WO2006027475A1

WO2006027475A1 - Signal compression method

Info

Publication number: WO2006027475A1
Application number: PCT/FR2005/002165
Authority: WO
Inventors: Michel Fliess; Cedric Join; Mamadou Mboup; Hebertt Sira-Ramirez
Original assignee: Ecole Polytechnique; Centre National De La Recherche Scientifique Cnrs
Priority date: 2004-08-31
Filing date: 2005-08-30
Publication date: 2006-03-16
Also published as: FR2874768A1; FR2874768B1

Abstract

The invention relates to a method for compressing an initial time-variable signal covering a time period consisting in defining in said time period at least one base of base digital functions whose variable is a time parameter, in defining in said time period at least one approximate function of the initial compressible signal by using said base digital functions and respective coefficients which are individually assigned thereto, and in producing a compressed version of said initial signal on the basis of said coefficients. A compressed signal, a method for compressing and a method for transmitting said signal are also disclosed.

Description

Signal compression method

The present invention generally relates to the compression of signals covering a time interval and variables over time.

More specifically, the invention relates to a method for compressing such signals, a compressed signal obtained by such a method, a method for decompressing such a compressed signal, and a method for transmitting a signal in compressed form.

The invention also relates to a compressed signal obtained by such a compression method, an associated decompression method and a compressed form transmission method of an initial signal.

The invention applies in particular to digital or analog signals of the audio type. However, this is in no way limiting and it will be seen that, on the contrary, the invention can be applied to all types of signals.

There already exist methods of compression of signals covering a time interval and variables in time (in particular digital audio signals).

These methods generally operate according to a frequency approach, that is to say that they treat differently certain frequency components of the signal to be compressed. In this regard, we can cite as an example the compression method corresponding to the MP3 compression standard (which we will refer to as the generic term of the MP3 method - it is specified that MP3 is a registered trademark).

This type of known method exploits the spectral distribution of the energy of the audio signal relative to the auditory capabilities of the human ear.

For example, in the compression of digital audio signals by a method such as the MP3 method, it is exploited that the energy of audio signals is always distributed according to a given general scheme, the energy levels of these signals in certain bands Frequency frequency being always higher than the energy levels of these same signals in other frequency bands - this being true regardless of the particular audio signal considered. For an MP3-type compression in fact, a spectral analysis of the audio signal is carried out and a psychoacoustic model is applied to retain only the audible sounds of this signal.

In such a compression, thus eliminates the inaudible frequencies for the human ear (on average the human ear is able to discern sounds between 20Hz and 2OkHz, with a maximum sensitivity in the band of 2 to 5 kHz).

These known compression methods thus exploit the prior knowledge of the average spectrum of an audio signal and the psychoacoustic properties of the human ear.

And these methods can thus distribute the "effort" of the compression coding (that is to say, to focus the coding bits) on the frequency bands where it is known that the sound is most noticeable.

But the limitation that arises from such an approach is that such known methods are therefore applicable only for a given type of signal (for example, audio signal for the MP3 method).

Moreover, if these known methods work well on signals - in particular audio - "clean", they do not as such eliminate the noise that may be associated with the initial signal to be compressed.

In particular, in the case of the compression of an initial audio signal which comprises before compression an audio noise, it is necessary to take into account this audio noise with specific treatments in order to reduce it - these specific treatments coming in addition to the implementation of the compression method itself, and thus weighing it down.

It is added that during the decompression of the signal, the reading of these files which correspond to the compressed signal (these files may be designated by the Anglo-Saxon word "buffer" file, for buffer file) mobilizes memory locations and means computer processing (ie typically CPU units) relatively important. In particular, the real-time decompression of signals compressed according to these known methods requires a high processing power.

It thus appears that the known methods are associated with certain limitations.

The object of the invention is to allow to overcome - at least to a certain extent - these limitations.

In order to achieve this goal, the invention proposes, according to a first aspect, a method of compressing an initial signal covering a temporal and variable time interval, characterized in that it comprises the steps of:

Defining on said time interval at least one base of basic digital functions whose variable is a time parameter,

Defining on said time interval at least one approximate function of the initial signal to be compressed, by using said basic digital functions and respective coefficients individually associated with said basic digital functions,

• develop a compressed version of said initial signal from said coefficients. Preferred, but not limiting, aspects of this compression method are the following:

Said approximate function is a linear combination of said basic digital functions,

Said irreversible compression is an irreversible compression with losses,

Said basic digital functions are polynomials,

Said coefficients are real numbers,

To identify an approximate function of the signal to be compressed, the following operations are performed: division of said time interval covered by the initial signal into time windows, > Definition for each time window:

^• a base of basic digital functions, and

- / local coefficients, each basic digital function being individually associated with one of the local coefficients on the window to generate by linear combination an approximate function of the initial signal for said window.

The union of said time windows covers the entirety of said time interval covered by the initial signal,

At least some of the time windows have different durations, to define said time windows the following operations are carried out:

Calculating the nth derivative with respect to the time of said initial signal over said time interval covered by the initial signal,

> Definition of each time window, said definition implying for a given time window the following operations:

S ^• Monitoring of the variation of said nth derivative of said initial signal in function of time from a start time of said given time slot, S Setting the end time of said time window given by identity with the instant at which the amplitude of said variation reaches a predetermined threshold value, ^• S Definition of the start time of the time window according to said given time window by identity with said end time of said given time window.

