EP2345029B1 - Method, computer program and device for decoding a digital audio signal - Google Patents

Description

The present invention relates to the field of coding digital signals.

The invention is advantageously applied to the coding of sounds with alternating speech and music.

To effectively code speech sounds, CELP (Code Excited Linear Prediction) techniques are recommended. To effectively encode musical sounds, transform coding techniques are preferred.

CELP coders are predictive coders. They aim to model the production of speech from various elements: a long-term prediction to model the vibration of vocal chords in voiced period, a stochastic excitation (white noise, algebraic excitation), and a short prediction -term to model vocal tract changes.

Transform coders use critical-sampling transforms to compact the signal in the transformed domain. A "critical-sampling transform" is a transform for which the number of coefficients in the transformed domain is equal to the number of coefficients of the digitized sound.

One solution for effectively coding a signal containing these two types of content is to select the best technique over time. This solution has been recommended by the 3GPP ("3rd Generation Partnership Project") standardization organization, and a technique called AMR WB + has been proposed in the document 3rd Generation Partnership Project; Technical Specification Group Service and System Aspects; Audio Codec processing functions; Extended AMR Wideband codec; Transcoding functions (Release 6) ", 3GPP standard; 3GPP TS 26.290, 3RD Geeneration Partnership Project (3GPP), 1 May 2004, pages 1-71 , XP050370247.

This technique is based on CELP technology of the AMR WB type and transform coding based on a Fourier transform overlay.

This solution suffers from insufficient quality on the music. This deficiency comes particularly from transform coding. Indeed, the overlapping Fourier transform is not a critical sampling transformation, and therefore, it is suboptimal.

In addition, the windows used in this encoder are not optimal vis-à-vis the concentration of energy: the frequency forms of these windows are relatively fixed.

Critical sampling transformations are known. For example, the transforms used in MP3 and AAC type music encoders. These transforms are based on the formalism called TDAC ("Time Domain Aliasing Cancellation").

The use of TDAC makes it possible to obtain excellent quality on the music. Nevertheless, this has the disadvantage of introducing temporal folds that make it difficult to combine with CELP type technologies.

Indeed, during a transition from TDAC to CELP the temporal folding of the TDAC portion is not canceled by the signal from the CELP which does not integrate any folding.

An object of the present invention is to propose a technique for reconstructing an audio signal, with good quality, by alternating transform coding techniques (for example with critical sampling) and predictive coding techniques (for example of the CELP type ).

• coder une première séquence d'échantillons du signal numérique audio selon un codage par transformée ;
• coder une deuxième séquence d'échantillons du signal numérique audio selon un codage prédictif ;

et dans lequel la deuxième séquence commence avant la fin de la première séquence, une sous-séquence commune aux première et deuxième séquences étant ainsi codée à la fois par codage prédictif et par codage par transformée.For this purpose, the present invention relates to a decoding method applied after encoding of a digital audio signal, said coding comprising the steps:

• encoding a first sequence of samples of the digital audio signal according to transform coding;
• encoding a second sequence of samples of the digital audio signal according to a predictive coding;

and wherein the second sequence begins before the end of the first sequence, whereby a subsequence common to the first and second sequences is encoded by both predictive and transform coding.

Thus, during the decoding of the digital audio signal, the folding created by the coding in the subsequence of the first sequence can be eliminated by means of samples of this subsequence resulting from the decoding of the sub-sequence within the second sequence. In addition, the second sequence can be decoded because the samples of the past, useful for predictive decoding, do not have this folding.

Advantageously, the transform coding is a critical sampling transform coding.

For example, transform coding is a TDAC type transform coding.

For example, predictive coding is a CELP encoding.

• une première partie nominale,
• une deuxième partie terminale sensiblement nulle,
• une troisième partie intermédiaire sensiblement continue entre les première et deuxième parties.

In an advantageous embodiment, the transform coding of the first sequence comprises the application of an analysis window making it possible to deduce from a perfect reconstruction relation of the digital signal a synthesis window comprising at least three parts:

• a first nominal part,
• a second terminal portion substantially zero,
• a third substantially continuous intermediate portion between the first and second portions.

It is then expected that at least the parts of the analysis window for respectively deducing the second and third parts of the synthesis window are applied to the subsequence common to both sequences.

"Substantially continuous" means that the third part makes it possible to have no discontinuity between the first and second parts. Indeed, this type of discontinuity reduces the quality of decoding by adding decoding noise.

The perfect reconstruction relation imposes a relation between the forms of the windows of analysis and synthesis. In addition, when switching between a transform coding and a predictive coding, it is possible to describe the analysis window or the synthesis window in an equivalent manner. Indeed, in this case, the relation of reconstruction makes appear a direct relation between the two forms.

With an analysis window (and therefore synthesis) thus chosen, it is possible to reduce the area in which the folding appears on the decoding of the first sequence.

With the window thus defined, it is possible to reduce the number of samples of the second sequence (predictive coding) to be transmitted for decoding.

In addition, the additional number of samples is related to the size of the intermediate part.

For example, the middle part is a sinus arch. For example again, the intermediate part is a function derived from "Kaiser-Bessel". In addition, it can come from a window optimization calculation and not have an explicit expression.

For example, the summary window is an asymmetric window.

Thus, it is possible to adapt the profile of the synthesis window (thus the analysis window) to the coding of the following sequence or preceding the first sequence.

In an advantageous embodiment, the synthesis window further comprises a fourth continuous initial portion between a substantially zero value and a non-zero value of the first part.

Thus, it is possible to minimize the impact of the transition between transform coding and predictive coding on transform coding.

For example, the fourth part of the synthesis window is a smooth transition between an initial value and a value of the nominal part, and the third part is an abrupt transition between a value of the nominal part and a value of the substantially zero part. .

Thus, a better concentration of the signal energy in the frequency domain is obtained for a better coding efficiency of the transformed part.

It can be provided that the first and second sequences belong to the same frame of the digital signal.

Thus, the encoding of the first sequence can be used as a transition encoding after the encoding of a frame by transform coding. This makes it possible to improve the coding efficiency by not disturbing this frame.

• recevoir un vecteur de transformée codant une première séquence d'échantillons du signal numérique audio selon un codage par transformée ;
• recevoir un vecteur de prédiction codant une deuxième séquence d'échantillons du signal numérique audio selon un codage de type CELP utilisant une prédiction à long terme;

dans lequel la deuxième séquence commence avant la fin de la première séquence, une sous-séquence commune aux première et deuxième séquences étant ainsi reçue codée à la fois par le codage de type CELP utilisant une prédiction à long terme et par le codage par transformée ; et qui comporte en outre les étapes :

1. a) appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder une sous-séquence de la première séquence non codée par codage de type CELP utilisant une prédiction à long terme;
2. b) décoder au moins dans le vecteur de prédiction la sous-séquence commune aux première et deuxième séquences au moins par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu de l'étape a);
3. c) décoder dans le vecteur prédictif par un décodage de type CELP utilisant une prédiction à long terme une sous-séquence de la deuxième séquence non codée par codage par transformée, en se fondant sur au moins un échantillon issu de l'une des étapes a) et b).

The present invention provides a method for decoding an audio digital signal, comprising the steps of:

• receiving a transform vector encoding a first sequence of samples of the digital audio signal according to transform coding;
• receiving a prediction vector encoding a second sequence of samples of the digital audio signal according to a CELP encoding using long-term prediction;

wherein the second sequence begins before the end of the first sequence, whereby a common subsequence to the first and second sequences is received encoded by both CELP encoding using long-term prediction and transform coding; and which further comprises the steps:

1. a) applying to the transform vector an inverse transformation of the transform coding for decoding a subsequence of the first uncoded coding sequence of CELP type using long-term prediction;
2. b) decoding at least in the prediction vector the subsequence common to the first and second sequences at least by a CELP type decoding using a long-term prediction based on at least one sample from step a);
3. c) decoding in the predictive vector by a CELP-type decoding using a long-term prediction a subsequence of the second non-coded sequence by transform coding, based on at least one sample from one of the steps a ) and B).

Thus, the folding present in the decoded subsequence can be eliminated by using decoded CELP-decoded samples using long-term prediction.

• b1) décoder dans le vecteur prédictif la sous-séquence commune aux première et deuxième séquences par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu de l'étape a) ;
• b2) appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder la sous-séquence commune aux première et deuxième séquences ; et b3) décoder la sous-séquence commune aux première et deuxième séquences par combinaison d'au moins un échantillon issu de l'étape b1) avec un échantillon correspondant issu de l'étape b2).

