DE19734905C1 - Electronic sound generator for re-synthesizer - Google Patents

Abstract

One or more oscillators (VCO K1...Kn) are activated or deactivated in accordance with control signals (K(x), A(x)) obtained from a signal characteristic to be reproduced based on pattern recognition (KL), so that the oscillators are actively switched for the oscillation waveform most similar to the one to be reproduced. A waveform memory (WT) stores several different waveforms. A method of electronic sound production is also claimed.

Description

The invention relates to an electronic sound generator ger with a plurality of different oscillators Waveform or with an oscillator with a waveform memory (WT) with a variety of different waveforms (W1 ... Wn).

A resynthesizer is understood by those skilled in the art the music electronics an electronic sound generator, with which has a real sound in the form of a waveform can be reproduced in an artificial way.

According to the known prior art is used to replicate Sounds primarily the desired waveform of the simulated sound in a resolution corresponding to the ge desired quality digitized and as so-called audio Sam ple saved as a digital waveform. This digital Waveform is then created using one of the tones to be generated corresponding frequency read from the memory and made audible by means of an oscillator driven thereby. Electronic sound generators that work on this principle are commonly referred to as sample player synthesizers.

Based on the functional principle of these well-known sound generators it is already clear that this is a purely quantitative Proceeds because the memory mentioned is without return view of the respective amplitudes and frequency requirements of the vibration curve to be simulated with a constant Sampling rate of usually 88,200 bytes at 44.1 kHz sampling rate and 16-bit resolution per second. It is about a kind of "snapshot" of the audio material, but the structure and structure of the sound event is not are known. Accordingly, when playing the clan no influence on the roots of the sound, to further educate them. It can only be done by reworking the stored sound event (e.g. by filtering  with a low pass, etc.) can be influenced, however Problems like the well-known "Mickey Mouse Effect" with the Trans ponation of the stored frequency in other audio frequency range due to compression or stretching of the original Timing and due to a shift in character static formants of the sound. This phenomenon is undesirable especially with natural sounds to be reproduced, so that a possibility of resynthesizing a sound event is desired, whose structure and structure can be influenced are.

An approach for this known from US Pat. No. 5,138,924 is based primarily on a process in which already digitized audio material (e.g. by the method described above) by a Analysis according to a physical model in a spectrum (Fre sequence composition of a sound), which is the Architecture corresponds to additive synthesis. When generating supply or resynthesis of the sound to be reproduced is then ent speaking of this model and the parameters obtained according to the additive synthesis of the sound generated. This happens because that an overall sound from a variety of sine waves according to the parameters obtained with the model is set. Based on Fourier's finding that any sound through the specific addition individually ner sine part tones can be mapped, which can be in the fundamental and Classify overtones (integer multiples of the fundamental) let the sine-part tones present in the output sound using additive synthesis to do exactly that for analysis used physical model (e.g. a Fourier analysis) reassembled.

The problems that arise with such a replica are obviously, because every musical sound changes in the Course of its duration, by volume, frequency and phase the positions of the individual partials vary permanently. To be a re synthesis according to this well-known principle in a musical to realize meaningful form in a resynthesizer the changes in real time for each individual partial tone are not  only calculated, but - with a view to a far-reaching Sound influencing - can also be manipulated. To authenticate Table results are to be achieved for complex sonic equipment Please consider at least 256, often even 1024 overtones The resynthesizer should then also enable polyphony Chen, there are up to 16 384 in a 16-voice polyphony Generate partials in real time. This is after today State of the art not feasible with reasonable effort.

The object of the present invention is therefore an electro African sound generator as well as an associated sound generator to create a process with which a resynthesis of a post sound is made possible in real time.

An electronic sound generator is used to solve this task further developed according to the preamble of claim 1, that one or more oscillators depending on with Mit for automatic pattern recognition from one control signals obtained in this way can be activated or can be deactivated that the or the oscillators with the the vibration pattern most similar to the vibration pattern to be simulated form are activated.

For an alternative solution to the task according to the present the invention is an electronic sound generator with a Oscillator with a waveform memory with a variety different waveforms det that the oscillator depending on means for au tomato pattern recognition from a replica Control signals obtained from the vibration curve, can be controlled so that the oscillator oscillates according to the vibration curve to be simulated most similar waveform or combination of waveforms generated.

In a first advantageous embodiment of the electronic Sound generator according to the present invention is as an agent a neural network for automatic pattern recognition hen is. Suitable control signals are therefore particularly good  derivable because of the learning ability of the neural network using its abstraction ability intelligent Control signals to activate the most suitable Oszilla gates can be generated.

According to further advantageous configurations of the electronic Sound generator according to the present invention are as means a geometric classic for automatic pattern recognition tor, in particular a Euclidean distance classifier or a city block distance classifier is provided.

In a further advantageous embodiment of the electronics rule sound generator according to the present invention is as A means for automatic pattern recognition a statistical Classifier, especially a maximum likelihood classifier tor or a Mahalanobis distance classifier.

In a further advantageous embodiment of the electronics rule sound generator according to the present invention is as Automatic pattern recognition means a polynomial classifier gate, in particular a linear polynomial classifier or a qua dratic polynomial classifier.

In a further advantageous embodiment of the electronics rule sound generator according to the present invention is as Means for automatic pattern recognition a heuristic Classifier, especially a nearest neighbor classifier with random sample or a nearest neighbor classifier with cleaners and reduction algorithm.

In a further advantageous embodiment of the electronics The sound generator according to the present invention is used Amplitude spectrum of the vibration curve to be simulated as Input signal for the means for automatic pattern recognition voltage, which in particular by means of a Fourier transformation is recoverable.

In a further advantageous embodiment of the electronics rule sound generator according to the present invention  Control signals at intervals of one whole or one egg half a period of oscillation of the person to be simulated predetermined course.

In a further advantageous embodiment of the electronics rule sound generator according to the present invention are Control signals can be stored in a storage medium and from it pro can be called up in proportion to the frequency of the pitch to be generated.

In a further advantageous embodiment of the electronics The sound generator according to the present invention is a Means for smoothing the transitions of different ac activated or deactivated oscillators generated output provided vibration path.

In a further advantageous embodiment of the electronics rule sound generator vibrates according to the present invention each oscillator activated by the respective control signals with the frequency of the pitch to be generated.

Furthermore, according to the present invention, a method for electronic sound generator created with which a Resyn thesis using an electronic sound generator at the beginning mentioned type is made particularly effective in real time. This is achieved by using one or more oscillators in Ab dependency of means for automatic pattern recognition tax obtained from a vibration curve to be simulated signals are activated or deactivated so that the or Oscillators with the waveform to be simulated most similar form of vibration is activated.

Advantages and embodiments of the invention are in the Drawings are shown and are explained in more detail below tert. Show:

Fig. 1 a block diagram according to the present Resynthesizers the invention having a plurality of oscillators under schiedlicher waveform,

Fig. 2 block diagram of a resynthesizer of FIG. 1 with optimized control signals for activating / deactivating the oscillators and

Fig. 3 is a block diagram of a Resynthesizers of FIG. 2 with waveform memory for implementing a plurality of oscillators of different waveform.

Elements with the same functionality are the same Reference numbers marked.

In the illustration of FIG. 1 is a block diagram of a possible Resynthesizers according shows ge of the present invention having a plurality of oscillators VCO_K1. . .VCO_Kn has different waveforms. The Schwingungsfor men are in particular sinusoidal, triangular, sawtooth, and rectangular vibrations with different pulse width ratio. Other spectra are also conceivable. All oscillators can be realized with discrete electronic components, with integrated oscillator circuits or as virtual oscillators in the form of digital models, in the latter case the desired output oscillation of the oscillator in one or more microprocessors (not shown) according to the digita len Model of the desired oscillator is calculated and converted into an electrical vibration signal via a digital / analog converter (not shown).

The VCO_K1 oscillators. . .VCO_Kn are via an interface MIDI controlled. This can advantageously be a Interface according to the "Music Instruments Digital Interface" Act standard, which on the keyboard KB of a keyboard / syn note information played by thesizers or digital pianos converts into a serial digital data stream. From these the Data representing note information becomes control signals for the VCO_K1 oscillators. . .VCO_Kn won, especially the other is that of the pitch played on the keyboard KB Note appropriate vibration frequency specified. Next to it  also generates a trigger signal T with which the attack time point of a key and associated note becomes. This also means the start of vibration for the VCO_K1 oscillators. . .VCO_Kn.

In addition, the trigger signal T also reaches a memory SP, preferably a volatile or non-fixed value memory, in the control signals A (x) and K (x) for activation or deactivation crossing of one or more oscillators VCO_K1. . .VCO_Kn are filed. With the trigger signal T is also a Counter (not shown) started and proportional to the frequency the desired pitch, which the parameter x in incremented and so different memory addresses of the memory chers SP addressed. In this way a sequence of im Control signals A (x) and K (x) stored in memory SP match the desired audio frequency.

