Patent Publication Number: US-2011051938-A1

Title: Method and apparatus for encoding and decoding stereo audio

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATION 
     This application claims priority from of Korean Patent Application No. 10-2009-0079770, filed on Aug. 27, 2009, in the Korean Intellectual Property Office, the disclosure of which is incorporated herein by reference in its entirety. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and apparatus for encoding and decoding stereo audio, and more particularly, to a method and apparatus for parametric-encoding and parametric-decoding stereo audio by minimizing the number of pieces of side information required for parametric-encoding and parametric-decoding the stereo audio. 
     2. Description of the Related Art 
     Generally, methods of encoding multi-channel (MC) audio include waveform audio coding and parametric audio coding. Examples of the waveform audio coding include moving picture experts group (MPEG)-2 MC audio coding, advanced audio coding (AAC) MC audio coding, and bit sliced arithmetic coding (BSAC)/audio video coding standard (AVS) MC audio coding. 
     In the parametric audio coding, an audio signal is encoded by analyzing a component of the audio signal, such as a frequency or amplitude, and parameterizing information about the component. When stereo audio is encoded by using the parametric audio coding, mono audio is generated by down-mixing right channel audio and left channel audio, and then the generated mono audio is encoded. Then, parameters about interchannel intensity difference (IID), interchannel correlation (IC), overall phase difference (OPD), and interchannel phase difference (IPD), which are required to restore the mono audio to the stereo audio, are encoded. Here, the parameters may also be called side information. 
     The parameters about IID and IC are encoded as information for determining the intensities of the left channel audio and the right channel audio, and the parameters about OPD and IPD are encoded as information for determining the phases of the left channel audio and the right channel audio. 
     SUMMARY OF THE INVENTION 
     The present invention provides a method and apparatus for parametric-encoding and parametric-decoding stereo audio by minimizing the number of pieces of side information required for performing parametric-encoding and parametric-decoding the stereo audio. 
     According to an aspect of the present invention, there is provided a method of encoding stereo audio, the method including: adding adjacent input audio signals to generate at least one beginning mono audio signal, the adjacent input audio signals being adjacent to each other among N received input audio signals of N channels of the stereo audio, if the at least one beginning mono audio signal is not a single final mono audio signal, consecutively adding adjacent mono audio signals to generate the final mono audio signal; generating side information for restoring each of the N input audio signals the mono audio signals obtained to generate the final mono audio signal and the final mono audio signal; and encoding the final mono audio signal and the side information, wherein the generating of the side information comprises: mapping an intensity of a first audio signal on an actual number axis of a vector space, the first audio signal being one of the adjacent input audio signals and the adjacent mono audio signals, and mapping an intensity of a second audio signal on an imaginary number axis of the vector space, the second audio signal being one of the input audio signal and the mono audio signal that is adjacent to the first audio signal, mapping a composed vector in the vector space, the composed vector being the sum of the intensity of the first audio signal and the intensity of the second audio signal, and calculating at least one of an angle between the composed vector and the actual number axis and an angle between the composed vector and the imaginary number axis, wherein the calculated angle is information for determining intensities of the first and second audio signals. 
     The method may further include: encoding the N input audio signals; decoding the encoded N input audio signals; and generating difference information about differences between the decoded N input audio signals and the N received input audio signals, wherein the encoding of the final mono audio signal and the side information comprises encoding the final mono audio signal, the side information, and the difference information. 
     The encoding of the side information may include: encoding information for determining intensities of the input audio signals and the mono audio signals obtained to generate the final mono audio signal; and encoding information about phase differences between the adjacent input audio signals and the adjacent mono audio signals. 
     In the generating of the final mono audio signal, the adding the adjacent input audio signals may include: if N is odd, selecting a first input audio signal among the N received input audio signals, creating two audio signals from the first input audio signal to generate an even number of input audio signals, and adding the adjacent input audio signals to generate the at least one beginning mono audio signal; and the consecutively adding adjacent mono audio signals to generate the single final mono audio signal may include: if the at least one beginning mono audio signal is not the single final mono audio signal, and if the at least one beginning mono audio signal is an odd number of mono audio signals, selecting a first beginning mono audio signal among the at least one beginning mono audio signal, creating two mono audio signals from the first beginning mono audio signal to generate an even number of mono audio signals, and consecutively adding the adjacent mono audio signals to generate the final mono audio signal. 
     According to another aspect of the present invention, there is provided a method of decoding stereo audio, the method including: extracting an encoded mono audio signal and encoded side information from received audio data; decoding the extracted mono audio signal and the extracted side information; and restoring at least two beginning restored audio signals from the decoded mono audio signal, if the at least two beginning restored audio signals are not N signals of the stereo audio, consecutively decoding the at least two beginning restored audio signals to generate the N final restored audio based on the decoded side information, wherein the decoded side information includes one of an angle between a composed vector and a first vector mapped to an actual number axis of a vector space and an angle between the composed vector and a second vector mapped to an imaginary number axis of the vector space as information for determining intensities of adjacent audio signals of the beginning restored audio signals and the final restored audio signals, wherein the composed vector is the sum of the first and second vectors, and the first vector represents an intensity of a first audio signal that is one of the adjacent input audio signals and the adjacent mono audio signals, and the second vector represents an intensity of a second audio signal being one of the input audio signal and the mono audio signal that is adjacent to the first audio signal. 
     The method may further include extracting difference information about differences between N decoded audio signals and N original audio signals from the audio data, wherein the N decoded audio signals are generated by decoding the N original audio signals, wherein the final restored audio signals are generated based on the decoded side information and the difference information. 
     The decoded side information may include information about phase differences between the adjacent beginning restored mono audio signals and adjacent final restored audio signals. 
     The restoring of the beginning restored audio signals may include: determining an intensity of at least one of a first beginning restored audio signal and a second beginning restored audio signal from among the adjacent beginning restored audio signals, by using at least one of the angle between the composed vector and the first vector and the angle between the composed vector and the second vector; calculating a phase of the first beginning restored audio signal and a phase of the second beginning restored audio signal based on information about a phase of the decoded mono audio and about a phase difference between the first beginning restored audio signal and the second beginning restored audio signal; and if the first beginning restored audio signal is restored based on the intensities and phases of the beginning restored audio signals, restoring the second beginning restored audio signal by subtracting the first beginning restored audio signal from the decoded mono audio, and if the second beginning restored audio signal is restored, restoring the first beginning restored audio signal by subtracting the second beginning restored audio signal from the decoded mono audio. 
