                               
                                 g 
                                 ⁢ 
                                 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                     γ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                           η 
                         
                         · 
                         
                           
                             
                               σ 
                               2 
                             
                             / 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                               ) 
                             
                           
                           
                             
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       β 
                                       n 
                                     
                                     ⁢ 
                                     
                                       exp 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             kn 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       | 
                       
                         
                           
                              
                             
                               X 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                           
                           η 
                         
                         
                           
                             ( 
                             
                               g 
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   γ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           η 
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   1 
                   
                     η 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ln 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       D 
                       IS 
                     
                     ( 
                     
                       
                         
                           
                             σ 
                             2 
                           
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           1 
                           
                             
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       β 
                                       n 
                                     
                                     ⁢ 
                                     
                                       exp 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                           ⁢ 
                                           π 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             kn 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       | 
                       
                         
                            
                           
                             X 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                         
                         η 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Conventional linear prediction analysis, that is, analysis in which the Levinson-Durbin algorithm is applied to what is obtained by performing inverse Fourier transform for a power spectrum is known as an operation of determining a linear prediction coefficient minimizing an Itakura Saito distance between the power spectrum and an all-pole spectral envelope. Therefore, as for the code length minimization problem described above, an optimal solution can be determined by applying the Levinson-Durbin algorithm to an amplitude spectrum raised to the power of η, that is, what is obtained by performing inverse Fourier transform for absolute values of an MDCT coefficient sequence raised to the power of η, similarly to the conventional method. 
     First Embodiment 
     (Encoding of First Embodiment) 
       FIG. 4  shows a configuration example of an encoding apparatus of the first embodiment. As shown in  FIG. 4 , the encoding apparatus of the first embodiment is, for example, provided with a frequency domain transforming portion  21 , a linear prediction analyzing portion  22 , an unsmoothed amplitude spectral envelope sequence generating portion  23 , a smoothed amplitude spectral envelope sequence generating portion  24 , an envelope normalizing portion  25  and an encoding portion  26 .  FIG. 5  shows an example of each process of an encoding method of the first embodiment realized by this encoding apparatus. 
     Each portion in  FIG. 4  will be described below. 
     &lt;Frequency Domain Transforming Portion  21 &gt; 
     A time domain sound signal is inputted to the frequency domain transforming portion  21 . An example of the sound signal is a voice digital signal or an acoustic digital signal. 
     The frequency domain transforming portion  21  transforms the inputted time domain sound signal to an MDCT coefficient sequence X(0), X(1), . . . , X(N−1) at a point N in a frequency domain for each frame with a predetermined time length (step A 1 ). Here, N is a positive integer. 
     The obtained MDCT coefficient sequence X(0), X(1), . . . , X(N−1) is outputted to the linear prediction analyzing portion  22  and the envelope normalizing portion  25 . 
     It is assumed that processes after that are performed for each frame unless otherwise stated. 
     In this way, the frequency domain transforming portion  21  determines a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a sound signal. 
     &lt;Linear Prediction Analyzing Portion  22 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  is inputted to the linear prediction analyzing portion  22 . 
     The linear prediction analyzing portion  22  generates linear prediction coefficients β 1 , β 2 , . . . , β p  by performing linear prediction analysis of  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1) defined by the following formula (12) using the MDCT coefficient sequence X(0), X(1), . . . , X(N−1), and encodes the generated linear prediction coefficients β 1 , β 2 , . . . , β p  to generate linear prediction coefficient codes and quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p , which are quantized linear prediction coefficients corresponding to the linear prediction coefficient codes (step A 2 ). 
     
       
         
           
             
               
                 
                   
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     The generated quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  are outputted to the unsmoothed spectral envelope sequence generating portion  23  and the smoothed amplitude spectral envelope sequence generating portion  24 . 
     Further, the generated linear prediction coefficient codes are transmitted to a decoding apparatus. 
     Specifically, by performing operation corresponding to inverse Fourier transform regarding absolute values of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) as a power spectrum, that is, the operation of the formula (12) first, the linear prediction analyzing portion  22  determines a pseudo correlation function signal sequence  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1), which is a time domain signal sequence corresponding to the MDCT coefficient sequence X(0), X(1), . . . , X(N−1). Then, the linear prediction analyzing portion  22  performs linear prediction analysis using the determined pseudo correlation function signal sequence  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1) to generate linear prediction coefficients β 1 , β 2 , . . . , β p . Then, by encoding the generated linear prediction coefficients β 1 , β 2 , . . . , β p , the linear prediction analyzing portion  22  obtains linear prediction coefficient codes and quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  corresponding the linear prediction coefficient codes. 
     The linear prediction coefficients β 1 , β 2 , . . . , β p  are linear prediction coefficients corresponding to a time domain signal when the absolute values of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) are regarded as a power spectrum. 
     Generation of the linear prediction coefficient codes by the linear prediction analyzing portion  22  is performed, for example, by a conventional encoding technique. As examples of the conventional encoding technique, for example, an encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to an LSP parameter, and a code corresponding to the LSP parameter is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to a PARCOR coefficient, and a code corresponding to the PARCOR coefficient is caused to be a linear prediction coefficient code, and the like are given. For example, the encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code is a technique in which a plurality of quantized linear prediction coefficient candidates are specified in advance; each candidates is stored being associated with a linear prediction coefficient code in advance; any of the candidates is decided as a quantized linear prediction coefficient corresponding to a generated linear prediction coefficient; and, thereby, the quantized linear prediction coefficient and the linear prediction coefficient code are obtained. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  23 &gt; 
     The quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  generated by the linear prediction analyzing portion  22  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  23 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), which is an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step A 3 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) is outputted to the encoding portion  26 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) defined by the following formula (13) as the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) using the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p . 
     
       
         
           
             
               
                 
                   
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     In this way, the unsmoothed amplitude spectral envelope sequence generating portion  23  obtains an unsmoothed amplitude spectral envelope sequence, which is a sequence of an amplitude spectral envelope corresponding to a sound signal. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  24 &gt; 
     The quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  generated by the linear prediction analyzing portion  22  are inputted to the smoothed amplitude spectral envelope sequence generating portion  24 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), which is a sequence obtained by reducing amplitude unevenness of an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step A 4 ). 
     The generated smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) is outputted to the envelope normalizing portion  25  and the encoding portion  26 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) defined by the following formula (14) as the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) using the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  and a correction coefficient γ. 
     
       
         
           
             
               
                 
                   
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     Here, the correction coefficient γ is a predetermined constant smaller than 1 and is a coefficient which reduces amplitude unevenness of the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), in other words, a coefficient which smoothes the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1). 
     In this way, the smoothed amplitude spectral envelope sequence generating portion  24  obtains a smoothed amplitude spectral envelope sequence, which is a sequence obtained by smoothing amplitude unevenness of an unsmoothed amplitude spectral envelope sequence. 
     &lt;Envelope Normalizing Portion  25 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  are inputted to the envelope normalizing portion  25 . 
     The envelope normalizing portion  25  generates a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by normalizing each coefficient of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) (step A 5 ). 
     The generated normalized MDCT coefficient sequence is outputted to the encoding portion  26 . 
     The envelope normalizing portion  25  generates each coefficient X N (k) of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by dividing each coefficient X(k) of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), for example, on the assumption of k=0, 1, . . . , N−1. That is, X N (k)=X(k)/^Hγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     In this way, the envelope normalizing portion  25  normalizes each sample of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, by a corresponding sample of a smoothed amplitude spectral envelope sequence and obtains a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence. 
     &lt;Encoding Portion  26 &gt; 
     The normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25 , the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  23  and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  are inputted to the encoding portion  26 . 
     The encoding portion  26  performs encoding by performing processes of steps A 6 - 1  to A 6 - 5  illustrated in  FIG. 8  (step A 6 ). That is, the encoding portion  26  determines a global gain g corresponding to the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) (step A 6 - 1 ), determines a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), which is a sequence of integer values obtained by quantizing a result of dividing each coefficient of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g (step A 6 - 2 ), determines Rice parameters r(0), r(1), . . . , r(N−1) corresponding to coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), respectively, from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (8) (step A 6 - 3 ), performs Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) using the Rice parameters r(0), r(1), . . . , r(N−1) to obtain integer signal codes (step A 6 - 4 ) and obtains a gain code corresponding to the global gain g (step A 6 - 5 ). 
     Here, the normalized amplitude spectral envelope sequence ^H N (0), ^H N (1), . . . , ^H N (N−1) in the above formula (8) is what is obtained by dividing each value of the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), that is, what is determined by the following formula (15). 
     
       
         
           
             
               
                 