Said nth derivative of said initial signal is a second derivative of this signal,

Said derivative calculation is carried out by means of a truncated Taylor development calculated on N time segments whose meeting covers the entirety of said time interval covered by the initial signal, Said temporal segments have overlaps between them, so as to make it possible to use only convergent estimates of the derivatives of said truncated Taylor development,

The size of said time segments is adjusted according to a compromise between denoising and reactivity to the dynamics of the signal,

Said time segments are defined independently of said time windows,

The value of said threshold can be adjusted according to a desired compromise between compression ratio and coding noise, the definition of local coefficients for each time window is performed by a least squares method,

The compressed signal comprises for each time window:

> The duration of said window,

> The coefficients making it possible to locally approach the initial signal by linear combination on said window.

The compressed signal comprises, for at least some of the time windows (from one to all), a basic identifier of functions.

Said initial signal is a digital signal; said initial signal is an audio signal.

The invention also proposes a compressed signal resulting from a compression of an initial signal produced by a compression method according to one of the aspects mentioned above. This signal can typically be composed mainly of tables of numbers. The invention also proposes a method for decompressing a compressed signal by a compression method according to one of the aspects mentioned above, characterized in that the decompression method comprises the steps of:

Extracting from the compressed signal at least one set of coefficients (the respective coefficients being individually associated with predefined basic digital functions), • Reconstitute an approximate function of an initial signal by linear combination of said coefficients with basic numerical functions.

A preferred but nonlimiting aspect of this process is as follows:

The decompression method also comprises the step of extracting from the compressed signal information making it possible to reconstruct time windows over which said basic digital functions are defined.

And the invention also proposes a method of transmission in compressed form of an initial signal characterized in that said transmission method comprises the steps of:

• Compress the initial signal to be transmitted by a compression process according to one of the aspects mentioned above,

• Transmit the signal in compressed form, • Decompress the compressed signal transmitted.

Preferred, but not limiting, aspects of this process are:

Said signal in compressed form comprises the coefficients associated with the basic digital functions that best approximate the initial signal,

Said signal in compressed form comprises only the coefficients associated with the basic digital functions that best approximate the initial signal,

Said signal in compressed form comprises only the coefficients associated with the basic digital functions that best approximate the initial signal in the least squares sense,

Said transmission method comprises the encryption of said compressed signal by a compression method according to one of the aspects mentioned above, said transmission method comprises the compression of a signal previously compressed according to a known compression method in itself, by a compression method according to one of the aspects mentioned above,

Said known compression method is a lossy compression method; said known compression method is a method of the MP3 (registered trademark) type.

Finally, the invention also relates to a computing device for compression and / or decompression. Such a device is characterized in that it comprises a memory capable of containing the expression of at least one function base, and at least one coefficient table, and at least one process capable of implementing the method referred to above. high (in all its variants) on a signal to compress (10) in order to establish a coefficient table and to draw a compressed signal. In particular, the memory may be further arranged to contain a definition of windows, and the device then comprises a preliminary process (not necessarily distinct from the first) capable of cutting the signal to be compressed into windows on the basis of a predefined criterion. Moreover, this preliminary process can operate by derivative computation on the basis of truncated Taylor development.

Other aspects, objects and advantages of the invention will appear better on reading the following description, and the appended drawings in which:

FIG. 1 schematically represents a signal to be compressed, the graph of FIG. 1 being a representation of the evolution as a function of time of the intensity of the signal over a time interval I ₁

FIG. 2 represents the same signal, the graph of FIG. 2 further showing time windows that cut out the time interval I. FIG. 3 shows the elements of FIG. 2, with the addition of the representation of temporal segments that cover the time interval I, FIG. 4 is a block diagram of the compression according to one embodiment of the invention,

Figure 5 is a block diagram of the decompression according to one embodiment of the invention, and Figure 6 is a timing diagram illustrating an aspect of the determination of the windows.

The description is followed by an appendix of non-text formatted forms which will be referred to below.

It is specified in advance to this description that Figures 1 to 3 - and the signal shown therein - do not correspond to a real case, but are intended only to illustrate the principle of the invention. It is also specified that the index integers, such as / or j, are contextual, and do not have a constant meaning in the present description.

Signal compression

Overview

Referring now to Figure 1, there is shown a time-varying signal over a time interval I. The signal 10 may be a digital or analog signal. It can in particular be an audio signal - but this signal can also be of any other type.

This section describes the main aspects of the signal compression method according to the invention.

And as will be seen, the compression method according to the invention adopts a different approach to known approaches, which have been mentioned in the introduction to this text. The development of a compressed signal is done in the case of the invention according to the following main steps:

Defining on said time interval at least one base of basic digital functions whose variable is a time parameter, defining on said time interval an approximate function of the initial signal to be compressed, by using said basic digital functions and respective coefficients individually associated with said basic digital functions,

Developing a compressed version of said initial signal from said coefficients.

These aspects will be detailed below.

Basic functions

The method thus comprises a step of defining at least one base B of so-called "basic" digital functions - because it will be seen that they typically form the elements of a base making it possible to generate (for example by linear combination) a function approximated signal 10.

According to FIG. 4, and as will be seen, a member 42 can cut the time interval I into time slots Fi, denoted 43-1 to 43-n in FIG. 4. And it is possible to define on each window Fi a function base individually associated with the window (member 45).