In an advantageous embodiment, step b) comprises the sub-steps:

• b1) decoding in the predictive vector the common subsequence to the first and second sequences by a CELP type decoding using long-term prediction based on at least one sample from step a);
• b2) applying to the transform vector an inverse transformation of the transform coding for decoding the common subsequence to the first and second sequences; and b3) decoding the common subsequence to the first and second sequences by combining at least one sample from step b1) with a corresponding sample from step b2).

For example, the combination is a linear combination. By thus combining the samples, a more robust decoding is obtained.

• b4) décoder dans le vecteur prédictif la sous-séquence commune aux première et deuxième séquences par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu de l'étape a);
• b5) créer à partir d'au moins un échantillon issu de l'étape b4) un échantillon contenant un repliement équivalent à un codage par transformée suivi d'un décodage par transformée ;
• b6) appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder la sous-séquence commune aux première et deuxième séquences ; et
• b7) décoder la sous-séquence commune aux première et deuxième séquences par combinaison d'au moins un échantillon issu de l'étape b5) avec un échantillon correspondant issu de l'étape b6).

In another advantageous embodiment, step b) comprises the sub-steps:

• b4) decoding in the predictive vector the common subsequence to the first and second sequences by a CELP type decoding using long-term prediction based on at least one sample from step a);
• b5) creating from at least one sample from step b4) a sample containing a folding equivalent to a transform coding followed by a transform decoding;
• b6) applying to the transform vector an inverse transformation of the transform coding for decoding the common subsequence to the first and second sequences; and
• b7) decoding the common subsequence to the first and second sequences by combining at least one sample from step b5) with a corresponding sample from step b6).

Thus, the folding created by step b5) corresponds exactly to the folding present in the decoded subsequence.

The creation of the folding can be done by applying a matrix representing direct and inverse transformation operations. Such a matrix may be equivalent to the application of transform coding immediately followed by transform decoding.

Of course, one can use the same predictive coding for all samples.

Similarly, the same coding / decoding by transform can be used, with the same windows of analysis and synthesis, each time such coding / decoding is performed.

• une première partie nominale,
• une deuxième partie terminale sensiblement nulle,
• une troisième partie intermédiaire continue entre les première et deuxième zones,

et au moins les deuxième et troisième parties sont appliquées à des échantillons codant la sous-séquence commune aux deux séquences.In one embodiment, step a) comprises the application of a synthesis window comprising at least three parts:

• a first nominal part,
• a second terminal portion substantially zero,
• a third intermediate portion continuous between the first and second zones,

and at least the second and third portions are applied to samples coding the subsequence common to both sequences.

The present invention also provides a computer program comprising instructions for implementing the decoding method as described, when the program is executed by a processor.

The present invention provides a decoding entity adapted to implement the decoding method as described.

• d'un vecteur de transformée codant une première séquence d'échantillons du signal numérique audio selon un codage par transformée ; et
• d'un vecteur de prédiction codant une deuxième séquence d'échantillons du signal numérique audio selon un codage de type CELP utilisant une prédiction à long terme;

dans lequel la deuxième séquence commence avant la fin de la première séquence, une sous-séquence commune aux première et deuxième séquences étant ainsi codée à la fois par le codage de type CELP utilisant une prédiction à long terme et par le codage par transformée ; et qui comporte en outre :

• un premier décodeur pour appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder une sous-séquence de la première séquence non codée par codage de type CELP utilisant une prédiction à long terme ;
• un deuxième décodeur pour décoder au moins dans le vecteur prédictif la sous-séquence commune aux première et deuxième séquences au moins par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu du premier décodeur par transformée ; et
• un troisième décodeur prédictif pour décoder dans le vecteur prédictif par un décodage de type CELP utilisant une prédiction à long terme une sous-séquence de la deuxième séquence non codée par codage par transformée, en se fondant sur au moins un échantillon issu de l'un des premier et deuxième codeurs.

It is possible to provide an audio digital signal decoding entity, comprising receiving means:

• a transform vector encoding a first sequence of samples of the digital audio signal according to transform coding; and
• a prediction vector encoding a second sequence of samples of the digital audio signal according to a CELP encoding using long-term prediction;

wherein the second sequence begins before the end of the first sequence, a common subsequence to the first and second sequences thus encoded by both CELP coding using long-term prediction and transform coding; and which further comprises:

• a first decoder for applying to the transform vector an inverse transformation of the transform coding for decoding a subsequence of the first uncoded coding sequence of the CELP type using long-term prediction;
• a second decoder for decoding at least in the predictive vector the subsequence common to the first and second sequences at least by a CELP type decoding using a long-term prediction based on at least one sample from the first transform decoder; and
• a third predictive decoder for decoding in the predictive vector by a CELP type decoding using a long-term prediction a subsequence of the second non-coded sequence by transform coding, based on at least one sample from one first and second encoders.

• des premiers moyens pour décoder dans le vecteur prédictif la sous-séquence commune aux première et deuxième séquences par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu du premier décodeur par transformée ;
• des deuxièmes moyens pour appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder la sous-séquence commune aux première et deuxième séquences; et
• des troisièmes moyens pour décoder la sous-séquence commune aux première et deuxième séquences par combinaison d'au moins un échantillon issu des premiers moyens avec un échantillon correspondant issu des deuxièmes moyens.

In an advantageous embodiment, the second decoder comprises:

• first means for decoding in the predictive vector the common subsequence of the first and second sequences by a CELP type decoding using long-term prediction based on at least one sample from the first transform decoder;
• second means for applying to the transform vector an inverse transformation of the transform coding for decoding the common subsequence to the first and second sequences; and
• third means for decoding the common subsequence to the first and second sequences by combining at least one sample from the first means with a corresponding sample from the second means.

• des premiers moyens pour décoder dans le vecteur prédictif la sous-séquence commune aux première et deuxième séquences par un décodage de type CELP utilisant une prédiction à long terme en se fondant sur au moins un échantillon issu du premier décodeur par transformée ;
• des quatrièmes moyens pour créer à partir d'au moins un échantillon restitué par les premiers moyens un échantillon contenant un repliement équivalent à un codage par transformée suivi d'un décodage par transformée ;
• des cinquièmes moyens pour appliquer au vecteur de transformée une transformée inverse du codage par transformée pour décoder la sous-séquence commune aux première et deuxième séquences ; et
• des sixièmes moyens pour décoder la sous-séquence commune aux première et deuxième séquences par combinaison d'au moins un échantillon issu des quatrièmes moyens avec un échantillon correspondant issu des cinquièmes moyens.

In another advantageous embodiment, the second decoder comprises:

• first means for decoding in the predictive vector the common subsequence of the first and second sequences by a CELP type decoding using long-term prediction based on at least one sample from the first transform decoder;
• fourth means for creating from at least one sample returned by the first means a sample containing folding equivalent to transform coding followed by transform decoding;
• fifth means for applying to the transform vector an inverse transformation of the transform coding for decoding the common subsequence to the first and second sequences; and
• sixth means for decoding the common subsequence to the first and second sequences by combining at least one sample from the fourth means with a corresponding sample from the fifth means.

Of course, all the means performing the same type of coding or decoding (predictive type CELP using long-term prediction or transform) can be combined in the same unit.

Similarly, one can provide a single unit (coding or decoding) to perform sometimes coding, respectively decoding, predictive and transform.

Of course, the encoders / decoders described may comprise a signal processor, storage elements, as well as means of communication between these elements.

For example, it is possible to alternate transformation coding techniques, for example with TDAC-type critical sampling, and predictive coding techniques, for example of the CELP type over time, in order to obtain a good quality of reconstruction.

For this purpose, particular temporal relationships can be used between the two types of coding: the time position of the CELP and transformed frames being temporally offset.

In an advantageous example, it is also proposed to extend the duration of the frames, or sequences covered by the CELP coding, by an overlap, during a transition from transform to CELP. This duration can be variable in time if the transform requires a good frequency concentration.

The duration of use of the CELP coding can be variable from one frame to another, in order to quickly adapt the coding technique to changes in the nature of the sounds.

According to an advantageous example, a frame of M samples can be subdivided into several subframes mixing portions encoded in CELP and others in the transformed domain.

The invention finds application in sound coding systems, in particular in standardized speech coders, in particular to the ITU ("International Telecommunication Union") or to the ISO ("International Standard Organization") for coding generic sounds, including speech signals.