                   
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     The generated integer signal codes and the gain code are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     The encoding portion  26  realizes a function of deciding such a global gain g that the number of bits of the integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible and generating a gain code corresponding to the decided global gain g and integer signal codes corresponding to the decided global gain by the above steps A 6 - 1  to A 6 - 5 . 
     Among steps A 6 - 1  to A 6 - 5  performed by the encoding portion  26 , it is step A 6 - 3  that a characteristic process is comprised. As for the encoding process itself which is for obtaining codes corresponding to the normalized MDCT coefficient sequence by encoding each of the global gain g and the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), various publicly-known techniques comprising the technique described in Non-patent literature 1 exist. Two specific examples of the encoding process performed by the encoding portion  26  will be described below. 
     [Specific Example 1 of Encoding Process Performed by Encoding Portion  26 ] 
     As a specific example 1 of the encoding process performed by the encoding portion  26 , an example which does not include a loop process will be described. 
       FIG. 6  shows a configuration example of the encoding portion  26  of the specific example 1. As shown in  FIG. 6 , the encoding portion  26  of the specific example 1 is, for example, provided with a gain acquiring portion  261 , a quantizing portion  262 , a Rice parameter deciding portion  263 , a Golomb-Rice encoding portion  264  and a gain encoding portion  265 . Each portion in  FIG. 6  will be described below. 
     &lt;Gain Acquiring Portion  261 &gt; 
     The gain acquiring portion  261  decides such a global gain g that the number of bits of integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible from an inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and outputs the global gain g (step S 261 ). For example, the gain acquiring portion  261  acquires and outputs a multiplication value of a square root of the total of energy of the inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and a constant which is in negative correlation with the number of allocated bits B as the global gain g. Otherwise, the gain acquiring portion  261  may tabulate a relationship among the total of the energy of the inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1), the number of allocated bits B and the global gain g in advance, and obtain and output a global gain g by referring to the table. 
     In this way, the gain acquiring portion  261  obtains a gain for performing division of all samples of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence. 
     &lt;Quantizing Portion  262 &gt; 
     The quantizing portion  262  obtains and outputs a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) as a sequence of the integer part of a result of dividing each coefficient of the inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g obtained by the gain acquiring portion  261  (step S 262 ). 
     In this way, the quantizing portion  262  determines a quantized normalized coefficient sequence by dividing each sample of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence by a gain and quantizing the result. 
     &lt;Rice Parameter Deciding Portion  263 &gt; 
     The Rice parameter deciding portion  263  obtains and outputs each Rice parameter of a Rice parameter sequence r(0), r(1), . . . , r(N−1) from the global gain g obtained by the gain acquiring portion  261 , an inputted unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and an inputted smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (8) (step S 263 ). 
     In this way, the Rice parameter deciding portion  263  determines a Rice parameter for performing Golomb-Rice encoding of a quantized normalized coefficient sequence, for each coefficient of the quantized normalized coefficient sequence, based on a smoothed amplitude spectral envelope sequence, an unsmoothed amplitude spectral envelope and a gain. 
     A Rice parameter decision device on the encoding side is a device which is at least provided with the Rice parameter deciding portion  263 . The Rice parameter decision device on the encoding side may be provided with other portions such as the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence generating portion  24 , the envelope normalizing portion  25 , the gain acquiring portion  261  and the quantizing portion  262 . 
     &lt;Golomb-Rice Encoding Portion  264 &gt; 
     The Golomb-Rice encoding portion  264  performs Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  262 , using each Rice parameter of the Rice parameter sequence r(0), r(1), . . . , r(N−1) obtained by the Rice parameter deciding portion  263  as a Rice parameter corresponding to each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), to obtain and output integer signal codes (step S 264 ). 
     &lt;Gain Encoding Portion  265 &gt; 
     The gain encoding portion  265  encodes the global gain g obtained by the gain acquiring portion  261  to obtain and output a gain code (step S 265 ). 
     The generated integer signal codes and gain code are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     That is, steps S 261  to S 265  of the present specific example correspond to the above steps A 6 - 1  to A 6 - 5 , respectively. 
     [Specific Example 2 of Encoding Process Performed by Encoding Portion  26 ] 
     As a specific example 2 of the encoding process performed by the encoding portion  26 , an example which comprises a loop process will be described. 
       FIG. 7  shows a configuration example of the encoding portion  26  of the specific example 2. As shown in  FIG. 7 , the encoding portion  26  of the specific example 2 is, for example, provided with a gain acquiring portion  261 , a quantizing portion  262 , a Rice parameter deciding portion  263 , a Golomb-Rice encoding portion  264 , a gain encoding portion  265 , a judging portion  266  and a gain updating portion  267 . Each portion in  FIG. 7  will be described below. 
     &lt;Gain Acquiring Portion  261 &gt; 
     The gain acquiring portion  261  decides such a global gain g that the number of bits of integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible from an inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and outputs the global gain g (step S 261 ). For example, the gain acquiring portion  261  acquires and outputs a multiplication value of a square root of the total of energy of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and a constant which is in negative correlation with the number of allocated bits B as the global gain g. 
     The global gain g obtained by the gain acquiring portion  261  becomes an initial value of a global gain used by the quantizing portion  262  and the Rice parameter deciding portion  263 . 
     In this way, the gain acquiring portion  261  obtains a gain for performing division of all samples of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence. 
     &lt;Quantizing Portion  262 &gt; 
     The quantizing portion  262  obtains and outputs a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) as a sequence of the integer part of a result of dividing each coefficient of the inputted normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g obtained by the gain acquiring portion  261  or the gain updating portion  267  (step S 262 ). 
     Here, a global gain g used when the process of the quantizing portion  262  is executed for the first time is the global gain g obtained by the gain acquiring portion  261 , that is, the initial value of the global gain. Further, a global gain g used when the process of the quantizing portion  262  is executed at and after the second time is the global gain g obtained by the gain updating portion  267 , that is, an updated value of the global gain. 
     In this way, the quantizing portion  262  determines a quantized normalized coefficient sequence by dividing each sample of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence by a gain and quantizing the result. 
     &lt;Rice Parameter Deciding Portion  263 &gt; 
     The Rice parameter deciding portion  263  obtains and outputs each Rice parameter of a Rice parameter sequence r(0), r(1), . . . , r(N−1) from the global gain g obtained by the gain acquiring portion  261  or the gain updating portion  267 , an inputted unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and an inputted smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (8) (step S 263 ). 
     Here, a global gain g used when the process of the Rice parameter deciding portion  263  is executed for the first time is the global gain g obtained by the gain acquiring portion  261 , that is, the initial value of the global gain. Further, a global gain g used when the process of the Rice parameter deciding portion  263  is executed at and after the second time is the global gain g obtained by the gain updating portion  267 , that is, an updated value of the global gain. 
     In this way, the Rice parameter deciding portion  263  determines Rice parameters for performing Golomb-Rice encoding of a quantized normalized coefficient sequence, for coefficients of the quantized normalized coefficient sequence, respectively, based on a smoothed amplitude spectral envelope sequence, an unsmoothed amplitude spectral envelope and a gain. 
     The Rice parameter decision device on the encoding side is a device which is at least provided with the Rice parameter deciding portion  263 . The Rice parameter decision device on the encoding side may be provided with other portions such as the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence generating portion  24 , the envelope normalizing portion  25 , the gain acquiring portion  261  and the quantizing portion  262 . 
     &lt;Golomb-Rice Encoding Portion  264 &gt; 
     The Golomb-Rice encoding portion  264  performs Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  262 , using each Rice parameter of the Rice parameter sequence r(0), r(1), . . . , r(N−1) obtained by the Rice parameter deciding portion  263  as a Rice parameter corresponding to each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) to obtain integer signal codes, and outputs the integer signal codes and the number of consumed bits C, which is the number of bits of the integer signal codes (step S 264 ). 
     &lt;Judging Portion  266 &gt; 
     The judging portion  266  makes a judgment based on the number of times of updating the gain. Specifically, when the number of times of updating the gain is a predetermined number of times, the judging portion  266  outputs the integer signal codes as well as outputting an instruction signal to encode the global gain g obtained by the gain updating portion  267  to the gain encoding portion  265 , and, when the number of times of updating the gain is smaller than the predetermined number of times, the judging portion  266  outputs the number of consumed bits C measured by the Golomb-Rice encoding portion  264  to the gain updating portion  267  (step S 266 ). 
     &lt;Gain Updating Portion  267 &gt; 
     When the number of consumed bits C measured by the Golomb-Rice encoding portion  264  is larger than the number of allocated bits B, the gain updating portion  267  updates the value of the global gain g to a larger value and outputs the value. When the number of consumed bits C is smaller than the number of allocated bits B, the gain updating portion  267  updates the value of the global gain g to a smaller value and outputs the updated value of the global gain g (step S 267 ). 
     &lt;Gain Encoding Portion  265 &gt; 
     The gain encoding portion  265  encodes the global gain g obtained by the gain updating portion  267  in accordance with the instruction signal outputted by the judging portion  266  to obtain and output a gain code (step S 265 ). 
     The integer signal codes outputted by the judging portion  266  and the gain code outputted by the gain encoding portion  265  are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     That is, in the present specific example, steps S 267 ,  262 ,  263  and  264  performed last correspond to the above steps A 6 - 1  to A 6 - 4 , respectively, and step S 265  corresponds to the above step A 6 - 5 . 
     The specific example 2 of the encoding process performed by the encoding portion  26  is described in more detail in International Publication No. WO2014/054556 and the like. 
     (Decoding of First Embodiment) 
       FIG. 9  shows a configuration example of the decoding apparatus corresponding to the encoding apparatus of the first embodiment. As shown in  FIG. 9 , the decoding apparatus of the first embodiment is, for example, provided with a linear prediction coefficient decoding portion  31 , an unsmoothed amplitude spectral envelope sequence generating portion  32 , a smoothed amplitude spectral envelope sequence generating portion  33 , a decoding portion  34 , an envelope denormalizing portion  35 , and a time domain transforming portion  36 .  FIG. 10  shows an example of each process of a decoding method of the first embodiment realized by this decoding apparatus. 
     At least codes corresponding to a normalized MDCT coefficient sequence and linear prediction coefficient codes outputted by the encoding apparatus are inputted to the decoding apparatus. 
     Each portion in  FIG. 9  will be described below. 
     &lt;Linear Prediction Coefficient Decoding Portion  31 &gt; 
     The linear prediction coefficient codes outputted by the encoding apparatus are inputted to the linear prediction coefficient decoding portion  31 . 
     For each frame, the linear prediction coefficient decoding portion  31  decodes the inputted linear prediction coefficient codes, for example, by a conventional decoding technique to obtain decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step B 1 ). 
     The obtained decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  are outputted to the unsmoothed amplitude spectral envelope sequence generating portion  32  and the smoothed amplitude spectral envelope sequence generating portion  33 . 
     Here, the conventional decoding technique is, for example, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized linear prediction coefficients, the linear prediction coefficient codes are decoded to obtain decoded linear prediction coefficients which are the same as the quantized linear prediction coefficients, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized LSP parameters, the linear prediction coefficient codes are decoded to obtain decoded LSP parameters which are the same as the quantized LS parameters, or the like. Further, the linear prediction coefficients and the LSP parameters are mutually transformable, and it is well known that a transformation process can be performed between the decoded linear prediction coefficients and the decoded LSP parameters according to inputted linear prediction coefficient codes and information required for subsequent processes. From the above, it can be said that what comprises the above linear prediction coefficient code decoding process and the above transformation process performed as necessary is “decoding by the conventional decoding technique”. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  32 &gt; 
     The decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  32 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  32  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), which is an amplitude spectral envelope sequence corresponding to the decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  by the above formula (13) (step B 2 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) is outputted to the decoding portion  34 . 
     In this way, the unsmoothed amplitude spectral envelope sequence generating portion  32  obtains an unsmoothed amplitude spectral envelope sequence, which is a sequence of an amplitude spectral envelope corresponding to linear prediction coefficients corresponding to inputted linear prediction coefficient codes. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  33 &gt; 
     The decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the smoothed amplitude spectral envelope sequence generating portion  33 . 
     The smoothed amplitude spectral envelope sequence generating portion  33  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), which is a sequence obtained by reducing amplitude unevenness of a sequence of an amplitude spectral envelope corresponding to the decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  by the above formula (14) (step B 3 ). 
     The generated smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) is outputted to the decoding portion  34  and the envelope denormalizing portion  35 . 
     In this way, the smoothed amplitude spectral envelope sequence generating portion  33  obtains a smoothed amplitude spectral envelope, which is a sequence obtained by smoothing amplitude unevenness of a sequence of an amplitude spectral envelope corresponding to linear prediction coefficients corresponding to inputted linear prediction coefficient codes. 
     &lt;Decoding Portion  34 &gt; 
     Codes corresponding to the normalized MDCT coefficient sequence outputted by the encoding apparatus, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  32  and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  are inputted to the decoding portion  34 . 
     The decoding portion  34  is provided with a Rice parameter deciding portion  341 . 
     The decoding portion  34  performs decoding by performing processes of steps B 4 - 1  to B 4 - 4  illustrated in  FIG. 11  (step B 4 ). That is, for each frame, the decoding portion  34  decodes a gain code comprised in the codes corresponding to the inputted normalized MDCT coefficient sequence to obtain a global gain g (step B 4 - 1 ). The Rice parameter deciding portion  341  of the decoding portion  34  determines each Rice parameter of a Rice parameter sequence r(0), r(1), . . . , r(N−1) from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (8) (step B 4 - 2 ). The decoding portion  34  obtains a decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by performing Golomb-Rice decoding of integer signal codes comprised in the codes corresponding to the normalized MDCT coefficient sequence using each Rice parameter of the Rice parameter sequence r(0), r(1), . . . , r(N−1) (step B 4 - 3 ) and generates a decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by multiplying each coefficient of the decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by the global gain g (step B 4 - 4 ). 
     The decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) which has been generated is outputted to the envelope denormalizing portion  35 . 
     In this way, the Rice parameter deciding portion  341  determines a Rice parameter for obtaining a decoded normalized coefficient sequence by Golomb-Rice decoding, for each coefficient of the above decoded quantized normalized coefficient sequence, based on a smoothed amplitude spectral envelope sequence, an unsmoothed amplitude spectral envelope sequence and a gain. 
     Further, the decoding portion  34  obtains a decoded normalized frequency domain sample sequence which is, for example, a decoded normalized MDCT coefficient sequence, by multiplying each coefficient of a decoded normalized coefficient sequence obtained by performing Golomb-Rice decoding of inputted integer signal codes by a gain obtained by decoding an inputted gain code. 
     A Rice parameter decision device on the decoding side is a device which is at least provided with the Rice parameter deciding portion  341 . The Rice parameter decision device on the decoding side may be provided with other portions such as the unsmoothed amplitude spectral envelope sequence generating portion  32 . 
     &lt;Envelope Denormalizing Portion  35 &gt; 
     The smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  and the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) generated by the decoding portion  34  are inputted to the envelope denormalizing portion  35 . 
     The envelope denormalizing portion  35  generates a decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by denormalizing the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) using the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) (step B 5 ). 
     The generated decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) is outputted to the time domain transforming portion  36 . 
     For example, the envelope denormalizing portion  35  generates the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by multiplying each coefficient ^X N (k) of the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by each envelope value ^Hγ(k) of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) on the assumption of k=0, 1, . . . , N−1. That is, ^X(k)=^X N (k)×^Hγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     In this way, the envelope denormalizing portion  35  obtains a decoded frequency domain sample sequence which is obtained by multiplying each coefficient of a normalized frequency domain sample sequence which is, for example, a decoded normalized MDCT coefficient sequence, by a corresponding coefficient of a smoothed amplitude spectral envelope sequence. 
     &lt;Time Domain Transforming Portion  36 &gt; 
     The decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) generated by the envelope denormalizing portion  35  is inputted to the time domain transforming portion  36 . 
     For each frame, the time domain transforming portion  36  transforms the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) obtained by the envelope denormalizing portion  35  to a time domain and obtains a sound signal (a decoded sound signal) for each frame (step B 6 ). 
     In this way, the time domain transforming portion  36  obtains a decoded sound signal corresponding to a decoded frequency domain sample sequence which is, for example, a decoded MDCT coefficient sequence. 
     Second Embodiment 
     (Encoding of Second Embodiment) 
       FIG. 4  shows a configuration example of an encoding apparatus of the second embodiment. As shown in  FIG. 4 , the encoding apparatus of the second embodiment is, for example, provided with a frequency domain transforming portion  21 , a linear prediction analyzing portion  22 , an unsmoothed amplitude spectral envelope sequence generating portion  23 , a smoothed amplitude spectral envelope sequence generating portion  24 , an envelope normalizing portion  25  and an encoding portion  26 .  FIG. 5  shows an example of each process of an encoding method of the second embodiment realized by this encoding apparatus. 
     Each portion in  FIG. 4  will be described below. 
     &lt;Frequency Domain Transforming Portion  21 &gt; 
     A time domain sound signal is inputted to the frequency domain transforming portion  21 . An example of the sound signal is a voice digital signal or an acoustic digital signal. 
     The frequency domain transforming portion  21  transforms the inputted time domain sound signal to an MDCT coefficient sequence X(0), X(1), . . . , X(N−1) at a point N in a frequency domain for each frame with a predetermined time length (step A 1 ). Here, N is a positive integer. 
     The obtained MDCT coefficient sequence X(0), X(1), . . . , X(N−1) is outputted to the linear prediction analyzing portion  22  and the envelope normalizing portion  25 . 
     It is assumed that processes after that are performed for each frame unless otherwise stated. 
     In this way, the frequency domain transforming portion  21  determines a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a sound signal. 
     &lt;Linear Prediction Analyzing Portion  22 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  is inputted to the linear prediction analyzing portion  22 . 
     The linear prediction analyzing portion  22  generates linear prediction coefficients α 1 , α 2 , . . . , α p  by performing linear prediction analysis of  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1) defined by the following formula (16) using the MDCT coefficient sequence X(0), X(1), . . . , X(N−1), and encodes the generated linear prediction coefficients α 1 , α 2 , . . . , α p  to generate linear prediction coefficient codes and quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p , which are quantized linear prediction coefficients corresponding to the linear prediction coefficient codes (step A 2 ). 
     