     The restoring of the beginning restored audio signals may include combining one of the restored audio signals that is restored based on the angle between the composed vector and first vector and the angle between the composed vector and the second vector, and one of the beginning restored audio signals that is generated by subtracting one of the beginning restored audio signals from the decoded mono signals, in a predetermined ratio. 
     The restoring of the beginning restored audio signals may include: calculating a phase of a second beginning restored audio signal based on information about a phase of the decoded mono audio and information about a phase difference between the beginning restored audio signals; and restoring the beginning restored audio signals based on information about the phase of the decoded mono audio signal, information about the phase of the second beginning restored audio signal, and information for determining intensities of the beginning restored audio signals. 
     According to another aspect of the present invention, there is provided a method of encoding stereo audio, the method including: generating a mono audio signal by adding a first channel audio signal and a second channel audio signal; mapping an intensity of the first channel audio signal on an actual number axis of a vector space; mapping intensity of the second channel audio signal on an imaginary number axis of the vector space; calculating at least one of an angle between a composed vector and the actual number axis and an angle between the composed vector and the imaginary number axis, wherein the composed vector is the sum of the first and second channel audio signals; and encoding the mono audio and the calculated angle. 
     The encoding may encode difference information about a phase difference between the first and second channel audio signals. 
     According to another aspect of the present invention, there is provided a method of decoding stereo audio, the method including: extracting an encoded mono audio signal and encoded side information from received audio data; decoding the encoded mono audio signal and the encoded side information; and restoring first and second channel audio signals from the decoded mono audio signal and the decoded side information, wherein the decoded side information includes one of an angle between a composed vector and a first vector mapped to an actual number axis of a vector space and an angle between the composed vector and a second vector mapped to an imaginary number axis of the vector space as information for determining intensities of adjacent audio signals of the first and second channel audio signals, wherein the composed vector is the sum of the first and second vectors, and wherein the first vector represents an intensity of the first channel audio signal, and the second vector represents an intensity of the second audio signal, the first and second channel audio signals being adjacent to each other. 
     The decoded side information may further include difference information about a phase difference between the first and second channel audio signals. 
     According to another aspect of the present invention, there is provided an apparatus for encoding stereo audio, the apparatus including: a mono audio generator that generates at least one beginning mono audio signal by adding adjacent input audio signals, the adjacent input audio signals begin adjacent to each other among N received input audio signals of N channels of the stereo audio, and, if the at least one beginning mono audio signal is not a single final mono audio signal, consecutively adds adjacent mono audio signals to generate the signal final mono audio signal; a side information generator that generates side information for restoring the N input audio signals and each of the mono audio signals obtained to generate the final mono audio signal, and the final mono audio signal; and an encoder that encodes the final mono audio signal and the side information, wherein the side information generator maps an intensity of a first audio signal on an actual number axis of a vector space, the first audio signal being one of the adjacent input audio signals and the adjacent mono audio signals, maps an intensity of a second audio signal on an imaginary number axis of the vector space, the second audio signal being one of the input audio signal and the mono audio signal that is adjacent to the first audio signal, maps a composed vector in the vector space, the composed vector being the sum of the intensity of the first audio signal and the intensity of the second audio signal, and calculates at least one of an angle between a composed vector and the actual number axis and an angle between the composed vector and the imaginary number axis, wherein the calculated angle is information for determining the intensities of the first and second audio signals. 
     The mono audio generator may include a plurality of down-mixers that add each two adjacent audio signals of at least one of the N input audio signals and the mono audio signals obtained to generate the final mono audio signal. 
     The apparatus may further include a difference information generator that encodes the N input audio signals, decodes the encoded N input audio signals, and generates difference information about differences between the N decoded input audio signals and the N received input audio signals, wherein the encoder encodes the difference information with the final mono audio signal and the side information. 
     According to another aspect of the present invention, there is provided an apparatus for decoding stereo audio, the apparatus including: an extractor that extracts an encoded mono audio signal and encoded side information from received audio data; a decoder that decodes the extracted mono audio signal and the extracted side information; and an audio restorer that restores at least one beginning restored audio signal from the decoded mono audio signal, and if the at least one beginning restored audio signal is at least one restored mono audio signal, generates N final restored audio signals by consecutively decoding the restored mono audio signal, based on the decoded side information, wherein the decoded side information includes one of an angle between a composed vector and a first vector mapped to an actual number axis of a vector space and an angle between the composed vector and a second vector mapped to an imaginary number axis of the vector space as information for determining intensities of adjacent audio signals of the beginning restored audio signals and the final restored audio signals, wherein the composed vector is the sum of the first and second vectors, and wherein the first vector represents an intensity of a first audio signal that is one of the adjacent input audio signals and the adjacent mono audio signals, and the second vector represents an intensity of the input audio signal and the mono audio signal that is adjacent to the first audio signal. 
     The audio restorer may include a plurality of up-mixers that generate two restored audio signals from at least one of the decoded mono audio signal and the beginning restored audio signals, based on the side information. 
     According to another aspect of the present invention, there is provided an apparatus for encoding stereo audio, the apparatus including: a mono audio generator that generates a mono audio signal by adding a first channel audio signal and a second channel audio signal; a side information generator that maps an intensity of the first channel audio signal on an actual number axis of a vector space, maps an intensity of the second channel audio signal on an imaginary number axis of the vector space, and calculates at least one of an angle between a composed vector and the actual number axis and an angle between the composed vector and the imaginary number axis, wherein the composed vector is the sum of the first and second audio signals; and an encoder that encodes the mono audio signal and the calculated angle. 