In the remainder of this text, we will consider that a given base Bi is associated with a given time window Fi. And the operations that will be described about a base Bi will be understood as being executed on the window Fi associated with the base Bi. In a limiting case, the interval I comprises only one temporal window, which is equal to it. The modalities for defining the Fi windows will be explained later in this text. As for the notations: - Bi globally designates the base of functions associated with the time window Fi, Bij denotes the set of functions of the base Bi associated with the time window Fi (j is the index of the elements in the base Bi).

if the base of functions is common to all the windows Fi, its set of functions can then be simply noted Bj.

The role of the digital functions of a given database Bi is to make it possible to construct on the associated window Fi to this base Bi an approximate function FA of the signal 10. By "digital function" is meant here any automatically calculable function. This construction can for example be made by linear combination of the basic functions, by multiplying each basic function Bij by a coefficient Cij and summing the products thus constituted. This corresponds to the members 48-1 to 48-n of FIG. 4, the basic functions Bij (or their identifiers) being stored at 45. For the current window Fi _1, the element 48-i provides a function approached Fai. The approximate function FA of the signal 10 is the sequence temporally of the functions FAi (if the windows Fi overlap, weights can be performed on the overlays)

In a preferred application of the invention, the basic functions are polynomials of increasing degrees. We can thus constitute for a window Fi a base Bij with the following polynomials as basic functions (the variable of the basic functions is the time t):

• BiO = 1, • BM = t,

• Bi2 = t ² ,

• etc.,

• Bin = t ⁿ .

In this case, the base Bi has (n + 1) basic functions. In the case of basic functions defined by polynomials of increasing order, the higher the order of these polynomials will be important: • the associated window can be large, but

• the more the number of coefficients to define and store for the window will be important.

A compromise must therefore be sought in this respect - and the Claimants have determined that order 2 gives quite satisfactory results.

The basic functions may be different for different Fi windows. It is also possible to retain the same basic functions - and therefore the same basis - for all Fi windows. In the remainder of this text, a particular example of implementation of the invention will be described in which the basic functions are the polynomials mentioned above, for all the windows F 1.

And, in this example, the coefficients Cij will be real numbers (that is, computer approximations of real numbers). They are used to construct by linear combination a function representing an approximate value of the signal 10, on each window (this function is called the "approximate function" - it is defined on a given window, each window Fi being associated with an approximate function FAi respectively). In this case (FIG. 4), for each window Fi, a number j of coefficients Cij equal to the number i of basic functions Bij defined on the window Fi will be defined. This is the role of the body 46.

Different coefficients can then be defined for each window: for each window Fi the values and the variations of the signal 10 will generally lead to choosing specific coefficients to construct, locally on the window, and by linear combination of the basic functions, a function representing an approximate value of the signal 10.

In any event, whatever the basic functions defined for windows Fi ₁ these functions are permanently set for a given implementation of the invention. A particular choice of basic functions Bij (that they are identical for all the windows Fi, or not) corresponds thus to a particular variant of setting implementation of the invention (that is to say a given choice of basic functions Bij corresponds to a particular protocol for implementing the invention).

In this respect, it is specified that the decompression of the compressed signal by using certain basic functions Bij implies the knowledge of these same basic functions Bij. This knowledge can be obtained by appropriate information (such as identifiers), incorporated in the compressed signal, or transmitted separately, or else determinable on the basis of other characteristics of the compressed signal. In the case where we establish a particular function base B ^ for a window F _n this can be done by examining the curvature presented by the signal in the window F ₁ -, or in a subset or sur¬ set containing it. As is known, the curvature can be determined in particular from the first derivative and the second derivative of the signal. The derivatives "debruised" which will be discussed later can be used for this purpose.

Definition of the approximate function

Once the basic functions Bij have been defined for all the windows, the step of definition of the approximate function on each window implies making a choice of coefficients Cij to construct the best possible approximated function (that is to say the function of which the values closely approach the signal 10 on the window Fi considered). This choice of coefficients can be achieved by various known optimization methods. Generically, the members 48 compare their input (signal in the concerned window Fi) and their sum of products of the form Cij * Bij to provide a difference signal allowing the member 46 to select the coefficients Cij. It is thus possible to implement a spline-type method. Applicants have observed that a least squares optimization method was particularly well the example of the polynomial basic functions discussed above). Of course, other optimization methods may be applied, depending on the context.

Whatever the method chosen to construct the approximate function on each window, an approximate function on each window has been developed from the basic functions.

And during this construction by approximation part of the signal has been lost, between the initial signal 10 and the compressed signal 50 which is defined by the approximate functions on the different windows. The compression process according to the invention is thus a method of irreversible compression, with losses. However, the tests conducted by the Applicants have shown that the performance of the method according to the invention - in particular in terms of quality of the compressed signal, compression ratio, speed of decompression, economy of calculation means, and compatibility with other methods - were extremely interesting.

Definition of windows - general principles

As has been explained above in this text, the implementation of the method according to the invention may involve the definition of time windows Fi. In practice, this definition of windows will very often be implemented.

The windows are defined so that their union covers the entire time interval I.

