Patent Publication Number: US-8126152-B2

Title: Method and arrangement for a decoder for multi-channel surround sound

Description:
This application claims the benefit of U.S. Provisional Application No. 60/743,871, filed Mar. 28, 2006, the disclosure of which is fully incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to decoding of a multi-channel surround audio bit stream. In particular, the present invention relates to a method and arrangement that uses spatial covariance matrix extrapolation for signal decoding. 
     BACKGROUND 
     In film theaters around the world, multi-channel surround audio systems have since long placed film audiences in the center of the audio spaces of the film scenes that are being played before them and are giving them a realistic and convincing feeling of “being there”. This audio technology has moved into the homes of ordinary people as home surround sound theatre systems and is now providing them with the sense of “being there” in their own living rooms. 
     The next field where this technology will be used includes mobile wireless units or terminals, in particular small units such as cellular phones, mp3-players (including similar music players) and PDAs (Personal Digital assistants). There the immersive nature of the surround sound is even more important because of the small screens. Moving this technology to the mobile terminal is, however, not a trivial matter. The main obstacles include that: 
     The available bit-rate is in many cases low especially in wireless mobile channels. 
     The processing power of the mobile terminal is rather limited. 
     Small mobile terminals generally have only two micro speakers and ear-plugs or headphones. 
     This means, in particular for mobile terminals such as cellular phones, that a surround sound solution on a mobile terminal has to use a much lower bit-rate than for example the 384 kbits/sec that is used in the Dolby Digital 5.1 system. Due to the limited processing power, the decoders of the mobile terminals must be computationally optimized and due to the speaker configuration of the mobile terminal the surround sound must be delivered through the earplugs or headphones. 
     A standard way of delivering multi-channel surround sound through headphones or earplugs is to perform a 3D audio or binaural rendering of the multichannel surround sound. 
     In general, in 3D audio rendering a model of the audio scene is used and each incoming monophonic signal is filtered through a set of filters that model the transformations created by the human head, torso and ears. These filters are called head related filters (HRF) having head related transfer functions (HRTFs) and if appropriately designed, they give a good 3D audio scene perception. 
     The diagram of  FIG. 1  illustrates a method of complete 3D audio rendering of a multichannel 5.1 audio signal. The six multi-channel signals are: 
     surround right (SR), right (R), center (C), low frequency element (LFE), left (L) and surround left (SL). 
     In the example illustrated in  FIG. 1  the center and low frequency signals are combined into one signal. Then, five different filters denoted: H I   B , H C   B , H C , H I   F  and H C   F  are needed in order to implement this method of head related filtering. The SR signal is input to filters H I   B  and H C   B , the R signal is input to filters H I   F  and H C   F , the C and LFE signals are jointly input to filter H C , the L signal is input to filters H I   F  and H C   F  and the SL signal is input to filters H I   B , H C   B . The signals output from the filters H I   B , H C   B , H C , H I   F  and H C   F  are summed in a right summing element  1 R to give a signal intended to be provided to the right headphone, not shown. The signals output from the filters H I   B , H C   B , H C , H C   F  and H C   F  are summed in a left summing element  1 L to give a signal intended to be provided to the left headphone, not shown. In this case a symmetric head is assumed, therefore the filters for the left ear and the right ear are assumed to be similar. 
     The quality in terms of 3D perception of such rendering depends on how closely the HRFs model or represent the listener&#39;s own head related filtering when she/he is listening. Hence, it may be advantageous if the HRFs can be adapted and personalized for each listener if a good or very good quality is desired. This adaptation and personalization step may include modeling, measurement and in general a user dependent tuning in order to refine the quality of the perceived 3D audio scene. 
     Current state-of-the-art standardized multi-channel audio codecs require a high amount of bandwidth in order to reach an acceptable quality and thus they prohibit the use of such codec for services such as wireless mobile streaming. 
     For instance, even if the Dolby Digital 5.1 (AC-3 codec) has very low complexity when compared to the AAC (Advanced Audio Coding) multi-channel codec, it requires much more bit-rate for similar quality. Both codecs, the AAC multi-channel codec and AC-3 codec remain until today unusable in the wireless mobile domain because of the high demands that they make on computational complexity and bit-rate. 
     New parametric multi-channel codecs based on the principles of binaural cue coding have been developed. The recently standardized MPEG parametric stereo tool is a good example of the low complexity/high quality parametric techniques for encoding stereo sound. The extension of parametric stereo to multi-channel coding is currently undergoing standardization in MPEG under the name Spatial Audio coding, and is also known as MPEG-surround. 
     The principles behind the parametric multi-channel coding can be explained and understood from the block diagram of  FIG. 2  that illustrates a general case. 
     The parametric surround encoder  3 , also referred to as a multi-channel parametric surround encoder, receives a multi-channel audio signal comprising the individual signals x I (n) to x N (n), where N is the number of input channels. The encoder  3  then forms in down-mixing unit  5  a down-mixed signal comprising the individual down-mixed signals z I (n) to z M (n). The number of down mixed channels M&lt;N is dependent upon the desired bit-rate, quality and the availability of an M-channel audio encoder  7 . One key aspect of the encoding process is that the down-mixed signal, typically a stereo signal but it could also be a mono signal, is derived from the multi-channel input signal, and it is this down mix signal that is compressed in the audio encoder  7  for transmission over the wireless channel  11  rather than the original multi-channel signal. In addition, the parametric surround encoder also comprises a spatial parameter estimation unit  9  that from the input signals x I (n) to x N (n) computes the spatial cues or spatial parameters such as inter-channel level differences, time differences and coherence. The compressed audio signal which is output from the M-channel audio encoder (main signal) is, together with the spatial parameters that constitute side information transmitted to the receiving side that in the case considered here typically is a mobile terminal. 
     On the receiving side, a parametric surround decoder  13  includes an M-channel audio decoder  15 . The audio decoder  15  produces signals {circumflex over (z)} I (n) to {circumflex over (z)} M (n) that the coded version of z I (n) to z M (n). These are together with the spatial parameters input to a spatial synthesis unit  17  that produces output signals {circumflex over (x)} I (n) to {circumflex over (x)} N (n). Because the decoding process is parametric in nature, the decoded signals {circumflex over (x)} I (n) to {circumflex over (x)} N (n) are not necessarily objectively close to the original multichannel signals x I (n) to x N (n) but are subjectively a faithful reproduction of the multichannel audio scene. 
     It is obvious, that depending on the bandwidth of the transmitting channel over the interface  11  that generally is relatively low there will be a loss of information and hence the signals {circumflex over (z)} I (n) to {circumflex over (z)} M (n) and {circumflex over (x)} I (n) to {circumflex over (x)} N (n) on the receiving side cannot be the same as their counterparts on the transmitting side. Even though they are not quite true equivalents of their counterparts, they may be sufficient good equivalents. 
     In general, such a surround encoding process is independent of the compression algorithm used in the units encoder  7  (core encoder) and the audio decoder  15  (core decoder) in  FIG. 2 . The core encoding process can use any of a number of high performance compression algorithms such as AMR-WB+ (extended adaptive multirate wide band), MPEG-1 Layer III (Moving Picture Experts Group), MPEG-4 AAC or MPEG-4 High Efficiency AAC, and it could even use PCM (Pulse Code Modulation). 
     In general, the above operations are done in the transformed signal domain, such as Fourier transform and in general on some time-frequency decomposition. This is especially beneficial if the spatial parameter estimation and synthesis in the units  9  and  17  use the same type of transform as that used in the audio encoder  7 . 