       
         
           
             
               
                 
                   
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     The generated quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  are outputted to the unsmoothed spectral envelope sequence generating portion  23  and the smoothed amplitude spectral envelope sequence generating portion  24 . 
     Further, the generated linear prediction coefficient codes are transmitted to a decoding apparatus. 
     Specifically, the linear prediction analyzing portion  22  determines a pseudo correlation function signal sequence  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1), which is a time domain signal sequence corresponding to the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by performing the operation of the formula (16). Then, the linear prediction analyzing portion  22  performs linear prediction analysis using the determined pseudo correlation function signal sequence  ˜ X(0),  ˜ X(1), . . . ,  ˜ X(N−1) to generate linear prediction coefficients α 1 , α 2 , . . . , α p . Then, by encoding the generated linear prediction coefficients α 1 , α 2 , . . . , α p , the linear prediction analyzing portion  22  obtains linear prediction coefficient codes and quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  corresponding the linear prediction coefficient codes. 
     Generation of the linear prediction coefficient codes by the linear prediction analyzing portion  22  is performed, for example, by a conventional encoding technique. As examples of the conventional encoding technique, for example, an encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to an LSP parameter, and a code corresponding to the LSP parameter is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to a PARCOR coefficient, and a code corresponding to the PARCOR coefficient is caused to be a linear prediction coefficient code, and the like are given. For example, the encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code is a technique in which a plurality of quantized linear prediction coefficient candidates are specified in advance; each candidates is stored being associated with a linear prediction coefficient code in advance; any of the candidates is decided as a quantized linear prediction coefficient corresponding to a generated linear prediction coefficient; and, thereby, the quantized linear prediction coefficient and the linear prediction coefficient code are obtained. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  23 &gt; 
     The quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  generated by the linear prediction analyzing portion  22  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  23 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), which is an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  (step A 3 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) is outputted to the encoding portion  26 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) defined by the following formula (17) as the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) using the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p . 
     
       
         
           
             
               
                 
                   
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     In this way, the unsmoothed amplitude spectral envelope sequence generating portion  23  obtains an unsmoothed amplitude spectral envelope sequence, which is a sequence of an amplitude spectral envelope corresponding to a sound signal. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  24 &gt; 
     The quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  generated by the linear prediction analyzing portion  22  is inputted to the smoothed amplitude spectral envelope sequence generating portion  24 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), which is a sequence obtained by reducing amplitude unevenness of an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  (step A 4 ). 
     The generated smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) is outputted to the envelope normalizing portion  25  and the encoding portion  26 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) defined by the following formula (18) as the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) using the quantized linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  and a correction coefficient γ. 
     