It is specified that each window may have a different duration. In a preferred (but not limiting) mode of implementation of the invention, the windows Fi are defined by carrying out the following operations:

• Calculation of the nth derivative (ie the derivative of order n) with respect to the time of the initial signal 10, on the time interval I,

• Definition of each time window Fi ₁ said definition implying for a given time window Fi the following operations: Monitoring the variation of the nth derivative of the signal as a function of time, starting from a start time of said time window F 1,

> Definition of the end time of said time window Fi by identity with the instant at which the amplitude of the variation of this nth derivative reaches a predetermined threshold value,

> Definition of the start time of the time window according to Fi + 1 by identity with the end time of the window Fi.

FIG. 2 represents, by way of illustration, windows F1 to F8 (it should be remembered that the figures do not correspond to real cases) and in this figure the windows F1 are not necessarily in correspondence with crossing thresholds of a derivative nth of the signal 10).

The Applicants have determined that the derivation order to be applied to the signal could be chosen as the second order (that is to say that the second derivative of the signal 10 is considered to define the windows F 1), which is well adapted. especially for the compression of audio signals. However, other derivation orders may be preferable, at least in some cases.

In this preferred embodiment, the windows F1 are thus defined from the monitoring of the variations of an nth derivative (for example the second derivative) of the initial signal 10. It is also possible to use derivative combinations, such as the derivatives of order (1) and (2), which represent the curvature, or other orders.

Nth derivative calculation

The calculation (it is specified that this "calculation" is not an exact calculation, but that it corresponds more precisely to an estimate) of the nth derivative of the signal can be achieved by means of a truncated Taylor development of said signal , this development being calculated on N time segments whose meeting covers the entirety of said time interval covered by the initial signal.

FIG. 3 thus represents time segments S1 to S10. It is specified that the segments Si are defined independently of the windows Fi. More precisely, these segments are defined prior to the definition of the windows Fi.

The Si segments may have overlaps between them (i.e., some neighboring segments may overlap a certain time domain), so as to allow only convergent estimates of the derivatives of said truncated Taylor development to be used (implementing typically sliding window type techniques). The size of the Si segments is defined so as to make it possible to construct an approximation of the signal on each segment, with the truncated Taylor development mentioned above.

And the size of these segments can be adjusted according to a compromise between denoising and reactivity to the dynamics of the signal. In this respect, the operators used for the derivative calculation are comparable to noise-resistant filters (anti-noise) - and the larger the sliding window mentioned above about truncated Taylor's development, the better the denoising).

A more detailed description of an exemplary embodiment is given below.

Adjustment of the threshold value

It is specified that the value of the threshold which is used to define the windows Fi can be adjusted according to a desired compromise between compression ratio and coding noise:

• the greater the value of the threshold, the greater the compression ratio will be: the compressed signal boils down to the definition of the time windows and the coefficients associated with each window, and a larger threshold value will tend to generate a number less important windows for the interval I, The smaller the threshold value, the smaller the windows will tend to be, and the closer the approximation of the signal 10 by the approximate function will be on each window.

Compressed signal

The compressed signal 50 - which corresponds to the approximate functions on the different windows Fi - is totally defined by the following information for each window Fi:

• The duration of said window, • The coefficients for locally approaching the initial signal by linear combination on said window.

The knowledge of this information makes it possible to reconstitute the compressed signal.

It is specified that the following elements are also known decompression means:

• Basic functions,

• mode of construction (linear combination for example) of the approximate functions on the windows Fi, from the coefficients associated with the window. These elements define a particular compression / decompression protocol according to the invention. Note that the value of the threshold (which made it possible to define the windows in the variant where these are defined as explained above) is not part of this protocol.

These protocol elements being considered as known, the compressed signal is completely defined by a matrix of which:

• each line corresponds to a time window Fi,

• each line contains the following information (in a predetermined order, which is also defined by the protocol considered):

> duration of the window,> coefficients to be applied to the basic functions of the window to reconstruct on this window the approximate function of the signal 10. For example, it is possible to envisage a protocol for implementing the invention according to which:

• the basic functions are identical on all the windows, and are three in number:> unit function which has a value of 1,

identity function that has not associated itself,

> square function which t associates t ² ,

• the construction of the approximate function is done on each window Fi by linear combination of these three basic functions, by associating them three respective coefficients (the coefficients being ordered for this purpose, in a first, a second and a third coefficient).

For such a protocol, each row of the matrix mentioned above will have a window duration and three coefficients, arranged in the order in which they are to be used for the linear combination of the three basic functions.

In all cases, the compression ratio of a sampled initial signal (in particular a digital signal) will be defined by the value: (Nf * Nd) / Ne, with Nf: number of windows Fi, Nd: number of coefficients per window,

Ne: number of sampling points of the initial signal on the interval I.

Transmission and decompression

In practice, it is possible to constitute with the compressed signal 50 a computer file, which in a preferred embodiment of the invention contains only the matrix explained above.

If several implementation protocols of the invention are likely to be used, the file containing the compressed signal may also contain the elements (basic functions, mode of construction approximate functions) to define the protocol used for compression.

Note that in any case the information defining the compressed signal form a file of extremely small size, since this information is reduced to the duration of each window Fi and associated coefficients.

This file can be transmitted to decompression means, by any means known per se (transmission of a physical medium containing a record of the file, remote transmission via a public link - for example the Internet - or private, etc.).