The control signals A (x) arrive at each oscillator VCO_K1. . .VCO_Kn downstream amplifier stages VCA. The tax The signals A (x) represent amplitude values and act on the Amplifier stages VCA. Preferably, the reinforcement kerst stage also as digitally controlled amplifier stages DCA characterized by what they directly by digital storage information mations can be influenced.

A switch follows each amplifier stage, for example an analog CMOS switch, which is controlled by the control signal K (x) is controlled. The control signals K (x) represent bi represent activation signals which are at a logical value from "zero" the respectively assigned oscillator VCO_K1. . .VCO_Kn Deactivate or activate with a logical value of "one". Of course there is also a corresponding negative logic conceivable.

The control signals A (x) and K (x) have the following form, for example:

A (x) = (A1 (x), A2 (x) ... An (x)) = (5.3 ... 10) and

K (x) = (K1 (x), K2 (x).. .Kn (x)) = (1.0.. .1)

The oscillator signals switched so active or inactive with the respective conditioned amplitude values are one Logic link VG supplied and there to a total vibration overlaid or mixed. The link VG can thus also viewed as a small mixer or mixer unit become. On the output side there is an output on the logic element vibration, which with others, explained in more detail below Sound shaping agents can be processed further.

In the following it will now be explained in more detail, according to which criteria series the control signals A (x) and stored in the memory SP K (x) can be obtained.

As explained at the beginning, it is a resynthesizer a sound generator with which an existing sound or Vibration course can be reproduced. The starting point is thus a waveform to be reproduced. This lies for example as a so-called audio sample SAMPLE in digitized Form before. When digitizing, please note that Quanti errors and aliasing effects (correct filtering in frame of the Shannon theorem: sampling frequency is double the highest signal frequency) can be avoided and high quality Analog / digital converter can be used. The desired sound The course is over as a result of amplitude values A (t) a time course t as an audio sample SAMPLE.

Building on this, a frequency analysis of this audio sample is carried out le SAMPLE. Fourier analysis methods are particularly useful for this suitable. After that, every complex waveform can be analyzed Recreate the superposition of sinusoidal vibrations, which differ in their frequency. Are particularly suitable for this Methods like the Fast Fourier Transform FFT or the Dis krete-Fourier transform DFT, with which a sound curve A (t) is transformed from the time domain t into the frequency domain can be. A harmonious analysis or digital Analysis performed and the amplitude spectrum SP of the audio samples SAMPLE determined.  

DFT is based on the assumption that an oscillation is repeated periodically. This makes the DFT particularly suitable for converting cyclical waveforms into a harmonic spectrum. The following basic calculation rule is used to carry out a DFT:

T describes a sampling interval over the time range t of the sound course A (t) and n the number of intervals.

Mathematical details for the implementation of a DFT algorithm can be found, for example, in "Signal Processing: Numerical Processing of digital signals / Schrüfer, E., Munich; Vienna: Hanser-Verlag 1990, pp. 147ff. ". Furthermore there is one Variety for the processing of digital audio material opti mated DFT and FFT algorithms.

With natural sounds, such a cyclical course is in the right gel not given. Therefore, you constantly change yourself changing sounds to the FFT, which in principle also in the It is able to analyze cyclical sound patterns. The basic thanks to the FFT is the decomposition of the amplitude sequence A (t) of the Audio samples follow SAMPLE into several smaller parts, which then can be transformed separately (overlay theorem of Sy stem theory). Particularly efficient and therefore saves computing time is a division into sequences with a number of amplitudes value A (t), which is a power to base 2 (so-called Radix-2 algorithms, cf. "Signal processing: numerical ver processing of digital signals / Schrüfer, E., Munich; Vienna: Han ser Verlag 1990, pp. 339ff. ").

In the present invention, the amplitude values A (t) of the Samples audio samples preferably divided into sections that a whole or, with higher accuracy, half a swin period corresponds to the waveform to be reproduced chen. In this way the audio sample SAMPLE with x Vibration periods in partial sequences A (t1) to A (tx), so-called  Time window, disassembled. The strings A (t1) lined up to A (tx) thus again describe the original Schwin course A (t), which is of course about the Time t can change in its waveform. Especially before It is also partial, the parts follow with the zero crossing every half wave (with half oscillation periods) or every po to start a sitative half-wave (with whole oscillation periods), however follow the length of each part and hence the time beyond that window with the sequence of amplitude values extend to a number that is a power to base 2 (see above explanations for FFT).

Each of the partial sequences A (t1) to A (tx) is now one on the other followed by a signal processing stage FFT in a audible amplitude spectrum SP1 to SPx transformed. Depending on The number of spectra is measured with the required accuracy which are determined by the FFT. The result rende amplitude spectrum SP be the amount of the spectral function stands for a fundamental tone (pure sine wave), also 1. Harmonics called H1 or Fundamentals, which are usually represents the loudest tone of a spectrum and therefore for the perceptible pitch is responsible. There are also more Harmonic H2 to Hy (also pure sine waves ent speaking Fourier approach), also called overtones, partials or partial tones, where y is the number of Harmonics and thus accuracy of the amplitude spectrum conditional. Values of y greater than or equal to 32 are particularly suitable. These Harmonics H2 to Hy are integer multiples of the fundamental H1 (e.g. the fundamental is at 440 Hz, the second Har is monical H2 or the first overtone at 880 Hz etc.). All fre sequences can be divided by the first harmonics H1 len, whereby the corresponding code H2, H3 etc. of the Harmo niches results.

From the respective amplitude spectrum it can be seen which sine tones in which volume to reproduce Subarea A (t1) to A (tx) of the sound or audio samples SAMP LE are involved.  

As an example, it should be pointed out that a sine tone consists of the corresponding first harmonic H1, the amount of H1 corresponding to the amplitude and thus the volume of the sine tone:

Sinusoidal vibration = H1 with full amplitude

The spectral structure of a sawtooth wave can be represented as follows:
Sawtooth wave = H1 with full amplitude, H2 with half amplitude, H3 with a third of the amplitude, H4 with a quarter of the amplitude, etc.

The spectral structure of a square wave stands out by from that versus the spectrum of the sawtooth vibration all even harmonics H2, H4 etc. are missing.

For each partial sequence A (t1) to A (tx) is thus in the signal processing stage FFT an associated amplitude spectrum SP1 to SPx with the harmonics H1 to Hy determined. The Values of these harmonics are now used in the further course Activation of a classification unit KL.

This classification unit KL can have different Classifiers and classification procedures are implemented the. All the classification procedures shown below is common that between a training phase or learning phase and a recall phase or real-time operating phase, which the Resynthesis enables, a distinction is made.

In the training phase, the classifier is conditioned in the one associated with the vibration pattern to be sensed Knowledge base is determined. The classifier is used for this Training data acted upon from the data for resynthesis provided oscillators VCO_K1. . .VCO_Kn according to the above outgoing signal preprocessing steps with FFT or DFT can be derived. This is for every oscillator VCO_K1. . .VCO_Kn and consequently for the associated one Waveform an associated amplitude spectrum SP with the  Harmonics H1 to Hy determined. Unless an oscillator VCO_K1. . .VCO_Kn creates a cyclic waveform - which is not necessary, but the rule is - it will only be one only partial sequence of amplitude values A (t) is required and corresponds accordingly only a single amplitude spectrum SP is determined.

Each classifier has the property of an input pattern assign one of several classes K1 to Kn on the output side, where n describes the number of classes. Therefore sol che classifiers also the technical field of automati pattern recognition. The individual classes are in the present invention, the oscillators VCO_K1. . .VCO_Kn. The input pattern represents the harmonics H1 to Hy each because of an oscillator VCO_K1. . .VCO_Kn associated ampli tudenspektrums SP. Generally speaking, the class forms fikator thus an input vector with the dimension y on ei NEN output vector with the dimension n, which is a multidimensional classification problem. The criteria s, which are assigned to classes K1 to Kn, are determined in the training phase and in the associated one Knowledge base saved.

Then in the recall phase or real-time operating phase via signal preprocessing FFT from the Schwin to be simulated development of the audio sample SAMPLE for each of the x partial sequences A (t1) to A (tx) ascertained amplitude spectra SP1 to SPx the respective harmonics H1 to Hy as the input vector of the Classification unit KL supplied and in KL based on the Wis basis for each of the x partial sequences of at least one class of n VCO_K1 oscillators. . .VCO_Kn assigned.

Unless there is a clear assignment to one of the n classes K1 to Kn and the oscillators VCO_K1 associated therewith. . .VCO_Kn is possible, this becomes noticeable by possibly assigning several classes K1 to Kn, each with a lower probability of assignment. Building on this, several strategies are conceivable in order to achieve the best possible allocation. In particular, it is conceivable to always assign only one class from K1 to Kn by selecting the class with the greatest probability of assignment (the largest absolute value in the output vector K of the classifier unit KL) (logical "one") and thus activating the associated oscillator only while All other classes are disregarded (logical "zero") or oscillators are deactivated. Another strategy, which is the basis of the exemplary embodiment according to FIG. 1, consists in choosing several classes according to their assignment probability by choosing a threshold value for the assignment probability and selecting all classes whose assignment probability is equal to or above the threshold value ( logical "one"), while such classes with an assignment probability below the threshold are not chosen (logical "zero"). This means that several classes are selected and the oscillators VCO K1 connected to them. . .VCO_Kn can be activated at the same time, the output vibrations of which are then superimposed or mixed by means of the link VG. The measures of the strategies described are part of the KL classifier unit.