       FIG. 3  is a detailed block diagram of an efficient parametric audio encoder. The N-channel discrete time input signal, denoted in vector form as x N (n), is first transformed to the frequency domain in a transform unit  21  that gives a signal  x   N (k, m). The index k is the index of the transform coefficients, or frequency sub-bands. The index m represents the decimated time domain index that is also related to the input signal possibly through overlapped frames. 
     The signal is thereafter down-mixed in a down-mixing unit  5  to generate the M-channel down mix signal z M (k, m), where M&lt;N. A sequence of spatial model parameter vectors p N (k, m) is estimated in an estimation unit  9 . This can be either done in an open-loop or closed loop fashion. 
     The spatial parameters consist of psycho-acoustical cues that are representative of the surround sound sensation. For instance, these parameters consist of inter-channel level differences (ILD), time differences (ITD) and coherence (IC) to capture the spatial image of a multi-channel audio signal relative to a transmitted down-mixed signal z M (k, m) (or if in closed loop, the decoded signal {tilde over (z)} M (k, m)). The cues p N (k, m) can be encoded in a very compact form such as in a spatial parameter quantization unit  23  producing the signal {tilde over (p)} N (k, m) followed by a spatial parameter encoder  25 . The M-channel audio encoder  7  produces the main bit stream which in a multiplexer  27  is multiplexed with the spatial side information produced by the parameter encoder. From the multiplexer the multiplexed signal is transmitted to a demultiplexer  29  on the receiving side in which the side information and the main bit stream are recovered as seen in the block diagram of  FIG. 4 . 
     On the receiving side the main bit stream is decoded to synthesize a high quality multichannel representation using the received spatial parameters. The main bit stream is first decoded in an M-channel audio decoder  31  from which the decoded signals {circumflex over (z)} M (k, m) are input to the spatial synthesis unit  17 . The spatial side information holding the spatial parameters is extracted by the demultiplexer  29  and provided to a spatial parameter decoder  33  that produces the decoded parameters {tilde over (p)} N (k, m) and transmits them to the synthesis unit  17 . The spatial synthesis unit produces the signal {tilde over (x)} N (k, m), that is provided to the signal Frequency-to-time transform unit  35  to produce the signal {circumflex over (x)} N (k, m), i.e. the multichannel decoded signal. 
     A personalized 3D audio rendering of a multi-channel surround sound can be delivered to a mobile terminal user by using an efficient parametric surround decoder to first obtain the multiple surround sound channels, using for instance the multi-channel decoder described above with reference to  FIG. 4 . Thereupon, the system illustrated in  FIG. 1  is used to synthesize a binaural 3D-audio rendered multichannel signal. This operation is shown in the schematic of  FIG. 5 . 
     Work has also been done in which spatial or 3D audio filtering has been performed in the subband domain. In C. A. Lanciani, and R. W. Schafer, “Application of Head-related Transfer Functions to MPEG Audio Signals”, Proc. 31st Symposium on System Theory, Mar. 21-23, 1999, Auburn, Ala., U.S.A., it is disclosed how an MPEG coded mono signal could be spatialized by performing the HR filtering operation in the subband domain. In A. B. Touimi, M. Emerit and J. M. Pernaux, “Efficient Method for Multiple Compressed Audio Streams Spatialization,” Proc. 3rd International Conference on Mobile and Ubiquitous Multimedia, pp. 229-235, Oct. 27-29, 2004, College Park, Md., U.S.A., it is disclosed how a number of individually MPEG coded mono signals can be spatialized by doing the Head Related (HR) filtering operations in the subband domain. The solution is based on a special implementation of the HR filters, in which all HR filters are modeled as a linear combination of a few predefined basis filters. 
     Applications of 3D audio rendering are multiple and include gamming, mobile TV shows, using standards such as 3GPP MBMS or DVB-H, listening to music concerts, watching movies and in general multimedia services, which contain a multi-channel audio component. 
     The methods described above of rendering multi-channel surround sound, although attractive since they allow a whole new set of services to be provided to wireless mobile units, have many drawbacks: 
     First of all, the computational demands of such rendering are prohibitive since both decoding and 3D rendering have to be performed in parallel and in real time. The complexity of a parametric multi-channel decoder even if low when compared to a full waveform multi-channel decoder is still quite high and at least higher than that of a simple stereo decoder. The synthesis stage of spatial decoding has a complexity that is at least proportional to the number of encoded channels. Additionally, the filtering operations of 3D rendering are also proportional to the number of channels. 
     The second disadvantage consists of the temporary memory that is needed in order to store the intermediate decoded channels. They are in fact buffered since they are needed in the second stage of 3D rendering. 
     Finally, one of the main disadvantages is that the quality of such 3D audio rendering can be very limited due to the fact that inter-channel correlations may be canceled. The inter-channel correlations are essential due to the way parametric multi-channel coding synthesizes the signals. 
     In MPEG surround, for instance, the correlations (ICC) and channel level differences (CLD) are estimated only between pairs of channels. The ICC- and the CLD-parameters are encoded and transmitted to the decoder. In the decoder, the received parameters are used in a synthesis tree as depicted in  FIG. 7  for one 5-1-5 configuration (in this case the 5-1-5 1  configuration).  FIG. 6  illustrates surround system configuration having 5-1-5 1  parameterization. From  FIG. 6  it can be seen that CLD and ICC parameters in the 5-1-5 1  configuration are estimated only between pairs of channels. 
     Due to that the correlations (ICC) and channel level differences (CLD) are only estimated between pairs of channels, not all single correlations are available. This in turn prohibits individual channel manipulation and re-use, as for instance, 3D rendering. In fact, if for instance two un-coded channels, for example RF and RS are uncorrelated and they are encoded by using the 5-1-5 1  configuration, then no control over their correlation is available since the correlation is simply not transmitted to the decoder as such but only the correlation on the second level of the tree is provided. At the decoder side, this in turn would lead to two correlated decoded channels. In fact, the decoder does not have access, nor does it have control over the correlation between certain individual channels. These channels belong to different third level boxes. In the example of  FIG. 6 , these are all pairs of channels which belong to different loudspeaker groupings. This can also be seen in  FIG. 7 . The pairs of channels are the ones which belong to different third-level tree boxes (OTT 3 , OTT 4  OTT 2 ) in the 5-1-5 1  configuration. This may not be a problem when listening in a loudspeaker environment; however it becomes a problem if the channels are combined together, as in 3D rendering, leading to possible unwanted channel cancellation or over-amplification. 
     SUMMARY 
     The object of the present invention is to overcome the disadvantages in parametric multichannel decoders related to possible unwanted cancellation and/or amplification of certain channels. That is achieved by rendering arbitrary linear combinations of the decoded multichannel signals by extrapolating a partially known covariance to a complete covariance matrix of all the channels and synthesizing based on the extrapolated covariance an estimate of the arbitrary linear combinations. 
     According to a first aspect of the present invention, a method for synthesizing an arbitrary predetermined linear combination of a multi-channel surround audio signal is provided. The method comprises the steps of receiving a description H of the arbitrary predetermined linear combination, receiving a decoded downmix signal of the multi-channel surround audio signal, receiving spatial parameters comprising correlations and channel level differences of the multi-channel audio signal, obtaining a partially known spatial covariance based on the received spatial parameters comprising correlations and channel level differences of the multi-channel audio signal, extrapolating the partially known spatial covariance to obtain a complete spatial covariance, forming according to a fidelity criterion an estimate of said arbitrary predetermined linear combination of the multi-channel surround audio signal based at least on the extrapolated complete spatial covariance, the received decoded downmix signal and the said description of the arbitrary predetermined linear combination, and synthesizing said arbitrary predetermined linear combination of a multi-channel surround audio signal based on said estimate of the arbitrary predetermined linear combination of the multi-channel surround audio signal. 
