Patent Publication Number: US-2007118370-A1

Title: Methods and apparatuses for variable dimension vector quantization

Description:
This is a divisional of application Ser. No. 10/379,201, filed on Mar.4, 2003, entitled “Methods and Apparatuses for Variable Dimension Vector Quantization,” and assigned to the corporate assignee of the present invention and incorporated herein by reference. 
    
    
     BACKGROUND  
      Speech analysis involves obtaining characteristics of a speech signal for use in speech-enabled and/or related applications, such as speech synthesis, speech recognition, speaker verification and identification, and enhancement of speech signal quality. Speech analysis is particularly important to speech coding systems.  
      Speech coding refers to the techniques and methodologies for efficient digital representation of speech and is generally divided into two types, waveform coding systems and model-based coding systems. Waveform coding systems are concerned with preserving the waveform of the original speech signal. One example of a waveform coding system is the direct sampling system which directly samples a sound at high bit rates (“direct sampling systems”). Direct sampling systems are typically preferred when quality reproduction is especially important. However, direct sampling systems require a large bandwidth and memory capacity. A more efficient example of waveform coding is pulse code modulation.  
      In contrast, model-based speech coding systems are concerned with analyzing and representing the speech signal as the output of a model for speech production. This model is generally parametric and includes parameters that preserve the perceptual qualities and not necessarily the waveform of the speech signal. Known model-based speech coding systems use a mathematical model of the human speech production mechanism referred to as the source-filter model.  
      The source-filter model models a speech signal as the air flow generated from the lungs (an “excitation signal”), filtered with the resonances in the cavities of the vocal tract, such as the glottis, mouth, tongue, nasal cavities and lips (a “synthesis filter”). The excitation signal acts as an input signal to the filter similarly to the way the lungs produce air flow to the vocal tract. Model-based speech coding systems using the source-filter model generally determine and code the parameters of the source-filter model. These model parameters generally include the parameters of the filter. The model parameters are determined for successive short time intervals or frames (e.g., 10 to 30 ms analysis frames), during which the model parameters are assumed to remain fixed or unchanged. However, it is also assumed that the parameters will change with each successive time interval to produce varying sounds.  
      The parameters of the model are generally determined through analysis of the original speech signal. Because the synthesis filter generally includes a polynomial equation including several coefficients to represent the various shapes of the vocal tract, determining the parameters of the filter generally includes determining the coefficients of the polynomial equation (the “filter coefficients”). Once the filter coefficients for the synthesis filter have been obtained, the excitation signal can be determined by filtering the original speech signal with a second filter that is the inverse of the synthesis filter (an “analysis filter”).  
      Methods for determining the filter coefficients include linear prediction analysis (“LPA”) techniques or processes. LPA is a time-domain technique based on the concept that during a successive short time interval or frame “N,” each sample of a speech signal (“speech signal sample” or “s[n]”) is predictable through a linear combination of samples from the past s[n-k] together with the excitation signal u[n]. The speech signal sample s[n] can be expressed by the following equation:  
               s   ⁡     [   n   ]       =         ∑     k   =   1     M     ⁢       a   k     ⁢     s   ⁡     [     n   -   k     ]           +     Gu   ⁡     [   n   ]                 (   1   )             
 
 where G is a gain term representing the loudness over a frame with a duration of about 10 ms, M is the order of the polynomial (the “prediction order”), and a k  are the filter coefficients which are also referred to as the “LP coefficients.” The filter is therefore a function of the past speech samples s[n] and is represented in the z-domain by the formula: 
 
 H[z]=G/A[z]   (2) 
 
 A[z] is an M order polynomial given by:  
               A   ⁡     [   z   ]       =     1   +       ∑     k   =   1     M     ⁢       a   k     ⁢     z     -   k                     (   3   )             
 
 The order of the polynomial A[z] can vary depending on the particular application, but a 10th order polynomial is commonly used with an 8 kHz sampling rate. 
 