As shown in FIG. 5, the decompression means can then, as a function of the protocol under consideration, decompress the signal by reconstructing the windows (53-1 to 53-n) from the window (52) and coefficient (56) information. ) the approximate functions (58-1 to 58-n), on each window, in the example using the basic functions Bij, which can be predetermined, or defined by identifiers included in the compressed signal 50. L set gives the decompressed signal 60.

The compressed signal being in the form of a matrix, it is particularly convenient to apply (for example for the transmission of the compressed signal) an encryption of this compressed signal, to transmit it in encrypted form.

And the compression method according to the invention can also be combined with a compression method known per se, by compressing according to the invention a signal previously compressed by a known method - for example a lossy compression method such as the method MP3.

This makes it possible to reduce - in a significant way as the Applicants have found - the size of files already already compressed by other methods. The very small size of the files containing the compressed signals according to the invention also allows extremely rapid decompression without mobilizing significant processing resources. This small size also makes it easy to store and transmit compressed files.

It is furthermore understood that the invention is not limited to any particular type of signal - for example audio - but applies to any type of signal.

It will also be appreciated that the invention permits - at least to some extent - denoising of audio signals, in the case of compression of such signals.

In fact, audio noise generally corresponds to rapid fluctuations. And in the embodiment of the invention where the approximate functions are constructed by linear combination of polynomials, these fast-fluctuating components are eliminated because the polynomials do not contain such components.

Sequence of steps

The steps of implementation of the invention have been explained above. In practice, and according to a particularly advantageous mode of implementation - but not limiting, the invention can be implemented according to the following sequence of steps:

• definition of the time interval I,

Calculating the nth derivative (for example, second derivative) of the initial signal 10,

• definition of the value of the threshold, • definition of the time windows Fi,

• approximation of the initial signal on each time window,

• constitution of the compressed signal, and a corresponding file,

• transmission of the compressed signal,

• decompression. Definition of windows - detailed description

It has been indicated above that the determination of the windows Fi is done on the basis of a derivative (in the sense of derived function, or derived signal) of the temporal signal x (t) to be compressed. It is recalled that this signal x (t) is available in the form of digital samples.

Made in a conventional manner, the estimation of a derivative is sensitive to the noise contained in the processed signal. Thus, although conventional methods can operate in certain cases, the Applicants currently prefer to use a particular technique, which makes it possible to estimate a derivative with a lower sensitivity to the noise contained in the initial signal x (t).

Derivative calculation - formulas in section 1 of the appendix

According to this particular technique, the derivative is computed at a point to, starting from integrals, calculated on a time segment of duration tf starting at time to.

One way of doing is illustrated by relation (11), in which:

- x ^(v) (0) represents the time derivative of order v at the instant origin to, with here to = 0.

- G _r , j is a determinable coefficient, which depends on r and i, as well as on parameters M and N, - the expression of the index α depends on the index n, on the summation terminal r, and only parameters M and N,

- Kα, _n is another determinable coefficient, which depends on r and n, via α, as well as on the parameter N,

- IT ⁴¹ Q _1n is a computable integral over the interval [t ₀₎ to + tf], on the basis of a determinable expression, which, via α, depends on r and n, as well as the difference between the parameters M and N. As will be seen, the parameter N can be associated with a truncation order in a Taylor development, as well as with an integration order of Taylor's truncated development. For its part, M is a parameter which we will see that we can associate with an order of integration applied after a series of derivations.

The general form of the expression \ T ^ϋ _a , n is given by the relation (12), in one embodiment. The magnitude α is related to MNr.

In relation (11), the unknowns are x ^(v) (0). The remainder represents coefficients, which can be calculated as follows:

- do the simple calculation of the member on the left,

- for the right-hand side, calculate ITVn.n using the simple integral of relation (12), and multiply the result by the coefficient K _r - _n , n.

The coefficients G _r , i and K _r - _n , n can be found for example in a table or tables where they are stored according to the indices.

Let Z be the order of a desired derivative x ^(Z) (0). We then consider a set of equations written according to the relation (11), corresponding respectively to the values of r from 0 to Z, taking the parameter N at least equal to Z. This provides a system of (Z + 1) equations. The coefficients of each equation are calculated as indicated above. It remains then to solve a system of (Z + 1) equations of the first degree to (Z + 1) unknown, which the current computing means allow easily. This gives access to the desired derivative x ^(Z) (0); we also have access to lower-order derivatives at the same time t ₀ = 0. For example, the derivative of order 0, ie x ⁽⁰⁾ (0), actually represents the denoised signal itself, which can be interesting in itself.

Having thus estimated the derivative x ^(Z) (0), for to = 0, it is possible to operate likewise at the following times, by temporal shift of the segment under consideration. The segment will therefore "slip", in a manner similar to the technique known as "sliding windows", frequently used in signal processing, for example for calculations of average, variance, instantaneous frequency, in particular.

Figure 6 presents more precisely the way in which one proceeds for the numerical implementation of the sliding segments. This is a segment Sj of size t _f , which can be written [tj, tj + t _f ]. From one segment to the next, we have _{j +} i = tj + kt _e , where te is the sampling period of the initial signal x (t) and k is in principle equal to 1 (k can be a number, in principle integer, greater than 1, if it is acceptable not to estimate the derivative at all points).