     According to a second aspect, an arrangement for synthesizing an arbitrary predetermined linear combination of a multi-channel surround audio signal is provided. The arrangement comprises a correlator for obtaining a partially known spatial covariance based on received spatial parameters comprising correlations and channel level differences of the multi-channel audio signal, an extrapolator for extrapolating the partially known spatial covariance to obtain a complete spatial covariance, an estimator for forming according to a fidelity criterion an estimate of said arbitrary predetermined linear combination of the multi-channel surround audio signal based at least on the extrapolated complete spatial covariance, a received decoded downmix signal m and a description of the coefficients giving the arbitrary predetermined linear combination, and a synthesizer for synthesizing said arbitrary predetermined linear combination of a multi-channel surround audio signal based on said estimate of the arbitrary predetermined linear combination of the multi-channel surround audio signal. 
     Thus, the invention allows a simple and efficient way to render surround sound, which is encoded by parametric encoders on mobile devices. The advantage consists of a reduced complexity and increased quality than that which is obtained by using a 3D rendering directly on the multi-channel signals. 
     In particular, the invention allows arbitrary binaural decoding of multichannel surround sound. 
     A further advantage is that the operations are performed in the frequency domain thus reducing the complexity of the system. 
     A further advantage is that signal samples do not have to be buffered, since the output is directly obtained in a single decoding step. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram illustrating a possible 3D audio or binaural rendering of a 5.1 audio signal, 
         FIG. 2  is a high level description of the principles of a parametric multi-channel coding and decoding system, 
         FIG. 3  is a detailed description of the parametric multi-channel audio encoder, 
         FIG. 4  is a detailed description of the parametric multi-channel audio decoder, 
         FIG. 5  is 3D-audio rendering of decoded multi-channel signal 
         FIG. 6  is a parameterization view of the spatial audio processing for the 5-1-5 1  configuration. 
         FIG. 7  is a tree structure view of the spatial audio processing for the 5-1-5 1  configuration. 
         FIG. 8  illustrates the relation between subbands k and hybrid subbands m and the relation between the time-slots n and the down-sampled time slot l. 
         FIG. 9   a  illustrates an OTT box showed in  FIG. 7  and  FIG. 9   b  illustrates the corresponding R-OTT box. 
         FIG. 10   a  illustrates the arrangement according to the present invention and  FIG. 10   b  illustrates an embodiment of the invention. 
         FIG. 11  is flowcharts illustrating the method according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     The basic concept of the present invention is to obtain a partially known spatial covariance of a multi-channel surround audio signal based on received spatial parameters and to extrapolate the obtained partially known spatial covariance to obtain a complete spatial covariance. Then, according to a fidelity criterion, a predetermined arbitrary linear combination of the multi-channel surround audio signal is estimated based at least on the extrapolated complete spatial covariance, a received decoded down mix signal m and a description H of the predetermined arbitrary linear combination to be able to synthesize the predetermined linear combination of the multi-channel surround audio signal based on said estimation. The predetermined arbitrary linear combination of the multichannel surround audio signal can conceptually be a representation of a filtering of the multichannel signals, e.g. head related filtering and binaural rendering. It can also represent other sound effects such as reverberation. 
     Thus, the present invention relates to a method for a decoder and an arrangement for a decoder. The arrangement is illustrated in  FIG. 10   a  and comprises a correlator  902   a , an extrapolator  902   b , an estimator  903  and a synthesizer  904 . The correlator  902   a  is configured to obtain a partially known spatial covariance matrix  911  based on received spatial parameters  901  comprising correlations ICC and channel level differences CLD of the multi-channel surround audio signal. The extrapolator  902   b  is configured to use a suitable extrapolation method to extrapolate the partially known spatial covariance matrix to obtain a complete spatial covariance matrix. Further, the estimator  903  is configured to estimate according to a fidelity criterion a linear combination  913  of the multi-channel surround audio signal by using the extrapolated complete spatial covariance matrix  912  in combination with a received decoded downmix signal and a matrix H k  of coefficients representing a description of the predetermined arbitrary linear combination. Finally the synthesizer  904  is configured to synthesize the linear combination  914  of the multi-channel surround audio signal based on said estimation  913  of the linear combination of the multi-channel surround audio signal. 
     A preferred embodiment of the present invention will now be described in relation to an MPEG surround decoder. It should be appreciated that although a preferred embodiment of the present information is described with reference to an MPEG surround decoder, other parametric decoders and systems may also suitable for use in connection with the present invention. 
     For sake of simplicity and without departing from the essence of the invention, the 5-1-5 1  MPEG surround configuration is considered, as depicted in  FIG. 7 . The configuration comprises a plurality of connected OTT (one-to-two) boxes. Side information such as res and of spatial parameters referred to as channel level differences (CLD) and correlations (ICC) are input to the OTT boxes. m is a downmix signal of the multichannel signal. 
     Synthesis of the multi-channel signals is done in the hybrid frequency domain. This frequency division is non linear which strives to a certain extent to mimic the time-frequency analysis of the human ear. In the following, every hybrid sub-band is indexed by k, and every time-slot is indexed by the index n. In order to lower the bit-rate requirements, the MPEG surround spatial parameters are defined only on a down-sampled time slot called the parameter time-slot l, and on a down-sampled hybrid frequency domain called the processing band m. The relations between the n and l and between the m and k are illustrated by  FIG. 8 . Thus the frequency band m 0  comprises the frequency bands k 1  and k 1  and the frequency band ml comprises the frequency bands k 2  and k 3 . Moreover, the time slots l is a downsampled version of the time slots n. The CLD and ICC parameters are therefore valid for that parameter time-slot and processing band. All processing parameters are calculated for every processing band and subsequently mapped to every hybrid band. 
     Thereafter, these are interpolated from the parameter time-slot to every time-slot n. 
     The OTT boxes of the decoder depicted in  FIG. 7  can be visualized as shown in  FIG. 9   a.    
     Based on this illustration, the output for an arbitrary OTT box strives to restore the correlation between the two original channels y 0   l,m  and y 1   l,m  into the two estimated channels ŷ 0   l,m  and ŷ 1   l,m . 
     This can be better understood by examination of the estimation part done in the encoder. The encoder comprises R-OTT boxes that are reversed OTT boxes as illustrated in  FIG. 9   b . The R-OTT boxes convert a stereo signal into a mono signal in combination with parameter extraction which represents the spatial cues between the respective input signals. Input signals to each of these R-OTT boxes are the original channels y 0   l,m  and y 1   l,m . Each R-OTT box computes the ratio of the powers of corresponding time/frequency tiles of the input signals (which will be denoted ‘Channel Level Difference’, or CLD), that is given by: 
               CLD   X     =     10   ⁢       log   10     ⁡     (         ∑     l   ,   m       ⁢       y   0     l   ,   m       ⁢     y   0     l   ,     m   *                 ∑     l   ,   m       ⁢       y   1     l   ,   m       ⁢     y   1     l   ,     m   *               )               
and a similarity measure of the corresponding time/frequency tiles of the input signals (which will be denoted ‘Inter-Channel Correlation’, or ICC), given by the cross correlation:
 