At the output of the classifier unit KL there is therefore an output vector K for each of the x partial sequences of the vibration pattern of the audio sample SAMPLE to be simulated, which is stored in the memory S as a control signal K (x) assigned to the respective partial sequence. In addition, according to the exemplary embodiment according to FIG. 1, the amplitude values A of the first harmonics H1, for example, are stored in the memory S for each of the x partial sequences as the control signal A (x) assigned to the respective partial sequence. These control signals are then used to control the oscillators VCO_K1 in the recall or real-time phase in the manner shown at the beginning. . .VCO_Kn.

In the following, different classifiers for the classifier unit KL are shown, with which one Realization of the invention described above in particular is advantageously possible.

The task of a classification system initially exists therein, an M-dimensional input vector, the feature vector  v to assign exactly one of K classes. To do this, a suitable one Mapping function f (v) can be determined, which is a feature vector assigns exactly one class k. The question in question classification algorithms should now be presented the, the number of different classes in the following - deviating from the previous explanations - denoted by K is not.

As input, the classification system receives the feature vector v and, if the class is known (monitors learning dimensioning), a target vector y. Starting from the feature vector v, as many decisions D (v) are calculated as there are classes K. In the subsequent minimum or maximum search (in the following) the distinctive value d _k (v) is isolated that allows the best possible allocation to one of the classes k. Furthermore, there is the possibility of going through a rejection threshold D ₀ . However, it can happen that none of the distinguishing values d _k (v) lies above this threshold and thus no suitable class for the assignment of the feature vector v is found. If you omit this rejection threshold, which is usually difficult to determine, you force the classification system to choose one of the K classes. Since it is desirable to get a result in each case, the level of rejection was not taken into account when implementing the classifiers.

The task of optimizing the recognition system, the construction of an optimal classifier, can therefore be broken down into the two building blocks, estimation function d _k (v) and minimum-maximum search. The central importance here is the determination of the estimation function, which transforms the feature vector v into an estimation vector d. The subsequent evaluation system now carries out an evaluation of the actual class assignment, expressed by the target vector y and the presumed meaning d. It is important in this context that the target vector indicating the true meaning (for the classification always a unit vector with the one for the class membership at the appropriate position) is immediately understood as a quantity to be estimated.

The optimization goal according to the so-called square mean approach is to minimize the amount square of the error vector | yd | ^2nd Since the minimization must take place over the entire process, the expected value E is used to formulate the following optimization condition.

S ² = E {| yd | ² } = MINIMUM (2)

The two vectors y and d have the same structure. The components of the target vector y only assume the values zero or one, since the class membership k is clearly known due to the dimensioning that is to be monitored. The component y _k is therefore a clear indicator of the class belonging to class k. In contrast, the component d _k (v) of the estimate vector d is a class membership estimate and should have the value one if possible, provided the feature vector v comes from the class k. Otherwise, d _k (v) should be zero if possible. The function d (v) referred to as the estimation function or also as the regression function thus provides probabilities of inference in its components. The square mean optimization leads to the conditional probability p (k | v) as a differentiating function.

In order to be able to implement an optimal classifier, at this point, the shape or parameters to be known that describe the probability of inference p (k | v). This is but not the case. For this reason, approximative Solutions for the true description of the  Inference probability p (k | v) can be found. Such approximate approaches are called classifications algorithms described below. Another Problem represents the fact that the learning samples in which the Feature vectors for dimensioning the classification system the properties of the associated pattern are determined should reproduce as completely as possible. In fact can by the random sample the actual position of the samples in the memo painting room can only be determined approximately. The ideal case that the realizations of the Mu classes are congruent with the actual population, will not be achievable in practice. For this The reason is the Realisa determined by the learning sample pattern class as an approximation of the actual population tion of the pattern classes used. A feature vector v becomes assigned to the object or event for whose Realisa tion of the model class is the most likely to be inferred speed delivers.

The following approaches are considered as possible approaches for determining classification methods:

• - statistical approach
• - geometric approach
• - polynomial approach
• - nonparametric approach

These approaches are presented in the further course.

The statistical approach

In the statistical approach to classification, the Sample classes in the characteristic space with the help of distribution functions NEN or distribution densities recorded. The feature vec tor v of a pattern is taken as the value of a random variable, whose observation then provides information about the present Pattern class supplies. The assignment is therefore made according to Ge viewpoints of the maximum probability. The estimation radio tion must therefore provide the probability that a  Feature vector v belongs to a certain class k. To do this however, the distribution density within a class is known be. The truth can then be determined using the distribution function Calculate the probability for all classes k and then ßende maximum search determine that the feature vector in the Class k lies.

Since there are many stochastic processes in nature and tech Let nik described by the normal distribution is common the class-specific distribution of M-dimensional randomness variable (in the form of the feature vector v) as a normal distribution described.

Μ describes the

Expected value E {v} = µ (6)

the continuous random variable v (feature vector) and K the corresponding symmetric

Covariance matrix E {(v - µ) (v - µ) ^T } = K (7)

The expected value µ is also referred to as the center of gravity or centroid vector. The coefficients of the covariance matrix represent a measure of the dependency of the random variable v, where KOV (v _x , v _y ) = 0 indicates that the xth random variable is uncorrelated with the yth random variable. For the one-dimensional case M = 1 one obtains the well-known "Gaussian bell curve". In the two-dimensional case M = 2, the contours of constant probability density p (v) describe concentric ellipsoids, the probability density, starting from the centroid vector µ, decreasing evenly from the center due to the exponential relationship of the normal distribution.

The coefficients of the inverse covariance matrix K ^{-1 are} responsible for the measure of the decrease in the probability density. In the M-dimensional case, the probability densities in the feature space are thus described by M-dimensional ellipsoids. It becomes clear that the number of parameters for describing the normal distribution increases rapidly with increasing dimension of the classification problem. This statement will concern us further in the further course.

However, the following interesting properties of the normal distribution can be recorded here:

• - The normal distribution can be determined by the two parameters Centroid value µ and covariance matrix K complete describe
• - by linear combination of two normally distributed variables a normally distributed stochastic variable is created again
• - If two normally distributed variables are uncorrelated (KOV [v _x , v _y ] = 0), they are also statistically independent

These properties are one for the following derivation Decision function for classification of great importance. Nevertheless, the question arises to what extent the classes specific objects or events possess to be normally distributed. It’s not very plausible that in all real applications the probability distribution lungs are so simple and regularly distributed normally. Also lets a normal distribution in the mostly large dimension of Classification problems are not easily caused by statistical Answer tests. For this reason, one goes in practice reverse way by using the normal distribution model as practicable approximation to reality, and then, if the classifier type gives a satisfactory result, concludes that the normal distribution hypothesis is correct was. As a rule, this procedure is followed in practice good classification results confirmed.

Maximum likelihood classifier

The maximum likelihood classifier is based on a statistical approach assuming the normal distribution. To derive a corresponding decision function or estimation equation d (v), some probabilities have to be defined:
p (k) a priori probability of the model class k;
p (v) probability of the feature vectors v;
R _k area of the feature space in which the feature vectors v are classified into the pattern class k;
p (v in the range R _k | k) probability that feature vector v is classified into the pattern class k;
C = [c _kj loss matrix (loss c _kj occurs if v comes from class k, but is classified to class j

This results in the probability that a feature vector v from class k occurs and is classified as k as follows

p (k) p (v in the range R _k | k) (8)

and analogously, the probability that a feature vector v from class k occurs and is incorrectly classified into class j is calculated according to

p (k) p (v in the range R _j | k) (9)

The loss to be expected when classifying a feature vector v to one of the K classes is thus given by

To derive an optimal decision function, it is therefore necessary to select the areas R of the feature space so that the expected loss is minimized. This is the case if a feature vector v lies in R _k and the following also applies:

A classifier that works according to this decision rule is also referred to as a "Bayesian classifier" (see also [3]). This general form is optimal if all statistical assumptions are correct and the loss matrix C is known. In practice, a so-called anti-proportional cost matrix is used for the latter, which means that if the classification is correct, there is no loss (c _ii = 0), but if the classification is incorrect, the same loss always occurs (c _1j = i << j). A classifier which estimates this as a maximum likelihood decision rule is accordingly referred to as a maximum likelihood classifier.

With the assumption of an anti-proportional cost matrix, the above condition is simplified:

p () - p (k) p (/ k) <p () - p (j) p (/ j), with j ε [0,. . ., K - 1], j ≠ k (12)

Since the probability p (v) is always the same, this term can be omitted and results in the estimation equation

d _k () = p (k) p (/ k) (13)

which is now only calculated from the product of the probability of occurrence of the associated class p (k) and the probability distribution within a class p (v | k). If one uses the terms determined in the preliminary considerations for the statistical approach with regard to the normal distribution, one obtains the estimation equation for the maximum likelihood classifier:

Due to the properties of the normal distribution and the monotony of the logarithm already mentioned above, it does not affect the decision if all the estimation equations d _k (v) of the same monotone mapping

subordinate open and logarithmic.  