       
         
           
             
               
                 
                   
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     Here, the correction coefficient γ is a predetermined constant smaller than 1 and is a coefficient which reduces amplitude unevenness of the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), in other words, a coefficient which smoothes the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1). 
     In this way, the smoothed amplitude spectral envelope sequence generating portion  24  obtains a smoothed amplitude spectral envelope sequence, which is a sequence obtained by smoothing amplitude unevenness of an unsmoothed amplitude spectral envelope sequence. 
     &lt;Envelope Normalizing Portion  25 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  are inputted to the envelope normalizing portion  25 . 
     The envelope normalizing portion  25  generates a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by normalizing each coefficient of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) (step A 5 ). 
     The generated normalized MDCT coefficient sequence is outputted to the encoding portion  26 . 
     The envelope normalizing portion  25  generates each coefficient X N (k) of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by dividing each coefficient X(k) of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), for example, on the assumption of k=0, 1, . . . , N−1. That is, X N (k)=X(k)/^Wγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     In this way, the envelope normalizing portion  25  normalizes each sample of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, by a corresponding sample of a smoothed amplitude spectral envelope sequence and obtains a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence. 
     &lt;Encoding Portion  26 &gt; 
     The normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25 , the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  23  and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  are inputted to the encoding portion  26 . 
     The encoding portion  26  performs encoding by performing processes of steps A 6 - 1  to A 6 - 5  illustrated in  FIG. 8  (step A 6 ). That is, the encoding portion  26  determines a global gain g corresponding to the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) (step A 6 - 1 ), determines a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), which is a sequence of integer values obtained by quantizing a result of dividing each coefficient of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g (step A 6 - 2 ), determines Rice parameters r(0), r(1), . . . , r(N−1) corresponding to coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), respectively, from the global gain g, the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) by the above formula (5) (step A 6 - 3 ), performs Golomb-Rice encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) using the Rice parameters r(0), r(1), . . . , r(N−1) to obtain an integer signal codes (step A 6 - 4 ) and obtains a gain code corresponding to the global gain g (step A 6 - 5 ). Here, the normalized amplitude spectral envelope sequence ^W N (0), ^W N (1), . . . , ^W N  in the above formula (5) is what is obtained by dividing each value of the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), that is, what is determined by the following formula (19). 
     
       
         
           
             
               
                 
                   
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     The generated integer signal codes and the gain code are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     The encoding portion  26  realizes a function of deciding such a global gain g that the number of bits of the integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible and generating a gain code corresponding to the decided global gain g and integer signal codes corresponding to the decided global gain by the above steps A 6 - 1  to A 6 - 5 . 
     Among steps A 6 - 1  to A 6 - 5  performed by the encoding portion  26 , it is step A 6 - 3  that a characteristic process is comprised. As for the encoding process itself which is for obtaining codes corresponding to the normalized MDCT coefficient sequence by encoding each of the global gain g and the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), various publicly-known techniques comprising the technique described in Non-patent literature 1 exist. 
     A specific example of the encoding process performed by the encoding portion  26  is as described in the description of the encoding portion  26  of the first embodiment. 
     In this way, the Rice parameter deciding portion  263  determines a Rice parameter for performing Golomb-Rice encoding of a quantized normalized coefficient sequence, for each coefficient of the quantized normalized coefficient sequence, based on a smoothed amplitude spectral envelope sequence, an unsmoothed amplitude spectral envelope and a gain. 
     A Rice parameter decision device on the encoding side is a device which is at least provided with the Rice parameter deciding portion  263 . The Rice parameter decision device on the encoding side may be provided with other portions such as the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence generating portion  24 , the envelope normalizing portion  25 , the gain acquiring portion  261  and the quantizing portion  262 . 
     (Decoding of Second Embodiment) 
       FIG. 9  shows a configuration example of the decoding apparatus corresponding to the encoding apparatus of the second embodiment. As shown in  FIG. 9 , the decoding apparatus of the second embodiment is, for example, provided with a linear prediction coefficient decoding portion  31 , an unsmoothed amplitude spectral envelope sequence generating portion  32 , a smoothed amplitude spectral envelope sequence generating portion  33 , a decoding portion  34 , an envelope denormalizing portion  35 , and a time domain transforming portion  36 .  FIG. 10  shows an example of each process of a decoding method of the second embodiment realized by this decoding apparatus. 
     At least codes corresponding to a normalized MDCT coefficient sequence and linear prediction coefficient codes outputted by the encoding apparatus are inputted to the decoding apparatus. 
     Each portion in  FIG. 9  will be described below. 
     &lt;Linear Prediction Coefficient Decoding Portion  31 &gt; 
     The linear prediction coefficient codes outputted by the encoding apparatus are inputted to the linear prediction coefficient decoding portion  31 . 
     For each frame, the linear prediction coefficient decoding portion  31  decodes the inputted linear prediction coefficient codes, for example, by a conventional decoding technique to obtain decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  (step B 1 ). 
     The obtained decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  are outputted to the unsmoothed amplitude spectral envelope sequence generating portion  32  and the smoothed amplitude spectral envelope sequence generating portion  33 . 
     Here, the conventional decoding technique is, for example, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized linear prediction coefficients, the linear prediction coefficient codes are decoded to obtain decoded linear prediction coefficients which are the same as the quantized linear prediction coefficients, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized LSP parameters, the linear prediction coefficient codes are decoded to obtain decoded LSP parameters which are the same as the quantized LS parameters, or the like. Further, the linear prediction coefficients and the LSP parameters are mutually transformable, and it is well known that a transformation process can be performed between the decoded linear prediction coefficients and the decoded LSP parameters according to inputted linear prediction coefficient codes and information required for subsequent processes. From the above, it can be said that what comprises the above linear prediction coefficient code decoding process and the above transformation process performed as necessary is “decoding by the conventional decoding technique”. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  32 &gt; 
     The decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  32 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  32  generates an unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1), which is an amplitude spectral envelope sequence corresponding to the decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  by the above formula (17) (step B 2 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) is outputted to the decoding portion  34 . 
     In this way, the unsmoothed amplitude spectral envelope sequence generating portion  32  obtains an unsmoothed amplitude spectral envelope sequence, which is a sequence of an amplitude spectral envelope corresponding to linear prediction coefficients corresponding to inputted linear prediction coefficient codes. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  33 &gt; 
     The decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the smoothed amplitude spectral envelope sequence generating portion  33 . 
     The smoothed amplitude spectral envelope sequence generating portion  33  generates a smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1), which is a sequence obtained by reducing amplitude unevenness of a sequence of an amplitude spectral envelope corresponding to the decoded linear prediction coefficients ^α 1 , ^α 2 , . . . , ^α p  by the above formula (18) (step B 3 ). 
     The generated smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) is outputted to the decoding portion  34  and the envelope denormalizing portion  35 . 
     In this way, the smoothed amplitude spectral envelope sequence generating portion  33  obtains a smoothed amplitude spectral envelope, which is a sequence obtained by smoothing amplitude unevenness of a sequence of an amplitude spectral envelope corresponding to linear prediction coefficients corresponding to inputted linear prediction coefficient codes. 
     &lt;Decoding Portion  34 &gt; 
     Codes corresponding to the normalized MDCT coefficient sequence outputted by the encoding apparatus, the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  32  and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  are inputted to the decoding portion  34 . 
     The decoding portion  34  is provided with a Rice parameter deciding portion  341 . 
     The decoding portion  34  performs decoding by performing processes of steps B 4 - 1  to B 4 - 4  illustrated in  FIG. 11  (step B 4 ). That is, for each frame, the decoding portion  34  decodes a gain code comprised in the codes corresponding to the inputted normalized MDCT coefficient sequence to obtain a global gain g (step B 4 - 1 ). The Rice parameter deciding portion  341  of the decoding portion  34  determines each Rice parameter of a Rice parameter sequence r(0), r(1), . . . , r(N−1) from the global gain g, the unsmoothed amplitude spectral envelope sequence ^W(0), ^W(1), . . . , ^W(N−1) and the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) by the above formula (5) (step B 4 - 2 ). The decoding portion  34  obtains a decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by performing Golomb-Rice decoding of integer signal codes comprised in the codes corresponding to the normalized MDCT coefficient sequence using each Rice parameter of the Rice parameter sequence r(0), r(1), . . . , r(N−1) (step B 4 - 3 ) and generates a decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by multiplying each coefficient of the decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by the global gain g (step B 4 - 4 ). 
     The decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) which has been generated is outputted to the envelope denormalizing portion  35 . 
     In this way, the Rice parameter deciding portion  341  determines a Rice parameter for obtaining a decoded normalized coefficient sequence by Golomb-Rice decoding, for each coefficient of the above decoded quantized normalized coefficient sequence, based on a smoothed amplitude spectral envelope sequence, an unsmoothed amplitude spectral envelope and a gain. 
     Further, the decoding portion  34  obtains a decoded normalized frequency domain sample sequence which is, for example, a decoded normalized MDCT coefficient sequence, by multiplying each coefficient of a decoded normalized coefficient sequence obtained by performing Golomb-Rice decoding of inputted integer signal codes by a gain obtained by decoding an inputted gain code. 
     A Rice parameter decision device on the decoding side is a device which is at least provided with the Rice parameter deciding portion  341 . The Rice parameter decision device on the decoding side may be provided with other portions such as the unsmoothed amplitude spectral envelope sequence generating portion  32 . 
     &lt;Envelope Denormalizing Portion  35 &gt; 
     The smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  and the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) generated by the decoding portion  34  are inputted to the envelope denormalizing portion  35 . 
     The envelope denormalizing portion  35  generates a decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by denormalizing the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) using the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) (step B 5 ). 
     The generated decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) is outputted to the time domain transforming portion  36 . 
     For example, the envelope denormalizing portion  35  generates the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by multiplying each coefficient ^X N (k) of the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by each envelope value ^Wγ(k) of the smoothed amplitude spectral envelope sequence ^Wγ(0), ^Wγ(1), . . . , ^Wγ(N−1) on the assumption of k=0, 1, . . . , N−1. That is, ^X(k)=^X N (k)×^Wγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     In this way, the envelope denormalizing portion  35  obtains a decoded frequency domain sample sequence which is obtained by multiplying each coefficient of a normalized frequency domain sample sequence which is, for example, a decoded normalized MDCT coefficient sequence, by a corresponding coefficient of a smoothed amplitude spectral envelope sequence. 
     &lt;Time Domain Transforming Portion  36 &gt; 
     The decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) generated by the envelope denormalizing portion  35  is inputted to the time domain transforming portion  36 . 
     For each frame, the time domain transforming portion  36  transforms the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) obtained by the envelope denormalizing portion  35  to a time domain and obtains a sound signal (a decoded sound signal) for each frame (step B 6 ). In this way, the time domain transforming portion  36  obtains a decoded sound signal corresponding to a decoded frequency domain sample sequence which is, for example, a decoded MDCT coefficient sequence. 
     Third Embodiment 
     (Encoding of Third Embodiment) 
       FIG. 4  shows a configuration example of an encoding apparatus of the third embodiment. As shown in  FIG. 4 , the encoding apparatus of the third embodiment is, for example, provided with a frequency domain transforming portion  21 , a linear prediction analyzing portion  22 , an unsmoothed amplitude spectral envelope sequence generating portion  23 , a smoothed amplitude spectral envelope sequence generating portion  24 , an envelope normalizing portion  25  and an encoding portion  26 .  FIG. 5  shows an example of each process of an encoding method of the third embodiment realized by this encoding apparatus. 
     Each portion in  FIG. 4  will be described below. 
     &lt;Frequency Domain Transforming Portion  21 &gt; 
     A sound signal, which is a time-series signal in a time domain, is inputted to the frequency domain transforming portion  21 . An example of the sound signal is a voice digital signal or an acoustic digital signal. 
     The frequency domain transforming portion  21  transforms the inputted time domain sound signal to an MDCT coefficient sequence X(0), X(1), . . . , X(N−1) at a point N in a frequency domain for each frame with a predetermined time length (step A 1 ). Here, N is a positive integer. 
     The obtained MDCT coefficient sequence X(0), X(1), . . . , X(N−1) is outputted to the linear prediction analyzing portion  22  and the envelope normalizing portion  25 . 
     It is assumed that processes after that are performed for each frame unless otherwise stated. 
     In this way, the frequency domain transforming portion  21  determines a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a sound signal. 
     &lt;Linear Prediction Analyzing Portion  22 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  is inputted to the linear prediction analyzing portion  22 . 
     The linear prediction analyzing portion  22  generates linear prediction coefficients β 1 , β 2 , . . . , β p  by performing linear prediction analysis of  ˜ R(0),  ˜ R(1), . . . ,  ˜ R(N−1) defined by the following formula (A 7 ) using the MDCT coefficient sequence X(0), X(1), . . . , X(N−1), and encodes the generated linear prediction coefficients β 1 , β 2 , . . . , β p  to generate linear prediction coefficient codes and quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p , which are quantized linear prediction coefficients corresponding to the linear prediction coefficient codes (step A 2 ). 
     