     
       
         
           
             
               ICC 
               X 
             
             = 
             
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               ( 
               
                 
                   
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     Additionally, the R-OTT box generates a mono signal which writes as
 
 x   l,m   =g   0   y   0   l,m   +g   1   y   1   l,m  
 
where g 0 , g 1  are appropriate gains. With g 0 =g 1 =½ a mono signal is generated. Another choice consists of choosing g 0 , g 1  such that
 
 E[x   l,m   x   l,m*   ]=E[y   0   l,m   y   0   l,m*   ]+E[y   1   l,m   y   1   l,m* ]
 
which can be realized using,
 
     
       
         
           
             
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                     + 
                     
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                         10 
                       
                     
                   
                   
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                         10 
                       
                     
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                         X 
                       
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                             CLD 
                             X 
                           
                           20 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     In the following, it is assumed that the above is true and that the energy of the output of the R-OTTx box is equal to the sum of the input energies. 
     The correlations (ICC) as well as the channel level differences (CLD) between any two channels that are input to an R-OFT box is quantized encoded and transmitted to the decoder. 
     This embodiment of the invention uses the CLD and the ICC corresponding to each (R)-OTT box in order to build the spatial covariance matrix, however other measures of the correlation and the channel level differences may also be used. 
     Conceptually the covariance matrix of any two channels is written as: 
     
       
         
           
             
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     Since only real correlations are available at the MPEG-surround decoder it is possible to assume real correlation matrices without loss of generality. Thus, each output channels of an OTT box (which is input to an R-OTT box) can be shown to have a covariance matrix as 
     
       
         
           
             
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                                     x 
                                   
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     2 
                                     , 
                                     x 
                                   
                                 
                                 ⁢ 
                                 
                                   ρ 
                                   x 
                                 
                               
                             
                             
                               
                                 c 
                                 
                                   2 
                                   , 
                                   x 
                                 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     Where σ OTT     X     2  denotes the energy of the input of the OTT X  (or alternatively the output of the R-OTT X ) box, the second term on the right-hand side of the equation is shown in order to simplify the notations. 
     If the channels vector corresponding to the output of OTT 3  and OTT 4  are denoted 
               v       OTT   3     ,     OTT             4           =     [         lf           rf           c           lfe         ]           
then, according to these notations, the spatial covariance matrix in the case of the 5-1-5 1  MPEG surround can be written with block matrices and the matrix is partially unknown which is shown below:
 
     
       
         
           
             
               Re 
               ⁢ 
               
                   
               
               ⁢ 
               
                 E 
                 ⁡ 
                 
                   [ 
                   
                     
                       [ 
                       
                         
                           
                             lf 
                           
                         
                         
                           
                             rf 
                           
                         
                         
                           
                             c 
                           
                         
                         
                           
                             lfe 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           lf 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           rf 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           lfe 
                           * 
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       C 
                       
                         OTT 
                         3 
                       
                     
                   
                   
                     ? 
                   
                 
                 
                   
                     ? 
                   
                   
                     
                       C 
                       
                         OTT 
                         4 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     The 2×2 matrices which are unknown are marked by “?”. Hence a partially known spatial covariance matrix is obtained based on the spatial parameters, CLD and ICC. 
     Furthermore, the input of OTT 3  and OTT 4  are related to each other and are represented by the covariance matrix C OTT     1   . It is easy in this case to relate both energies, i.e. σ OTT     3     2  and σ OTT     4     2  as follows,
 
σ OTT     3   =c 1,1   2 σ OTT     1     2 ,
 
σ OTT     4   =c 2,1   2 σ OTT     1    
 
     Therefore the covariance matrix for the first four channels can be written as 
     
       
         
           
             
               Re 
               ⁢ 
               
                   
               
               ⁢ 
               
                 E 
                 ⁡ 
                 
                   [ 
                   
                     
                       [ 
                       
                         
                           
                             lf 
                           
                         
                         
                           
                             rf 
                           
                         
                         
                           
                             c 
                           
                         
                         
                           
                             lfe 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           lf 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           rf 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           lfe 
                           * 
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
             
             = 
             
               
                 σ 
                 
                   OTT 
                   1 
                 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           
                             1 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             3 
                           
                           2 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             1 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             x 
                           
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             3 
                           
                         
                         ⁢ 
                         
                           ρ 
                           3 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           lfe 
                         
                       
                     
                   
                   
                     
                       
                         
                           c 
                           
                             1 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             3 
                           
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             3 
                           
                         
                         ⁢ 
                         
                           ρ 
                           3 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             1 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             3 
                           
                           2 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           lfe 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             2 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             4 
                           
                           2 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             2 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             4 
                           
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             4 
                           
                         
                         ⁢ 
                         
                           ρ 
                           4 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             2 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             1 
                             , 
                             4 
                           
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             4 
                           
                         
                         ⁢ 
                         
                           ρ 
                           4 
                         
                       
                     
                     
                       
                         
                           c 
                           
                             2 
                             , 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           
                             2 
                             , 
                             4 
                           
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     In the MPEG surround standard, the value of ρ 4 =ICC 4  does not exist and is conceptually assumed to be equal to 1, i.e. center and LFE are identical except for a scale factor. However, for the sake of a generic development, this assumption will not be made. 
     The last matrix equation shows that a number of unknown spatial inter-channel correlations are present. Namely, R lf,c , R lf,lfe , R rf,c , R rf,lfe , however it is known that, the cross correlation of the two inputs to OTT 3  and OTT 4  is equal to ICC 1 =ρ 1 . Given that, according to the previous matrix equation: 
     
       
         
           
             
               
                 
                   Re 
                   ⁢ 
                   E 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       [ 
                       
                         
                           
                             
                               lf 
                               + 
                               rf 
                             
                           
                         
                         
                           
                             
                               c 
                               + 
                               lfe 
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           lf 
                           * 
                         
                         + 
                         
                           
                             rf 
                             * 
                           
                           ⁢ 
                           
                             c 
                             * 
                           
                         
                         + 
                         
                           lfe 
                           * 
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                   
                   
               
             
             = 
             
                 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           
                             1 
                             , 
                             1 
                           
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               
                                 1 
                                 , 
                                 3 
                               
                               2 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 c 
                                 
                                   1 
                                   , 
                                   3 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 
                                   2 
                                   , 
                                   3 
                                 
                               
                               ⁢ 
                               
                                 ρ 
                                 3 
                               
                             
                             + 
                             
                               c 
                               
                                 2 
                                 , 
                                 3 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           R 
                           
                             lf 
                             , 
                             c 
                           
                         
                         + 
                         
                           R 
                           
                             lf 
                             , 
                             lfe 
                           
                         
                         + 
                         
                           R 
                           
                             rf 
                             , 
                             c 
                           
                         
                         + 
                         
                           R 
                           
                             rf 
                             , 
                             lfe 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           R 
                           