• la figure 1 illustre deux fenêtres de synthèse d'un codage par transformée,
• la figure 2 illustre des fenêtres de synthèse d'un mode de réalisation de l'invention,
• la figure 3 illustre des trames de donnée traitées par des fenêtres de synthèse,
• la figure 4 illustre des vecteurs d'échantillons obtenus par application des fenêtres de synthèse,
• la figure 5 illustre le cas d'un codage TDAC suivi par un codage AMR WB, puis suivi d'un codage TDAC selon un mode de réalisation de l'invention,
• la figure 6 illustre le même cas de codage avec une fenêtre asymétrique avantageuse,
• la figure 7 illustre un contexte général d'un problème résolu par l'invention,
• la figure 8 illustre un schéma général de résolution de ce problème par la présente invention,
• la figure 9 illustre les étapes d'un mode de réalisation d'un procédé de codage,
• la figure 10 illustre la composition d'une fenêtre de synthèse selon un mode de réalisation de l'invention,
• la figure 1 1 illustre les étapes d'un mode de réalisation d'un procédé de décodage selon la présente invention,
• la figure 12 illustre un décodage avantageux utilisé dans le procédé de décodage,
• la figure 13 illustre une variante de ce décodage avantageux,
• la figure 14 illustre un codeur,
• la figure 15 illustre un décodeur selon un mode de réalisation de l'invention,
• la figure 16 illustre un dispositif matériel adapté pour réaliser un exemple de codeur ou un décodeur selon un mode de réalisation de la présente invention.

Other features and advantages of the invention will appear on examining the detailed description below, and the appended figures among which:

• the figure 1 illustrates two windows of synthesis of a transform coding,
• the figure 2 illustrates synthetic windows of an embodiment of the invention,
• the figure 3 illustrates frames of data processed by synthesis windows,
• the figure 4 illustrates sample vectors obtained by application of synthesis windows,
• the figure 5 illustrates the case of a TDAC coding followed by an AMR WB coding, followed by a TDAC coding according to one embodiment of the invention,
• the figure 6 illustrates the same case of coding with an advantageous asymmetric window,
• the figure 7 illustrates a general context of a problem solved by the invention,
• the figure 8 illustrates a general scheme of solving this problem by the present invention,
• the figure 9 illustrates the steps of an embodiment of a coding method,
• the figure 10 illustrates the composition of a synthesis window according to one embodiment of the invention,
• the figure 1 1 illustrates the steps of an embodiment of a decoding method according to the present invention,
• the figure 12 illustrates an advantageous decoding used in the decoding method,
• the figure 13 illustrates a variant of this advantageous decoding,
• the figure 14 illustrates an encoder,
• the figure 15 illustrates a decoder according to one embodiment of the invention,
• the figure 16 illustrates a hardware device adapted to perform an exemplary encoder or decoder according to an embodiment of the present invention.

In the following, we begin by describing a perfect reconstruction TDAC transformation, then we present a technique to make it compatible with critical sampling. Finally, a CELP coding and a combination of this coding with TDAC coding are described.

TDAC and perfect reconstruction

(Fe étant la fréquence d'échantillonnage). Pour une trame donnée d'indice t, les échantillons sont notés x _n+tM pour chaque instant n+tM.A digitized sound signal is considered according to a sampling period $\frac{1}{F_e}$

(Fe being the sampling frequency). For a given frame of index t, the samples are denoted x _{n + tM} for each moment n + tM .

• M représente la taille de la transformée,
• X_l,k sont les échantillons dans le domaine transformé pour la trame t,
• $p_k\left(n\right)=h_a\left(n\right)C_{n,k}=\sqrt{\frac{2}{M}}{h}_a\left(n\right)\cos \left[\frac{\pi }{4M}\left(2n+1+M\right)\left(2k+1\right)\right]$
  
  est une fonction de base de la transformée dont :
  • le terme h_a (n) est appelé filtre prototype ou "fenêtre de pondération d'analyse" et couvre 2M échantillons,
  • et dont le terme C_n,k définit la modulation.

The expression of the TDAC transform to the frame encoding is shown below: $$X_{t,k}={\displaystyle \underset{}{\overset{2M-1}{<figure-callout\; id="2"\; label="\Sigma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\Sigma </figure-callout>}}}{x}_{\mathrm{not}+\mathit{tM}}{p}_k\left(\mathrm{not}\right)0\le k<M,$$

• M represents the size of the transform,
• X _{l, k} are the samples in the transformed domain for the frame t,
• $p_k\left(\mathrm{not}\right)=h_{\mathrm{at}}\left(\mathrm{not}\right){\mathrm{VS}}_{\mathrm{not},k}=\sqrt{\frac{2}{M}}{h}_{\mathrm{at}}\left(\mathrm{not}\right)\cos \left[\frac{\pi }{4M}\left(2\mathrm{not}+1+M\right)\left(2k+1\right)\right]$
  
  is a basic function of the transform including:
  • the term h _a ( n ) is called a prototype filter or "analysis weight window" and covers 2 M samples,
  • and whose term C _{n, k} defines the modulation.

où $p_k^s\left(n\right)=h_s\left(n\right)C_{n,k}$

définit la transformée de synthèse, la fenêtre de pondération de synthèse étant notée h_s (n) et couvrant également 2M échantillons.To restore the initial time samples, the following inverse transform, at decoding, is applied in order to reconstruct the samples 0 ≤ n < M which are then in an overlapping area of two consecutive transforms. The decoded samples are then given by: $${\hat{x}}_{\mathrm{not}+\mathit{tM}+M}={\displaystyle \underset{k=0}{\overset{M-1}{\Sigma }}}\left[X_{t+1,k}{p}_k^s\left(\mathrm{not}\right)+X_{t,k}{p}_k^s\left(\mathrm{not}+M\right)\right],$$

or $p_k^s\left(\mathrm{not}\right)=h_s\left(\mathrm{not}\right){\mathrm{VS}}_{\mathrm{not},k}$

defines the synthesis transform, the synthetic weighting window being denoted h _s ( n ) and also covering 2 M samples.

The reconstruction equation giving the decoded samples can also be written in the following form: $$\begin{array}{ll}{\hat{x}}_{\mathrm{not}+\mathit{tM}+M}& ={\displaystyle \underset{k=0}{\overset{M-1}{\Sigma }}}\left[X_{t+1,k}{h}_s\left(\mathrm{not}\right){\mathrm{VS}}_{k,\mathrm{not}}+X_{t,k}{h}_s\left(\mathrm{not}+M\right){\mathrm{VS}}_{k,\mathrm{not}+M}\right]\\ & =h_s\left(\mathrm{not}\right){\displaystyle \underset{k=0}{\overset{M-1}{\Sigma }}}{X}_{t+1,k}{\mathrm{VS}}_{k,\mathrm{not}}+h_s\left(\mathrm{not}+M\right){\displaystyle \underset{k=0}{\overset{M-1}{\Sigma }}}{X}_{t,k}{\mathrm{VS}}_{k,\mathrm{not}+M}\end{array}$$

This other presentation of the reconstruction equation amounts to considering that two inverse cosine transforms can be carried out successively on the samples in the transformed domain X _{t, k} and X _{t + 1, k} , their result being then combined by an operation of weighting and addition.

$$\left[\begin{array}{c}{X}_{1,0}\\ X_{1,1}\\ ⋮\\ X_{1,M-1}\end{array}\right]=\left[\begin{array}{cccc}{C}_{0,0}& C_{0,1}& \cdots & C_{0,2M-1}\\ C_{1,0}& C_{1,1}& \cdots & C_{1,2M-1}\\ ⋮& ⋮& \ddots & ⋮\\ C_{M-1,0}& C_{M-1,1}& \cdots & C_{M-1,2M-1}\end{array}\right]\cdot \left[\begin{array}{cccc}{h}_{a1}\left(0\right)& 0& \cdots & 0\\ 0& h_{a1}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{a1}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{c}{x}_M\\ x_{M+1}\\ ⋮\\ x_{3M-1}\end{array}\right]$$