For each segment, we calculate the estimate of the derivative of order Z of the signal at the origin tj of this segment. For example, for the segment S2, we compute the derivative of order Z at the origin t ₂ of this segment, that is: x ^(Z) (t ₂ ) = x ^(Z) (2 kt _e )

The set of these estimates x ^(Z) (tj), arranged temporally (directly or by indexing), forms the derivative signal of order Z at each point of x (t).

It will be observed that simplifications are possible. For example, it is not necessary to completely recalculate the integrals

with each slip of the segment. After having determined an integral IT ¹ Vn _1n for the first segment S ₀ , it is then sufficient to subtract the contribution of the sample or samples that leave the segment upstream side, and to add that of the sample or samples that enter the segment, side downstream.

In fact, since the signal is sampled, these integrals are summations on successive samples. It is therefore possible to calculate the IT ¹ Vn _1n by proceeding sample by sample, so without summation at this stage. For the first segment, we sum up on the p samples corresponding to the duration t _f ; then we proceed by deducting / adding contributions, as indicated above. Other simplifications of the calculation are possible.

It is by following the variations of this signal derived from order Z that we can determine the consecutive windows Fi, as indicated above.

The width t _f of the segments Sj can be chosen according to an appreciation of a duration ε, related to an approximation, contained in the equation (11) and on which we will return.

It is conceivable to use segments of variable length, and / or a variable pitch between segments, at least in certain cases.

Thus, the size of the segment Si "in line" can be adapted as a function, for example, of the estimation error, obtained by comparing point by point the original signal x (ti) and the derivative of order 0 denoised x ⁽⁰⁾ (ti), already mentioned. The sliding segment is then of variable length.

Mathematical Basics - Forms in Section 2 of the Appendix

In general, a physical signal is piecewise analytic. This signal x (t) can therefore be written in the form of a Taylor series development, according to the appended relation (21), in which: - 1 represents the time; and

x ^(v) (0) represents the time derivative of order v at the instant origin t = 0.

In detail :

- x ⁽⁰⁾ (0) represents the value of x (t) at time t ≈ 0 (derivative of order 0), - x ⁽¹⁾ (0) represents the first derivative of x (t) to l moment t = 0,

x ^(N) (0) represents the derivative of order N of x (t) at time 0. An interesting approximation of the signal is obtained using an N-truncated Taylor series expansion, as shown in relation (22). This relation (22) can be analyzed as defining a polynomial of order N as a function of time t.

The truncated Taylor series expansion to the order N gives an approximate value of the signal x (t), valid over a short time interval [0, ε], with ε> 0. The value of the order N can be chosen according to the following considerations: - what is the desired Z derivation order,

- If the original signal is highly noisy, the size of the sliding segment can not be too small (the greater the horizon of integration, the less noise has influence).

- having a good representation of the original signal on a fairly large segment requires a polynomial of sufficient degree.

- a compromise must be found between the desired precision for the approximation and the allowable complexity for the calculation.

Consequently, the noise level, the common size of the segments (possibly their average size), the order N of the polynomial, and the desired derivation order, in particular, thus play an important role in the choice of the calculation mode of the derivative.

In principle, the estimation of the derivative at the origin of each segment can be done in several ways. One can, for example, try to directly estimate the N + 1 coefficients of the polynomial corresponding to the Taylor truncated development of the relation (22), and to retain only those which are of interest.

It is currently preferred to use a particular method, which will now be described. It should be noted that this method is not unique, and may admit different variants, particularly as regards the choice of integration and / or derivation orders.

It is a question of estimating the N + 1 coefficients of the polynomial corresponding to the Taylor truncated development of the relation (22), or at least some of them.

The theoretical explanation of this estimate is more direct by working in the operational field, for example by working on the Laplace transform. By this transformation, we go from equation (22) to equation (23), where s is the Laplace variable. Multiplying the two members of equation (23) by s ^{N + 1} yields equation (24). In principle, we will have N - v greater than or equal to 0.

We then consider a succession of successive derivations with respect to the Laplace variable s. Each derivation can be written d ^m / ds ^m , with m being the values of the sequence {0, 1, ..., N}. The degree of derivation m = 0 corresponds to equation (24) as such.

This set of derivations according to s corresponds to the system of equations (25), in which:

- on the left-hand side, there appears first a unicolon matrix of derivation operators, of which, at each line of this matrix, the derivation operator is applied to the expression s ^{N + 1} XN (S) - to Right member, P is a matrix of Laplace operators.

The matrix P is an upper triangular matrix (N + 1, N + 1). We can note P (u, v) its current element, with, in lines, ue [0, N], and in columns, ve [0, N]. The current element P (u, v) is then written according to the following table I: Table I

We then proceed to an integration of order M, with M> N. In the operational domain, the integration of order M corresponds to the system of equations (26). Like the matrix P, the product Q = 1 / s ^M P remains a triangular matrix of operators. Its current element is then written according to the following Table II:

Table II

Note that the coefficients are the same in the P and Q matrices.

Left-hand member of equation (26) - Formulas in section 3 of the appendix

Equation (31) defines a convenience notation, where the left-hand side of the relation (26) is represented by a column matrix of elements {AU, Al_i ... AI_N}. The "L" recalls that we are in the Laplace area. We denote by AL _r a current element of this matrix, and by AT _r the time-domain equivalent of AL _r .