                             lf 
                             , 
                             c 
                           
                         
                         + 
                         
                           R 
                           
                             lf 
                             , 
                             lfe 
                           
                         
                         + 
                         
                           R 
                           
                             rf 
                             , 
                             c 
                           
                         
                         + 
                         
                           R 
                           
                             rf 
                             , 
                             lfe 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           
                             2 
                             , 
                             1 
                           
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               
                                 1 
                                 , 
                                 4 
                               
                               2 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 c 
                                 
                                   1 
                                   , 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 
                                   2 
                                   , 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 ρ 
                                 4 
                               
                             
                             + 
                             
                               c 
                               
                                 2 
                                 , 
                                 4 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Thus, it is immediately seen that the missing quantities have to satisfy
 
 R   lf,c   +R   lf,lfe   +R   rf,c   +R   rf,lfc =ρ 1   ·c   1.1   c   1.2 √{square root over (( c   1.3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}{square root over (( c   1.3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}
 
     It is also clear that this constraint alone cannot determine all the missing spatial variables. 
     In order to manipulate further the individual channels. This embodiment of the present invention extrapolates the missing correlation quantities while maintaining the correlation sum constraint. It should be noted that extrapolation of such a matrix must also be such that the resulting extrapolated matrix is symmetric and positive definite. This is in fact a requirement for any matrix to be admissible as a covariance matrix. 
     Several techniques can be used from the literature in order to extrapolate the partially known covariance matrix to obtain a complete covariance matrix. The use of one method or another is within the scope of the invention. 
     According to the preferred embodiment the Maximum-Entropy principle is used as extrapolation method. This leads to an easy implementation and has shown quite good performance in terms of audio quality. 
     Accordingly, the extrapolated correlation quantities are chosen such that they maximize the determinant of the covariance matrix, i.e. 
     
       
         
           
             det 
             ⁡ 
             
               [ 
               
                 
                   
                     
                       
                         c 
                         
                           1 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           3 
                         
                         2 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           1 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           x 
                         
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           3 
                         
                       
                       ⁢ 
                       
                         ρ 
                         3 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       
                         c 
                         
                           1 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           3 
                         
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           3 
                         
                       
                       ⁢ 
                       
                         ρ 
                         3 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           1 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           3 
                         
                         2 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           2 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           4 
                         
                         2 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           2 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           4 
                         
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           4 
                         
                       
                       ⁢ 
                       
                         ρ 
                         4 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           2 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           1 
                           , 
                           4 
                         
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           4 
                         
                       
                       ⁢ 
                       
                         ρ 
                         4 
                       
                     
                   
                   
                     
                       
                         c 
                         
                           2 
                           , 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         c 
                         
                           2 
                           , 
                           4 
                         
                         2 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     Under the constraint that,
 
 R   lf,c   +R   lf,lfe   +R   rf,c   +R   rf,lfe =ρ 1   ·c   1,1   c   1,2 √{square root over (( c   1,3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}{square root over (( c   1,3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}
 
     This is a convex optimization problem and a closed form solution exists. In order to simplify the notation we will derive the solution for a generic covariance matrix, 
     
       
         
           
             Γ 
             = 
             
               [ 
               
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         lf 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         lfe 
                         , 
                         lfe 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     First it should be noted that maximizing the determinant of Γ is also equivalent to maximizing the determinant of the following matrix 
     
       
         
           
             
               Γ 
               ′ 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ] 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             R 
                             
                               lf 
                               , 
                               lf 
                             
                           
                         
                         
                           
                             R 
                             
                               lf 
                               , 
                               rf 
                             
                           
                         
                         
                           
                             R 
                             
                               lf 
                               , 
                               c 
                             
                           
                         
                         
                           
                             R 
                             
                               lf 
                               , 
                               lfe 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               lf 
                               , 
                               rf 
                             
                           
                         
                         
                           
                             R 
                             
                               rf 
                               , 
                               rf 
                             
                           
                         
                         
                           
                             R 
                             
                               rf 
                               , 
                               c 
                             
                           
                         
                         
                           
                             R 
                             
                               rf 
                               , 
                               lfe 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               lf 
                               , 
                               c 
                             
                           
                         
                         
                           
                             R 
                             
                               rf 
                               , 
                               c 
                             
                           
                         
                         
                           
                             R 
                             
                               c 
                               , 
                               c 
                             
                           
                         
                         
                           
                             R 
                             
                               c 
                               , 
                               lfe 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               lf 
                               , 
                               lfe 
                             
                           
                         
                         
                           
                             R 
                             
                               rf 
                               , 
                               lfe 
                             
                           
                         
                         
                           
                             R 
                             
                               c 
                               , 
                               lfe 
                             
                           
                         
                         
                           
                             R 
                             
                               lfe 
                               , 
                               lfe 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                       
                       
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ] 
                   
                 
               
               = 
               
                   
                 
                   [ 
                   
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             fm 
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             cm 
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             cm 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             cm 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             cm 
                           
                         
                       
                       
                         
                           R 
                           
                             cm 
                             , 
                             cm 
                           
                         
                       
                       
                         
                           R 
                           
                             cm 
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fs 
                             , 
                             cs 
                           
                         
                       
                       
                         
                           R 
                           
                             rf 
                             , 
                             lfe 
                           
                         
                       
                       
                         
                           R 
                           
                             cm 
                             , 
                             cs 
                           
                         
                       
                       
                         
                           R 
                           
                             cs 
                             , 
                             cs 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     This is also equivalent to evaluating the covariance matrix of the mono and side channel obtained from the center channels (C and LFE) and the front channels (FL,FR), namely, 
     
       
         
           
             
               [ 
               
                 
                   
                     fm 
                   
                 
                 
                   
                     fs 
                   
                 
                 
                   
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                   
                 
                 
                   
                     cs 
                   
                 
               
               ] 
             
             = 
             
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                 
                 ] 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       lf 
                     
                   
                   
                     
                       rf 
                     
                   
                   
                     
                       c 
                     
                   
                   
                     
                       lfe 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Now clearly the constraint on the matrix Γ easily translates to
 
 R   fm,cm =ρ 1   ·c   1,1   c   1,2 √{square root over (( c   1,3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}{square root over (( c   1,3   2 +2 c   1,3   c   2,3 ρ 3   +c   2,3   2 )( c   1,4   2 +2 c   1,4   c   2,4 ρ 4   +c   2,4   2 ))}
 
     The remaining unknown correlations are R fm,cs , R fs,cm  and R fm,cs  are extrapolated by using the maximization of the determinant of Γ′, the computation steps are quite cumbersome, but the results are in the end quite simple and lead to the following closed-form formulas: 
     
       
         
           
             
               
                 R 
                 
                   fm 
                   , 
                   cs 
                 
               
               = 
               
                 
                   
                     R 
                     
                       fm 
                       , 
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                     
                   
                   ⁢ 
                   
                     R 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                       , 
                       cs 
                     
                   
                 
                 
                   R 
                   
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                     , 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 R 
                 
                   fs 
                   , 
                   
                     c 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                   
                 
               
               = 
               
                 
                   
                     R 
                     
                       fm 
                       , 
                       fs 
                     
                   
                   ⁢ 
                   
                     R 
                     
                       fm 
                       , 
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                     
                   
                 
                 