     According to another aspect of the present invention, there is provided an apparatus for decoding stereo audio, the apparatus including: an extractor that extracts an encoded mono audio signal and encoded side information from received audio data; a decoder that decodes the encoded mono audio signal and the encoded side information; and a restorer that restores first and second channel audio signals from the decoded mono audio signal and the decoded side information, wherein the decoded side information includes one of an angle between a composed vector and a first vector mapped to an actual number axis of a vector space and an angle between the composed vector and an imaginary number axis of the vector space as information for determining intensities of adjacent audio signals of the first and second channel audio signals, wherein the composed vector is the sum of the first and second vectors, and wherein the first vector represents an intensity of the first channel audio signal, and the second vector represents an intensity of a second audio signal being one of the second audio signal, the first and second channel audio signals being adjacent to each other. 
     According to another aspect of the present invention, there is provided a computer readable recording medium recorded thereon a program for executing a method of encoding stereo audio, the method including: adding adjacent input audio signals to generate at least one beginning mono audio signal, the adjacent input audio signals being adjacent to each other among N received input audio signals of N channels of stereo audio, if the at least one beginning mono audio signal is not a single final mono audio signal, consecutively adding adjacent mono audio signals to generate the final mono audio signal; generating side information for restoring each of the mono audio signals obtained to generate the final mono audio signal and the final mono audio signal, while generating the final mono audio signal; and encoding the final mono audio signal and the side information, wherein the generating of the side information comprises: mapping an intensity of a first audio signal on an actual number axis of a vector space, the first audio signal being one of the adjacent input audio signals and the adjacent mono audio signals, and mapping an intensity of a second audio signal on an imaginary number axis of the vector space, the second audio signal being one of the input audio signal and the mono audio signal that is adjacent to the first audio signal, mapping a composed vector in the vector space, the composed vector being the sum of the intensity of the first audio signal and the intensity of the second audio signal, and calculating at least one of an angle between the composed vector and the actual number axis and an angle between the composed vector and the imaginary number axis, wherein the calculated angle is information for determining intensities of the first and second audio signals. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       
           
           
             
               The above and other aspects of the present invention will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings in which: 
             
           
         
      
         FIG. 1  is a diagram illustrating an apparatus for encoding audio, according to an exemplary embodiment of the present invention; 
         FIG. 2  is a diagram illustrating sub-bands in parametric audio coding; 
         FIG. 3A  is a diagram for describing a method of generating information about intensities of a first channel input audio signal and a second channel input audio signal, according to an exemplary embodiment of the present invention; 
         FIG. 3B  is a diagram for describing a method of generating information about intensities of a first beginning mono audio signal and a second beginning mono audio signal, according to an exemplary embodiment of the present invention; 
         FIG. 4  is a flowchart illustrating a method of encoding side information, according to an exemplary embodiment of the present invention; 
         FIG. 5  is a flowchart illustrating a method of encoding audio, according to an exemplary embodiment of the present invention; 
         FIG. 6  is a diagram illustrating an apparatus for decoding audio, according to an exemplary embodiment of the present invention; 
         FIG. 7  is a flowchart illustrating a method of decoding audio, according to an exemplary embodiment of the present invention; 
         FIG. 8  is a diagram illustrating an apparatus for encoding 5.1-channel stereo audio, according to an exemplary embodiment of the present invention; 
         FIG. 9  is a diagram illustrating an apparatus for decoding 5.1-channel stereo audio, according to an exemplary embodiment of the present invention; and 
         FIG. 10  is a diagram for describing an operation of an up-mixer, according to an exemplary embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, the present invention will be described more fully with reference to the accompanying drawings, in which exemplary embodiments of the invention are shown. 
       FIG. 1  is a diagram for describing an apparatus for encoding audio, according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 1 , the apparatus  100  includes a mono audio generator  110 , a side information generator  120 , and an encoder  130 . 
     The mono audio generator  110  receives first through nth channel input audio signals Ch 1  through Chn from N channels, generates first through mth beginning mono audio signals BM 1  through BMm by adding adjacent input audio signals among the received first through nth channel input audio signals Ch 1  through Chn, and generates a final mono audio signal FM. The final mono audio signal FM may be generated by performing the same adding method used to generate the first through mth beginning mono audio signals BM 1  through BMm, where n and m are positive integers. 
     Here, since the two adjacent input audio signals among the signals Ch 1  through Chn are added to generate the first through mth beginning mono audio signals BM 1  through BMm, the adding method for obtaining the first through mth beginning mono audio signals BM 1  through BMm is also performed on adjacent beginning mono audio signals BM 1  through BMm. Also, as will be described later, if phases of the two adjacent input audio signals of the first through nth channel input audio signals Ch 1  through Chn are adjusted to be the same while generating the first through mth beginning mono audio signals BM 1  through BMm, the same adding method for obtaining the first through mth beginning mono audio signals BM 1  through BMm is also performed after adjusting phases of two adjacent input audio signals among the mono audio signals BM 1  through BMm. 
     Here, the mono audio generator  110  generates first through jth transient mono audio signals TM 1  through TMj from the first through mth beginning mono audio signals BM 1  through BMm, and the final mono audio signal FM, where j is a positive integer. 
     Also, as illustrated in  FIG. 1 , the mono audio generator  110  includes a plurality of down-mixers  111 - 119  that add adjacent audio signals of the first through nth channel input audio signals Ch 1  through Chn, adjacent audio signals of the first through mth beginning mono audio signals BM 1  through BMm, and adjacent audio signals of the first through jth transient mono audio signals TM 1  through TMj. The final mono audio signal FM is generated through the plurality of down-mixers  111 - 119 . 
     For example, a down-mixer  111 , which received a first channel input audio signal Ch 1  and a second channel input audio signal Ch 2 , generates a first beginning mono audio signal BM 1  by adding the first and second channel input audio signals Ch 1  and Ch 2 . Then, a down-mixer  115 , which received the first beginning mono audio signal BM 1  and a second beginning mono audio signal BM 2 , generates a first transient mono audio signal TM 1 . 
     Here, the down-mixers  111 - 119  may adjust a phase of one of two adjacent audio signals received as an input to be identical to a phase of the other of the two adjacent audio signals received as an input before adding the two adjacent audio signals. Accordingly, the down-mixers  111 - 119  may add phase adjusted adjacent audio signals, instead of adding the two adjacent audio signals as they are received. For example, before adding the first and second channel input audio signals Ch 1  and Ch 2 , a phase of the second channel input audio signal Ch 2  may be adjusted to be identical to a phase of the first channel input audio signal Ch 1 , thereby adding the phase-adjusted second channel input audio signal Ch 2 ′ with the first channel input audio signal Ch 1 . The details thereof will be described later. 