      The LP coefficients a 1  . . . a M  are computed by analyzing the actual speech signal s[n]. The LP coefficients are approximated as the coefficients of a filter used to reproduce s[n] (the “synthesis filter”). The synthesis filter uses the same LP coefficients as the analysis filter and when driven by an excitation signal, produces a synthesized version of the speech signal. The synthesized version of the speech signal may be estimated by a predicted value of the speech signal {tilde over (s)}[n]. {tilde over (s)}[n] is defined according to the formula:  
                 s   ~     ⁡     [   n   ]       =     -       ∑     k   =   1     M     ⁢       a   k     ⁢     s   ⁡     [     n   -   k     ]                     (   4   )             
 
      Because s[n] and {tilde over (s)}[n] are not exactly the same, there will be an error associated with the predicted speech signal {tilde over (s)}[n] for each sample n referred to as the prediction error e p [n], which is defined by the equation:  
                 e   p     ⁡     [   n   ]       =         s   ⁡     [   n   ]       -       s   ~     ⁡     [   n   ]         =       s   ⁡     [   n   ]       +       ∑     k   =   1     M     ⁢       a   k     ⁢     s   ⁡     [     n   -   k     ]                       (   5   )             
 
 Interestingly enough, the prediction error e p [n] is also equal to the excitation signal scaled by the gain. Where the sum of all the prediction errors defines the total prediction error E p : 
 
E p =Σe p   2 [k]  (6) 
 
 where the sum is taken over the entire speech signal. The LP coefficients a 1  . . . a M  are generally determined so that the total prediction error E p  is minimized (the “optimum LP coefficients”). 
 
      One common method for determining the optimum LP coefficients is the autocorrelation method. The basic procedure consists of signal windowing, autocorrelation calculation, and solving the normal equation leading to the optimum LP coefficients. Windowing consists of breaking down the speech signal into frames or intervals that are sufficiently small so that it is reasonable to assume that the optimum LP coefficients will remain constant throughout each frame. During analysis, the optimum LP coefficients are determined for each frame. These frames are known as the analysis intervals or analysis frames. The LP coefficients obtained through analysis are then used for synthesis or prediction inside frames known as synthesis intervals. However, in practice, the analysis and synthesis intervals might not be the same.  
      When windowing is used, assuming for simplicity a rectangular window of unity height including window samples w[n], the total prediction error Ep in a given frame or interval may be expressed as:  
               E   p     =       ∑     k   =     n   ⁢           ⁢   1         n   ⁢           ⁢   2       ⁢       e   p   2     ⁡     [   k   ]                 (   7   )             
 
 where n1 and n2 are the indexes corresponding to the beginning and ending samples of the window and define the synthesis frame. 
 
      Once the speech signal samples s[n] are isolated into frames, the optimum LP coefficients can be found through autocorrelation calculation and solving the normal equation. To minimize the total prediction error, the values chosen for the LP coefficients must cause the derivative of the total prediction error with respect to each LP coefficients to equal or approach zero. Therefore, the partial derivative of the total prediction error is taken with respect to each of the LP coefficients, producing a set of M equations. Fortunately, these equations can be used to relate the minimum total prediction error to an autocorrelation function:  
                 E   p     =         R   p     ⁡     [   0   ]       -       ∑     i   =   1     M     ⁢       a   i     ⁢     R     p   [       ⁢   k           ]           (   8   )             
 
 where M is the prediction order and R p (k) is an autocorrelation function for a given time-lag l which is expressed by:  
               R   ⁡     [   l   ]       =       ∑     k   =   l       N   -   1       ⁢       w   ⁡     [   k   ]       ⁢     s   ⁡     [   k   ]       ⁢     w   ⁡     [     k   -   l     ]       ⁢     s   ⁡     [     k   -   l     ]                   (   9   )             
 
 where s[k] is a speech signal sample, w[k] is a window sample (collectively the window samples form a window of length N expressing in number of samples) and s[k-l] and w[k-l] are the input signal samples and the window samples lagged by l. It is assumed that w[n] may be greater than zero only from k=0 to N-1. Because the minimum total prediction error can be expressed as an equation in the form Ra=b (assuming that R p [0] is separately calculated), the Levinson-Durbin algorithm may be used to solve the normal equation in order to determine for the optimum LP coefficients. 
 