                   R 
                   
                     fm 
                     , 
                     fm 
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 R 
                 
                   fs 
                   , 
                   cs 
                 
               
               = 
               
                 
                   
                     R 
                     
                       fm 
                       , 
                       fs 
                     
                   
                   ⁢ 
                   
                     R 
                     
                       fm 
                       , 
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                     
                   
                   ⁢ 
                   
                     R 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                       , 
                       cs 
                     
                   
                 
                 
                   
                     R 
                     
                       fm 
                       , 
                       fm 
                     
                   
                   ⁢ 
                   
                     R 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                       , 
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                     
                   
                 
               
             
           
         
       
     
     These quantities can therefore be extrapolated quite easily from the available data. Finally, the complete extrapolated covariance matrix Γ a simple matrix multiplication, is needed; 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         lf 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         rf 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         c 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         lfe 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         lf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         rf 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         c 
                         , 
                         lfe 
                       
                     
                   
                   
                     
                       R 
                       
                         lfe 
                         , 
                         lfe 
                       
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   
                     1 
                     4 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ] 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             fm 
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                           
                         
                       
                       
                         
                           R 
                           
                             fm 
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             fs 
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fm 
                             , 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                           
                         
                       
                       
                         
                           R 
                           
                             fs 
                             , 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                           
                         
                       
                       
                         
                           R 
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             , 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                           
                         
                       
                       
                         
                           R 
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             , 
                             cs 
                           
                         
                       
                     
                     
                       
                         
                           R 
                           
                             fs 
                             , 
                             cs 
                           
                         
                       
                       
                         
                           R 
                           
                             rf 
                             , 
                             lfe 
                           
                         
                       
                       
                         
                           R 
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             , 
                             cs 
                           
                         
                       
                       
                         
                           R 
                           
                             cs 
                             , 
                             cs 
                           
                         
                       
                     
                   
                   ] 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     These steps are also be applied in order to extrapolate the total covariance matrix of the additional two channels, i.e. KS and RS. Leading to the total extrapolated covariance matrix: 
     
       
         
           
             Re 
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 E 
                 [ 
                 
                   
                     [ 
                     
                       
                         
                           lf 
                         
                       
                       
                         
                           rf 
                         
                       
                       
                         
                           c 
                         
                       
                       
                         
                           lfe 
                         
                       
                       
                         
                           ls 
                         
                       
                       
                         
                           rs 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   [ 
                   
                     
                       lf 
                       * 
                     
                     ⁢ 
                     
                       rf 
                       * 
                     
                     ⁢ 
                     
                       c 
                       * 
                     
                     ⁢ 
                     
                       lfe 
                       * 
                     
                     ⁢ 
                     
                       ls 
                       * 
                     
                     ⁢ 
                     
                       rs 
                       * 
                     
                   
                   ] 
                 
                 ] 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           lf 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           rf 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           lf 
                           , 
                           rs 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           rf 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           rf 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           rs 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           c 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           rs 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           fle 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           lfe 
                           , 
                           lfe 
                         
                       
                     
                     
                       
                         R 
                         
                           lfe 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           lfe 
                           , 
                           rs 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           lfe 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           ls 
                           , 
                           ls 
                         
                       
                     
                     
                       
                         R 
                         
                           ls 
                           , 
                           rs 
                         
                       
                     
                   
                   
                     
                       
                         R 
                         
                           lf 
                           , 
                           rs 
                         
                       
                     
                     
                       
                         R 
                         
                           rf 
                           , 
                           rs 
                         
                       
                     
                     
                       
                         R 
                         
                           c 
                           , 
                           rs 
                         
                       
                     
                     
                       
                         R 
                         
                           lfe 
                           , 
                           rs 
                         
                       
                     
                     
                       
                         R 
                         
                           ls 
                           , 
                           rs 
                         
                       
                     
                     
                       
                         R 
                         
                           rs 
                           , 
                           rs 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     By using the same approach, i.e. converting the channels to virtual mono and side channels, it is quite easy to derive closed form formulas for the extrapolated covariance matrices. 
     So far, what has presented is a two step approach where the partial covariance matrix of the channels [lf rf c lfe] is first extrapolated and then the total covariance matrix of all channels is then extrapolated. However, another approach would consist in computing the total incomplete covariance matrix and then to globally extrapolate all correlations. The two approaches are conceptually equivalent. The second approach is however more effective since it globally extrapolates all possible correlations while the former implies a two step approach. 
     Both approaches are similar in implementation and are based on the maximum entropy (i.e. determinant maximization) approach. 
     It should be noted that all quantities depend both on time and frequency. The indexing was omitted for sake of clarity. The time index corresponds to the parameter time-slot l, while the frequency index to the processing band index m. Finally it should also be pointed out that all the resulting correlations will be defined relatively to the energy of the mono down mix signal, which is represented by a σ OTT     0   . This is in fact true for any OTT x  box, due to the presence of the term σ OTT     X     2 . 
     In the following, in order to simplify the notation the mono downmix energy normalized extrapolated covariance matrix is defined as 
     
       
         
           
             
               
                 C 
                 ~ 
               
               
                 l 
                 , 
                 m 
               
             
             = 
             
               
                 1 
                 
                   
                     σ 
                     
                       OTT 
                       0 
                     
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       l 
                       , 
                       m 
                     
                     ) 
                   
                 
               
               ⁢ 
               Re 
               ⁢ 
               
                   
               
               ⁢ 
               
                 E 
                 [ 
                 
                   
                     [ 
                     
                       
                         
                           lf 
                         
                       
                       
                         
                           rf 
                         
                       
                       
                         
                           c 
                         
                       
                       
                         
                           lfe 
                         
                       
                       
                         
                           ls 
                         
                       
                       
                         
                           rs 
                         
                       
                     
                     ] 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         lf 
                         * 
                       
                       ⁢ 
                       
                         rf 
                         * 
                       
                       ⁢ 
                       
                         c 
                         * 
                       
                       ⁢ 
                       
                         lfe 
                         * 
                       
                       ⁢ 
                       
                         ls 
                         * 
                       
                       ⁢ 
                       
                         rs 
                         * 
                       
                     
                     ] 
                   
                 
                 ] 
               
             
           
         
       
     
     The estimation and the synthesis of arbitrary channels based on extrapolated covariance matrix is described below. 
     Suppose that arbitrary channels defined as a predetermined arbitrary linear combination of the original channels are to be decoded/synthesized, for example 
     
       
         
           
             
               a 
               
                 n 
                 , 
                 k 
               
             
             = 
             
               
                 H 
                 k 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         lf 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         rf 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         c 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         lfe 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         ls 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         rs 
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Where the matrix H k  denotes a matrix of coefficients representing a description of predetermined arbitrary linear combination and a n,k , is the desired linear combination, i.e. desired output signal. The prior art direct technique would directly compute â n,k  as a simple linear combination of the output of the decoder, i.e. to apply the matrix H k  in the frequency domain to the decoded channels l{circumflex over (f)} k,n , r{circumflex over (f)} k,n , ĉ k,n , l{circumflex over (f)}e k,n , {circumflex over (l)}s k,n , {circumflex over (r)}s k,n , formally this would write as 
     
       
         
           
             
               
                 a 
                 ^ 
               
               
                 n 
                 , 
                 k 
               
             
             = 
             
               
                 H 
                 k 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           lf 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         
                           rf 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         
                           c 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         
                           lfe 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         
                           ls 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                   
                     