     Meanwhile, according to the current exemplary embodiment of the present invention, the first through nth channel input audio signals Ch 1  through Chn transmitted to the mono audio generator  110  are considered to be digital signals. However, when the first through nth channel input audio signals Ch 1  through Chn may be analog signals according to another embodiment of the present invention, and the analog first through nth channel input audio signals Ch 1  through Chn may be converted to digital signals before being input to the mono audio generator  110 . The conversion may be accomplished by performing sampling and quantization on the first through nth channel input analog audio signals Ch 1  through Chn. 
     The side information generator  120  generates side information for restoring each of the first through nth channel input audio signals Ch 1  through Chn, the first through mth beginning mono audio signals BM 1  through BMm, and the first through jth transient mono audio signals TM 1  through TMj. 
     Here, whenever the down-mixers  111 - 119  included in the mono audio generator  110  add adjacent audio signals, the side information generator  120  generates side information required to restore the adjacent audio signals based on the result of adding the adjacent audio signals. Here, for convenience of description, the side information input from each down-mixer  111 - 119  to the side information generator  120  is not illustrated in  FIG. 1 . 
     Here, the side information includes information for determining intensities of each of the first through nth channel input audio signals Ch 1  through Chn, intensities of the first through mth beginning mono audio signals BM 1  through BMm, and intensities of the first through jth transient mono audio signals TM 1  through TMj. The side information may also include information about phase differences between adjacent audio signals of the first through nth channel input audio signals Ch 1  through Chn, adjacent audio signals of the first through mth beginning mono audio signals BM 1  through BMm, and adjacent audio signals of the first through jth transient mono audio signals TM 1  through TMj. The phase difference between the two adjacent audio signals denotes a difference between phases of audio signals that are added in a down-mixer. 
     According to another embodiment of the present invention, each down-mixer  111 - 119  may include the side information generator  120  in order to add the adjacent audio signals while generating the side information about the adjacent audio signals. 
     A method of generating the side information, wherein the method is performed by the side information generator  120 , will be described in detail later with reference to  FIGS. 2 through 4 . 
     The encoder  130  encodes the final mono audio signal FM generated by the mono audio generator  110  and the side information generated by the side information generator  120 . 
     Here, a method of encoding the final mono audio signal FM and the side information may be any general method used to encode mono audio and side information. 
     According to another exemplary embodiment of the present invention, the apparatus  100  may further include a difference information generator (not shown) which encodes the first through nth channel input audio signals Ch 1  through Chn, decodes the encoded first through nth channel input audio signals Ch 1  through Chn, and generates information about differences between the decoded first through nth channel input audio signals Ch 1  through Chn and the original first through nth channel input audio signals Ch 1  through Chn. 
     As such, when the apparatus includes the difference information generator, the encoder  130  may encode the information about differences along with the final mono audio signal FM and the side information. When the encoded mono audio signal generated by the apparatus is decoded, the information about differences is added to the decoded mono audio signal, so that audio signals that are similar to the original first through nth channel input audio signals Ch 1  through Chn are generated. 
     According to another exemplary embodiment of the present invention, the apparatus  100  may further include a multiplexer (not shown), which generates a final bitstream by multiplexing the final mono audio signal FM and the side information that are encoded by the encoder  130 . 
     A method of generating side information and a method of encoding the generated side information will now be described in detail. For convenience of description, the side information generated while the down-mixers  111 - 119  included in the mono audio generator  110  generate the first beginning mono audio signal BM 1  by receiving the first and second channel input audio signals Ch 1  and Ch 2  will be described. Also, a case of generating information for determining intensities of the first and second channel input audio signals Ch 1  and Ch 2 , and a case of generating information for determining phases of the first and second channel input audio signals Ch 1  and Ch 2  will be described. 
     (1) Information for Determining Intensity 
     According to parametric audio coding, each channel audio signal is changed to the frequency domain, and information about the intensity and phase of each channel audio signal is encoded in the frequency domain, as will be described in detail with reference to  FIG. 2 . 
       FIG. 2  is a diagram illustrating sub-bands in parametric audio coding. 
     In detail,  FIG. 2  illustrates a frequency spectrum in which an audio signal is converted to the frequency domain. When a fast Fourier transform is performed on the audio signal, the audio signal is expressed with discrete values in the frequency domain. In other words, the audio signal may be expressed as a sum of a plurality of sine curves. 
     In the parametric audio coding, when the audio signal is converted to the frequency domain, the frequency domain is divided into a plurality of sub-bands. Information for determining intensities of the first and second channel input audio signals Ch 1  and Ch 2 , and information for determining phases of the first and second channel input audio signals Ch 1  and Ch 2  are encoded in each sub-band. Here, side information about intensity and phase in a sub-band k is encoded, and then side information about intensity and phase in a sub-band k+1 is encoded. As such, the entire frequency band is divided into sub-bands, and the side information is encoded according to each sub-band. 
     An example of encoding side information of the first and second channel input audio signals Ch 1  and Ch 2  in a predetermined frequency band, i.e., in the sub-band k, will now be described in relation to encoding and decoding of stereo audio having first through nth channel input audio signals. 
     When side information about stereo audio is encoded according to conventional parametric audio coding, information about interchannel intensity difference (IID) and information about interchannel correlation (IC) is encoded as information for determining intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k, as described above. Here, in the sub-band k, the intensity of the first channel input audio signal Ch 1  and the intensity of the second channel input audio signal Ch 2  are each calculated, and a ratio of the intensity of the first channel input audio signal Ch 1  to the intensity of the second channel input audio signal Ch 2  is encoded as the information about IID. However, the ratio alone is not sufficient to determine the intensities of the first and second channel input audio signals Ch 1  and Ch 2 , and thus the information about IC is encoded as side information along with the ratio, and inserted into a bitstream. 