It is the addition of two consecutive frames which makes it possible to suppress the so-called folded components of the transformation. Indeed, if we write the direct and inverse transformation operations in matrix form for frames t = 0 and t = 1 we have: $$\left[\begin{array}{c}{X}_{0,0}\\ X_{0,1}\\ ⋮\\ X_{0,M-1}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{VS}}_{0,0}& {\mathrm{VS}}_{0,1}& \cdots & {\mathrm{VS}}_{0,2M-1}\\ {\mathrm{VS}}_{1,0}& {\mathrm{VS}}_{1,1}& \cdots & {\mathrm{VS}}_{1,2M-1}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{VS}}_{M-1,0}& {\mathrm{VS}}_{M-1,1}& \cdots & {\mathrm{VS}}_{M-1,2M-1}\end{array}\right]\cdot \left[\begin{array}{cccc}{h}_{\mathrm{at}0}\left(0\right)& 0& \cdots & 0\\ 0& h_{\mathrm{at}0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{\mathrm{at}0}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{2M-1}\end{array}\right]$$

$$\left[\begin{array}{c}{X}_{1,0}\\ X_{1,1}\\ ⋮\\ X_{1,M-1}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{VS}}_{0,0}& {\mathrm{VS}}_{0,1}& \cdots & {\mathrm{VS}}_{0,2M-1}\\ {\mathrm{VS}}_{1,0}& {\mathrm{VS}}_{1,1}& \cdots & {\mathrm{VS}}_{1,2M-1}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{VS}}_{M-1,0}& {\mathrm{VS}}_{M-1,1}& \cdots & {\mathrm{VS}}_{M-1,2M-1}\end{array}\right]\cdot \left[\begin{array}{cccc}{h}_{\mathrm{at}1}\left(0\right)& 0& \cdots & 0\\ 0& h_{\mathrm{at}1}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{\mathrm{at}1}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{c}{x}_M\\ x_{M+1}\\ ⋮\\ x_{3M-1}\end{array}\right]$$

$$\left[\begin{array}{c}{\tilde{x}}_{0,0}\\ {\tilde{x}}_{0,0}\\ ⋮\\ {\tilde{x}}_{0,2M-1}\end{array}\right]=\left[\begin{array}{cccc}{h}_{s0}\left(0\right)& 0& \cdots & 0\\ 0& h_{s0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{s0}\left(2M-1\right)\end{array}\right]\cdot S\cdot \left[\begin{array}{cccc}{h}_{a0}\left(0\right)& 0& \cdots & 0\\ 0& h_{a0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{a0}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{2M-1}\end{array}\right]$$

At the synthesis, we obtain: $$\left[\begin{array}{c}{\tilde{x}}_{0,0}\\ {\tilde{x}}_{0,0}\\ ⋮\\ {\tilde{x}}_{0,2M-1}\end{array}\right]=\left[\begin{array}{cccc}{h}_{s0}\left(0\right)& 0& \cdots & 0\\ 0& h_{s0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{s0}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{cccc}{\mathrm{VS}}_{0,0}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,0}& \cdots & {\mathrm{VS}}_{M-1,0}\\ {\mathrm{<figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{0,1}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,1}& \cdots & {\mathrm{VS}}_{M-1,1}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{VS}}_{0,2M-1}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,2M-1}& \cdots & {\mathrm{<figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout>}}_{2M-1,M-1}\end{array}\right]<figure-callout\; id="0"\; label="\cdot \; X"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>\left[\begin{array}{c}{X}_{}\end{array}\right]\left[\begin{array}{c}{}_{0,0}\\ {\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="X"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{0,1}\\ ⋮\\ x_{0,M-1}\end{array}\right]$$

$$\left[\begin{array}{c}{\tilde{x}}_{0,0}\\ {\tilde{x}}_{0,0}\\ ⋮\\ {\tilde{x}}_{0,2M-1}\end{array}\right]=\left[\begin{array}{cccc}{h}_{s0}\left(0\right)& 0& \cdots & 0\\ 0& h_{s0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{s0}\left(2M-1\right)\end{array}\right]\cdot S\cdot \left[\begin{array}{cccc}{h}_{\mathrm{at}0}\left(0\right)& 0& \cdots & 0\\ 0& h_{\mathrm{at}0}\left(1\right)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & h_{\mathrm{at}0}\left(2M-1\right)\end{array}\right]\cdot \left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{2M-1}\end{array}\right]$$

$$S=\left[\begin{array}{cc}{I}_M-J_M& 0_M\\ 0_M& I_M+J_M\end{array}\right]$$

• I_M étant la matrice carrée identité de taille M,
• J_M étant la matrice carrée anti-identité de taille M, qui à une série de valeurs d'indices croissants, renvoie la même série de valeur avec les indices décroissants,
• 0 _M est une matrice carrée de taille M ne contenant que des zéros.

With $$S=\cdot \left[\begin{array}{cccc}{\mathrm{<figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{0,0}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,0}& \cdots & {\mathrm{VS}}_{M-1,0}\\ {\mathrm{<figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{0,1}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,1}& \cdots & {\mathrm{VS}}_{M-1,1}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{VS}}_{0,2M-1}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,2M-1}& \cdots & {\mathrm{<figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout>}}_{2M-1,M-1}\end{array}\right]\cdot \left[\begin{array}{cccc}{\mathrm{VS}}_{0,0}& {\mathrm{<figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{0,1}& \cdots & {\mathrm{<figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{0,2M-1}\\ {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="0"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,0}& {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,1}& \cdots & {\mathrm{<figure-callout\; id="1"\; label="VS"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="VS"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">VS</figure-callout></figure-callout>}}_{1,2M-1}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{VS}}_{M-1,0}& {\mathrm{VS}}_{M-1,1}& \cdots & {\mathrm{VS}}_{M-1,2M-1}\end{array}\right]$$

$$S=\left[\begin{array}{cc}{I}_M-J_M& 0_M\\ 0_M& I_M+J_M\end{array}\right]$$

• I _M being the square matrix identity of size M,
• J _M being the anti-identity square matrix of size M, which at a series of increasing index values, returns the same series of values with decreasing indices,
• 0 _M is a square matrix of size M containing only zeros.

et par analogie en utilisant la trame t=1 : $$\{\begin{array}{l}{\tilde{x}}_{1,n}=h_{s1,n}\left[h_{a1,n}{x}_{M+n}-h_{a1,M-1-n}{x}_{2M-1-n}\right]\\ {\tilde{x}}_{1,M+n}=h_{s1,M+n}\left[h_{a1,M+n}{x}_{2M+n}+h_{a1,2M-1-n}{x}_{3M-1-n}\right]\end{array}\mathrm{.}$$

So, he comes: $$\{\begin{array}{l}{\tilde{x}}_{0,\mathrm{not}}=h_{s0,\mathrm{not}}\left[h_{\mathrm{at}0,\mathrm{not}}{x}_{\mathrm{not}}-h_{\mathrm{at}0,M-1-\mathrm{not}}{x}_{M-1-\mathrm{not}}\right]\\ {\tilde{x}}_{0,M+\mathrm{not}}=h_{s0,M+\mathrm{not}}\left[h_{\mathrm{at}0,M+\mathrm{not}}{x}_{M+\mathrm{not}}+h_{\mathrm{at}0,2M-1-\mathrm{not}}{x}_{2M-1-\mathrm{not}}\right]\end{array},$$

and by analogy using the t = 1 frame: $$\{\begin{array}{l}{\tilde{x}}_{1,\mathrm{not}}=h_{s1,\mathrm{not}}\left[h_{\mathrm{at}1,\mathrm{not}}{x}_{M+\mathrm{not}}-h_{\mathrm{at}1,M-1-\mathrm{not}}{x}_{2M-1-\mathrm{not}}\right]\\ {\tilde{x}}_{1,M+\mathrm{not}}=h_{s1,M+\mathrm{not}}\left[h_{\mathrm{at}1,M+\mathrm{not}}{x}_{2M+\mathrm{not}}+h_{\mathrm{at}1,2M-1-\mathrm{not}}{x}_{3M-1-\mathrm{not}}\right]\end{array}\mathrm{.}$$

$${\hat{x}}_{M+n}={\tilde{x}}_{0,M+n}+{\tilde{x}}_{1,n}=x_{M+n}\left[h_{a0,M+n}{h}_{s0,M+n}+h_{a1,n}{h}_{s1,n}\right]+x_{2M-1-n}\left[h_{a0,2M-1n}{h}_{s0,M+n}-h_{a1,M-1-n}{h}_{s1,n}\right]$$