       
         
           
             
               
                 
                   
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     In the above formula, is a shape parameter, which is decided in advance. For example, η is assumed to be a predetermined integer except 2. 
     The generated quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  are outputted to the unsmoothed spectral envelope sequence generating portion  23  and the smoothed amplitude spectral envelope sequence generating portion  24 . During the linear prediction analysis process, predictive residual energy σ 2  is calculated. In this case, the calculated predictive residual energy σ 2  is outputted to a variance parameter deciding portion  268 . 
     Further, the generated linear prediction coefficient codes are transmitted to a decoding apparatus. 
     Specifically, by performing operation corresponding to inverse Fourier transform regarding absolute values of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) raised to the power of η as a power spectrum, that is, the operation of the formula (A 7 ) first, the linear prediction analyzing portion  22  determines a pseudo correlation function signal sequence  ˜ R(0),  ˜ R(1), . . . ,  ˜ R(N−1), which is a time domain signal sequence corresponding to the absolute values of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) raised to the power of η. Then, the linear prediction analyzing portion  22  performs linear prediction analysis using the determined pseudo correlation function signal sequence  ˜ R(0),  ˜ R(1), . . . ,  ˜ R(N−1) to generate linear prediction coefficients β 1 , β 2 , . . . , β p . Then, by encoding the generated linear prediction coefficients β 1 , β 2 , . . . , β p , the linear prediction analyzing portion  22  obtains linear prediction coefficient codes and quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  corresponding the linear prediction coefficient codes. 
     The linear prediction coefficients β 1 , β 2 , . . . , β p  are linear prediction coefficients corresponding to a time domain signal when the absolute values of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) to the power of are regarded as a power spectrum. 
     Generation of the linear prediction coefficient codes by the linear prediction analyzing portion  22  is performed, for example, by a conventional encoding technique. As examples of the conventional encoding technique, for example, an encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to an LSP parameter, and a code corresponding to the LSP parameter is caused to be a linear prediction coefficient code, an encoding technique in which a linear prediction coefficient is transformed to a PARCOR coefficient, and a code corresponding to the PARCOR coefficient is caused to be a linear prediction code, and the like are given. For example, the encoding technique in which a code corresponding to a linear prediction coefficient itself is caused to be a linear prediction coefficient code is a technique in which a plurality of quantized linear prediction coefficient candidates are specified in advance; each candidates is stored being associated with a linear prediction coefficient code in advance; any of the candidates is decided as a quantized linear prediction coefficient corresponding to a generated linear prediction coefficient; and, thereby, the quantized linear prediction coefficient and the linear prediction coefficient code are obtained. 
     In this way, the linear prediction analyzing portion  22  performs linear prediction analysis using a pseudo correlation function signal sequence obtained by performing inverse Fourier transform regarding absolute values of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, raised to the power of as a power spectrum, and generates coefficients transformable to linear prediction coefficients. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  23 &gt; 
     The quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  generated by the linear prediction analyzing portion  22  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  23 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), which is an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step A 3 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) is outputted to the encoding portion  26 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  23  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) defined by the following formula (A 2 ) as the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) using the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p . 
     
       
         
           
             
               
                 
                   
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     In this way, the unsmoothed amplitude spectral envelope sequence generating portion  23  performs spectral envelope estimation by obtaining an unsmoothed spectral envelope sequence, which is a sequence obtained by raising a sequence of an amplitude spectral envelope corresponding to coefficients transformable to linear prediction coefficients generated by the linear prediction analyzing portion  22  to the power of 1/η. Here, when c is assumed to be an arbitrary number, a sequence obtained by raising a sequence constituted by a plurality of values to the power of c refers to a sequence constituted by values each of which is obtained by raising each of the plurality of values to the power of c. For example, a sequence obtained by raising a sequence of an amplitude spectral envelope to the power of 1/η refers to a sequence constituted by values obtained by raising each coefficient of the amplitude spectral envelope to the power of 1/η. 
     The process of raise to the power of 1/η by the unsmoothed amplitude spectral envelope sequence generating portion  23  is due to the process performed by the linear prediction analyzing portion  22  in which absolute values of a frequency domain sample sequence raised to the power of η are regarded as a power spectrum. That is, the process of raise to the power of 1/η by the unsmoothed amplitude spectral envelope sequence generating portion  23  is performed in order to return values raised to the power of by the process performed by the linear prediction analyzing portion  22  in which absolute values of a frequency domain sample sequence raised to the power of η are regarded as a power spectrum, to the original values. 
     Though Hγ(k) [k=0, 1, . . . , N−1] defined by the formula (13) is used in the first and second embodiments, Hγ(k) [k=0, 1, . . . , N−1] defined by the formula (A 2 ) is used in the third embodiment. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  24 &gt; 
     The quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  generated by the linear prediction analyzing portion  22  are inputted to the smoothed amplitude spectral envelope sequence generating portion  24 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), which is a sequence obtained by reducing amplitude unevenness of an amplitude spectral envelope sequence corresponding to the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step A 4 ). 
     The generated smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) is outputted to the envelope normalizing portion  25  and the encoding portion  26 . 
     The smoothed amplitude spectral envelope sequence generating portion  24  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) defined by the following formula (A 3 ) as the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) using the quantized linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  and a correction coefficient γ. 
     
       
         
           
             
               
                 
                   
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                       η 
                     
                   
                 
               
               
                 
                   ( 
                   A3 
                   ) 
                 
               
             
           
         
       
     
     Here, the correction coefficient γ is a predetermined constant smaller than 1 and is a coefficient which reduces amplitude unevenness of the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), in other words, a coefficient which smoothes the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1). 
     Though ^Hγ(k) [k=0, 1, . . . , N−1] defined by the formula (14) is used in the first and second embodiments, ^Hγ(k) [k=0, 1, . . . , N−1] defined by the formula (A 3 ) is used in the third embodiment. 
     &lt;Envelope Normalizing Portion  25 &gt; 
     The MDCT coefficient sequence X(0), X(1), . . . , X(N−1) obtained by the frequency domain transforming portion  21  and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  are inputted to the envelope normalizing portion  25 . 
     The envelope normalizing portion  25  generates a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by normalizing each coefficient of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) (step A 5 ). 
     The generated normalized MDCT coefficient sequence is outputted to the encoding portion  26 . 
     The envelope normalizing portion  25  generates each coefficient X N (k) of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by dividing each coefficient X(k) of the MDCT coefficient sequence X(0), X(1), . . . , X(N−1) by the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), for example, on the assumption of k=0, 1, . . . , N−1. That is, X N (k)=X(k)/^Hγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     &lt;Encoding Portion  26 &gt; 
     The normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25 , the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24  and average residual energy σ 2  calculated by the linear prediction analyzing portion  22  are inputted to the encoding portion  26 . 
     The encoding portion  26  performs encoding, for example, by performing processes of steps A 61  to A 65  shown in  FIG. 13  (step A 6 ). 
     The encoding portion  26  determines a global gain g corresponding to the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) (step A 61 ), determines a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), which is a sequence of integer values obtained by quantizing a result of dividing each coefficient of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g (step A 62 ), determines variance parameters φ(0), φ(1), . . . , φ(N−1) corresponding to coefficients of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), respectively, from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) and the average residual energy σ 2  by the above formula (A 1 ) (step A 63 ), performs arithmetic encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) using the variance parameters φ(0), φ(1), . . . , φ(N−1) to obtain an integer signal codes (step A 64 ) and obtains a gain code corresponding to the global gain g (step A 65 ). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       31 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ϕ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       η 
                       
                         1 
                         / 
                         η 
                       
                     
                     ⁢ 
                     
                       B 
                       ⁡ 
                       
                         ( 
                         η 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           H 
                           ^ 
                         
                         N 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         σ 
                         
                           2 
                           / 
                           η 
                         
                       
                       g 
                     
                   
                 
               
               
                 
                   ( 
                   A1 
                   ) 
                 
               
             
           
         
       
     
     Here, the normalized amplitude spectral envelope sequence ^H N (0), ^H N (1), . . . , ^H N (N−1) in the above formula (A 1 ) is what is obtained by dividing each value of the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) by a corresponding value of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), that is, what is determined by the following formula (A 8 ). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       32 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           H 
                           ^ 
                         
                         N 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           H 
                           ^ 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       
                         
                           
                             H 
                             ^ 
                           
                           γ 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     N 
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   A8 
                   ) 
                 
               
             
           
         
       
     