     A method of encoding audio, according to an exemplary embodiment of the present invention, uses a vector representing the intensity of the first channel input audio signal Ch 1  in the sub-band k and a vector representing the intensity of the second channel input audio signal Ch 2  in the sub-band k, in order to minimize the number of pieces of side information encoded as the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k. Here, an average value of intensities in frequencies f 1  through fn in the frequency spectrum, in which the first channel input audio signal Ch 1  is converted to the frequency domain, is the intensity of the first channel input audio signal Ch 1  in the sub-band k, and also is a size of a vector Ch 1  that will be described later. 
     Similarly, an average value of intensities in frequencies f 1  through fn in the frequency spectrum, in which the second channel input audio signal Ch 2  is converted to the frequency domain, is the intensity of the second channel input audio signal Ch 2  in the sub-band k, and also is a size of a vector Ch 2 , as will be described in detail with reference to  FIGS. 3A and 3B . 
       FIG. 3A  is a diagram for describing a method of generating information about intensities of the first channel input audio signal Ch 1  and the second channel input audio signal Ch 2 , according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 3A , the side information generator  120  generates a vector space, wherein the vector Ch 1 , i.e., the vector representing the intensity of the first channel input audio signal Ch 1  in the sub-band k, is mapped on an imaginary number axis in a complex space, and the vector Ch 2 , i.e., the vector representing the intensity of the second channel input audio signal Ch 2  in the sub-band k, is mapped on an actual number axis in the complex space. Also, a vector BM 1 , which is a vector representing the intensity of the first beginning mono audio signal BM 1  generated by adding the Ch 1  vector and the Ch 2  vector, is illustrated in  FIG. 3A . 
     The side information generator  120  generates information about an angle θm 1  between the vector BM 1  and the vector Ch 2  or information about an angle θm 2  between the vector BM 1  and the vector Ch 1  instead of information about IID and IC, as information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k. 
     Alternatively, the side information generator  120  may generate cosine values, such as cos θm 1  or cos θm 2 , instead of the angle θm 1  or the angle θm 2 . In order to generate and encode information about an angle, a quantization process may be performed. Here, a cosine value of the angle may be generated and encoded in order to minimize the loss occurring during the quantization process. 
     The method of generating the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  has been described above. Hereinafter, a method of generating information for determining intensities of first and second beginning mono audio signals BM 1  and BM 2  will be described with reference to  FIG. 3B . 
       FIG. 3B  is a diagram for describing the method of generating the information about the intensities of the first beginning mono audio signal BM 1  and the second beginning mono audio signals BM 2 , according to an embodiment of the present invention. 
     Referring to  FIG. 3B , the side information generator  120  generates a vector space, wherein a vector BM 1 , i.e., a vector about the intensity of the first beginning mono audio signal BM 1  in the sub-band k, is mapped on an actual number axis, and a vector BM 2 , i.e., a vector about the intensity of the second beginning mono audio signals BM 2  in the sub-band k, is mapped on an imaginary number axis. In other words, |BM 1 |, which is a size of the vector BM 1  generated by adding the vector Ch 1  and the vector Ch 2  in  FIG. 3A , is mapped on the actual number axis, and |BM 2 |, which is a size of the vector BM 2 , is mapped on the imaginary number axis. Alternatively, the |BM 1 | may be mapped on the imaginary number axis and the |BM 2 | may be mapped on the actual number axis. 
     Also, a vector TM 1 , which is a vector about intensity of a first transient mono audio signal TM 1  generated by adding the vector BM 1  and the vector BM 2 , is illustrated in  FIG. 3B . 
     The side information generator  120  may generate information about an angle θ L1  between the TM 1  vector and the BM 1  vector or information about an angle θ L2  between the vector TM 1  and the vector BM 2  instead of the information about IID and IC, as the information for determining the intensities of the first and second beginning mono audio signals BM 1  and BM 2  in the sub-band k. 
     Alternatively, the side information generator  120  may generate cosine values, such as cos θ L1  or cos θ L2 , instead of the information about the angle θ L1  or the angle θ L2 . 
     (2) Information for Determining Phase 
     It has been described above that in the conventional parametric audio coding, information about overall phase difference (OPD) and information about interchannel phase difference (IPD) is encoded as information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k. 
     In other words, conventionally, the information about OPD is generated and encoded by calculating a phase difference between the first channel input audio signal Ch 1  in the sub-band k and the first beginning mono audio signal BM 1  generated by adding the first channel input audio signal Ch 1  and the second channel input audio signal Ch 2  in the sub-band k. Similarly, the information about IPD is generated and encoded by calculating a phase difference between the first channel input audio signal Ch 1  and the second channel input audio signal Ch 2  in the sub-band k. The phase difference may be obtained by calculating each of the phase differences at the frequencies f 1  through f n  included in the sub-band and calculating the average of the calculated phase differences. 
     However, the side information generator  120  only generates information about a phase difference between the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k, as information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2 . 
     According to an exemplary embodiment of the present invention, the down-mixer  111 - 119  generates the phase-adjusted second channel input audio signal Ch 2 ′ by adjusting the phase of the second channel input audio signal Ch 2  to be identical to the phase of the first channel input audio signal Ch 1 , and then adds the phase-adjusted second channel input audio signal Ch 2 ′ with the first channel input audio signal Ch 1 . Thus, the phases of the first and second channel input audio signals Ch 1  and Ch 2  are each calculated only based on the information about the phase difference between the first and second channel input audio signals Ch 1  and Ch 2 . 
     As an example of audio of the sub-band k, the phases of the second channel input audio signal Ch 2  in the frequencies f 1  through fn are each respectively adjusted to be identical to the phases of the first channel input audio signal Ch 1  in the frequencies f 1  through fn. An example of adjusting the phase of the second channel input audio signal Ch 2  in the frequency f 1  will now be described. When the first channel input audio signal Ch 1  is expressed as |Ch 1 |e i(2πf1t+θ1)  in the frequency f 1 , and the second channel input audio signal Ch 2  is expressed as |Ch 2 |e i(2πf1t+θ2)  in the frequency f 1 , the phase-adjusted second channel input audio signal Ch 2 ′ in the frequency f 1  may be obtained as Equation 1 below. Here, θ 1  denotes the phase of the first channel input audio signal Ch 1  in the frequency f 1  and θ 2  denotes the phase of the second channel input audio signal Ch 2  in the frequency f 1 . 