Moreover, equations (32) and (33) give the first and second derivatives of s ^{N + 1} XN (S), respectively. By dividing the two members of equation (33) by s ^M , we obtain equation (34), which gives element AL2 of equation (31) in the Laplace domain.

It is recalled that the passage of the domain of Laplace to the time domain admits the correspondences of the following table III: Table III

Thus, starting from equation (34), and in the case M = N + 1, equation (35) gives the equivalent time domain AT2 of equation AL2 in the Laplace domain.

In equation (35), the right-hand side is the sum of three terms: - the first is a double integral (iterated twice) over a time interval [0, tf],

the second is a simple integral over a time interval [0, tf], and

the third is a term related to the time t _f , to the square.

Over successive derivations (in s, in the operational domain), this structure will be preserved:

- in the Laplace domain, we will have expressions of the form IL given to equation (36)

their temporal equivalent has the form IT given by the relation (37). It will be noted that the iterated integrals of the relation (37) all relate to the same duration t, which is equal to t _f in the example; the index that is added to the variable ta only to distinguish the different integration variables. Moreover, the Applicants have observed that iterated integrals according to relation (37) can be reduced to the simple integral of the aforementioned relation (12), since these iterated integrals all originate at the same time t = 0 ( more generally, the time tj of the beginning of the segment Sj concerned).

By generalizing, the current element AL _r has in the time domain an equivalent AT _r that can be expressed by the relation (38). It has the form of a sum of integrals IT _r - _n , n, computable over a time interval t (here, t = t _f ), and multiplied by coefficients K _r - _n , n respective, determinable in advance from the expression of the derivative of order r of s ^{N + 1} XN (S), divided by s ^M , and converted into the time domain.

If we now return to equation (11), it appears that the right-hand side of equation (11) corresponds exactly to AT _r .

Right-hand member of equation (26) - Formulas in section 4 of the appendix

In turn, the right-hand side of equation (26) can be represented by a matrix of elements BL _n as indicated by relation (41). The equivalent of BL _r in the time domain will be noted BT _r . Moreover, BL _r is obtained by a matrix multiplication which concerns the line r of the matrix Q (u, v). Equation (42) expresses this current term BT _r in the time domain, with u = r and v = i, where i is the summation index. We reach the null elements beyond i = N - r.

The transition to the time domain, for a non-zero current element of the matrix Q, can be better understood using the following Table IV: Table IV

The coefficient G ₁ -, - of equation (42) is therefore expressed according to equation (43).

If we return again to equation (11), it appears that the left-hand side of equation (11) corresponds to BT ₁ - according to equation (42).

Although they are not necessary to implement equation (11), the explanations above make it possible to better understand its origin, and also to determine a priori the values of the coefficients, in particular of the coefficients K _r - _n, n and G _r, j according to the parameters N and M chosen a priori, and values of their indices. Determination and / or trial-and-error verification of the coefficients K _r - _n , n and G _r , j would also be conceivable.

As regards the computation of the integrals contained in AT _n the relation (12) proposes a particularly advantageous simple integral. However, it would be possible to proceed to the same calculation by iterated integrals calculated according to equation (36).

In summary, the set of derivatives at t = 0 can be estimated using integrals of the signal to be processed x (t), having the form of the expression (37), or, better, of the expression (12).

Choice of parameter M - Formulas in section 5 of the appendix

We have seen that the order M of integration was at least equal to N + 1, and could be greater than N + 1. The basics of choosing M can be as follows: the integration filters the fast fluctuations of the original signal. The more we integrate, the more we will filter; in return, it also reduces the informative nature of the signal. To minimize the calculations to be performed, we can use the bare minimum integration, ie M = N + 1.

It will be observed, however, that equation (32) contains on the far right a term in d / ds [X _N (s)]. It follows the expression of Al_i, according to equation (51). For M = N + 1, the equivalent ATi in the time domain is given by equation (52). Its right-hand term, of the form (-1) t _f XNO), is dependent on the signal x _N (t) itself (and, in this case, it is the only one). It may be preferable to show only integrations of the signal, and not the expression in XN (I) of the signal itself. In this case, an additional integration, ie with M = N + 2, is to be performed. The expression of ATi then takes the form of the relation (53).

More generally, it is considered as accessible to those skilled in the art to determine the coefficients of all the aforementioned equations in the time domain for any order of integration M greater than N.

In the above, the system of equations taken from equation (11) for different values of r corresponds to one and the same value of the parameters M and N. Since the matrices P and Q are triangular, so always invertible the system of equations thus obtained always admits solutions.

One variant would consist in establishing several systems of equations for different values of M, with for example a system of equations for M = N + 1, and another for M = N + 2. For a given signal, or for a portion I of a given signal, one can then choose the system of equations applied, or a particular combination of them (linear combination for example), according to an appreciation of the characteristics of the signal. input to compress. We could still mix the equations of the two systems.

The technique proposed above for calculating a selected order derivative signal can be considered as a filter, since it reduces the effect of noise. This filtering is non-frequency, in that it is not completely comparable to conventional low-pass filtering, associated with a cut-off frequency. The filtering proposed here proceeds from an estimate of iterated integral derivatives, since it filters the signal itself and / or one or more of its derivatives from local integrals of the initial signal.