     The generated integer signal codes and the gain code are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     The encoding portion  26  realizes a function of deciding such a global gain g that the number of bits of the integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible and generating a gain code corresponding to the decided global gain g and integer signal codes corresponding to the decided global gain by the above steps A 61  to A 65 . 
     Among steps A 61  to A 65  performed by the encoding portion  26 , it is step A 63  that a characteristic process is comprised. As for the encoding process itself which is for obtaining codes corresponding to the normalized MDCT coefficient sequence by encoding each of the global gain g and the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), various publicly-known techniques comprising the technique described in Non-patent literature 1 exist. Two specific examples of the encoding process performed by the encoding portion  26  will be described below. 
     [Specific Example 1 of Encoding Process Performed by Encoding Portion  26 ] 
     As a specific example 1 of the encoding process performed by the encoding portion  26 , an example which does not include a loop process will be described. 
       FIG. 14  shows a configuration example of the encoding portion  26  of the specific example 1. As shown in  FIG. 14 , the encoding portion  26  of the specific example 1 is, for example, provided with a gain acquiring portion  261 , a quantizing portion  262 , a variance parameter deciding portion  268 , an arithmetic encoding portion  269  and a gain encoding portion  265 . Each portion in  FIG. 14  will be described below. 
     &lt;Gain Acquiring Portion  261 &gt; 
     A normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25  is inputted to the gain acquiring portion  261 . 
     The gain acquiring portion  261  decides such a global gain g that the number of bits of integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible from the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and outputs the global gain g (step S 261 ). For example, the gain acquiring portion  261  acquires and outputs a multiplication value of a square root of the total of energy of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and a constant which is in negative correlation with the number of allocated bits B as the global gain g. Otherwise, the gain acquiring portion  261  may tabulate a relationship among the total of energy of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1), the number of allocated bits B and the global gain g in advance, and obtain and output a global gain g by referring to the table. 
     In this way, the gain acquiring portion  261  obtains a gain for performing division of all samples of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence. 
     The obtained global gain g is outputted to the quantizing portion  262  and the variance parameter deciding portion  268 . 
     &lt;Quantizing Portion  262 &gt; 
     A normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25  and the global gain g obtained by the gain acquiring portion  261  are inputted to the quantizing portion  262 . 
     The quantizing portion  262  obtains and outputs a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) as a sequence of the integer part of a result of dividing each coefficient of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g (step S 262 ). 
     In this way, the quantizing portion  262  determines a quantized normalized coefficient sequence by dividing each sample of a normalized frequency domain sample sequence which is, for example, a normalized MDCT coefficient sequence by a gain and quantizing the result. 
     The obtained quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is outputted to the arithmetic encoding portion  269 . 
     &lt;Variance Parameter Deciding Portion  268 &gt; 
     The global gain g obtained by the gain acquiring portion  261 , the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24 , and predictive residual energy σ 2  obtained by the linear prediction analyzing portion  22  are inputted to the variance parameter deciding portion  268 . 
     The variance parameter deciding portion  268  obtains and outputs each variance parameter of a variance parameter sequence φ(0), φ(1), . . . , φ(N−1) from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) and the predictive residual energy &amp; by the above formulas (A 1 ) and (A 8 ) (step S 268 ). 
     The obtained variance parameter sequence φ(0), φ(1), . . . , φ(N−1) is outputted to the arithmetic encoding portion  269 . 
     &lt;Arithmetic Encoding Portion  269 &gt; 
     The quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  262  and the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) obtained by the variance parameter deciding portion  268  are inputted to the arithmetic encoding portion  269 . 
     The arithmetic encoding portion  269  performs arithmetic encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) using each variance parameter of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) as a variance parameter corresponding to each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) to obtain and output integer signal codes (step S 269 ). 
     At the time of performing arithmetic encoding, the arithmetic encoding portion  269  configures such arithmetic codes that each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) becomes optimal when being in accordance with generalized Gaussian distribution f GG (X|φ(k),η) and performs encoding with arithmetic codes based on this configuration. As a result, an expected value of bit allocation to each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is decided with the variance parameter sequence φ(0), φ(1), . . . , φ(N−1). 
     The obtained integer signal codes are outputted to the decoding apparatus. 
     Arithmetic encoding may be performed over a plurality of coefficients in the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1). In this case, since each variance parameter of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) is based on the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) as seen from the formulas (A 1 ) and (A 8 ), it can be said that the arithmetic encoding portion  269  performs such encoding that bit allocation substantially changes based on an estimated spectral envelope (an unsmoothed amplitude spectral envelope). 
     &lt;Gain Encoding Portion  265 &gt; 
     The global gain g obtained by the gain acquiring portion  261  is inputted to the gain encoding portion  265 . 
     The gain encoding portion  265  encodes the global gain g to obtain and output a gain code (step S 265 ). 
     The generated integer signal codes and the gain code are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     Steps S 261 , S 262 , S 268 , S 269  and S 265  of the present specific example 1 correspond to the above steps A 61 , A 62 , A 63 , A 64  and A 65 , respectively. 
     [Specific Example 2 of Encoding Process Performed by Encoding Portion  26 ] 
     As a specific example 2 of the encoding process performed by the encoding portion  26 , an example which comprises a loop process will be described. 
       FIG. 15  shows a configuration example of the encoding portion  26  of the specific example 2. As shown in  FIG. 15 , the encoding portion  26  of the specific example 2 is, for example, provided with a gain acquiring portion  261 , a quantizing portion  262 , a variance parameter deciding portion  268 , an arithmetic encoding portion  269 , a gain encoding portion  265 , a judging portion  266  and a gain updating portion  267 . Each portion in  FIG. 15  will be described below. 
     &lt;Gain Acquiring Portion  261 &gt; 
     A normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25  is inputted to the gain acquiring portion  261 . 
     The gain acquiring portion  261  decides such a global gain g that the number of bits of integer signal codes is equal to or smaller than the number of allocated bits B, which is the number of bits allocated in advance, and is as large as possible from the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and outputs the global gain g (step S 261 ). For example, the gain acquiring portion  261  acquires and outputs a multiplication value of a square root of the total of energy of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) and a constant which is in negative correlation with the number of allocated bits B as the global gain g. 
     The obtained global gain g is outputted to the quantizing portion  262  and the variance parameter deciding portion  268 . 
     The global gain g obtained by the gain acquiring portion  261  becomes an initial value of a global gain used by the quantizing portion  262  and the variance parameter deciding portion  268 . 
     &lt;Quantizing Portion  262 &gt; 
     A normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) generated by the envelope normalizing portion  25  and the global gain g obtained by the gain acquiring portion  261  or the gain updating portion  267  are inputted to the quantizing portion  262 . 
     The quantizing portion  262  obtains and outputs a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) as a sequence of the integer part of a result of dividing each coefficient of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g (step S 262 ). 
     Here, a global gain g used when the quantizing portion  262  is executed for the first time is the global gain g obtained by the gain acquiring portion  261 , that is, the initial value of the global gain. Further, a global gain g used when the quantizing portion  262  is executed at and after the second time is the global gain g obtained by the gain updating portion  267 , that is, an updated value of the global gain. 
     The obtained quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) is outputted to the arithmetic encoding portion  269 . 
     &lt;Variance Parameter Deciding Portion  268 &gt; 
     The global gain g obtained by the gain acquiring portion  261  or the gain updating portion  267 , the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  23 , the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  24 , and predictive residual energy σ 2  obtained by the linear prediction analyzing portion  22  are inputted to the variance parameter deciding portion  268 . 
     The variance parameter deciding portion  268  obtains and outputs each variance parameter of a variance parameter sequence φ(0), φ(1), . . . , φ(N−1) from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) and the predictive residual energy σ 2  by the above formulas (A 1 ) and (A 8 ) (step S 268 ). 
     Here, a global gain g used when the variance parameter deciding portion  268  is executed for the first time is the global gain g obtained by the gain acquiring portion  261 , that is, the initial value of the global gain. Further, a global gain g used when the variance parameter deciding portion  268  is executed at and after the second time is the global gain g obtained by the gain updating portion  267 , that is, an updated value of the global gain. 
     The obtained variance parameter sequence φ(0), φ(1), . . . , φ(N−1) is outputted to the arithmetic encoding portion  269 . 
     &lt;Arithmetic Encoding Portion  269 &gt; 
     The quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) obtained by the quantizing portion  262  and the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) obtained by the variance parameter deciding portion  268  are inputted to the arithmetic encoding portion  269 . 
     The arithmetic encoding portion  269  performs arithmetic encoding of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) using each variance parameter of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) as a variance parameter corresponding to each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) to obtain and output integer signal codes and the number of consumed bits C, which is the number of bits of the integer signal codes (step S 269 ). 
     At the time of performing arithmetic encoding, the arithmetic encoding portion  269  performs such bit allocation that that each coefficient of the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1) becomes optimal when being in accordance with the generalized Gaussian distribution f GG (X|φ(k),η) by performing arithmetic encoding and performs encoding with arithmetic codes based on the performed bit allocation. 
     The obtained integer signal codes and the number of consumed bits C are outputted to the judging portion  266 . 
     Arithmetic encoding may be performed over a plurality of coefficients in the quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1). In this case, since each variance parameter of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) is based on the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) as seen from the formulas (A 1 ) and (A 8 ), it can be said that the arithmetic encoding portion  269  performs such encoding that bit allocation substantially changes based on an estimated spectral envelope (an unsmoothed amplitude spectral envelope). 
     &lt;Judging Portion  266 &gt; 
     The integer signal codes obtained by the arithmetic encoding portion  269  are inputted to the judging portion  266 . 
     When the number of times of updating the gain is a predetermined number of times, the judging portion  266  outputs the integer signal codes as well as outputting an instruction signal to encode the global gain g obtained by the gain updating portion  267  to the gain encoding portion  265 , and, when the number of times of updating the gain is smaller than the predetermined number of times, the judging portion  266  outputs the number of consumed bits C measured by the arithmetic coding portion  269  to the gain updating portion  267  (step S 266 ). 
     &lt;Gain Updating Portion  267 &gt; 
     The number of consumed bits C measured by the arithmetic coding portion  269  is inputted to the gain updating portion  267 . 
     When the number of consumed bits C is larger than the number of allocated bits B, the gain updating portion  267  updates the value of the global gain g to a larger value and outputs the value. When the number of consumed bits C is smaller than the number of allocated bits B, the gain updating portion  267  updates the value of the global gain g to a smaller value and outputs the updated value of the global gain g (step S 267 ). 
     The updated global gain g obtained by the gain updating portion  267  is outputted to the quantizing portion  262  and the gain encoding portion  265 . 
     &lt;Gain Encoding Portion  265 &gt; 
     An output instruction from the judging portion  266  and the global gain g obtained by the gain updating portion  267  are inputted to the gain encoding portion  265 . 
     The gain encoding portion  265  encodes the global gain g to obtain and output a gain code in accordance with an instruction signal (step S 265 ). 
     The integer signal codes outputted by the judging portion  266  and the gain code outputted by the gain encoding portion  265  are outputted to the decoding apparatus as codes corresponding to the normalized MDCT coefficient sequence. 
     That is, in the present specific example 2, step S 267  performed last corresponds to the above step A 61 , and steps S 262 , S 263 , S 264  and S 265  correspond to the above steps A 62 , A 63 , A 64 , and A 65 , respectively. 
     The specific example 2 of the encoding process performed by the encoding portion  26  is described in more detail in International Publication No. WO2014/054556 and the like. 
     [Modification of Encoding Portion  26 ] 
     The encoding portion  26  may perform such encoding that bit allocation is changed based on an estimated spectral envelope (an unsmoothed amplitude spectral envelope), for example, by performing the following process. 
     The encoding portion  26  determines a global gain g corresponding to a normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) first, and determines a quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), which is a sequence of integer values obtained by quantizing results of dividing coefficients of the normalized MDCT coefficient sequence X N (0), X N (1), . . . , X N (N−1) by the global gain g. 
     As for quantized bits corresponding to each coefficient of this quantized normalized coefficient sequence X Q (0), X Q (1), . . . , X Q (N−1), distribution of X Q (k) is assumed to be uniform in a certain range, and the range can be decided based on estimated values of an envelope. Though it is also possible to encode estimated values of an envelope for each of a plurality of samples, the encoding portion  26  can decide the range of X Q (k) using values ^H N (k) of a normalized amplitude spectral envelope sequence based on linear prediction, for example, like the following formula (A 9 ). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       33 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ϕ 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             H 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         
                           