         Ch 2 ′=Ch 2 ×e   i(θ1−θ2)   =|Ch 2 |e   i(2πf1t+θ1)   Equation 1
 
     According to Equation 1, the phase of the second channel input audio signal Ch 2  in the frequency f 1  is adjusted to be identical to the phase of the first channel input audio signal Ch 1 . The phases of the second channel input audio signal Ch 2  are repeatedly adjusted in other frequencies f 2  through fn in the sub-band k, thereby generating the phase-adjusted second input audio signal Ch 2 ′ in the sub-band k. 
     Since the phase of the phase-adjusted second channel input audio signal Ch 2 ′ is identical to the phase of the first channel input audio signal Ch 1  in the sub-band k, a decoding unit for the first beginning mono audio signal BM 1  can obtain the phase of the second channel input audio signal Ch 2  when only the phase difference between the first and second channel input audio signals Ch 1  and Ch 2  is encoded. Since the phase of the first channel input audio signal Ch 1  and the phase of the first beginning mono audio signal BM 1  generated by the down-mixer are the same, information about the phase of the first channel input audio signal Ch 1  does not need to be separately encoded. 
     Accordingly, when only the information about the phase difference between the first and second channel input audio signals Ch 1  and Ch 2  is encoded, the decoding unit can calculate the phases of the first and second channel input audio signals Ch 1  and Ch 2  by using the encoded information. 
     Meanwhile, the method of encoding the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  by using intensity vectors of channel audio signals in the sub-band k, and the method of encoding the information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k by adjusting the phases may be used independently or in combination. In other words, the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  is encoded by using a vector according to the present invention, and the information about OPD and IPD may be encoded as the information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2  according to the conventional technology. Alternatively, the information about IID and IC may be encoded as the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  according to the conventional technology, and only the information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2  may be encoded by using phase adjustment according to the present invention. Here, the side information may be encoded by using both methods according to the present invention. 
       FIG. 4  is a flowchart illustrating a method of encoding side information, according to an exemplary embodiment of the present invention. 
     A method of encoding the information about the intensities and phases of the first and second channel input audio signals Ch 1  and Ch 2  in a predetermined frequency band, i.e., in the sub-band k, will now be described with reference to  FIG. 4 . 
     In operation  410 , the side information generator  120  maps the intensity of the first channel input audio signal Ch 1  in the sub-band k on the imaginary number axis in the complex space, and maps the intensity of the second channel input audio signal Ch 2  in the sub-band k on the actual number axis in the complex space. 
     Here, the mapping of the intensity of the first channel input audio signal Ch 1  on the imaginary number axis means that a vector about the intensity of the first channel input audio signal Ch 1  is mapped, and the mapping of the intensity of the second channel input audio signal Ch 2  on the actual number axis means that a vector about the intensity of the second channel input audio signal Ch 2  is mapped. 
     According to another exemplary embodiment of the present invention, the intensity of the first channel input audio signal Ch 1  may be mapped on the actual number axis, and the intensity of the second channel input audio signal Ch 2  may be mapped on the imaginary number axis. 
     In operation  420 , information about an angle between a composed vector and the actual number axis or between the composed vector and the imaginary number axis is generated, wherein the composed vector is generated by adding the vectors representing the first and second channel input audio signals Ch 1  and Ch 2 . 
     In operation  430 , information about a phase difference between the first and second channel input audio signals Ch 1  and Ch 2  is generated. 
     Here, the information about the angle is the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k. Also, the information about the angle may be information about a cosine value of the angle, instead of the angle itself. 
     Here, the first beginning mono audio signal BM 1  may be generated by adding the first and second channel input audio signals Ch 1  and Ch 2 , or by adding the first channel input audio signal Ch 1  and the phase-adjusted second channel input audio signal Ch 2 ′. Here, the phase of the phase-adjusted second channel input audio signal Ch 2 ′ is identical to the phase of the first channel input audio signal Ch 1  in the sub-band k. 
     In operation  440 , the information about the angle between the composed vector and the actual number axis or information about the angle between the composed vector and the imaginary number axis, and the information about the phase difference between the first and second channel input audio signals Ch 1  and Ch 2  are encoded. 
     The method of generating and encoding side information described above with reference to  FIGS. 2 through 4  may be identically applied to generate side information for restoring audio signals that are added in each of the first through nth channel input audio signals Ch 1  through Chn, the first through mth beginning mono audio signals BM 1  through BMm, and the first through jth transient mono audio signals TM 1  through TMj illustrated in  FIG. 1 . 
       FIG. 5  is a flowchart illustrating a method of encoding audio, according to an exemplary embodiment of the present invention. In operation  510 , beginning mono audio signals are generated by adding adjacent input audio signals of N received input audio signals, and one final mono audio signal, which is generated by performing the same adding method on the beginning mono audio signals a plurality of times, where N is a positive integer. 
     In operation  520 , side information for restoring the input audio signals, the beginning mono audio signals, and transient mono audio signals is generated. 
     Here, intensity of a first audio signal from among two adjacent audio signals of each of the input audio signals, the beginning mono audio signals, and the transient mono audio signals is mapped on an actual number axis, and the intensity of the second audio signal from among the two adjacent audio signals is mapped on an imaginary number axis. Then, information about an angle between a vector and the actual number axis or information about an angle between the vector and the imaginary number axis is generated as information for determining the intensity of each of the mapped audio signals, wherein the vector is generated by adding the two mapped audio signals. 
     In operation  530 , the final mono audio signal and the side information are encoded. 
       FIG. 6  is a diagram illustrating an apparatus for decoding audio, according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 6 , the apparatus  600  includes an extractor  610 , a decoder  620 , and an audio restorer  630 . 
     The extractor  610  extracts encoded mono audio EM and encoded side information ES from received audio data. Here, the extractor  610  may also be called a demultiplexer. 
     According to another exemplary embodiment of the present invention, the encoded mono audio signal EM and the encoded side information ES may be received instead of the audio data, and in this case, the extractor  610  may not be included in the apparatus  600 . 
     The decoder  620  decodes the encoded mono audio signal EM and the encoded side information ES extracted by the extractor  610  to produce decoded side information DS and a decoded mono audio signal DM, respectively. 