      Unfortunately, no matter how well the model parameters are represented, the quality of the synthesized speech produced by speech coders will suffer if the excitation signal u[n] is not adequately modeled. In general, the excitation signal is modeled differently for voiced segments and unvoiced segments. While the unvoiced segments are generally modeled by a random signal, such as white noise, the voiced segments generally require a more sophisticated model. One known model used to model the voiced segments of the excitation signal is the harmonic model.  
      The harmonic model models periodic and quasi-periodic signals, such as the voiced segments of the excitation signal u[n] as the sum of more than one sine wave according to the following equation:  
               u   ⁡     [   n   ]       =       ∑     j   =   1       N   ⁡     (   T   )         ⁢       x   j     ⁢     cos   ⁡     (         ω   j     ⁢   n     +     θ   j       )                   (   10   )             
 
 where each sine wave x j  cos(ω j n+θ j ) is known as a harmonic component, and each harmonic component has a frequency value that is an integer multiple “j” of a fundamental frequency ω o ; ω j  is the frequency of the j-th harmonic component (the “harmonic frequency”); x j  is the magnitude of the j-th harmonic component (the “harmonic magnitude”); θ j  is the phase of the j-th harmonic component (the “harmonic phase”); and N(T) is the number of harmonic components. The harmonic frequency ω j  is defined according to the following equation:  
                   ω   j     =       2   ⁢   π   ⁢           ⁢   j     T       ;     ⁢     
     ⁢       j   =   1     ,   2   ,   …   ⁢           ,     N   ⁡     (   T   )                 (   11   )             
 
 where T is the pitch period representing the periodic nature of the signal and is related to the fundamental frequency according to the following equation:  
             T   =       2   ⁢   π       ω   o               (   12   )             
 
 Together, all the harmonic magnitude components x j , j=1, 2, . . . , N(T) form a vector (a “harmonic magnitude vector” or “harmonic magnitude”) according to the following equation: 
 
x T =[x 1  x 2  x j  . . . x N(T) ]  (13) 
 
 where the number of harmonic components (also referred to as the “harmonic magnitude vector dimension”) N(T) is defined according to the following equation:  
               N   ⁡     (   T   )       =       α   ⁢           ⁢   T     2             (   14   )             
 
 where α is a constant (the “period constant”) and is often selected to be slightly lower than one so that the harmonic component at the frequency ω=π is excluded. As indicated in equation (14), the number of harmonic components N(T) is a function of the pitch period T. The typical range of values for T in speech coding applications is [20, 147] and is generally encoded with 7 bits. Under these circumstances and with α=0.95, N(T) ε [9,69]. 
 
      Together, the fundamental frequency or pitch period, harmonic magnitudes and harmonic phases comprise the three harmonic parameters used to represent the voiced excitation signal. The harmonic parameters are determined once per analysis frame using a group of techniques, where each techniques is referred to as “harmonic analysis.” In the harmonic model, if the analysis frame is short enough so that it can be assumed that the pitch or pitch period does not change within the frame, it can also be assumed that the harmonic parameters do not change over the analysis frame. Additionally, in speech coding applications, it can be assumed that only the phase continuity and not the harmonic phases of the harmonic components are needed to create perceptually accurate synthetic speech signals. Therefore, for speech coding applications, harmonic analysis generally refers only to the procedures used to extract the fundamental frequency and the harmonic magnitudes.  
      An example of a known harmonic analysis process used to extract the harmonic parameters of the excitation signal of a speech signal is shown in  FIG. 1 . The harmonic analysis process  200  is performed on a frame-by-frame basis for each frame of the excitation signal u[n] and generally includes: windowing and converting the excitation signal into the frequency domain  206 ; and performing spectral analysis  207 . Windowing and converting the excitation signal into the frequency domain  206  includes windowing a frame of the excitation signal to produce a windowed excitation signal and transforming the windowed excitation signal into the frequency domain using the fast Fourier transform (“FFT”). The window used to window the excitation signal frame may be a Hamming or other type of window. If the window is longer than the frame, the frame is padded with samples having zero magnitude.  
      Performing spectral analysis  207  basically includes, estimating the pitch period  208 ; locating the magnitude peaks  210 ; and extracting the harmonic magnitudes from the magnitude peaks  212 . Estimating the pitch period  208  includes determining the pitch period T or the fundamental frequency ω o  using known pitch extraction techniques. The pitch period may be estimated from either the excitation signal or the original speech signal. Locating the magnitude peaks  210  is accomplished using the pitch period and gives the location of the harmonic components. The harmonic magnitudes are then extracted from the magnitude peaks in step  212 .  
      There are many known speech coders that use the harmonic model as the basis for modeling the voiced segments of the excitation signal (the “voiced excitation signal”). These coders represent the harmonic parameters with varying levels of complexity and accuracy and include coders that use the following techniques: constant magnitude approximations such as that used by some linear prediction (“LPC”) coders; partial harmonic magnitude techniques such as that used by mixed excitation linear prediction-type (“MELP-type”)of coders; vector quantization techniques including, variable to fixed dimension conversion techniques such as that used by harmonic vector excitation coders (“HVXC”); and variable dimension vector quantization techniques.  
      In order to compare the performance of these coders, spectral distortion (“SD”) is often used as a performance indicator for both models and, as will be discussed later, quantizers. SD provides a measure of the distortion caused by representing a value f(x j ) (through modeling and/or quantizing) with another value f(y j ), and is determined according to the following equation:  
             SD   =           1     N   ⁡     (   T   )         ⁢       ∑     j   =   1       N   ⁡     (   T   )         ⁢       (       f   ⁡     (     x   j     )       -     f   ⁡     (     y   i     )         )     2           .             (   15   )             
 