                       
                         
                           rs 
                           ^ 
                         
                         
                           k 
                           , 
                           n 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Which would limit the quality on the output and may cause unwanted channel correlations as well as possible cancellations. 
     As stated earlier, the output of each R-OTT box leads to a linear combination. Thus, it is easily seen that the downmix signal is in fact a linear combination of all channels. 
     The downmix signal denoted m k,n  can therefore be written as: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   m 
                   
                     n 
                     , 
                     k 
                   
                 
                 = 
                 
                   
                     
                       W 
                       
                         n 
                         , 
                         k 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               lf 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               rf 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               c 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               lfe 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               ls 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               rs 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           w 
                           lf 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           w 
                           rf 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           w 
                           c 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           w 
                           lfe 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           w 
                           ls 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           w 
                           rs 
                           
                             n 
                             , 
                             k 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               lf 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               rf 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               c 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               lfe 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               ls 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               rs 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               , 
             
           
         
       
     
     The W n,k  matrix of coefficients is known and is dependent only on the received CLDx parameters. In the case of a single channel downmix, i.e. the downmix signal consists of a mono only signal, then the matrix W n,k  is indeed a row vector as shown in the above equation. The problem can then be stated in terms of a least mean squares problem, or in general as a weighted least mean squares problem. 
     Given the mono down mix signal m n,k , a linear estimate of the channels A n,k  can be formed as: 
     â n,k =Q n,k m n,k , where Q n,k  is a matrix which needs to be optimized such as when it is applied to the downmix channels, in this case the mono channel m n,k , it should provide a result as close as the one obtained with the original linear combination, a n,k . 
     The objective is therefore to minimize the error e n,k =a n,k −â n,k  with respect to some fidelity criterion, in this case the mean square error criterion. This leads to minimization of 
     
       
         
           
             
               e 
               
                 n 
                 , 
                 k 
               
             
             = 
             
               
                 
                   
                     H 
                     k 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             lf 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             rf 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             lfe 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             ls 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             rs 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 - 
                 
                   
                     Q 
                     
                       n 
                       , 
                       k 
                     
                   
                   ⁢ 
                   
                     
                       W 
                       
                         n 
                         , 
                         k 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               lf 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               rf 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               c 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               lfe 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               ls 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               rs 
                               
                                 k 
                                 , 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               = 
               
                 
                   ( 
                   
                     
                       H 
                       k 
                     
                     - 
                     
                       
                         Q 
                         
                           n 
                           , 
                           k 
                         
                       
                       ⁢ 
                       
                         W 
                         
                           n 
                           , 
                           k 
                         
                       
                     
                   
                   ) 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           lf 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           rf 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           lfe 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           ls 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           rs 
                           
                             k 
                             , 
                             n 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Assuming that the matrices are stationary, i.e. that they can be factored out of the averaging operator, the mean squares solution to this problem can easily be solved with respect to Q n,k  resulting in 
     
       
         
           
             
               Q 
               
                 n 
                 , 
                 k 
               
             
             = 
             
               
                 
                   H 
                   k 
                 
                 ⁢ 
                 
                   C 
                   
                     n 
                     , 
                     k 
                   
                 
                 ⁢ 
                 
                   W 
                   
                     n 
                     , 
                     
                       k 
                       * 
                     
                   
                 
               
               
                 
                   W 
                   
                     n 
                     , 
                     k 
                   
                 
                 ⁢ 
                 
                   C 
                   
                     n 
                     , 
                     k 
                   
                 
                 ⁢ 
                 
                   W 
                   
                     n 
                     , 
                     
                       k 
                       * 
                     
                   
                 
               
             
           
         
       
     
     The matrix C n,k  denotes the covariance matrix of the channels, i.e. 
     
       
         
           
             
               C 
               
                 n 
                 , 
                 k 
               
             
             = 
             
               E 
               ⁡ 
               
                 [ 
                 
                   
                     [ 
                     
                       
                         
                           
                             lf 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
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     Which, as discussed earlier, may not be available at the decoder but which is extrapolated according to the technique described previously. Here the covariance matrix is shown to be complex. However, since only the real correlations are used, it can be easily shown that the result is still valid with real covariance matrices. 
     So far it have been shown that the least mean square is estimated for every hybrid sub-band k and every time slot n. In reality, a substantial amount of complexity reduction can be made by computing the mean square estimate on a certain number of time slots and then use interpolation in order to extend this to all time slots. For instance, it is beneficial to map the estimation onto the same time slots as those used for the parameters, i.e. to compute the covariance matrix only for the parameters time-slots, index l. The same technique for complexity reduction could be used by mapping the mean square estimate to be computed only for the parameter bands, index m. However, in general this is not as straightforward as for the time index since a certain amount of frequency resolution may be needed in order to efficiently represent the action of the matrix H k . In the following the sub-sampled parameter domain, i.e. l,m, is considered. 
     As already stated earlier, the covariance matrix C l,m  is known only relatively to the energy of the mono downmix signal, i.e. σ OTT     0     2 (l,m). Because of this constraint, it can be easily shown that W l,m C l,m W l,m* =σ OTT     0     2 (l,m) for all l,m. The least mean square estimate can therefore be written as
 
 Q   l,m   =H   m{tilde over (C)}   l,m   W   l,m*  
 
     It should be noted that Q l,m  depends only on know quantities which are available in the decoder. In fact, H m  is an external input, a matrix, describing the desired linear combination, while {tilde over (C)} l,m  and W l,m  are derived from the spatial parameters contained in the received bit stream. 
     The least squares estimate inherently introduces a loss in energy that can have negative effects on the quality of the synthesized channels. The loss of energy is due to the mismatch between the model when applied to the decoded signal and the real signal. In least squares terminology, this is called the noise subspace. In spatial hearing this term is called the diffuse sound field, i.e. the part of the multichannel signal which is uncorrelated or diffuse. In order to circumvent this, a number of decorrelated signals are used in order to fill the noise subspace and diffuse sound part and therefore to get an estimated signal which is psycho-acoustically similar to the wanted signal. 
     Because of the orthogonal properties of least mean squares, the energy of the desired signal can be expressed as:
 
 E[a   n,k   a   n,k*   ]=E[â   n,k   â   n,k*   ]+E[e   n,k   e   n,k* ]
 
     Thus the normalized covariance matrix of the error in the l, m domain can be expressed as
 
H m {tilde over (C)} l,m H m* −Q l,m W l,m {tilde over (C)} l,m W l,m* Q l,m*  
 
     In order to generate an estimated signal, ã n,k , which has the same psycho-acoustical characteristics as the desired signal a n,k  an error signal independent from â n,k  is generated. The error signal must have a covariance matrix which is close to that of the true error signal E[e n,k e n,k* ] and it also has to be uncorrelated from the mean squares estimate â n,k . 
     The artificial error signal, denoted by {tilde over (e)} n,k  is then added to the mean square error estimate in order to form the final estimate, ã n,k =â n,k +{tilde over (e)} n,k . 
     One way of generating a signal similar to the error signal is through the use of the decorrelation applied to the mono down-mix signal, This guarantees that the error signal is uncorrelated from the mean square estimate since â n,k  is directly dependent on the mono downmix signal. However this is insufficient in itself, the decorrelators need to be spatially shaped such that their covariance matrix matches the correlation of the true error signal E[e n,k e n,k* ]. 
     A simple way to do this is to force the generated decorrelated signals to be uncorrelated also between themselves and then to apply a correlation shaping matrix referred to as Z n,k . If d n,k  is denoted to be the vector output of the decorrelators, then the shaping matrix Z n,k  has to fulfill,
 