     The audio restorer  630  restores two beginning restored audio signals BR 1  and BR 2  from the decoded mono audio signal DM based on the decoded side information DS. The audio restorer  630  generates N final restored audio signals Ch 1  through Chn by consecutively applying the restoring method on the two beginning restored audio signals BR 1  and BR 2 , where n is a positive integer. 
     Here, the audio restorer  630  generates transient restored audio signals TR 1  through TRs+m while generating the final restored audio signals Ch 1  through Chn from the beginning restored audio signals BR 1  and BR 2 . 
     Also, as illustrated in  FIG. 6 , the audio restorer  630  includes a plurality of up-mixers  631 - 637 , which generate two restored audio signals from each one of the beginning restored audio signals BR 1  and BR 2 . The up-mixers  631 - 637  generate the transient restored audio signals TR 1  through TRs+m from the beginning restored audio signals BR 1  and BR 2 , and generate the final restored audio signals Ch 1  through Chn from the transient restored audio signals TR 1  through TRs+m. 
     In  FIG. 6 , the decoded side information DS is transmitted to the up-mixers  631 - 637  included in the audio restorer  630  through the decoder  620 , but for convenience of description, the decoded side information DS transmitted to each of the up-mixers  631 - 637  is not illustrated. 
     Meanwhile, according to another exemplary embodiment of the present invention, if the extractor  610  further extracts information about differences between N decoded audio signals, which are generated by encoding and decoding N original audio signals that are to be restored from the audio data through the N final restored audio signals, and the N original audio signals, the information about the differences is decoded by using the decoder  620 . The decoded information about the differences may be added to each of the final restored audio signals Ch 1  through Chn generated by the audio restorer  630 . Accordingly, the final restored audio signals Ch 1  through Chn are similar to the N original audio signals. 
     Operations of an up-mixer  634  will now be described in detail. Here, for convenience of description, the up-mixer  634  receives an s+1th transient restored audio signal TRs+1 and restores the first and second channel input audio signals Ch 1  and Ch 2  as final restored audio signals from the transient restored audio signal TRs+1. 
     Referring to the vector space illustrated in  FIG. 3A , the up-mixer  634  uses information about an angle between a vector BM 1  and a vector Ch 1  or information about an angle between the vector BM 1  and a vector Ch 2  as information for determining intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k, wherein the vector BM 1  represents the intensity of the s+1th transient restored audio signal TRs+1, the vector Ch 1  represents the intensity of the first channel input audio signal Ch 1 , and the vector Ch 2  represents the intensity of the second channel input audio signal Ch 2 . The up-mixer  634  may use information about a cosine value of the angle between the vector BM 1  and the vector Ch 1  or between the BM 1  vector and the vector Ch 2 . 
     For example, in  FIG. 3A , the intensity of the first channel input audio signal Ch 1 , i.e., the size of the vector Ch 1  may be calculated according to |Ch 1 |=|BM 1 |×sin θm 1 . Here, |BM 1 | denotes the intensity of the s+1th transient restored audio signal TRs+1, i.e., the size of the vector BM 1 . Similarly, the intensity of the second channel input audio signal Ch 2 , i.e., the size of the vector Ch 2  may be calculated according to |Ch 2 |=|BM 1 |×cos θm 1 . 
     Also, the up-mixer  634  may use information about a phase difference between the first and second channel input audio signals Ch 1  and Ch 2  as information for determining phases of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k. When the phase of the second channel input audio signal Ch 2  is already adjusted to be identical to the phase of the first channel input audio signal Ch 1  while encoding the s+1th transient restored audio signal TRs+1, the up-mixer  634  may calculate the phases of the first and second channel input audio signals Ch 1  and Ch 2  by using only the information about the phase difference between the first and second channel input audio signals Ch 1  and Ch 2 . 
     Meanwhile, the method of decoding the information for determining the intensities of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k by using a vector, and the method of decoding the information for determining the phases of the first and second channel input audio signals Ch 1  and Ch 2  in the sub-band k by using phase adjustment as described above may be used independently or in combination. 
       FIG. 7  is a flowchart illustrating a method of decoding audio, according to an exemplary embodiment of the present invention. 
     In operation  710 , an encoded mono audio signal EM and encoded side information ES are extracted from received audio data. 
     In operation  720 , the extracted mono audio signal EM and the extracted side information ES are decoded. 
     In operation  730 , two beginning restored audio signals BR 1  and BR 2  are restored from the decoded mono audio signal DM, and N final restored audio signals Ch 1  through Chn are restored by consecutively applying the same decoding method on the two beginning restored audio signals BR 1  and BR 2 , based on the decoded side information DS. 
     Here, transient restored audio signals TR 1  through TRs+m are generated from the beginning restored audio signals BR 1  and BR 2 . 
     According to another exemplary embodiment of the present invention, when the final restored audio signals Ch 1  through Chn are generated, the generated final restored audio signals Ch 1  through Chn may be converted and output as analog signals. 
       FIG. 8  is a diagram illustrating an apparatus for encoding 5.1-channel stereo audio, according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 8 , the apparatus  800  includes a mono audio generator  810 , a side information generator  820 , and an encoder  830 . Audio signals input to the apparatus  800  include a left channel front audio signal L, a left channel rear audio signal Ls, central audio signal C, a sub-woofer audio signal Sw, a right channel front audio signal R, and right channel rear audio signal Rs. 
     Operations of the mono audio generator  810  will now be described. 
     The mono audio generator  810  includes a plurality of down-mixers  811 - 816 . A first down-mixer  811  generates a signal LV 1  by adding the left channel front audio signal L and the left channel rear audio signal Ls, a second down-mixer  812  generates a signal CSw by adding the central audio signal C and the sub-woofer audio signal Sw, and a third down-mixer  813  generates a signal RV 1  by adding the right channel front audio signal R and the right channel rear audio signal Rs. 
     Here, the first through third down-mixers  811  through  813  may adjust phases of audio signals received as inputs to be identical before adding the audio signals. 