 where, x j  and y j  each represent a set of harmonic magnitudes, and f(•)=20 log 10 (•) converts the harmonic magnitudes to the decibel domain (dB). 
 
      Constant magnitude approximations use a very crude approximation of the harmonic magnitudes to model the excitation signal (referred to herein as the “constant magnitude approximation”). In the constant magnitude approximation, used by some standard LPC coders (for example, see T. Tremain, “The Government Standard Linear Predictive Coding Algorithm: LPC-10,” Speech Technology Magazine, pp. 40-49, April 1982), the voiced excitation signal is represented by a series of periodic uniform-amplitude pulses. These pulses have a harmonic structure in the frequency domain which roughly approximates the harmonic magnitudes x j  of the voiced excitation signal. The constant magnitude approach thus represents the voiced excitation signal by a constant value “a” for each of its harmonic magnitudes x j , where the modeled or approximated harmonic magnitudes (each “y j ”) are generally expressed in the log domain f(y j )=20 log(y j ), according to the following equation: 
 
 f ( y   j )= a; j =1, 2 , . . . , N ( T )   (16) 
 
 To minimize the SD, “a” is determined as the arithmetic mean of the harmonic magnitudes in the log domain, according to the equation:  
             a   =       1     N   ⁡     (   T   )         ⁢       ∑     j   =   1       N   ⁡     (   T   )         ⁢     f   ⁡     (     x   j     )                   (   17   )             
 
 where each f(x j )=20 log(x j ), and N(T) is the number of harmonic magnitudes. Although LPC coders using the constant magnitude approximation can produce intelligible synthesized speech at low bit rates, the quality is generally considered poor. 
 
      Quality improvements can be achieved by modeling only some of the harmonic components with a constant value. In a partial harmonic magnitude technique, a specified number of harmonic magnitudes are preserved while the rest are modeled by a constant value. The rationale behind this technique is that the perceptually important components of the excitation signal are often located in the low frequency region. Therefore, even by preserving only the first few harmonic magnitudes, improvements over LPC coders can be achieved.  
      In one example, where the partial harmonic magnitude technique is implemented in the federal standard version of an MELP-type coder (see A. W. McCree et al, “MELP: the New Federal Standard at 2400 BPS,” IEEE ICASSP, pp. 1591-1594, 1997), the first ten (10) modeled harmonic magnitudes in the log domain f(y j ) are made equal to the actual harmonic magnitudes in the log domain f(x j ), but the remaining N(T)-10 harmonic magnitudes are set equal to a constant value “a” according to the following equations: 
 
 f ( y   j )= f ( x   j );  j =1, 2, . . . , 10   (18) 
 
 f ( y   j )= a; j =11 , . . . , N ( T )   (19) 
 
             a   =       1       N   ⁡     (   T   )       -   10       ⁢       ∑     j   =   11       N   ⁡     (   T   )         ⁢     f   ⁡     (     x   j     )                   (   20   )             
 
 assuming N(T)&gt;10. If equations (18), (19) and (20) are satisfied, the SD is minimized. However, in practice, equation (18) cannot be satisfied because representing the harmonic magnitude exactly would require an infinite number of bits (infinite resolution) which cannot be stored or transmitted in actual physical systems. The partial harmonic magnitude technique works best for encoding speech signals with a low pitch period, such as those produced by females or children, because a smaller amount of distortion is introduced when the number of harmonics is small. However, when encoding speech signals produced by males, the distortion is higher because this type of speech signal possesses a greater number of harmonics. 
 