 Z   n,k   E[d   n,k   d   n,k*   ]Z   n,k*   =E[e   n,k   e   n,k* ]
 
     However, because E[e n,k e n,k* ] is defined only as the normalized covariance matrix, (relative to the energy of the mono downmix signal) the decorrelators have also to have a covariance matrix which is relatively defined to that of the mono downmix energy. 
     In accordance with prior art, a simple way to ensure this is to use all-pass filtering decorrelation thus leading to a normalized (with respect to the mono signal energy) covariance matrix which writes as, E[d n,k d n,k* ]=I, i.e. the identity matrix and then apply a shaping matrix Z n,k . 
     It can be easily seen that a simple Cholesky factorization of E[e n,k e n,k* ]=Z n,k Z n,k*  can produce a suitable matrix Z n,k . Of course another factorization is also possible, e.g. by using the Eigen-vectors and Eigen-values of the normalized error covariance matrix. In addition, an advantage is obtained by evaluating the matrix Z n,k  only in the parameter domain, i.e. l,m. 
     Finally, the total synthesis can be written as:
 
 ã   n,k   =Q   n,k   m   n,k   +Z   n,k   d   n,k  
 
     Where the matrix Q n,k  is obtained by interpolating the matrix Q l,m =H m {tilde over (C)} l,m W l,m*  in the time domain (i.e. from l to n) and by mapping the sub-band parameter bands to the hybrid bands (i.e. from m to k). 
     And similarly the matrix Z n,k  is obtained by interpolating and mapping the matrix Z l,m  defined by the equation
 
 Z   n,k   Z   n,k*   =H   m   {tilde over (C)}   l,m   H   m*   −Q   l,m   W   l,m   {tilde over (C)}   l,m   W   l,m*   Q   l,m*  
 
       FIG. 10   b , summarizes and illustrates the arrangement used in order to synthesize arbitrary channels according to an embodiment of the present invention described above. The reference signs correspond to the reference signs of  FIG. 10   a . In this embodiment the estimator  903  comprises a unit  905  configured to determine a matrix Q by minimizing a mean square error (i.e. e n,k =a n,k −â n,k ) between the estimated linear combination of the multi-channel surround audio signal and the arbitrary predetermined linear combination of the multi-channel surround audio signal. It should be noted that one does not have to have access to the arbitrary predetermined linear combination of the multichannel surround sound signal, it is enough to have knowledge of the covariance matrix of the original multichannel signals in order to form an estimate of the said linear combination of the multichannel surround sound signal. The latter is obtained from the received bit stream through forming a partially known covariance matrix and then extrapolating it by the use of principles such as the maximum entropy principle. 
     Moreover, the estimator  903  comprises a further unit  907  configured to multiply Q n,k  with the downmix signal to obtain the estimate  913  of the linear combination of a multi-channel surround audio signal. The estimator  913  further comprises a unit  905  adapted to determine a decorrelated signal shaping matrix Z n,k  indicative of the amount of decorrelated signals. In this embodiment, the synthesizer  904  is configured to synthesize the linear combination by computing  908 ,  909  Z n,k* d n,k , and then ã n,k =Q n,k m n,k +Z n,k d n,k , where d n,k  is “a decorrelation signal”, for each frequency band and each time slot to compensate for energy losses. Further, the arrangement also comprises an interpolating and mapping unit  906 . This unit can be configured to interpolate the matrix Q l,m  in the time domain and to map downsampled frequency bands m to hybrid bands k and to interpolate the matrix Z l,m  in the time domain and to map downsampled frequency bands m to hybrid bands k. The extrapolator  902   b  may as stated above use the Maximum-Entropy principle by selecting extrapolated correlation quantities such that they maximize the determinant of the covariance matrix under a predetermined constraint. 
     Turning now to  FIG. 11  showing a flowchart of an embodiment of the present invention. The method comprises the steps of: 
       1000 . Receive a description H of the arbitrary predetermined linear combination. 
       1001 . Receive a decoded downmix signal of the multi-channel surround audio signal. 
       1002 . Receive spatial parameters comprising correlations and channel level differences of the multi-channel audio signal. 
       1003 . Obtain a partially known spatial covariance matrix based on the received spatial parameters comprising correlations and channel level differences of the multi-channel audio signal. 
       1004 . Extrapolate the partially known spatial covariance matrix to obtain a complete spatial covariance matrix, 
       1005 . Form according to a fidelity criterion an estimate of said arbitrary predetermined linear combination of the multi-channel surround audio signal based at least on the extrapolated complete spatial covariance matrix, the received decoded downmix signal and the said description of the arbitrary predetermined linear combination. 
       1006 . Synthesize said arbitrary predetermined linear combination of a multi-channel surround audio signal based on said estimate of the arbitrary predetermined linear combination of the multi-channel surround audio signal. 
     Step  1005  may comprise the further steps of: 
       1005   a . Determine a matrix Q by minimizing a mean square error between the estimated linear combination of the multi-channel surround audio signal and the arbitrary predetermined linear combination of the multi-channel surround audio signal. 
       1005   b . Multiply Q with the downmix signal to obtain the estimate of the arbitrary predetermined linear combination of a multi-channel surround audio signal. 
       1005   c . Determine a decorrelated signal shaping matrix Z indicative of the amount of decorrelated signals. 
       1005   d , Interpolate Q and Z in the time domain. 
       1005   e . Map downsampled frequency bands m to hybrid bands k. 
     The method may be implemented in a decoder of a mobile terminal. 
     The present invention is not limited to the above-described preferred embodiments. Various alternatives, modifications and equivalents may be used. Therefore, the above embodiments should not be taken as limiting the scope of the invention, which is defined by the appending claims. 
     Abbreviations 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 AAC 
                 Advanced Audio Coding 
               
               
                   
                 AMR−WB+ 
                 extended adaptive multirate wide band 
               
               
                   
                 C 
                 Center 
               
               
                   
                 CLD  
                 channel level differences 
               
               
                   
                 HR 
                 Head Related 
               
               
                   
                 HRF 
                 Head Related Filters 
               
               
                   
                 HRTF 
                 Head Related Transfer Function 
               
               
                   
                 IC 
                 inter-channel coherence 
               
               
                   
                 ICC 
                 correlation 
               
               
                   
                 ILD 
                 inter-channel level differences 
               
               
                   
                 ITD 
                 inter-channel time differences 
               
               
                   
                 L 
                 left 
               
               
                   
                 LFE 
                 low frequency element 
               
               
                   
                 MPEG 
                 Moving Picture Experts Group 
               
               
                   
                 OTT 
                 One-to-two 
               
               
                   
                 PCM 
                 Pulse Code Modulation 
               
               
                   
                 PDA 
                 Personal Digital assistant 
               
               
                   
                 R 
                 right 
               
               
                   
                 R-OTT 
                 Reversed one-to-two 
               
               
                   
                 SL 
                 surround left 
               
               
                   
                 SR 
                 Surround Right