Thus, if we add x _{0, M + n} and x _{l, n} term term we obtain: $${\hat{x}}_{M+\mathrm{not}}={\tilde{x}}_{0,M+\mathrm{not}}+{\tilde{x}}_{1,\mathrm{not}}=h_{s0,M+\mathrm{not}}\left[h_{\mathrm{at}0,M+\mathrm{not}}{x}_{M+\mathrm{not}}+h_{\mathrm{at}0,2M-1-\mathrm{not}}{x}_{2M-1-\mathrm{not}}\right]+h_{s1,\mathrm{not}}\left[h_{\mathrm{at}1,\mathrm{not}}{x}_{M+\mathrm{not}}-h_{\mathrm{at}1,M-1-\mathrm{not}}{x}_{2M-1-\mathrm{not}}\right]$$

$${\hat{x}}_{M+\mathrm{not}}={\tilde{x}}_{0,M+\mathrm{not}}+{\tilde{x}}_{1,\mathrm{not}}=x_{M+\mathrm{not}}\left[h_{\mathrm{at}0,M+\mathrm{not}}{h}_{s0,M+\mathrm{not}}+h_{\mathrm{at}1,\mathrm{not}}{h}_{s1,\mathrm{not}}\right]+x_{2M-1-\mathrm{not}}\left[h_{\mathrm{at}0,2M-1\mathrm{not}}{h}_{s0,M+\mathrm{not}}-h_{\mathrm{at}1,M-1-\mathrm{not}}{h}_{s1,\mathrm{not}}\right]$$

c'est-à-dire $$\{\begin{array}{ccc}{h}_{a1}\left(M-1-n\right)& =& D\left(n\right)h_{s0}\left(n+M\right)\\ h_{a0}\left(2M-1-n\right)& =& D\left(n\right)h_{s1}\left(n\right)\end{array},$$

avec $$D\left(n\right)=h_{a0}\left(n+M\right)\cdot h_{a1}\left(M-1-n\right)+h_{a1}\left(n\right)\cdot h_{a0}\left(2M-1-n\right)\mathrm{.}$$

If we want to ensure x _{M + n} = x _{M + n} and thus obtain the perfect reconstruction, we obtain the following necessary conditions on the analysis and synthesis filters: $$\{\begin{array}{r}{h}_{\mathrm{at}0,M+\mathrm{not}}{h}_{s0,M+\mathrm{not}}+h_{\mathrm{at}1,\mathrm{not}}{h}_{s1,\mathrm{not}}=1\\ h_{\mathrm{at}0,2M-1-\mathrm{not}}{h}_{s0,M+\mathrm{not}}-h_{\mathrm{at}1,M-1-\mathrm{not}}{h}_{s1,\mathrm{not}}=0\end{array},$$

that is to say $$\{\begin{array}{ccc}{h}_{\mathrm{at}1}\left(M-1-\mathrm{not}\right)& =& D\left(\mathrm{not}\right)h_{s0}\left(\mathrm{not}+M\right)\\ h_{\mathrm{at}0}\left(2M-1-\mathrm{not}\right)& =& D\left(\mathrm{not}\right)h_{s1}\left(\mathrm{not}\right)\end{array},$$

with $$D\left(\mathrm{not}\right)=h_{\mathrm{at}0}\left(\mathrm{not}+M\right)\cdot h_{\mathrm{at}1}\left(M-1-\mathrm{not}\right)+h_{\mathrm{at}1}\left(\mathrm{not}\right)\cdot h_{\mathrm{at}0}\left(2M-1-\mathrm{not}\right)\mathrm{.}$$

It appears that to ensure perfect reconstruction, the forms of analysis and synthesis are constructed by temporal reversal and weighting. Consequently, if h _s contains zeros at n, then h _a will contain some in the symmetrical part around M / 2, that is to say at the index M-1-n.

The synthesis is illustrated by an example on the figure 1 . In this example, two inverse transforms of size M h _s0 and h _{s1 are passed} .

To reconstruct the samples between M and 2M-1, the samples covered by the common part are added between h _s0 and h _s1 . The reconstruction will be perfect if the windows verify the perfect reconstruction conditions stated above.

The usual case of reconstruction is therefore when, in a decoder, two consecutive spectra are received, for example X _t and X _{t + 1} , resulting from direct transformations and the inverse transformations are applied to them to obtain x ₀ and x _{1, respectively} . The original signal will be perfectly reconstructed by adding the last M samples of the first set with the first M of the second.

It can also be considered that X _t alone has been transmitted. The perfect reconstruction can be obtained if we know how to construct the signal x _{l , n} . This will be possible if we know the samples x _M to x _2M-1 . In this way it will be possible, by weighting by windows h _s1 and h _a1 , to construct the vector for removing the folding from the vector x ₀ .

In the foregoing, it has been considered that the signals X _t and x _M at x 2 _{M -1 are available} .

If we now consider that the next frame is transmitted in the frequency domain (X _{t + 2} ), we do not remove the folding located between x _2M to x _3M-1 . To do this, it would have been necessary to have received these samples beforehand. Nevertheless, this trivial solution is suboptimal from the point of view of critical sampling.

In the following, we present a way to overcome this disadvantage.

Effective time coding

It is proposed to choose particular windows for transmitting the coded signal in time when desired without losing the critical sampling (ie the same number of samples transmitted and reconstructed). This is illustrated on the figure 2 .

• hs0 = 0 pour n compris entre M+ (M+Mo)/2 et 2M-1, et
• hs1 = 0 pour n compris entre 0 et (M-Mo)/2,

avec M_o une valeur entière donnée comprise entre 1 et M-1.By construction, as illustrated on the figure 2 , we choose :

• hs0 = 0 for n between M + (M + Mo) / 2 and 2M-1, and
• hs1 = 0 for n between 0 and (M-Mo) / 2,

with M _o a given integer value between 1 and M-1.

• h_s1(n) =sin (pi * (0.5+n - ((M-Mo)/2))/2/Mo) pour n compris entre (M-Mo)/2 et (M+Mo)/2.

For example, the descending and ascending portions of hs0 and hs1 around the M + M / 2 sample consist of sinus arches given by the equation:

• h _s1 (n) = sin (pi * (0.5 + n - ((M-Mo) / 2)) / 2 / Mo) for n between (M-Mo) / 2 and (M + Mo) / 2.

h _s0 (n) will be taken as symmetric in this area of h _s1 to obtain the perfect reconstruction.

h _s1 can also be defined by a derived "Kaiser Bessel" function used for example in AAC coders.

Thus defined, the forms of h _s0 and h _s1 make it possible to ensure perfect reconstruction.

As illustrated on the figure 3 , a first frame T30 (windowed by h _s0 ) combined with the frame T31 (windowed by hs1) makes it possible to reconstruct the segment from M to 2M-1, the frames T31 and T33 making it possible to obtain the samples 2M to 3M-1 etc. .

In the case where the signal of the frame T31 is transmitted in frequency, the critical sampling is respected and the reconstruction is perfect insofar as the analysis and synthesis filters verify the necessary condition.

Since the sample x _{3M / 2 + n} (n <Mo / 2) is transmitted in the frame T31 then the sample x _{3M / 2-1} -n can be generated based on the knowledge of x _{0 , M + M / 2 + n} from the frame T30. We will build on the relationship: $${\tilde{x}}_{0,M+\mathrm{not}}=h_{s0,M+\mathrm{not}}\left[h_{\mathrm{at}0,M+\mathrm{not}}{x}_{M+\mathrm{not}}+h_{\mathrm{at}0,2M-1-\mathrm{not}}{x}_{2M-1-\mathrm{not}}\right]\mathrm{for\; n}=\mathrm{M}/2.$$

We will then have: $x_{3M/2-1-\mathrm{not}}=\frac{1}{h_{\mathrm{at}0,3M/2-1-\mathrm{not}}}\left[\frac{{\tilde{x}}_{0,3M/2+\mathrm{not}}}{h_{s0,3M/2+\mathrm{not}}}-h_{\mathrm{at}0,3M/2+\mathrm{not}}{x}_{3M/2+\mathrm{not}}\right]\mathrm{.}$

This can be reiterated to find the samples in the overlap zone, ie between the samples (M-Mo) / 2 and M / 2.

Using predefined relationships: $$\{\begin{array}{ccc}{h}_{\mathrm{at}1}\left(M-1-\mathrm{not}\right)& =& D\left(\mathrm{not}\right)h_{s0}\left(\mathrm{not}+M\right)\\ h_{\mathrm{at}0}\left(2M-1-\mathrm{not}\right)& =& D\left(\mathrm{not}\right)h_{s1}\left(\mathrm{not}\right)\end{array}\mathrm{.}$$

Because h _s0 contains zeros between M + (M + M _o ) / 2 and 2M-1, h _a1 contains zeros between 0 and (MM _o ) / 2.

Similarly, because h _s1 contains only zeros between 0 and (M-Mo) / 2, h _a0 contains only zeros between M + (M + Mo) / 2 and 2M-1.

hs0 = 0 for n = M + (M + Mo) / 2 ... 2M-1,

hs1 = 0 for n = 0 ... (M-Mo) / 2,

ha1 = 0 for n = 0 ... (M-Mo) / 2,

ha0 = 0 for n = M + (M + Mo) / 2 and 2M-1.