This results in the following final estimation equation for determining the maximum inference probability:

Because this estimation equation is a quadratic function with respect to of the feature vector v, you get a quadra table classifier, the decision-making function of which accordingly geometrically as a paraboloid over the feature space lets interpret. There is such a para for each sample class boloid, the classification result being all the more clear will fall, the further the paraboloid in the feature space others are distant. The probability of inference increases with increasing proximity to the centroid vector. In the M-dimensional The different sample classes become classification problems separated by hyperparaboloids. The areas are not necessarily coherent and especially when the Class areas can be a fairly complicated limitation arise.

Above all, the classification is problematic Constellation shows that individual class areas overlap and a feature vector lies within this intersection. Such penetrations involve a high risk of failure classification in the course of the statistical decision theory cannot be further minimized. Such problems can only be achieved by a suitable feature extraction avoid. To this day, however, no metho can be used and establish rules that follow a systematic approach would enable. This task is therefore high Measures the expertise and ingenuity of the engineer required.

Another problem is the fact that the maximum likelihood classifier is a quadratic classifier. The first two terms of the preceding estimation equation (15) ln (P _k ) and -1/2 ln (detK _k ) are constant and can therefore be calculated before the classification, so that they have only an insignificant influence on the computational effort. As will be clear from the experimental part of this diploma thesis in a later chapter, the computation effort (and also the storage space requirement) of the last term is much higher. Due to the required matrix multiplications for the formation of the quadratic form, which must be carried out for each classification of a feature vector v, the computing effort increases with the dimension M by the order O (M ² ). For this reason, measures to simplify the classification and reduce the computing effort are described below.

Mahalanobi's distance classifier

A significant simplification of the maximum likelihood classifier can be achieved by assuming that the class-specific covariance matrices K _{k are} identical and that the class-specific occurrence probabilities p _{k are the} same:

p (v | k) = N (µ _k , K) and p (k) = P _k = 1 / k (16)

In this case, the decision function according to equation (3-14) can be represented by a monotonous mapping

simplify considerably to the following estimation equation of the Mahalanobis distance classifier:

The class-specific probability of occurrence p (k) can be dispensed with, since this is identical for all classes when calculating the probability of inference. This gives a linear estimator with respect to the feature vector v, so that the number of required numerical operations can be significantly reduced compared to the quadratic classifier. In addition, the second term is constant and can be calculated before the classification. In addition, the effort for parameter storage in the learning phase is reduced enormously, since it is no longer necessary to save a covariance matrix for each class, but only the precalculated vector µ _k ^T K ^-1 .

For the statistical assumption of identical covariance matrices read are particularly useful for communications classifications tasks achieve good results because cases often occur there ten in which a feature vector v results from the additive transfer storage of a given setpoint (signal) and a random genetically distributed disturbance varieties that are the same for all classes blen (noise).

The Mahalanobis distance classifier's equation lets themselves geometrically - in continuation of the considerations of the quadra table classifier - interpret as follows. The class limit are now offered by (M-1) -dimensional linear subspaces separated. In the three-dimensional case, these are separation planes or in the two-dimensional case, dividing lines between the individual Classes.

As the term distance classifier already indicates, the linear classifier can also be interpreted geometrically as the distance from the feature vector v to the centroid vector µ. If one does not remove the constant factor 1/2 v ^T K ^-1 v in the course of simplifying the decision function according to equation (14), the result is an estimation equation

d _k () = (- _k ) ^T K ^-1 (- _k ) (17)

Since the figure for deriving the latter estimation equation mono is falling, the classification is now given by a minimal decision. The geometric location of all features vectors v that can also be used as Mahalanobis distance the same distance from a sample class k sit, describe an M-dimensional ellipsoid. in the  this distance is accordingly two-dimensional feature space shown as an ellipse.

This is the transition to another classifi cation strategy, the geometric approach. This Approach will be pursued in more detail below.

The geometric approach

The idea on which the geometric approach is based exists in the calculation of separation functions between the individual Sample classes. Because - as in the previous section became clear - aprio intended for the statistical approach ri information regarding the distribution density and the up Probabilities of occurrence must exist and the Calculation of an M-dimensional distribution density as very has proven to be complex, the statistical classifi kator not for everyone due to the restrictions mentioned use any classification problem. We are looking for half a classification strategy without the necessary Knowledge of distribution densities. This request he fills the geometric approach, in which a feature vector v as a point in the M-dimensional feature space is understood. A Classification is achieved by finding a hypersurface possible, which are the feature vectors of a class of which one separates another class. With the geometric approach, the Ent divorce function thus by separating surfaces or separating function NEN or realized by combinations thereof. However, too this approach is not without problems. So they can Dividing surfaces, similar to the distribution densities in the statistical approach, theoretically take any form. Also at In practice, geometric approaches therefore have limitations be made by looking at simple forms of separation limited functions. In addition, the way of separation interpre differently between the classes in the geometric approach be animals. While the statistical approach is a maxim the probability of inference leads to classification, the geometric interpretation requires a minimization of the  Distance between a feature vector and the individual classes focus, the already introduced centroid vectors. First, the feature space can only be in because of the interface two areas can be divided. Leave for multiple partitions however, several classes also differ.

Euclidean distance classifier

As became clear from the Mahalanobis distance classifier, geometric classifiers can be derived from the statistical classifiers. The requirement of not using distribution densities associated with the geometric approach can be met by the special case of uncorrelated features. This clearly means that all classes have the same rotationally symmetrical distribution and now differ only in their focal points, the centroid vectors p. The covariance matrix K can be dispensed with by assuming in its place a unit matrix I multiplied by the scatter:

K = σ ² I (19)

Inserted into the initial form of the estimation equations (14) it is obtained after a few simplifications:

With this simplification, from equation (3-20), the monotonically falling figure shows

derive the following simple estimation equation for the Euclidean distance classifier:

d _k () = | - _k | ² = || ² + | _k | ² - 2 _k ^T (21)

The estimator therefore describes nothing other than the Eu klid'sche square between the feature vector v and the Centroid vector µ. The classification can be defined geometrically  thus simply as a minimization of the distance between the Class midpoints (centroid vectors) and the to be classified interpret the characteristic vector. In the two-dimensional feature space the estimation functions are now described by circles, the dividing surfaces remain levels because the classifier continues is linear with respect to the feature vectors v. The parting surfaces are now perpendicular to the connecting lines between the individual major subjects.

Transferred to the M-dimensional feature space, the Ent So divorce functions to form hyperspheres that separate partitions Hyperplanes.

If we take a closer look at equation 21, it is easy to know that the term | v | ^{2 is} constant and can therefore be omitted. The Euclidean distance classifier is therefore much easier to implement than its statistical predecessor. It also offers considerably more favorable conditions in terms of computing effort.

City block classifier

By simply modifying the geometric distance calculation, a further geometric classifier can be derived from the estimation equation. By calculating the geometric distance instead of the squaring by forming an amount, one obtains the estimation equation for the city block classifier:

d _k () = | - _k | (22)

This notation is descriptive as a component Add the amounts to be interpreted across all characteristics M. There a realization of this classifier for the learning phase if only the calculation of the centroid vectors requires µ, be there may be no additional routines to implement the Learning phase of the city block classifier because of the training data (knowledge base) of the Euclidean distance classifier can be used.  

The polynomial approach

Since the separating surfaces considered in the geometrical approach to Separation of the pattern classes in the characteristic space is not arbitrary complex shapes can be taken with the Polynoman set another way. As with the preliminary considerations for the un different classification strategies based on the optimal Classification system has been shown, the difference can take any shape. One way for classification can therefore consist in ver seeks to approximate this optimal distinctive function. However, in order to obtain a mathematically comprehensible form, some restrictions are made. Basically everyone can any mathematical function in the form of a series development be described. This method of representation is ever but unsuitable for an implementation. It is advisable, restrict yourself to a certain class of functions. A a possible approximation approach consists in the description ei ner distinctive function in the form of polynomials, as one can also used in methods of compensation calculation. Theoretically can be any mathematical by an M-th degree polynomial cal function can be approximated. As a complete in practice polynomial approach is hardly feasible, one limits oneself mostly on 3rd order polynomial approximations.

In the following, the general form of the polynomial approach for the description of the differentiation functions will be presented:

As is generally known, the polynomial is composed of the product terms of the coefficients a and the components of the feature vector v. The feature vector v has the dimension M, after which the general polynomial approach can be described from the constant feature components and their combinations to the degree of the desired order. In order to obtain a more descriptive form of the polynomial approach, it is advisable to rename the feature components according to the following scheme:

x ₀ = 1, x ₁ = v ₁ , x ₂ = v _2,. . ., x _M = v _M , x _{M + 1} = v ₁ ² , x _{M + 2} = v ₁ v _2,. . . (24)

This enables the estimation equation of the polynomial approach to be formulated in the more compact vector notation:

The polynomial structure is therefore shown by Fig. X (v) wrote the feature vek extended by the polynomial terms gate v reflects. There is also a for each class k Coefficient vector a, which for the respective estimator parameters to be optimized in a class.