Annex 1

Section 1

Section 2

Section 3

Section 4

Section 5

Claims

1. A method of compressing a signal, characterized in that it comprises the following steps: a. defining on a time interval (I) at least one base of digital functions (Bj; Bij) whose variable is a time parameter (t), b. defining on said time interval (I) at least one approximate function (FA) of a signal to be compressed (10), by using said basic digital functions (Bj; Bij) and respective coefficients (Cij) individually associated with said digital functions Basic (Bj; Bij), c. developing a compressed version (50) of the signal (10) from said coefficients.

2. Method according to claim 1, characterized in that said approximate function (FA) is a linear combination of said basic digital functions (Bj; Bij).

3. Method according to one of claims 1 and 2, characterized in that said basic digital functions (Bj; Bij) are polynomials.

4. Method according to one of claims 1 to 3, characterized in that said coefficients (Cij) are real numbers.

5. Method according to one of the preceding claims characterized in that step b. includes the following operations: b1. cutting off said time interval (I) covered by the initial signal in time windows (Fi), b2. define a base of basic digital functions (Bj; Bij) for each time window (Fi) ₁ b3. for said window Fi, determining local coefficients (Cij) forming, by linear combination with the base of basic digital functions (Bj; Bij), an approximate function (FAi) of the initial signal on this window.

6. Method according to claim 5, characterized in that the time windows cover together the entirety of said time interval (I) covered by the initial signal.

7. Method according to one of claims 5 and 6, characterized in that at least some of the time windows have different durations.

8. Method according to one of claims 5 to 7, characterized in that the operation b1. in turn includes the following operations:

Calculation of the nth derivative with respect to the time of said initial signal, over said time interval covered by the initial signal,

Definition of each time window, said definition implying for a given time window the following operations:

Monitoring the variation of said nth derivative of said initial signal as a function of time, starting from a start time of said given time window,

> Definition of the end time of said given time window by comparison of the amplitude of said variation with a threshold value,

> Definition of the start time of the time window according to said given time window by identity with said end time of said given time window.

9. The method of claim 8, characterized in that said nth derivative of said initial signal is a second derivative of this signal.

10. Method according to one of claims 8 and 9, characterized in that said derivative calculation is carried out on the basis of a truncated Taylor development calculated on a set of time segments included in said time interval (I).

11. Method according to claim 10, characterized in that said calculation of derivative is carried out point by point using a system of equations whose unknowns are the derivatives, these equations having for coefficients on the one hand values determinable in advance and secondly calculable integrals on the segment concerned.

12. Method according to claim 11, characterized in that the writing of the system of equations is parameterizable according to at least one parameter representative of a degree of integration and / or of derivation (N; M).

13. Method according to one of claims 11 and 12, characterized in that said computable integrals have the general form of equation (12).

14. Method according to one of claims 11 to 13, characterized in that said temporal segments have overlaps between them.

15. Method according to one of claims 10 to 14, characterized in that the segments are slippery with a pitch equal to or multiple of the sampling period of the signal to be compressed (10).

16. Method according to one of claims 10 to 15, characterized in that the size of said time segments is adjustable according to a compromise between denoising and reactivity to the dynamics of the signal.

17. Method according to one of claims 8 to 16, characterized in that the value of said threshold can be adjusted according to a desired compromise between compression ratio and distortion admissible after compression

18. Method according to one of claims 5 to 18, characterized in that step b3. comprises determining (46-48) local coefficients by a spline method.

19. Method according to one of claims 5 to 19, characterized in that step b3. comprises a determination (46-48) of local coefficients by a least squares method.

20. Method according to one of the preceding claims, characterized in that the compressed signal (50) comprises for each time window:

• The duration of said window,

• The coefficients allowing to approach the initial signal locally by linear combination on said window.

21. The method of claim 20, characterized in that the compressed signal (50) comprises, for at least some time windows, a basic function identifier.

22. Method according to one of the preceding claims, characterized in that the compressed signal (50) is further encrypted.

23. Method according to one of the preceding claims, characterized in that said initial signal is a signal previously compressed according to another compression method.

24. Method according to one of the preceding claims, characterized in that said initial signal is an audio-digital signal.

25. Method according to one of the preceding claims, characterized in that said initial signal is an audio-digital signal previously compressed according to a method of type MP3 (trademark).

26. Compressed signal (50) resulting from a compression of an initial signal produced by a compression method according to one of the preceding claims.

27. Computer device, characterized in that it comprises:

a memory capable of containing the expression of at least one function base, and at least one coefficient table, and

at least one process capable of implementing the method according to one of claims 1 to 25 on a signal to be compressed (10) in order to establish a coefficient table and to draw a compressed signal therefrom.

28. Computer device according to claim 27, characterized in that the memory is further arranged to contain a definition of windows, and in that it comprises a preliminary process capable of cutting the signal to be compressed into windows on the basis of a predefined criterion.

The computer apparatus of claim 28, characterized in that the preliminary process operates by calculating a derivative based on a truncated Taylor expansion.

30. A method of decompressing a signal according to claim 26, characterized in that it comprises the following steps:

Extracting (56) from the compressed signal (50) at least one set of coefficients,

• Reconstituting (58) an approximate function of an initial signal by linear combination of said coefficients with basic digital functions.

31. The decompression method as claimed in claim 30, characterized in that the decompression method also comprises the following step: extracting (52) from the compressed signal information making it possible to reconstitute time windows on which said basic digital functions are defined .