                             
                               H 
                               ^ 
                             
                             γ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             H 
                             ^ 
                           
                           N 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                   , 
                   
                     ( 
                     
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         N 
                         - 
                         1 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   A9 
                   ) 
                 
               
             
           
         
       
     
     In order to minimize a square error of X Q (k) at the time of quantizing X Q (k) for a certain k, it is possible to set the number of bits b(k) to be allocated under the restriction of the following formula: 
     
       
         
           
             
               [ 
               
                 Formula 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 34 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             B 
             = 
             
               
                 ∑ 
                 
                   j 
                   = 
                   0 
                 
                 
                   j 
                   = 
                   
                     N 
                     - 
                     1 
                   
                 
               
               ⁢ 
               
                 ϕ 
                 ⁡ 
                 
                   ( 
                   j 
                   ) 
                 
               
             
           
         
       
     
     The number of bits b(k) to be allocated can be represented by the following formula (A 10 ): 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       35 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       b 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         B 
                         N 
                       
                       + 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             log 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               0 
                             
                             
                               j 
                               = 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               log 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ϕ 
                                   ⁡ 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     ( 
                     
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         N 
                         - 
                         1 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   A10 
                   ) 
                 
               
             
           
         
       
     
     Here, B is a positive integer specified in advance. At this time, the encoding portion  26  may perform a process for readjusting b(k) by performing rounding off so that b(k) becomes an integer, setting b(k)=0 when b(k) is smaller than 0, or the like. 
     Further, the encoding portion  26  may decide the number of allocated bits not for each sample but for a plurality of collected samples and may perform not scalar quantization for each sample but quantization for each vector of a plurality of collected samples. 
     When the number of quantized bits b(k) of X Q (k) of a sample k is given as described above, and encoding is performed for each sample, X Q (k) can take 2 b(k)  kinds of integers from −2 b(k)−1  to 2 b(k)−1 . The encoding portion  26  encodes each sample with b(k) bits to obtain an integer signal code. 
     The generated integer signal codes are outputted to the decoding apparatus. For example, the generated b(k)-bit integer signal codes corresponding to X Q (k) are sequentially outputted to the decoding apparatus, with k=0 first. 
     If X Q (k) exceeds the range from −2 b(k)−1  to 2 b(k)−1  described above, it is replaced with a maximum value or a minimum value. 
     When g is too small, quantization distortion is generated by the replacement. When g is too large, a quantization error is increased, and it is not possible to effectively utilize information because the range X Q (k) can take is too small in comparison with b(k). Therefore, optimization of g may be performed. 
     The encoding portion  26  encodes the global gain g to obtain and output a gain code. 
     The encoding portion  26  may perform encoding other than arithmetic encoding as done in this modification of the encoding portion  26 . 
     (Decoding of Third Embodiment) 
       FIG. 16  shows a configuration example of the decoding apparatus corresponding to the encoding apparatus of the third embodiment. As shown in  FIG. 16 , the decoding apparatus of the third embodiment is, for example, provided with a linear prediction coefficient decoding portion  31 , an unsmoothed amplitude spectral envelope sequence generating portion  32 , a smoothed amplitude spectral envelope sequence generating portion  33 , a decoding portion  34 , an envelope denormalizing portion  35 , and a time domain transforming portion  36 .  FIG. 10  shows an example of each process of a decoding method of the third embodiment realized by this decoding apparatus. 
     At least codes corresponding to a normalized MDCT coefficient sequence and linear prediction coefficient codes outputted by the encoding apparatus are inputted to the decoding apparatus. 
     Each portion in  FIG. 16  will be described below. 
     &lt;Linear Prediction Coefficient Decoding Portion  31 &gt; 
     The linear prediction coefficient codes outputted by the encoding apparatus are inputted to the linear prediction coefficient decoding portion  31 . 
     For each frame, the linear prediction coefficient decoding portion  31  decodes the inputted linear prediction coefficient codes, for example, by a conventional decoding technique to obtain decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  (step B 1 ). 
     The obtained decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  are outputted to the unsmoothed amplitude spectral envelope sequence generating portion  32  and the unsmoothed amplitude spectral envelope sequence generating portion  33 . 
     Here, the conventional decoding technique is, for example, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized linear prediction coefficients, the linear prediction coefficient codes are decoded to obtain decoded linear prediction coefficients which are the same as the quantized linear prediction coefficients, a technique in which, when the linear prediction coefficient codes are codes corresponding to quantized LSP parameters, the linear prediction coefficient codes are decoded to obtain decoded LSP parameters which are the same as the quantized LS parameters, or the like. Further, the linear prediction coefficients and the LSP parameters are mutually transformable, and it is well known that a transformation process can be performed between the decoded linear prediction coefficients and the decoded LSP parameters according to inputted linear prediction coefficient codes and information required for subsequent processes. From the above, it can be said that what comprises the above linear prediction coefficient code decoding process and the above transformation process performed as necessary is “decoding by the conventional decoding technique”. 
     In this way, the linear prediction coefficient decoding portion  31  generates coefficients transformable to linear prediction coefficients corresponding to a pseudo correlation function signal sequence obtained by performing inverse Fourier transform regarding absolute values of a frequency domain sample sequence corresponding to a time-series signal raised to the power of η as a power spectrum, by decoding inputted linear prediction codes. 
     &lt;Unsmoothed Amplitude Spectral Envelope Sequence Generating Portion  32 &gt; 
     The decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the unsmoothed amplitude spectral envelope sequence generating portion  32 . 
     The unsmoothed amplitude spectral envelope sequence generating portion  32  generates an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1), which is an amplitude spectral envelope sequence corresponding to the decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  by the above formula (A 2 ) (step B 2 ). 
     The generated unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) is outputted to the decoding portion  34 . 
     In this way, on the assumption that η is a predetermined positive number other than 2, the unsmoothed amplitude spectral envelope sequence generating portion  32  obtains an unsmoothed spectral envelope sequence, which is a sequence obtained by raising a sequence of an amplitude spectral envelope corresponding to coefficients transformable to linear prediction coefficients generated by the linear prediction coefficient decoding portion  31  to the power of 1/η. Here, η used by the unsmoothed amplitude spectral envelope sequence generating portion  32  is specified in advance and is the same as η specified in advance in the corresponding encoding apparatus. 
     &lt;Smoothed Amplitude Spectral Envelope Sequence Generating Portion  33 &gt; 
     The decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  obtained by the linear prediction coefficient decoding portion  31  are inputted to the smoothed amplitude spectral envelope sequence generating portion  33 . 
     The smoothed amplitude spectral envelope sequence generating portion  33  generates a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1), which is a sequence obtained by reducing amplitude unevenness of a sequence of an amplitude spectral envelope corresponding to the decoded linear prediction coefficients ^β 1 , ^β 2 , . . . , ^β p  by the above formula A(3) (step B 3 ). 
     The generated smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) is outputted to the decoding portion  34  and the envelope denormalizing portion  35 . 
     &lt;Decoding Portion  34 &gt; 
     Codes corresponding to the normalized MDCT coefficient sequence outputted by the encoding apparatus, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) generated by the unsmoothed amplitude spectral envelope sequence generating portion  32  and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  are inputted to the decoding portion  34 . 
     The decoding portion  34  is provided with a variance parameter deciding portion  342 . 
     The decoding portion  34  performs decoding, for example, by performing processes of steps B 41  to B 44  shown in  FIG. 17  (step B 4 ). That is, for each frame, the decoding portion  34  decodes a gain code comprised in the codes corresponding to the inputted normalized MDCT coefficient sequence to obtain a global gain g (step B 41 ). The variance parameter deciding portion  342  of the decoding portion  34  determines each variance parameter of a variance parameter sequence φ(0), φ(1), . . . , φ(N−1) from the global gain g, the unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (A 1 ) (step B 42 ). The decoding portion  34  obtains a decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by performing arithmetic decoding of integer signal codes comprised in the codes corresponding to the normalized MDCT coefficient sequence in accordance with an arithmetic decoding configuration corresponding to the variance parameters of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1) (step B 43 ) and generates a decoded normalized MDCT coefficient sequence ^X N (0), ^H N (1), . . . , ^H N (N−1) by multiplying each coefficient of the decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by the global gain g (step B 44 ). Thus, the decoding portion  34  may decode inputted integer signal codes in accordance with bit allocation which substantially changes based on an unsmoothed spectral envelope sequence. 
     When encoding is performed by the process described in [Modification of encoding portion  26 ], the decoding portion  34  performs, for example, the following process. For each frame, the decoding portion  34  decodes a gain code comprised in the codes corresponding to an inputted normalized MDCT coefficient sequence to obtain a global gain g. The variance parameter deciding portion  342  of the decoding portion  34  determines each variance parameter of a variance parameter sequence φ(0), φ(1), . . . , φ(N−1) from an unsmoothed amplitude spectral envelope sequence ^H(0), ^H(1), . . . , ^H(N−1) and a smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) by the above formula (A 9 ). The decoding portion  34  can determine b(k) by the formula (A 10 ), based on each variance parameter φ(k) of the variance parameter sequence φ(0), φ(1), . . . , φ(N−1). The decoding portion  34  obtains a decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by sequentially decoding values of X Q (k) with the number of bits b(k), and generates a decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by multiplying each coefficient of the decoded normalized coefficient sequence ^X Q (0), ^X Q (1), . . . , ^X Q (N−1) by the global gain g. Thus, the decoding portion  34  may decode inputted integer signal codes in accordance with bit allocation which changes based on an unsmoothed spectral envelope sequence. 
     The decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) which has been generated is outputted to the envelope denormalizing portion  35 . 
     &lt;Envelope Denormalizing Portion  35 &gt; 
     The smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) generated by the smoothed amplitude spectral envelope sequence generating portion  33  and the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) generated by the decoding portion  34  are inputted to the envelope denormalizing portion  35 . 
     The envelope denormalizing portion  35  generates a decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by denormalizing the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) using the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) (step B 5 ). 
     The generated decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) is outputted to the time domain transforming portion  36 . 
     For example, the envelope denormalizing portion  35  generates the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) by multiplying each coefficient ^X N (k) of the decoded normalized MDCT coefficient sequence ^X N (0), ^X N (1), . . . , ^X N (N−1) by each envelope value ^Hγ(k) of the smoothed amplitude spectral envelope sequence ^Hγ(0), ^Hγ(1), . . . , ^Hγ(N−1) on the assumption of k=0, 1, . . . , N−1. That is, ^X(k)=^X N (k)×^Hγ(k) is satisfied on the assumption of k=0, 1, . . . , N−1. 
     &lt;Time Domain Transforming Portion  36 &gt; 
     The decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) generated by the envelope denormalizing portion  35  is inputted to the time domain transforming portion  36 . 
     For each frame, the time domain transforming portion  36  transforms the decoded MDCT coefficient sequence ^X(0), ^X(1), . . . , ^X(N−1) obtained by the envelope denormalizing portion  35  to a time domain and obtains a sound signal (a decoded sound signal) for each frame (step B 6 ). 
     In this way, the decoding apparatus obtains a time-series signal by decoding in the frequency domain. 
     [Modifications and the Like] 
     When the linear prediction analyzing portion  22  and the unsmoothed amplitude spectral envelope sequence generating portion  23  are grasped as one spectral envelope estimating portion  2 A, it can be said that this spectral envelope estimating portion  2 A performs estimation of a spectral envelope (an unsmoothed amplitude spectral envelope sequence) regarding absolute values of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a time-series signal raised to the power of η as a power spectrum, on the assumption that η is a predetermined positive number other than 2. For example, it can be said that a process in a case where η is 1 is performed in the first embodiment. It can be said that a process in a case where η is 2 is performed in the second embodiment. Further, it can be said that a process in a case where η is a predetermined positive number other than 2 is performed in the third embodiment. Here, “regarding . . . as a power spectrum” means that a spectrum raised to the power of η is used where a power spectrum is usually used. 
     In this case, it can be said that, on the assumption that η is a predetermined integer other than 2, the linear prediction analyzing portion  22  of the spectral envelope estimating portion  2 A performs linear prediction analysis using a pseudo correlation function signal sequence obtained by performing inverse Fourier transform regarding absolute values of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, raised to the power of η as a power spectrum, and generates coefficients transformable to linear prediction coefficients. Further, it can be said that, on the assumption that η is a predetermined integer other than 2, the unsmoothed amplitude spectral envelope sequence generating portion  23  of the spectral envelope estimating portion  2 A performs estimation of a spectral envelope by obtaining an unsmoothed spectral envelope sequence, which is a sequence obtained by raising a sequence of an amplitude spectral envelope corresponding to coefficients transformable to linear prediction coefficients obtained by the linear prediction analyzing portion  22  to the power of 1/η. 
     Further, when the smoothed amplitude spectral envelope sequence generating portion  24 , the envelope normalizing portion  25  and the encoding portion  26  are grasped as one encoding portion  2 B, it can be said that this encoding portion  2 B performs such encoding that changes bit allocation or that bit allocation substantially changes based on a spectral envelope (an unsmoothed amplitude spectral envelope sequence) estimated by the spectral envelope estimating portion  2 A, for each coefficient of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a time-series signal. 
     The process of the spectral envelope estimating portion  2 A (that is, the processes of the linear prediction analyzing portion  22  and the unsmoothed amplitude spectral envelope sequence generating portion  23 ) and the process of the encoding portion  2 B (that is, the processes of the smoothed amplitude spectral envelope sequence generating portion  24 , the envelope normalizing portion  25  and the encoding portion  26 ) described in the first to third embodiments are mere examples. The spectral envelope estimating portion  2 A may perform a process for estimating a spectral envelope (an unsmoothed amplitude spectral envelope sequence) regarding absolute values of a frequency domain sample sequence which is, for example, an MDCT coefficient sequence, corresponding to a time-series signal raised to the power of η as a power spectrum, other than the processes described in the first to third embodiments. Further, the encoding portion  2 B may perform a process for performing such encoding that changes bit allocation or that bit allocation substantially changes based on a spectral envelope (an unsmoothed amplitude spectral envelope sequence) estimated by the spectral envelope estimating portion  2 A, for each coefficient of a frequency domain sample sequence, which is, for example, an MDCT coefficient sequence, corresponding to a time-series signal, other than the processes described in the first to third embodiments. 
     For example, the encoding portion  2 B may not be provided with the smoothed amplitude spectral envelope sequence generating portion  24 . In this case, the MDCT coefficient sequence normalization process by the envelope normalizing portion  25  is not performed, but the encoding portion  26  performs an encoding process similar to the above process for an MDCT coefficient sequence instead of a normalized MDCT coefficient sequence. Further, in this case, the variance parameter deciding portion  268  of the encoding portion  26  decides variance parameters based on the following formula (A 11 ) instead of the formula (A 1 ). 
     