If all these coefficient vectors of all K classes are combined into a matrix and all estimated values d are combined into one estimated vector, the complete polynomial approach can be formulated compactly:

() = ^AT () (26)

The determination poses another problem in the following of the optimal coefficients a. Like the para to be estimated meters, centroid vectors µ for the geometric approach and additional Lich the covariance matrix K in the statistical approach, so must also the optimal coefficients a in for the polynomial classifier during a learning phase. The coefficients are then optimal if they have the true decision function in the form of Reflect polynomials. A first limitation is in In this regard, the restriction to polynomials up to one be agreed degrees. This question of the influence of the order of the bottom However, lynoms should only be discussed in the following two sections be discussed. To derive an equation for  the solution is therefore again the complete polynomial approach considered because the optimal equation is independent of the order of the polynomial approach.

The starting point again is the square mean approach | y - d | ² , i.e. minimizing the amount square of the error vector over the entire process:

S ² = E {| - ^AT () |} (27)

S as a function of the sought coefficients a must be optimized. The extreme of S is at the point where the partial derivatives vanish according to the coefficients. Since the 2nd derivative of the matrix A after the coefficients is always positive (matrix is not a negative definite), you get the minimum you are looking for. For the sake of clarity, the following formulas are shown using a single class k:

By solving these equations and placing y _k on the right side, one obtains the so-called normal equations, which form the following system of linear equations with respect to the coefficients a _k for given polynomial vectors x:

This system of equations can also be formulated in vector writing to determine the coefficients for a specific class k as follows:

E {() () ^T } _k = E {() _k } (30)

Combining the individual coefficient vectors a _k into a matrix A again gives a compact notation for the optimal determination equations over the entire recognition problem:

E {() () ^T } A = E {() ^T } (31)

This notation is to be interpreted in such a way that the left term describes the quadratic matrix from the empirical values over the entire learning sample - regardless of the class k. It is multiplied by the matrix A, which supplies the sought coefficient vectors a _k . The right term is particularly important. The notation is to be understood in such a way that the target vector y has a blanking property with regard to the values of the polynomial vector x. This means that the right side always represents the empirical value over the polynomial vectors x of a certain class k. Of particular note is the fact that only those polynomial vectors that belong to the respective class k are included in the calculation of the empirical value. However, the expected value is calculated over the entire sample size (!). This linear system of equations can e.g. B. solve with the Gauss-Jordan algorithm. To achieve an unambiguous solution (there is always the possibility of a variety of solutions) and a short computing time, special pivot strategies are recommended for practical handling of the polynomial approach.

The coefficients a calculated in this way describe an approximate mapping of the real differentiation function depending on the degree of polynomial implemented and are classified in the estimation equation according to Eq. 25, 26 used. A clearer interpretation of the polynomial approach is obtained if instead of the target vector y the probability p (v) = E {y | v} is used in the optimization condition. The estimate d _{k can} thus be interpreted as the probability of inference. The result of this is that the sum of the estimated values d _k over all classes K must always result in the value one:

Ideally, the estimate of the correct class will provide that Value one and the values of the other classes are zero. In in practice, a feature vector v will always be one Assigned to class k, which provides the greatest estimated value d. Or if one estimates the estimates as the probability of inference in interpreted, the estimated value with the highest true classified as probability.

Linear polynomial classifier

After the detailed considerations of the complete polynomial approach, what is relevant in practice should now be considered depending on the order of the polynomial. The simplest approach is the linear polynomial classifier. The M components of the feature vector v and a constant term v _{0 are included} in the calculation. The polynomial vector of the linear approach thus formed has the following form:

The feature vector v is thus around the zero degree element in This approach of order one expands and forms the bottom lynomial vector x.

The linear polynomial classifier thus clearly sets linear ones Separation functions in the feature space. In two-dimensional space the classes are thus separated by a straight line, in M-di dimensional feature space through hyperplanes. Regarding the The linear polynomial classifier also involves computing effort to handle high-dimensional classification problems well the main arithmetic routines focus on the solution of the linea limit their system of equations of order M + 1. However are the possibilities to approximate the real distinction limited functions, since, as mentioned, only approximate straight lines can be heard. Complex forms of differentiation Therefore, the linear approach is insufficient describe.

Square polynomial classifier

Similar to the linear approach, the feature vector v must be extended by certain terms. In addition to the constant term v _0, these are now also all second-degree combinations that can be formed from the M feature components v. According to the approach for calculating the possible combinations with repetition from the statistics, the following general determination equation results for the dimension of the polynomial vector x for higher degree polynomial approaches:

In this case, M denotes the number of signal pre-ver processing and feature extraction of the resulting components, G stands for the degree of the polynomial approach. It becomes clear that the effort in terms of computing speed and memory space increases enormously. The approximation matrix A z. B. increases quadratic with the dimension of the polynomial vector x. That has to Consequence that all polynomial approaches from the third order on Classification problems with a higher number of features components can no longer be mastered in practice. Though would be a higher degree of the polynomial approach with respect to one best possible approximation of the real distinction function desirable, but above all makes the case that at favorable hardware implementation according to classification results aspires to a higher order than the quadratic approach almost impossible.

For the quadratic approach, the polynomial vector x is calculated as follows after the preliminary considerations:

While in the case of the linear polynomial classifier, the classes areas in the two-dimensional feature space by straight lines sepa square curves are now possible. There through can do much more complex distinctive functions to be approached. A problem can also be solved by the Do not yet use the polynomial approach. These are the cases in which the area of a model class in the feature space is not ge is closed, but occupies several regions or in several re cluster disintegrates. Let such subdivided class areas neither geometrically nor statistically and with the help of the buttocks describe the lynomial approach. For this reason, the following a completely different classification strategy is presented that takes this problem into account.

The nonparametric approach

The nonparametric approach has the advantage that none Knowledge of distribution density or form of the difference The functions of the individual sample classes must be available. It is therefore a model-free classifier. Of the The basic idea behind this classification strategy is the assumption that clusters whose feature vectors in the Feature space are close together, ge to a sample class Listen. This is exactly the constellation that is a pattern class composed of several class areas. The non-pa Rametric approach is therefore very close to the geometric approach sentence connected to minimize the distance to the The focus of each class is based. However, in the way look at a possible occurrence of several clusters of a Mu accordingly also one for each individual area Class a custom centroid vector can be determined. You designate therefore such a nonparametric classifier also as a nearest neighbor classifier. A classification is simply based on that centroid vector of a cluster to determine the closest to the feature vector in the feature space most lies. One way of calculating the distance was already presented with the Euclidean distance. However, since it it is often not known how the clusters in the feature space  are distributed and therefore no statement about the corresponding appropriate class membership, one fails direct calculation of the cluster focus.

One way to determine reference vectors (i.e., Vek gates that, like the centroid vectors, form a cluster of be represent agreed class k in the feature space) in, simply the information of a randomly drawn learning sample directly as a knowledge base for the nearest neighbor class save the specification. This process of a so-called Nearest neighbor classifier with random sample can give quite satisfactory results, but it exists always the risk that the random sample is not "random" in each case at least one feature vector from all available Clustering. This is an optimal classification Determination of the next neighbor is incomplete and there insufficiently possible. Or else one would have to take this risk to minimize a very large proportion of the existing memo Include paint vectors in the reference set. Such an approach however, negatively affects storage space requirements as well the computational speed for classification, since a big one Number of redundant feature vectors saved in the reference set and calculates when calculating the nearest neighbor must be net. Given this unsatisfactory result It is also intended to be a systematic process for extraction of an optimal reference vector set.

The reduction algorithm

The reduction algorithm for determining a reference vector set that is optimal in terms of number and distribution in the feature space is based on the fact that only those feature vectors are included in the reference list that are located directly at the class boundary. For this purpose, all training vectors are initially arranged in a list and a second list for recording the reference vectors is initialized. Now, in a first step, the first training vector is included in the reference set and in the following, each training vector is successively classified using the current reference set. If the corresponding training vector is classified incorrectly, it is included in the reference set. This process clearly means that only those vectors are included that lie on the class boundary and are therefore easily assigned to a wrong class. Once all training vectors have been processed in this way, a second step is used to examine whether individual reference vectors have become superfluous as a result of vectors later added to the reference list. For this purpose, in a second step, all reference vectors are classified against each other and those reference vectors that are classified correctly are removed from the reference set. This prevents the list of reference vectors from becoming unnecessarily large. Then repeat steps 1 and 2 as long as either further training vectors are included in the reference set or in step 2 are removed from the reference set. This functionality of the reduction algorithm is listed again in the following overview.