• x̃ _0,M+n =0 de n= (M+Mo)/2...M-1,
• x̃ _0,M+n ne contient pas de composantes repliées entre n=0 et n=(M-Mo)/2, et
• la zone centrale autour de M+M/2 pour laquelle il existe des composantes repliées.

Therefore, as illustrated on the figure 4 , the vector x _{0, M + n} contains 3 zones:

• x _{0, M + n} = 0 of n = (M + Mo) / 2 ... M-1,
• x _{0, M + n} does not contain folded components between n = 0 and n = (M-Mo) / 2, and
• the central zone around M + M / 2 for which there are folded components.

• x̃ _l,n =0 entre n=0 et n=(M-Mo)/2,
• x̃ _l,n ne contient pas de composantes de repliement entre (M+Mo)/2 et M-1, et
• la zone centrale autour de M/2 pour laquelle il existe des composantes repliées.

Likewise:

• x _{l, n} = 0 between n = 0 and n = (M-Mo) / 2,
• x _{l, n} does not contain folding components between (M + Mo) / 2 and M-1, and
• the central zone around M / 2 for which there are folded components.

Thanks to these properties, we can recover the segment x _{M ...} x _2M-1 by ensuring a perfect reconstruction.

• par transmission dans le domaine transformé du vecteur X₁,
• par transmission dans le domaine temporel des échantillons x_3M/2̇···x_5M/2-1

This perfect reconstruction can be obtained:

• by transmission in the transformed domain of the vector X ₁ ,
• by time-domain transmission of samples x _{3M / 2̇ ···} x _{5M / 2-1}

From the foregoing, one can now perform critical sampling TDAC coding while avoiding problems related to aliasing. In the following, a CELP coding is described, allowing an advantageous combination with the previously described TDAC coding.

TDAC + CELP

It is recalled that one places oneself within the framework of an operation of the type presented in the AMR WB + specification. A converted type coding using TDAC is alternated with a time-type coding which consists of a CELP coder (for example according to AMR recommendation WB).

Without loss of generality, with reference to figure 5 , we take the case of a coding of a T51 frame by TDAC (windowed by h ₅₁ ) followed by a T52 frame in AMR WB and then a T53 frame again in TDAC (windowed by h ₅₃ ).

In order to reconstruct the samples, AMR WB coding is based on a prediction of the periodicity of the signal, termed long-term prediction. As such, he builds his samples as follows: $$r_{\mathrm{not}}=\mathrm{at}\cdot r_{\mathrm{not}-T}+b\cdot w_{\mathrm{not}}\mathrm{.}$$

The signal r is constructed with respect to old samples taken upstream of T samples weighted by a gain, transmitted and updated periodically, and a so-called stochastic part w _n assigned a gain b, transmitted and updated also over time. T represents the pitch. The AMR encoder WB estimates the components a, b and T and the part w _n to be added according to the flow rate considered.

Thus, in order to effectively achieve long-term prediction, the CELP decoder uses past samples that should not have artifacts. However, since the frame T51 is coded in TDAC, there will be folding in the samples between M + (MM ₀ ) / 2 and M + (M + M ₀ ) / 2 as long as the frame T52 will not be restored. with the folding to delete that of the T51 frame.

In order to allow the return of the samples of the CELP-encoded frame T52 without folding, the area of coverage of the samples transmitted by this coding is widened to cover the initial transition zone completely.

The duration of the CELP is extended to the content of index M + (M-Mo) / 2 ... 5M / 2.

In this sense, there is no critical sampling for the coded part by the predictive coding.

On the other hand, the zone M _o is limited in duration in order to avoid transmitting too much additional information.

For example, M _o is around 1 to 2 ms for a frame of duration M corresponding to 20 ms. The number of samples is calculated based on the sampling frequency. We can also choose Mo / 2 as a duration proportional to a subframe of CELP, that is to say the usual duration of updating pitch / gain values and stochastic vector, where a size adapted to fast algorithms for the search of the stochastic vector and its transmission effectively. For example, we take a power of 2.

To reconstruct the samples of the area between M and 2M-1, the period between M and M + (M-Mo) / 2 is first reconstructed using the inverse transform of a frame T50 (not shown) preceding the frame T51. Then the area between M + (M-Mo) / 2 and 2M-1 is reconstructed with the CELP alone which is based for the long term on the samples returned by the transformed part.

A variant for obtaining the samples between M + (M-Mo) / 2 and M + (M + Mo) / 2-1 consists in combining the CELP samples with the samples containing folding originating from the T51 frame. In this case, we can achieve a linear combination of samples from the CELP and the previously determined equation $$x_{3M/2-1-\mathrm{not}}=\frac{1}{h_{\mathrm{at}0,3M/2-1-\mathrm{not}}}\left[\frac{{\tilde{x}}_{0,3M/2+\mathrm{not}}}{h_{s0,3M/2+\mathrm{not}}}-h_{\mathrm{at}0,3M/2+\mathrm{not}}{x}_{3M/2+\mathrm{not}}\right]\mathrm{.}$$

The linear combination works according to the model below: $$x_{3M/2-1-\mathrm{not}}={\alpha }_{\mathrm{not}}\underset{\mathrm{from\; the\; celp}}{\underset{\}}{x_{3M/2-1-\mathrm{not}}}}+\left(1-{\alpha }_{\mathrm{not}}\right)\underset{\mathrm{from\; the\; transformed}}{\underset{\}}{x_{3M/2-1-\mathrm{not}}}}\mathrm{.}$$

With α _n set of positive or zero coefficients less than or equal to one.

Portion 2M, ... 3M-1 is decoded using the end of the CELP samples transmitted between the indices 2M to 5M / 2. Then, based on this decoded result, the samples from the following transform are reconstructed in the overlap area, which contains folding in a similar manner to the overlap area between the frames T51 and T52. The difference with the other direction of transition lies in the fact that the CELP will not supply all the samples of the transition zone of the transform, but only a half (ie M'o / 2 = M / 8 in our example for a transition size of M'o = M / 4). However, only half of this transition zone is necessary to be able to cancel the time folding of the transform.

The window h ₅₁ may be asymmetrical. Thus, the overlap area between the CELP part and TDAC, noted M _o ', may be different from M _o .

Transmission of CELP

Several alternatives are described hereinafter for transmitting the CELP frame.

In one example, the CELP frame covers a duration equal to the size M + Mo / 2 as presented on the figure 4 . According to AMR standard WB, this frame is divided into sub segments, of size noted Mc on the figure 5 , allowing a frequent update of the parameters making it possible to synthesize a signal CELP of quality. Thus the values of pitch, gain and the stochastic part are initially transmitted and updated optionally.

The length of the first sub-segment (Mc ') immediately following the transform may be different if it is desired to use an arbitrary length Mo' with a standard CELP encoder with Mc imposed by this standard.

The pitch can be estimated on the decoded part before the sample of index M + (M-Mo) / 2. Thus, it is possible to avoid transmitting the initial pitch, only the pitch gain that is estimated according to the common method presented in AMR recommendation WB is transmitted.

In a variant of this example, the pitch gain is not transmitted. It is estimated on the decoded signal in the transformed part.

In an alternative example, the pitch estimate can be performed by including the period M + (M-Mo) / 2 to M + (M + Mo) / 2 which contains folded components.

The stochastic part is transmitted in preamble, or ignored. And this, especially if it is considered negligible because of its low power, or if during the reconstruction is based on the version using the weighting α _n .

Indeed, a stochastic part is implicitly present in the signal coming from the folded components coming from the transformed part.

The portion of duration Mo / 2 covered by the CELP can therefore be a specialized part, in that it can benefit from the information resulting from the complete decoding of the part resulting from the previous transform.

Mo / 2 can be equal to Mc if one looks for a particular compatibility with an existing coder. For example, in the context of an embodiment including a CELP AMR type WB, one can choose Mo / 2 = Mc = 5 ms.

An alternative example is presented on the figure 6 . In this example, the CELP encoding covers a shorter length than the base frame of length M. The covered part by the samples M + (MM / 2) / 2 to 2M + M / 16 is encoded from a transform of a size shorter than the initial size (M / 2).

In the figure 6 only the T63 frame is coded in CELP. The frames T61, T62 and T64 are represented in the transformed domain of the TDAC. The frames T61 and T64 are encoded with transformations of length M (windows h ₆₁ and h ₆₄ ), the frame T62 being coded with a transform of size M / 2 (window h ₆₂ ).