       
         
           
             
               
                 
                   
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     Further, in this case, the decoding apparatus may not be provided with the smoothed amplitude spectral envelope sequence generating portion  33 . In this case, the decoded normalized MDCT coefficient sequence denormalization process by the envelope denormalizing portion  35  is not performed, but a decoded MDCT coefficient sequence is obtained by decoding by the decoding portion  34 . Further, in this case, the variance parameter deciding portion  342  of the decoding portion  34  decides variance parameters based on the above formula (A 11 ) instead of the formula (A 1 ). Furthermore, in this case, the time domain transforming portion  36  performs time domain transformation similar to the time domain transformation described above for the decoded MDCT coefficient sequence instead of a decoded normalized MDCT coefficient sequence. 
     Further, for example, the encoding portion  2 B may not be provided with the envelope normalizing portion  25 . In this case, the encoding portion  26  performs an encoding process similar to the encoding process described above for an MDCT coefficient sequence instead of a normalized MDCT coefficient sequence. Further, in this case, the decoding apparatus may not be provided with the envelope denormalizing portion  35 . In this case, a decoded MDCT coefficient sequence is obtained by decoding by the decoding portion  34 , and the time domain transforming portion  36  performs time domain transformation similar to the time domain transformation described above for the decoded MDCT coefficient sequence instead of a decoded normalized MDCT coefficient sequence. 
     When the decoding portion  34  and the envelope denormalizing portion  35  are grasped as one decoding portion  3 A, it can be said that this decoding portion  3 A obtains a frequency domain sample sequence corresponding to a time-series sequence signal by performing decoding of inputted integer signal codes in accordance with such bit allocation that changes or substantially changes based on an unsmoothed spectral envelope sequence. 
     The processes described above are not only executed in order of description but also may be executed in a time-series in parallel or individually according to processing capacity of an apparatus to execute the processes or as necessary. 
     Further, the process of each portion in each apparatus may be realized by a computer. In that case, process content of each apparatus is written by a program. Then, by executing the program on the computer, the process of each portion in each apparatus is realized on the computer. 
     The program in which the process content is written can be recorded in a computer-readable recording medium. As the computer readable recording medium, any recording medium, such as a magnetic recording apparatus, an optical disk, a magneto-optical medium and a semiconductor memory, is possible. 
     Further, distribution of this program is performed, for example, by sales, transfer, lending and the like of a portable recording medium such as a DVD and a CD-ROM in which the program is recorded. Furthermore, the program may be distributed by storing the program in a storage apparatus such as a server computer and transferring the program from the server computer to another computer via a network. 
     For example, a computer which executes such a program stores the program stored in the portable recording medium or transferred from the server computer into its storage portion once. Then, at the time of executing a process, the computer reads the program stored in its storage portion and executes a process in accordance with the read program. Further, as another embodiment of this program, the computer may directly read the program from the portable recording medium and execute a process in accordance with the program. Furthermore, it is also possible for the computer to, each time the program is transferred from the server computer to the computer, execute a process in accordance with the received program one by one. Further, a configuration is also possible in which transfer of the program from the server computer to the computer is not performed, but the processes described above are executed by a so-called ASP (Application Service Provider) type service for realizing a processing function only by an instruction to execute the program and acquisition of a result. It is assumed that the program comprises information provided for processing by an electronic calculator and equivalent to a program (such as data which is not a direct instruction to a computer but has properties defining processing of the computer). 
     Further, though it is assumed that each apparatus is configured by executing a predetermined program on a computer, at least a part of process content of the apparatus may be realized by hardware.