• 1: Organize all training vectors in a list
• 2: Initialize reference list (initially empty)
• 3: Select the first vector of the training list
• 4: Training vector based on the vectors of the reference list classify
• 5: In case of incorrect classification (or empty reference list) the Include the training vector in the reference list
• 6: Select the next vector in the training list
• 7: Go to 4 as long as training list doesn't end
• 8: Select the first vector of the reference list
• 9: Selected vector based on the other vectors of the refe Classify border list
• 10: If misclassified go to 12
• 11: Remove the selected vector from the reference list
• 12: Select the next vector in the reference list
• 13: Go to 9 as long as reference list doesn't end
• 14: Go to 3 as long as reference list changes at 5 or 11 was changed
• 15: Delete all former training vectors
• 16: end

As a result of the reduction, a set of reference vectors is obtained tors, which enables the training vectors to be a hundred pro to classify correctly. If you look at the spatial distribution If the reference vectors are considered in more detail, it is confirmed  the assumption that reference vector, especially at class boundaries ren are located. Especially when the clusters in Merk painting space far apart, is the number of be required reference vectors low. It is different in the case that the Clu are very close to each other. Here is the number of refes reference vectors of a cluster in the reference set are considerably larger because the individual class areas more precisely delimited from each other Need to become.

Cleaning the training vectors

At this point, however, a problem is touched that has hardly been mentioned in previous considerations. This is the frequently occurring overlap of clusters of different classes in the characteristic space. As a result, almost all training vectors in the overlap area are included in the reference list to avoid misclassifications. In order to limit this effect, a method is proposed by which the training vector set is freed up in the run-up to the reduction of individual particularly exposed vectors in the border area. In order to carry out this process, known as the cleaning algorithm, two parameters must be specified for each class, which determine the intensity of the cleaning process. The first parameter p ₁ influences the width of the area to be cleaned in the feature space. It determines the number of nearest neighbors to be examined. With the second parameter p ₂ , the number of the nearest neighbors to be classified correctly among the p _{1 is} determined. It determines when a vector is removed from the training vector set. The cleaning algorithm therefore runs as follows. One selects the first training vector from the complete training list, determines the p ₁ nearest neighbors and leaves it in the training vector set, provided that p _{2 of} the p ₁ determined nearest neighbors belong to the class of the selected training vector. The same applies to the other training vectors. For a better overview, the cleaning algorithm is shown again in the following overview.

• 1: Organize all training vectors in a list
• 2: Initialize list for cleaned vectors (to Beginning empty)
• 3: Select the first training vector from the list
• 4: Determine the nearest p ₁ neighbors from the training list
• 5: If at least p _{2 of} these p ₁ neighbors belong to the class of the selected vector, enter the vector in the list of cleaned vectors
• 6: Select the next element in the training list
• 7: Go to 4 as long as training list doesn't end
• 8: Delete all old training vectors
• 9: end

The advantage of cleaning before the subsequent reduction is that there is a much better generalization and so achieved with a further reduction in the reference vector number can be.

Now that the theoretical basics of the individual classi fiction strategies are set out in the following Ver first run exemplary implementations of the various Classifiers are described.

Euclidean distance classifier functionality

In the learning phase, the centroid vector is calculated from the feature vectors of the learning sample for each individual class. To do this, the available memory is dynamically requested, the maximum number of vectors to be processed is calculated and the characteristics file is then read in. A sequential processing of the characteristics file could not be realized due to the possibility of scaling. Such sequential processing is however ruled out from the outset in the following more complicated classifiers. The centroid vectors are then calculated using the following formula:

The centroid vectors are written into the training file and the allocated memory is released again.

The tutorial for the Euclidean distance classifier also forms the basis for the City Block class sificator, since this is only in the calculation of the geometri distance, that in the classification phase he follows.

Maximum likelihood classifier functionality

The learning phase follows the same pattern as before. The calculation of the centroid vectors for determining the covariance matrices is also carried out according to equation 35. The covariance matrices are calculated according to the following rule:

The for the later calculation of the estimation equation for the classi required inversion of the covariance matrices using the Gauss-Jordan algorithm with full pivot by the covariance matrix by solving the equation system is successively converted into a uniform matrix.

The calculation of the determinant of each covariance matrix is carried out by bringing the square matrix K _k into its upper triangular shape, so that all elements below the main diagonal become zero. The corresponding determinant then results from the product of the elements over the main diagonal. The transformation into the upper triangular shape takes place with the help of the CROUT algorithm in the course of an LR decomposition. Then each square matrix A can be broken down into two matrices L and R, so that:

A = LR (37)

L are a triangular matrix of the form

and R has the following form:

In the Eq. 36 there are M ² equations from which the unknowns r _ij and l _ij can be calculated:

i <j: l _i0 r _0j + l _i1 r _1j +. . . + r _ij = a _ij
i = j: l _i0 r _0j + l _i1 r _1j +. . . + r _jj = a _ij
i <: l _i0 r _0j + l _i1 r _1j +. . . + l _ij r _jj = a _ij (40)

With the Crout algorithm already mentioned, these equations can be solved using the following procedure:

Since the determinant has to be log arithmetic anyway for later classification, this is done directly when determining the determinant. On the one hand, this saves computing time in the classification phase, and on the other hand, "overflow errors" are avoided, which can occur in the course of multiplication via the main dialogs, especially with large matrices. The log arithm of the determinant is thus calculated according to:

ln (det K _k ) = ln (r ₀₀ ) + ln (r ₁₁ ) +. . . + ln (r _MM ) (43)

Mahalanobi's distance classifier functionality

With regard to the calculation, the same applies under the maximum like lihood classifier said. However, in the course of Ver simplification which is the equality of the covariance matrices of all Requires classes, only one inverse across all classes partial covariance matrix formed. The inversion is again with Help the Gauss-Jordan algorithm performed by the Kova Rianzmatrix by solving the system of equations in one unit matrix is transferred. Due to preprocessing for the classification is the storage space requirement of the training data Part of this linear statistical classifier is low.

Linear polynomial classifier functionality

The dimension of a coefficient vector is increased by one compared to the dimension of the classification problem, since the constant term with the value one must be added. The extension by the constant term takes place directly when reading in the feature vectors. According to the approach for determining the optimal coefficient matrix A, the following system of equations has to be solved:

E { ^T } _k = E { _{k k} } (44)

Such a linear system of equations must be resolved for each class k according to the coefficient vector a _k . The left term describes a matrix that contains the expected value from the vector multiplication of the mean vector over the entire learning sample with its transposed counterpart. All feature vectors are included here regardless of their class affiliation. The term on the right is different. This describes the mean vector of a certain class k. However, the expected value is in turn formed over the entire learning sample, i.e. all feature vectors of the training file. The target vector y _{k has} a kind of blanking property in that only those feature vectors are summed up that belong to the processed class k. The mathematical solution of the K linear equation systems is carried out with the Gauss-Jordan algorithm with full pivoting.

The vector a _k is accordingly expanded by the matrix Z and E {x _k y _k ^T } by the unit matrix. The coefficients a and the elements of the matrix Z must therefore be determined, the latter representing the inverse of the matrix E {xx ^T }. The matrix E {xx ^T } is transferred column by column into the unit matrix. This is achieved by continued interchanging of rows and columns and replacing rows of the linear system of equations by linear combinations with other rows. After these conversions have been completed, the right-hand side contains the solution vector with the coefficients a _k in its first column and the remaining columns describe the inverse of E {xx ^T }.

For the exact procedure: first the first equation is divided by the element x ₀₀ and thus itself one. Then the remaining lines i = (1, 2, _... , M) are each replaced by the difference between the line i being processed and the first line multiplied by x _i0 . In this way, all elements x _i0 become zero and thus the first column of E {xx ^T } is identical to the first column of the unit matrix. The same procedure is followed for the other columns, so that the following algorithm results:

• - for all lines i = 0, 1,. . ., M
• - divide line i by diagonal element x _ii
• - replace all lines j = 0, 1,. . ., M; j << i by linear combinations of lines i and j according to the rule: line j = line j - x _ji line i

These changes are both on the left and on the right side. For stabilization and in the rear looking at the receipt of a clear solution that becomes full constant pivoting applied. For this the element, wel ches by swapping rows and columns on the main slide gonal can be brought and which has the largest amount, chosen as pivot element. To columns that are already with the one agreement matrix, may not be changed again only lines below the current pivot line or to the right of current pivot column can be swapped. Despite this stabilization However, it can happen that the matrix sin is valid and the calculations have to be canceled. The The program displays a corresponding message on the screen. However, such cases are very rare. Occasionally helps with sol problems, one of the two scaling options use. This avoids calculation accuracy those that can lead to a diagonal element becoming zero and thereby the matrix becomes singular (determinant = 0).

Square polynomial classifier functionality

The operation of the quadratic polynomial classifier ent speaks essentially the one for solving the linear approach. However, the feature vectors need not only be around the constant Term, but also to be expanded by the quadratic terms.  