      Although, in some cases, it is possible for the harmonic model to produce high quality synthesized speech signals, the harmonic parameters, particularly the harmonic magnitudes, can require a great many bits for their representation. The harmonic magnitudes can, however, be represented in a much more efficient manner if their possible values are limited through quantization. Once the possible values are defined and limited, each harmonic magnitude can be rounded-off or “quantized” to the most appropriate of these limited values. A group of techniques for defining a limited set of possible harmonic magnitudes and the rules for mapping harmonic magnitudes to a possible harmonic magnitude in this limited set are collectively referred to as vector quantization techniques.  
      Vector quantization techniques include the methods for finding the appropriate codevector for a given harmonic magnitude (“quantization”), and generating a codebook (“codebook generation”). In vector quantization, a codebook Y lists a finite number N c  of possible harmonic magnitudes. Each of these N c  possible harmonic magnitudes y i  is referred to as a “codebook entry,” “entry” or “codevector” and are defined according to the following equation: 
 
y l   T =[y i,0  y i,1  . . . y i,Nv-1 ]  (21) 
 
 where each y ij  is one of N v  components of the i-th codevector (each y ij  a “codevector component”); N v  is the codevector dimension; and “i” is a codevector index. Using the codebook to encode the harmonic magnitudes of the excitation signal involves finding the appropriate entry, and determining the codevector index associated with that entry. This enables each harmonic magnitude to be quantized to one of a finite number of values and represented solely by the corresponding codevector index. It is this codevector index that, along with the pitch period and other parameters, represents the harmonic magnitude for storage and/or transmission. Because the codebook is known to both the encoder and the decoder, the codevector index can also be used to recreate the harmonic magnitude. 
 
      However, before any harmonic magnitudes can be quantized, the vector quantization technique must generate a codebook, which includes determining the codevectors and the rule or rules for mapping all possible harmonic magnitudes to an appropriate codevector (“partitioning”). Codebook generation generally includes determining a finite set of codevectors in order to reduce the number of bits needed to represent the harmonic magnitudes. Partitioning defines the rules for quantization, which are basically the rules that govern how each potential harmonic magnitude is “quantized” or rounded-off.  
      There are several known methods for codebook generation (“codebook generation methods”), which, in general, include defining a partition rule and initial values for the codevectors; and using an iterative approach to optimize these codevectors for a given training data set according to some performance measure. The training data set is a finite set of vectors (“input vectors”) that represent all the possible harmonic magnitudes that may require quantization, which is used to create a codebook. A finite training data set is used to create the codebook because determining a codebook based on all possible harmonic magnitudes would be too computationally intensive and time consuming.  
      One example of a known codebook generation method is the generalized Lloyd algorithm (“GLA”) which is shown in  FIG. 2  and indicated by reference number  250 . The GLA  250  generally includes, collecting a training data set  252 ; defining a codebook  254 ; defining a partition rule  256 ; partitioning the training data set according to the partition rule and the codebook  258 ; optimizing the codebook for the partition using centriod computation  260 ; and determining whether an optimization criterion has been met  262 , where if the optimization criterion has not been met, repeating partitioning the training data set according to the partition rule and the codebook  258 ; optimizing the codebook for the partition using centriod computation  260 ; and determining whether an optimization criterion has been met  262  until the optimization criterion has been met.  
      Collecting a training data set  252  includes defining a set of input vectors containing N t  vectors as representative of the possible harmonic magnitude vectors, where each input vector x k  is associated with a pitch period T k  for k=0 to N t -1, and denoted according to the following equation: 
 
{x k , T k }  (22) 
 
 Defining a codebook  254  generally includes selecting initial values for the codevectors in the codebook by random selection or other known method. Additionally, the steps  252 ,  254  and  265  can be performed in any order, simultaneously, or any combination of the foregoing. 
 