This coding is effective because the window h ₆₁ is relatively soft, which allows to obtain a better concentration of energy in the frequency domain. The h ₆₂ window, on the other hand, has a steeper transition in the vicinity of the 2M sample, but this steep window does not penalize the quality of the coding too much because temporally the duration affected is short. The T63 is coded in CELP as shown above, here Mo = M / 8.

Thus a frame of length M can be subdivided into subparts coded in CELP or TDAC of variable size.

Once the samples are restored in the time domain, LPC synthesis filters can optionally be applied to restore the sound signal if necessary.

In a particular example, the transform is operated in a weighted domain, that is to say that the transform is performed on the filtered signal by a weighting filter of type W ( z ) = A ( z / γ ₁ ) H _de-emph ( z ) with A (z) the linear prediction filter (LPC) and gamma a flattening factor of this filter, the filter H _de-emph (z) is a high-frequency de-emphasis filter. The CELP coder operates it, that is to say that the excitation signal r _n will be well calculated in the residual domain of a linear prediction filter A ( z ). Special attention will be paid for the signal synthesized by the first inverse transform, and which is therefore in a perceptually weighted domain, is returned to the field of CELP excitation, so that the long-term portion of the CELP excitation can be calculated.

In the following an example of coding method is described.

With reference to the figure 7 , we illustrate the problem of switching between a type of coding transformed with a predictive type coding.

We consider a signal x to code and then decode. Samples from 0 to 3M-1 are considered to be transform coded, whereas samples from 3M to 4M-1 should be coded by predictive coding, as indicated by the double arrows T and P.

According to the prior art, the samples are coded from 0 to 2M-1 by transform coding according to a transform vector. $X_0^T\mathrm{.}$

The decoding of this transform vector gives the samples from 0 to 2M-1 of a decoded signal x . This decoding reveals REP1 folding, especially in samples from M to 2M-1.

Furthermore, samples of M to 3M-1 are coded by transform coding according to a transform vector. $X_1^T\mathrm{.}$

Il fait également apparaître du repliement REP2 dans les échantillons de 2M à 3M-1 dans x.Decoding of this transform vector gives the samples from M to 3M-1 of the decoded signal x . This decoding shows the same folding with a sign opposite to REP1 in the samples from M to 2M-1 than during the decoding of $X_0^T\mathrm{.}$

It also shows REP2 folding in samples from 2M to 3M-1 in x .

et $X_1^T$

de supprimer (SUPPR_REP) le repliement REP1.Thus, it is possible by combination of the samples from M to 2M-1 respectively derived from the decoding of $X_0^T$

and $X_1^T$

to delete (DELETE_REP) the REP1 folding.

We then code the samples from 3M to 4M-1 of x by predictive coding according to the prediction vector $X_2^p\mathrm{.}$

néanmoins ils sont inutilisables du fait de la présence du repliement REP2.To be decoded, this vector requires knowledge of previous samples. That is, samples from 2M to 3M-1. These samples are available for decoding $X_1^T,$

nevertheless they are unusable because of the presence of REP2 folding.

ne peut être décodé.So, $X_2^p$

can not be decoded.

In addition, the removal of REP2 folding requires knowledge of x samples from 2M to 3M-1 to recreate the folding and delete it by combination. However, these samples are not available for decoding.

n'est pas terminé.Thus, the decoding of $X_1^T$

is not finished.

To solve these difficulties, the prior art proposes to communicate to the decoder the samples it needs in addition to the vectors from the transform and the prediction part. Nevertheless, this solution is not optimal from the point of view of flow.

The present invention proposes the solution illustrated on the figure 8 .

et le vecteur de prédiction $X_2^p\mathrm{.}$

In this figure, we find the signal x, the transform vector $X_1^T,$

and the prediction vector $X_2^p\mathrm{.}$

code un nombre M d'échantillons comportant une partie des échantillons codés par $X_1^T\mathrm{.}$

However, according to the present invention, the prediction vector $X_2^p$

code a number M of samples comprising part of the samples coded by $X_1^T\mathrm{.}$

This arrangement makes it possible to reconstruct the signal x at decoding.

sont utilisés pour le décodage des premiers échantillons que le décodage de $X_2^p$

va permettre d'obtenir. C'est-à-dire, ceux qu'il a en commun avec $X_1^T\mathrm{.}$

Indeed, the samples preceding the REP folding creates the decoding of $X_1^T$

are used for the decoding of the first samples that the decoding of $X_2^p$

will allow to get. That is to say, those he has in common with $X_1^T\mathrm{.}$

Thus, samples of x are retrieved to recreate REP folding. For example, the samples of x corresponding to REP are subjected to coding followed by a decoding identical to those of the samples from M to 3M-1.

et $X_1^T$

peut ainsi être complètement décodé.This folding thus created is combined with that present in the samples resulting from the decoding of $X_1^T,$

and $X_1^T$

can be completely decoded.

Then we can use the completely decoded 3M-1 M samples to decode $X_2^p\mathrm{.}$

In the following, with reference to figure 9 a coding method is described which incorporates the principles described above.

In step S90, samples of a signal to be encoded are received. Then, in step S91, two sample sequences are delimited, so that the second sequence begins before the end of the first sequence. A first sequence SEQ1 and a second sequence SEQ2 are thus obtained.

Each of these sequences is then coded according to transform coding in step S93 for SEQ1, and according to a predictive coding in step S94 for SEQ2.

With reference to the figure 10 an example is described in which the transform coding is done by applying an analysis window, making it possible to determine a synthesis window, by means of a perfect reconstruction relation, adapted to the present coding.

Since the analysis and synthesis windows are linked by the perfect reconstruction relation, it is equivalent to describe one or the other.

On the figure 10 the window of synthesis H is described. This window has four particular parts.

INIT corresponds to the initial part of the filter, one chooses this part according to the coding of the preceding samples. For example, here, H makes it possible to reconstitute a portion of SEQ1 (samples 0 to M-1). If the samples preceding SEQ1 are coded by transform, it is advantageous to choose INIT as a smooth transition. This makes it possible not to disturb these previous samples.

NOMI corresponds to a nominal part. Advantageously, this portion takes a substantially constant value.

NL corresponds to a substantially zero portion of the window. The duration of NL (or the number of coefficients of NL) can advantageously be chosen as a function of the duration (or number of coefficients) of NOMI.

Finally, the INTER part is a continuous part between NOMI and NL. This part can have a shape adapted to the transition between the transform coding of SEQ1 and the predictive encoding of SEQ2. For example, it's a relatively abrupt transition.

Thus, INIT and NOMI are applied to the subsequence S-SEQ1 of SEQ1 which does not include a sample of S-SEQ, the subsequence common to SEQ1 and SEQ2. INTER is applied to S-SEQ. And NL is applied to S-SEQ2, the subsequence of SEQ2 that does not have an S-SEQ sample.

With reference to the figure 11 , an advantageous decoding method for decoding a digital signal according to the principles described above is described.

In steps S110 and S111, a transform vector comprising S-SEQ1 * samples encoding S-SEQ1, and a prediction vector comprising S-SEQ * samples encoding S-SEQ and S-SEQ2 * samples encoding S, are respectively received. -SEQ2.

In step S112, an inverse transform is applied to the S-SEQ1 * samples. For example, it is a window of the H type. For example, it is also possible to provide a step S113 comprising additional decoding operations to obtain S-SEQ1.

In step S114, S-SEQ decoded by step S113, and S-SEQ * are received. At least S114 S-SEQ is decoded, at least by predictive decoding.

Finally, in step S115, decoded S-SEQ is received in step S114 and S-SEQ2 * and then S-SEQ2 is decoded by predictive decoding. If necessary, it is also possible to use S-SEQ1 decoded in step S113.

One embodiment of step S114 is described with reference to the figure 12 .

In this embodiment, both transform decoding and predictive decoding are involved.

In step S120, S-SEQ1 (from S114) and S-SEQ * are received, then S-SEQ is decoded by predictive decoding. We obtain S-SEQ '.

In step S121, an inverse transform (for example that already applied to S-SEQ1 * to obtain S-SEQ1) is applied to S-SEQ1 *. We obtain S-SEQ ".

Finally, in step S 122, a linear combination of the samples S-SEQ 'and S-SEQ "is performed to obtain S-SEQ.

With reference to the figure 13 another embodiment of step S114 is described.

In this embodiment, the decoding of the opposite sign generated by the decoding of S-SEQ * by transform (S-SEQ ") is recreated from S-SEQ * decoded by predictive decoding.