According to the explanations for the derivation of the quadratic approach in the previous chapter, the dimension of the quadratic polynomial vector results from:

The extension by the constant term and by the quadratic combinations without repetition was achieved by multiplying the feature vector v by its transposed counterpart. A square matrix is obtained. If you take the elements of the upper or lower triangular shape including the main diagonal elements, you have the second-degree combinations you are looking for. This procedure is explained using the upper triangular shape.

In this way, the feature vectors are expanded by the second-order terms. The extension will be done directly at Reading the feature vectors from the file with the learning stitch sample made. To make similar calculations in the classifica avoidance phase, the dimension of the polynomial vectors written in the training file. Do the two formulas clear how quickly the storage and computing effort Classification problems of higher order can increase.

Nearest neighbor classifier with random sample functionality

Using the standard function for generating random numbers under C the desired number of feature vectors for each class included in the reference set. To generate the Zu if numbers are entered, a start value is entered. This starting value  has the advantage that it is always the same coincidence number series can be reproduced. For this reason, the Start value also saved in the training file with later to be able to understand the selection of the reference vectors.

Nearest neighbor classifier with reduction algorithm functionality

How it works is based on the explanations in the derivation of the algorithms for cleaning and reducing. The class The calculation in the course of the calculations is based on the formula of the EU Klidian distance (Eq. 21). Instead of zen However, the corresponding reference vector becomes troidvector used.

Neural Networks

There are approximately 10 billion nerve cells in the human brain act, each with an average of 1000 to 10,000 others connected is. On this enormous parallelism and connectivity human mental abilities. The properties such as self-organization, trainability, fault tolerance and above the ability to generalize such a system one tries to take advantage of it also in electronic systems do. But you do not try the circuit diagram of the Ge brain, but you formally grasp the properties a single component, the neuron, and thus receives a simple one mathematical model of a neuron. Generally there are neurona le networks of knots and edges. The latter connect the individual a knot with each other. A network node contains a summation link, which adds all input signals, and a non-link linear transfer function. The mesh edges contain evaluators weights that amplify or weaken a signal. First if the sum of the weighted input signals is a certain one Exceeds the threshold value, the neuron emits an output signal.

Here you can see the analogy to the biological neuron. The Neurons correspond to the network nodes, the dentrites and the  Synapses the network edges and the action potentials beyond the axon is passed on to the subsequent neurons Chen input and output values. For this replica of a biological neurons, the term PERCEPTRON is also used citizens. They are characteristic of a neural network Network topology and the learning process. Except for the topology is the way the network stores information chert, crucial. In this process, which is also called learning or training, are the thresholds and weights set. They represent the network's knowledge base.

The multi-layer perceptron with error back propagation

There are now quite a number of different networks structures with different properties. One of them is the multilayer perceptron with error back propagation algo rhythm. It is a recursive learning process, which is now referred to as a modern standard process can be.

Mathematically speaking, neural networks form input vectors (Feature vectors) on output vectors. The procedure be rests on the consideration that the network is responding correctly learns a feature vector from its own mistake learn. The choice of shares is decisive for learning behavior vation function. Activation is the reaction on the applied signals. There has been a linear act tion function was found to be unsatisfactory because it is a Has a number of disadvantages. So is a network of linear per ceptronen simply unable to solve the standard problem of Learn EXOR function. Even if you have any number of linear This is how neurons are interconnected to form a complex network Result a linear network and as such by a single one Neuron replaceable. For this reason, a logisti is used cal activation function (sigmoid function).

You can see that it can only take values between 0 and 1, is limited up and down. In addition, she owns an activation threshold, namely the turning point. Since one  the most favorable location of this threshold is not a priori knows, this must be learned just like the edge weights.

Basically, weights and threshold values can be handled immediately deln if the latter as signal connections from constantly ak activated neurons (signal 1 is constantly present). First, the weights and thresholds with random who initialized, which must not be the same, however otherwise the system cannot learn. Now put an on gangsvector to the input layer of the network, this is with by means of a matrix multiplication on the hid layer ben. The Hidden layer only on any intermediate layers that may still be present and finally mapped onto the output layer. This before course of forward propagation is also after its English Name called "forward propagation". The resulting one Output vector initially deviates due to initialization by random values from the desired target vector. Then ßend-is the backward correction, which is the network type Has given names, namely BACKPROPAGATION. To do this, calculate by forming the difference between the desired and the received out put an error value, the so-called delta. The weights and Thresholds of all signal connections to the treated new ron must be changed depending on the delta. This is done by multiplying the delta by all Signa len of the neurons from the intermediate layer proportional to the Er result. So you go for all neurons between Hiddenschicht and output layer and in this way achieves a correction of the entire weight matrix. This process is called the Delta rule.

Do the same for the layers in front of you until you has worked its way to the entrance layer. It turns out but the following problem. For the middle layer to be changed there are no concrete setpoints! The solution is imagining recursive approach, the so-called generalized delta rule. The new delta values are calculated using the del tas of the output layer. For larger than three-layer  Networking is done exactly the same way, except that you have the delta values the last working shift in the calculation must bring. It can be seen that it is a recursive loading invoice rule.

With this learning algorithm according to the gradient descent method becomes the direction in which the error changes the fastest dert, determined. A special parameter, the learning rate, be agrees the step size and should strong oscillations of the mis prevent reading terms. However, there is no guarantee that the System really came to the smallest possible error. It can namely accidentally getting into a so-called local minimum, whose error value is above the smallest possible. The system then no longer gets out of this "trap". A very beginning vivid example of this phenomenon is that of a mountain trip. If the starting point is random and the algorithm is to always go down the steepest path, that's how it is no guarantee of finding your way out of the mountains. Man perhaps advises in a basin. This can be done under certain circumstances you can only reach it if you cross other hills, so also going uphill. If you realize that the neural Network is a "multidimensional" mountain tour Dimension of this problem clearly. Knowledge of the local Minima is important and will help us later when evaluating the Encounter simulation results again.

The mathematical implementation

Not all formulas and equations are to be found here completely be directed. Only the calculation rules in as far as explained how they are important for the implementation. A three-layer network with an input, egg serves as a model a hidden and an output layer.

Total errors for a learning pattern p:

t _pk = desired target value
O _pk = activation of the corresponding neuron
p = learning pattern
k = output element k

Total error of the neural network:

Correction value for the weighting factors between element j of the hidden and element k of the output layer (delta):

Δw _ij = β.δ _k .O _j (50)

Β is the learning rate, a proportionality factor between 0 and 1, which be the speed of the learning process Right. The smaller it is, the slower it is learned, however the overall system is all the more stable (typical value: 0.2-0.7).

When using the sigmoid activation function (1 / (1 + e ^input )), the error term δ _k for the element k of an output layer is calculated according to:

δ _k = O _k . (1 - O _k ). (t _k - O _k ) (51)

O _j = output value of the upstream layer
t _k = desired target value

There is no target value to compare with. Therefore, a recursive calculation takes into account the delta of the previous shift.

w _ij (L _s ) = β.δ _j .O _i + α.Δw _ij (L _s - 1) (52)

The second term of the formula takes into account the past of the learning process. Changes in the weighting factor in the previous learning step (L _s - 1) are included. This can dampen the oscillation of the output values, since the correction of the weighting factors does not follow every small change in the error gradient, not the 10405 00070 552 001000280000000200012000285911029400040 0002019734905 00004 10286t. The term is often referred to as momentum.

The summation over k determines the error that all downstream neurons introduce proportionately.
α = weighting factor of the momentum term (factor between 0 and 1; typical value: <0.9)
L _s = number of learning steps

In the following, reference should be made again to the designation used can be used.

A measure that helps reduce circuitry is outsourcing the training phase to an external NEN computer to then use the knowledge base determined there take and only the circuit elements to carry out the recall phase in real time on the resynthesizer. In all cases, an option to reload is advantageous of knowledge bases via an external storage medium such as a floppy disk or CD-ROM in the classifier unit KL. With that he becomes possible, with different training results a re Synthesize the output sound, giving flexibility of the resynthesizer according to the present invention increases.

In the illustration of FIG. 2 is a modified form of exporting is approximately Resynthesizers according to the present invention is shown which differs only in details from the block diagram of FIG. 1. The changes concern the control signals for activating / deactivating the oscillators VCO_K1. . .VCO_Kn. The control signals A (x) and K (x) are combined to form a single control signal K (x), thus reducing the circuitry and memory requirements in the memory S. This is made possible by dispensing with downstream switching elements on each amplifier stage VCA and activating / deactivating the oscillators VCO_K1. . .VCO_Kn al lein takes place via the amplifier stages VCO by these being controlled by the new control signal K (x) and an amplification of ZERO corresponds to a deactivation of an oscillator.

The control signals K (x) no longer have the binary form explained with reference to FIG. 1, but correspond to a multiplication of the original control signals A (x) and K (x). This results in the example mentioned above:

K '(x) = A (x) .K (x) =
(A1 (x), A2 (x)... An (x)). (K1 (x), K2 (x).. .Kn (x)) =
(5.3 ... 10). (1.0 ... 1) = (5.0 ... 10) (54)

In the illustration of FIG. 3 shows a modified form of exporting is approximately Resynthesizers according to the present invention is shown, which in terms of the structure of the plurality of oscillators from the block diagram of Fig. 1 and Fig. 2 differs. There is no longer a separate oscillator VCO_K1 for each waveform and thus class K1 to Kn. . .VCO_Kn is used, but a waveform memory WT is provided, in which the individual waveforms are stored as waveforms W1 to Wn in digital form as audio samples. A single amplifier stage VCA is driven by these waveforms W1 to Wn.