      Defining a partition rule  256  generally includes adopting the nearest-neighbor condition and defining a distortion measure. Under the nearest-neighbor condition, an input vector is mapped to the codevector with which the input vector minimizes some measure of distortion. The distortion measure is generally defined by some measure of distance between an input vector x k  and a codevector y j  (the “distance measure d(y j , x k ) ”). It is this distance measure d(y j , x k ) that, along with the partition rule, is then used in step  258  to partition the training data set.  
      Partitioning the training data set  258  includes mapping each input vector in the training data set to a codevector according to the nearest-neighbor condition and the distance measure. This essentially amounts to dividing the training data into cells (creating a “partition”), where each cell includes a codevector and all the input vectors that are mapped to that codevector. The partition is determined so that within each cell the average distance measure, as determined between each input vector in the cell and the codevector in the cell, is minimized, yielding the optimum partition. Determining the optimum partition includes determining to which codevector each input vector should be mapped so that the distance between a given input vector and the codevector to which it is mapped is smaller than the distance between that input vector and any of the other codevectors. In other words, an input vector is said to be mapped to the i-th cell if the following equation is satisfied for all j≠i: 
 
 d ( y   i   , x   k )≦ d ( y   j   , x   k )   (23) 
 
 Because satisfying the nearest-neighbor condition is generally accomplished using an exhaustive search method, it is sometime known as the “nearest neighbor search.”
 
      Once the optimum partition is known, the codebook is then optimized using centroid computation  260 . Optimizing the codebook  260  generally includes, determining the optimum codevectors, which are the codevectors that minimize the sum of the distortions at each cell. Because the distortion measure is generally defined in step  256  as some distance measure d(y j , x k ), the sum of the distance measures at each cell is expressed according to the following equation:  
               D   t     =       ∑     k   ,       i   k     =   i         ⁢     d   ⁡     (       x   k     ,     y   i       )                 (   24   )             
 
 where i k  is the index of the cell to which x k  pertains. The sum of the distance measure is minimized by the centroid of the cell. In the present context, a centroid is the point in the cell from which the average distance to all the other vectors in the cell is the lowest, which can be determined using a centroid computation. Therefore, the optimum codevectors are the centroids for their respective cells as determined by centroid computation, where the exact manner in which the centroid computation is performed is determined by the distance measure defined in step  256 . 
 
      Because the GLA  250  produces an approximation of the optimum partition and the optimum codebook, it is determined in step  260  whether the optimum partition and optimum codebook are sufficiently optimized by determining if some optimization criterion has been met. One example of an optimization criterion is reaching the saturation of the total sum of distances for all cells, which is the point at which the total sum of distances for all cells remains constant or decreases by less than a predetermined value. If the criterion has not been met, steps  258 ,  260  and  261  are repeated until the optimization criterion has been met. When the optimization criterion has been met, the most recent codebook is defined as the optimum codebook.  
      Once the codebook has been generated, harmonic magnitudes can then be quantized. Quantization in vector quantization is the process by which a harmonic magnitude vector x (with harmonic magnitude elements, each “x k ”) in k-dimensional Euclidean space (“R k ”), is mapped into one of N c  codevectors. A harmonic magnitude is mapped to the appropriate codevector according to the partition rule. If the partition rule is the nearest-neighbor condition, the appropriate codevector for a given harmonic magnitude is the codevector that, together with that harmonic magnitude, provides the lowest distortion between that harmonic magnitude and each of the codevectors. Therefore, to quantize a harmonic magnitude, the distortion between the harmonic magnitudes and each codevector in the codebook is determined according to the distance measure, and the harmonic magnitude is then represented by the codevector that, together with that harmonic magnitude, created the smallest distortion.  
      Although vector quantization reduces the distortion inherent in the MELP-type coders, it introduces its own errors because vector quantization can only be used in cases where the harmonic magnitude dimension N(T) equals the codevector dimension N v , and harmonic magnitudes generally do not have a fixed dimension. Therefore, if the harmonic magnitude vectors have a variable dimension, another vector quantization technique must be used that can map variable dimension harmonic magnitudes to the fixed-dimension codebook entries. There are several known vector quantization techniques that may be used including: variable to fixed dimension conversion using interpolation (“variable to fixed conversion techniques”) and variable dimension vector quantization techniques (“VDVQ techniques”).  
      Variable to fixed conversion techniques generally include converting the variable dimension harmonic magnitude vectors to vectors of fixed dimension using a transformation that preserves the general shape of the harmonic magnitude. One example of a variable to fixed dimension conversion technique is the one implemented in the harmonic vector excitation coding (“HVXC”) coder (see M. Nishiguchi, et al. “Parametric Speech Coding-HVXC at 2.0-4.0 KBPS,” IEEE Speech Coding Workshop, pp. 84-86, 1999). The variable to fixed conversion technique used by the HVXC coder relies on a double interpolation process, which includes converting the original dimension of the harmonic magnitude, which is in the range of [9, 69] to a fixed dimension of 44. When a speech signal encoded using this technique is subsequently reproduced, a similar double-interpolation procedure is applied to the encoded 44 dimension harmonic magnitude vectors to convert them back into their original dimensions. On the encoding side, the HVXC coder uses a multi-stage vector quantizer having four bits per stage with a total of 13 bits (including 5 bits used to quantize the gain) to encode the harmonic magnitudes. With the previously described configuration, the HVXC coder is used for 2 kbit/s operation. It can also be used for 4 kbit/s operation by adding enhancements to the encoded harmonic magnitudes.  
      VDVQ is a vector quantization technique that uses an actual codevector to determine to which fixed dimension codevector a variable dimension harmonic magnitude vector should be mapped. This process is shown in more detail in  FIG. 3 . The VDVQ procedure  300  includes extracting an actual codevector for each codevector in a codebook  302 ; computing the distortion between the harmonic magnitude vector and each actual codevector  304 ; and choosing the codevector corresponding to the optimum actual codevector  306 .  
      An actual codevector u i  is a vector that is extracted from a codevector in a codebook but that has the same dimension N(T) (the “variable actual codevector dimension”) as the harmonic magnitude vector being quantized, and is expressed according to the following equation: 
 