Thus, in this embodiment, step S130 S-SEQ1 and S-SEQ * are received and S-SEQ is decoded. We obtain S-SEQ '.

Then, during step S131, the same folding as S-SEQ "in S-SEQ 'is created, for which purpose the matrix S described above is applied to it.

S-SEQ "corresponds to the decoding of S-SEQ * by transform in step S132.

Finally, S-SEQ '' and S-SEQ '' are combined in step S133 to obtain S-SEQ.

With reference to the figure 14 , a coding entity COD adapted to implement the exemplary coding method described above is described.

This coding entity comprises a processing unit 140 adapted to receive a digital signal GIS and to determine two sequences of samples: a first sequence comprising an S-SEQ subsequence common to both sequences, and a subsequence S-SEQ1, and a second sequence which begins before the end of the first sequence and which contains S-SEQ and an S-SEQ2 subsequence.

The coding entity also comprises a transform coder 141, and a predictive coder 142. These coders are adapted to implement the steps of the coding method described above, and respectively deliver a transformed vector V_T encoding the first sequence and a prediction vector V_P encoding the second sequence.

Communication means (not shown) may be provided for exchanging signals between the encoders.

With reference to the figure 15 , a decoding entity is described for implementing the decoding method described above.

This decoding entity DECOD comprises reception units 150 and 151 for respectively receiving a transform vector V_T comprising S-SEQ1 * samples encoding S-SEQ1, and a prediction vector V_P comprising S-SEQ * samples encoding S-SEQ. and S-SEQ2 * samples encoding S-SEQ2.

The unit 150 provides S-SEQ1 * to an inverse transform application unit 152. For example, it is also possible for the unit 152 to provide a result for a transform decode unit 153 to perform additional decoding operations. and provide S-SEQ1.

Once decoded by the unit 153, the decoding unit 154 receives S-SEQ1 decoded by the unit 153, and S-SEQ * provided by the unit 151. The unit 154 decodes, at least by predictive decoding S -SEQ, and provided S-SEQ.

Finally, DECOD includes a predictive decoding unit 155 for receiving S-SEQ provided by the unit 154, and S-SEQ2 * provided by the unit 151, then decoding S-SEQ2 by predictive decoding and providing S-SEQ2. If necessary, the unit 153 also provides S-SEQ1 previously decoded by the unit 153.

A computer program for comprising instructions for implementing the exemplary coding method described above could be established according to a general algorithm described by the figure 9 .

This computer program could be executed in a processor of a coding entity as described above, to encode a signal with at least the same advantages as those provided by the coding method.

In the same way, a computer program for comprising instructions for implementing the decoding method described above could be established according to a general algorithm described by the figure 11 .

This computer program could be executed in a processor of a decoding entity as described above, for decoding a signal with at least the same advantages as those provided by the decoding method.

With reference to the figure 16 a hardware device suitable for making an example of a decoder according to one embodiment of the present invention is described.

This device DISP comprises an input E to receive a digital signal SIG. The device also comprises a processor PROC of digital signals adapted to carry out coding / decoding operations in particular on a signal coming from the input E. This processor is connected to one or more memory units MEM adapted to store information necessary for driving. of the device for coding / decoding. For example, these memory units include instructions for implementing the coding / decoding method described above. These memory units may also include calculation parameters or other information. The processor is also adapted to store results in these memory units. Finally, the device comprises an output S connected to the processor to provide an output signal SIG *.

Of course, it is advantageous to combine one or more features described above in the context of the object defined by the following claims.

Claims (8)

1. Method for decoding a digital audio signal, comprising the steps:

   - receiving (S110) a transform vector coding a first sequence of samples of the digital audio signal according to a transform-based coding;

   - receiving (S101) a prediction vector coding a second sequence of samples of the digital audio signal according to a coding of CELP type using a long-term prediction;

   the method being characterized in that the second sequence begins before the end of the first sequence, a sub-sequence common to the first and second sequences thus being received coded both by the coding of CELP type using a long-term prediction and by the transform-based coding; and in that it furthermore comprises the steps:

   a) applying (S112) to the transform vector a transform inverse to the transform-based coding so as to decode a sub-sequence of the first sequence not coded by coding of CELP type using a long-term prediction;

   b) decoding (S114) at least in the prediction vector the sub-sequence common to the first and second sequences at least by a decoding of CELP type using a long-term prediction, based on at least one sample arising from step a);

   c) decoding (S115) in the predictive vector by a decoding of CELP type using a long-term prediction a sub-sequence of the second sequence not coded by transform-based coding, based on at least one sample arising from one of steps a) and b).
2. Method according to Claim 1, characterized in that step b) comprises the sub-steps:

   b1) decoding (S120) in the predictive vector the sub-sequence common to the first and second sequences by a decoding of CELP type using a long-term prediction, based on at least one sample arising from step a);

   b2) applying (S121) to the transform vector a transform inverse to the transform-based coding so as to decode the sub-sequence common to the first and second sequences; and

   b3) decoding (S122) the sub-sequence common to the first and second sequences by combining at least one sample arising from step b1) with a corresponding sample arising from step b2).
3. Method according to Claim 1, characterized in that step b) comprises the sub-steps:

   b4) decoding (S130) in the predictive vector the sub-sequence common to the first and second sequences by a decoding of CELP type using a long-term prediction, based on at least one sample arising from step a);

   b5) creating (S131) on the basis of at least one sample arising from step b4) a sample containing an aliasing equivalent to a transform-based coding followed by a transform-based decoding;

   b6) applying (S132) to the transform vector a transform inverse to the transform-based coding so as to decode the sub-sequence common to the first and second sequences; and

   b7) decoding (S133) the sub-sequence common to the first and second sequences by combining at least one sample arising from step b5) with a corresponding sample arising from step b6).
4. Method according to Claim 1, characterized in that step a) comprises the application of a synthesis window comprising at least three parts:

   - a first nominal part,

   - a second substantially zero terminal part,

   - a third continuous intermediate part between the first and second zones,

   and in that at least the second and third parts of the synthesis window are applied to samples coding the sub-sequence common to the two sequences.
5. Computer program comprising instructions for the implementation of the method according to Claim 1 when the program is executed by a processor.
6. Entity (DECOD) for decoding a digital audio signal, comprising means (150, 151) for receiving:

   - a transform vector (V_T) coding a first sequence of samples of the digital audio signal according to a transform-based coding; and

   - a prediction vector (V_P) coding a second sequence of samples of the digital audio signal according to a coding of CELP type using a long-term prediction;

   the decoding entity being characterized in that the second sequence begins before the end of the first sequence, a sub-sequence common to the first and second sequences thus being coded both by coding of CELP type using a long-term prediction and by transform-based coding; and in that it furthermore comprises:

   - a first decoder (152, 153) for applying to the transform vector a transform inverse to the transform-based coding so as to decode a sub-sequence of the first sequence not coded by coding of CELP type using a long-term prediction;

   - a second decoder (154) for decoding at least in the predictive vector the sub-sequence common to the first and second sequences at least by a decoding of CELP type using a long-term prediction, based on at least one sample arising from the first transform-based decoder; and

   - a third predictive decoder (155) for decoding in the predictive vector by a decoding of CELP type using a long-term prediction a sub-sequence of the second sequence not coded by transform-based coding, based on at least one sample arising from one of the first and second decoders.
7. Decoding entity according to Claim 6, characterized in that the second decoder comprises:

   - first means for decoding in the predictive vector the sub-sequence common to the first and second sequences by a decoding of CELP type using a long-term prediction, based on at least one sample restored by the first transform-based decoder;

   - second means for applying to the transform vector a transform inverse to the transform-based coding so as to decode the sub-sequence common to the first and second sequences; and

   - third means for decoding the sub-sequence common to the first and second sequences by combining at least one sample arising from the first means with a corresponding sample arising from the second means.
8. Decoding entity according to Claim 6, characterized in that the second decoder comprises:

   - first means for decoding in the predictive vector the sub-sequence common to the first and second sequences by a decoding of CELP type using a long-term prediction, based on at least one sample restored by the first transform-based decoder;

   - fourth means for creating an aliasing on the basis of at least one sample arising from the first means equivalent to a transform-based coding followed by a transform-based decoding;

   - fifth means for applying to the transform vector a transform inverse to the transform-based coding so as to decode the sub-sequence common to the first and second sequences; and

   - sixth means for decoding the sub-sequence common to the first and second sequences by combining at least one sample arising from the fourth means with a corresponding sample arising from the fifth means.

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