An oscillator is realized in that, depending on the control signal K (x), the assigned class K1 to Kn or the assigned oscillator VCO_K1. . .VCO_Kn corresponding waveform W1 to Wn is read out from the waveform memory WT in proportion to the desired audio frequency and these digital audio data are converted into an analog oscillation signal via a digital / analog converter (not shown), which is then subsequently amplified with the amplifier stage VCA. The respective control signal K (x) requires activation of several classes K1 to Kn or oscillators VCO_K1. . .VCO_Kn, the corresponding waveforms W1 to Wn are read out in parallel and implemented digitally / analogously, with a link (not shown) of the parallel waveform data then being made before the digi / analog conversion according to the possibilities shown with reference to FIG. 1 . With such a structure, the circuit complexity is considerably reduced and, moreover, there is an easier possibility to include non-cyclic wave forms or wave forms with inharmonic signal components as separate classes K1 to Kn.

Common to all three advantageous embodiments according to FIGS. 1 to 3 is the problem that if, according to the control signals A (x) or only K (x), a "switchover" from one class K1 to Kn to another class takes place, the transition of the oscillator signals VCO_K1. . .VCO_Kn or waves form W1. . .Wn not always running continuously at zero crossing. For this reason, the resynthesized output oscillation curve can be smoothed. Such a possible smoothing level GL is inserted after the linkage VG of partial vibrations to the overall vibration, as is also shown in FIGS. 1 to 3.

Possible smoothing methods are in particular averaging over a predeterminable interval, linear compensation by at for example three points or the determination of an end same-order 3rd order polynomial (see, inter alia, "Signal Processing tion: numerical processing of digital signals / Schrüfer, E., Munich; Vienna: Hanser-Verlag 1990, p. 110ff. ").

Based on the smoothing process mentioned it is clear that the se particularly advantageous on the digital level by means of a suitably programmed microprocessor or digital  Signal processor can be performed. For this reason it is recommended that all switching elements up to and including the Link VG to perform at the digital level by both the generation of waveforms by the oscillators VCO_K1. . .VCO_Kn or the waveforms W1. . .Wn and their Reinforcement and possible linking or mixing purely digi Tal occur and only after smoothing the resynthesized Output waveform through a digital / analog conversion to be led. This shows that the resynthesizer according to the present invention particularly advantageous and ef fectively implemented as a digital resynthesizer.

Furthermore, there is the possibility of storing the memory S cyclically read out, started by the trigger signal T. Frame made during the game on the keyboard KB pitch bending (so-called pitch bending) the readout frequency the control signals A (x) or K (x) from the memory S via the variable counter mentioned above.

In all of the exemplary embodiments, the reproduced waveform at the output of the link VG or in the case of smoothing at the smoothing stage GL can be further processed or modified by sound processing stages according to the taste of the musician. Such sound processing stages can be carried out after a digital / analog conversion as analog circuits with discrete electronics or also on a digital level, in the latter case a digital / analog conversion only taking place at the output of the resynthesizer. In the block diagrams according to FIGS . 1 to 3, the output is symbolically marked with a loudspeaker.

In particular, for sound processing in a filter stage F the cutoff frequency of the sound can be varied, e.g. B. the upper one Cut-off frequency through a low-pass filter or the lower limit frequency through a high-pass filter. Also bandpass and bandpass re-filters can be used. Furthermore, a subsequent one Amplitude, filter frequency or pitch envelope ADSR pre will be seen with the timeline of the corresponding  Sound parameters in a suitable manner on the type of reproduce th sound pattern (e.g. flat sound, percussive sound, piano sound etc.) can be set. Furthermore there are modules the pitch, the amplitude or the filter frequency possible.

The resynthesizer according to the present invention can thus be used to learn sounds in the training phase and to resynthesize them in the recall phase or real-time playback phase with a low memory requirement. Even if it is often not possible to reproduce the output sound 100% identically, the resynthesized sound is extremely similar to the output sound and can be suitably reshaped by the musician with the other sound processing options, thereby creating diverse possibilities for creating new, resynthesized output sound building sounds is made possible. In addition, there is the possibility, especially inharmonic signal components (partials of a spectrum, which are not integer multiples of the fundamental; especially breathy sound material), which are provided in specially designed oscillator models or in the embodiment according to FIG. 3 are provided as wave forms , zuzumi rule the resynthesized signal, whereby the tonal diversity of the resynthesizer according to the present invention can be further increased.

All mentioned in the above description or in the Fi guren shown features should, provided the known state the technology allows this, alone or in combination are considered to be within the scope of the invention.

The foregoing description of preferred embodiments according to the invention are for the purpose of illustration give. These are not exhaustive. The invention is also not limited to the exact form specified, but it are numerous modifications and changes under the Technical teaching specified above possible. One before preferred embodiment was chosen and described to principal details of the invention and practical applications  to clarify in order to enable the expert to Realize invention. A variety of preferred designs Forms and other modifications come with special Areas of application into consideration.

Claims (13)

1. Electronic sound generator with a plurality of oscillators of different waveforms, characterized in that one or more oscillators (VCO_K1... VCO_Kn) are obtained as a function of control signals obtained with means for automatic pattern recognition (KL) from a vibration curve (A (t)) to be simulated (K (x), A (x)) can be activated or deactivated in such a way that the oscillator (s) (VCO_K1 ... VCO_Kn) are activated with the most similar form of oscillation to the oscillation curve (A (t)) to be simulated. 2. Electronic sound generator with an oscillator with one Waveform memory (WT) with a variety different waveforms (W1 ... Wn) characterized in that the oscillator depending on with means for automatic pattern recognition (KL) from one reproduced waveform (A (t)) obtained Control signals (K (x), A (x)) can be controlled so that the An oscillator corresponding to the oscillation the most similar vibration pattern (A (t)) Waveform or combination of waveforms. 3. Electronic sound generator according to claim 1 or 2, characterized in that as a means for automatic pattern recognition (KL) neural network is provided. 4. Electronic sound generator according to claim 1 or 2, characterized in that as a means for automatic pattern recognition (KL) geometric classifier, especially an Euclidean Distance classifier or a city block Distance classifier is provided.   5. Electronic sound generator according to claim 1 or 2, characterized in that as a means for automatic pattern recognition (KL) statistical classifier, especially a Maximum likelihood classifier or a Mahalanobis Distance classifier is provided. 6. Electronic sound generator according to claim 1 or 2, characterized in that as a means for automatic pattern recognition (KL) Polynomial classifier, especially a linear one Polynomial classifier or a quadratic Polynomial classifier is provided. 7. Electronic sound generator according to claim 1 or 2, characterized in that as a means for automatic pattern recognition (KL) heuristic classifier, especially a Nearest neighbor classifier with random sample or a nearest neighbor classifier with cleaners and Reduction algorithm is provided. 8. Electronic sound generator according to one of claims 1 to 7, characterized in that an amplitude spectrum (SP) of the model Vibration curve (A (t)) as an input signal for the Means for automatic pattern recognition (KL), which in particular by means of a Fourier transformation (FFT) is obtainable. 9. Electronic sound generator according to one of claims 1 to 8th, characterized in that Control signals (K (x), A (x)) for activation or Deactivation of the majority of oscillators (VCO_K1. .VCO_Kn) or to control the oscillator with Waveform memory (WT) in intervals (x) of each  a whole or half an oscillation period of the to be reproduced vibration curve (A (t)) can be specified. 10. Electronic sound generator according to one of claims 1 to 9, characterized in that the control signals (K (x), A (x)) for activation or Deactivation of the majority of oscillators (VCO_K1. .VCO_Kn) or to control the oscillator with Waveform memory (WT) in a memory means (S) depositable and hence proportional to the frequency of the generating pitch are available. 11. Electronic sound generator according to one of claims 1 to 10, characterized in that a means for smoothing (GL) the transitions of the different activated or deactivated Oscillators (VCO_K1 ... VCO_Kn or WT) generated Output vibration curve is provided. 12. Electronic sound generator according to one of claims 1 to 11, characterized in that each by the respective control signals (K (x), A (x)) activated oscillator (VCO_K1 ... VCO_Kn or WT) with the Frequency of the pitch to be generated oscillates. 13. Process for electronic sound generation with a Majority of oscillators different Waveform, characterized in that one or more oscillators (VCO_K1 ... VCO_Kn) in Dependence on using means for automatic Pattern recognition (KL) from a replica Vibration curve (A (t)) obtained control signals (K (x), A (x)) can be activated or deactivated so that the or the oscillators (VCO_K1 ... VCO_Kn) with the  the most similar vibration pattern (A (t)) Waveform is activated.