     Meanwhile, the second down-mixer  812  generates signals C 1  and Cr from the signal CSw. This is because the number of audio signals output from the first to third down mixers  811  to  813 , which are to be input to fourth and fifth down-mixers  814  and  815 , is  3 , i.e., an odd number. Accordingly, the second down-mixer  812  divides the signal CSw into the signals C 1  and the Cr so that the fourth and fifth down-mixers  814  and  815  each receive two audio signals. Here, the signals C 1  and the Cr each have a size obtained by multiplying CSw by 0.5, but the sizes of the signals C 1  and the Cr are not limited thereto and any value may be used for the multiplication. 
     The fourth down-mixer  814  generates a signal LV 2  by adding the signals LV 1  and C 1 , and the fifth down-mixer  815  generates a signal RV 2  by adding the signals RV 1  and Cr. 
     A sixth down-mixer  816  generates a final mono audio signal FM by adding the signals LV 2  and the RV 2 . 
     Here, the signals LV 1 , the RV 1 , and the signal CSw correspond to the beginning mono audio signals BMs described above, and the signals LV 2  and the RV 2  correspond to the transient mono audio signals TMs described above. 
     The side information generator  820  receives side information SI 1  through SI 6  from the first through sixth down mixers  811  through  816 , or reads the side information SI 1  through SI 6  from the first through sixth down-mixers  811  through  816  and then outputs the side information SI 1  through SI 6  to the encoder  830 . Here, dotted lines in  FIG. 8  indicate that the side information S 1  through SI 6  is transmitted from the first through sixth down-mixers  811  through  816  to the side information generator  820 . 
     The encoder  830  encodes the final mono audio signal FM and the side information SI 1  through SI 6 . 
       FIG. 9  is a diagram illustrating an apparatus for decoding 5.1-channel stereo audio, according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 9 , the apparatus  900  includes an extractor  910 , a decoder  920 , and an audio restorer  930 . The operations of the extractor  910  and the decoder  920  of  FIG. 9  are respectively similar to those of the extractor  610  and the decoder  620  of  FIG. 6 , and thus details thereof are omitted herein. The operations of the audio restorer  930  will now be described in detail. 
     The audio restorer  930  includes a plurality of up-mixers  931 - 936 . A first up-mixer  931  restores signals LV 2  and RV 2  from decoded mono audio signal DM. 
     Here, first through sixth up-mixers  931  through  936  perform restoration based on decoded side information SI 1  through SI 6  received from the decoder  920 . 
     The second up-mixer  932  restores signals LV 1  and C 1  from signal the LV 2 , and the third up-mixer  933  restores signals RV 1  and Cr from the signal RV 2 . 
     The fourth up-mixer  934  restores signals L and Ls from the signal LV 1 , the fifth up-mixer  935  restores signals C and Sw from the signal CSw, which is generated by combining the signals C 1  and Cr, and the sixth up-mixer  936  restores signals R and Rs from the signal RV 1 . 
     Here, the signals LV 2  and the RV 2  correspond to the beginning restored audio signals BRs described above, and the signals LV 1 , CSw, and RV 1  correspond to the transient restored audio signals TRs described above. 
     A method of restoring audio signals performed by the first through sixth up mixers  931  through  936  will now be described in detail. Hereinafter, the operations of the fourth up-mixer  934  will be described with reference to  FIG. 10 . 
       FIG. 10  is a diagram for describing the operations of the fourth up-mixer  934 , according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 10 , a vector L, which is a vector representing the intensity of the left channel front audio signal L in the sub-band k, is mapped on an imaginary number axis of a two-dimensional vector space. A vector Ls, which is a vector representing the intensity of the left channel rear audio Ls in the sub-band k, is mapped on an actual number axis, is generated in the vector space. Last, a vector LV 1 , which is a vector representing the intensity of the beginning mono audio signal LV 1 , which is generated by adding the left channel front audio signal L and the left channel rear audio Ls, is also illustrated. 
     Various methods of restoring the left channel front audio signal L and the left channel rear audio Ls will now be described. 
     A first method is to restore the left channel front audio signal L and the left channel rear audio signal Ls by using an angle between the vector LV 1  and the vector Ls as described above. In other words, the size of the vector Ls is calculated according to |LV 1 |cos θm and the size of the vector L is calculated according to |LV 1 |sin θm so as to determine the intensity of the left channel front audio signal L and the intensity of the left channel rear audio signal Ls. Then, the phases of the left channel front audio signal L and the left channel rear audio signal Ls are calculated based on side information. Accordingly, the left channel front audio signal L and the left channel rear audio signal Ls are restored. 
     In a second method, when the left channel front audio signal L or the left channel rear audio signal Ls are restored according to the first method, the left channel front audio signal L is restored by subtracting the left channel rear audio signal Ls from the beginning mono audio signal LV 1 , and the left channel rear audio signal Ls is restored by subtracting the left channel front audio signal L from the beginning mono audio signal LV 1 . 
     A third method is to restore audio signals by combining audio signals restored according to the first method and audio signals restored according to the second method in a predetermined ratio. 
     In other words, when the left channel front audio signal L and the left channel rear audio signal Ls restored according to the first method are respectively referred to as Ly and Lsy, and the left channel front audio signal L and the left channel rear audio signal Ls restored according to the second method are respectively referred to as Lz and Lsz, the intensities of the left channel front audio signal L and the left channel rear audio signal Ls are respectively determined according to |LS=a×|Ly|+(1−a)×|Lz| and |Ls|=a×|Lsy|+(1−a)×|Lsz|. The phases of the left channel front audio signal L and the left channel rear audio signal Ls are calculated based on side information, thereby restoring the left channel front audio signal L and the left channel rear audio signal Ls. Here, “a” is a value between 0 and 1. 
     The embodiments of the present invention can be written as computer programs and can be implemented in general-use digital computers that execute the programs using a computer readable recording medium. Examples of the computer readable recording medium include magnetic storage media (e.g., ROM, floppy disks, hard disks, etc.), optical recording media (e.g., CD-ROMs, or DVDs), and storage media. 
     While this invention has been particularly shown and described with reference to preferred embodiments thereof, it will be understood by those of ordinary skill in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims. The exemplary embodiments should be considered in descriptive sense only and not for purposes of limitation. Therefore, the scope of the invention is defined not by the detailed description of the invention but by the appended claims, and all differences within the scope will be construed as being included in the present invention.