u i   t =[u i,1  u i,2  . . . u i,N(T) ]  (25) 
 
 The actual codevectors are related to the codevectors according to the following equation: 
 
 u   i   =C ( T ) y   i    (26) 
 
 where C(T) is a selection matrix associated with the pitch period T and defined according to the following equation: 
 
 C ( T )= c   j,m   T ; for all  j= 1 , . . . ,N ( T ) and  m= 0 , . . . ,N   v −1   (27) 
 
 where each element of the selection matrix (each a “selection matrix element” or “c j,m   T ”) is defined according to the following equations: 
 
 c   j,m   T =1; if index( T,j )= m    (28a) 
 
c j,m   T =0; otherwise   (28b) 
 
 Each actual codevector includes codevector elements, where each actual codevector element u i,j  is related to a corresponding codevector element y i,j  as a function of a codevector index index(T,j) and according to the following equation: 
 
 u   i,j   =y   i,index(T,j)   ; j= 1 , . . . , N ( T )   (29) 
 
      The step of extracting the actual codevector  302  includes determining the appropriate codevector element y i,j  to extract for each actual codevector element u i,j . Step  302  is shown in more detail in  FIG. 4  and includes, defining a codevector index  320  and determining the actual codevectors  322 . Defining a codevector index  320  includes defining an index relationship and determining a value for the codevector index index(T,j) according to the index relationship. Generally, the index relationship defines the codevector index index(T,j) as a function of the pitch period T and according to the following equation:  
                     index   ⁢           ⁢     (     T   ,   j     )       =     round   ⁢           ⁢     (         (       N   v     -   1     )     ⁢     ω   j       π     )                       =     round   ⁢           ⁢     (       2   ⁢     (       N   vv     -   1     )     ⁢   j     T     )         ;     j   =   1       ,     …   ⁢           ⁢     N   ⁡     (   T   )                       (   30   )             
 
 where round(x) converts x to the nearest integer either by rounding up or rounding down and if x is a non-integer multiple of 0.5, round (x) may be defined to either round up or round down.  FIG. 5  shows an example of the inverse dependence of index(T,j) defined by the index relationship with the pitch period T as indicated by equation (30). As the pitch period increases, the vertical separation between the dots in the graph gets smaller. Once the codevector index index(T,j) has been defined, the actual codevectors are determined in step  322  according to equations (25) and (29). 
 
      Returning to  FIG. 3 , once the actual codevectors are extracted from each codevector in a codebook, the distortion measure between the harmonic magnitude vector and each actual codevector is computed  304 . The distortion measure is the distortion measure defined by the partition rule chosen during codebook generation. Generally, the distortion measure is a distance measure, which is defined as a distance between the actual codevector u i  as defined in equation (26) and the harmonic magnitude being quantized x, as expressed according to the following equation: 
 
 d ( x,u   i )= d ( x, C ( T ) y   i );  i =0 to  N   c −1   (31) 
 
 The step of choosing the codevector corresponding to the optimum actual codevector  306  includes designating the actual codevector with which the distortion measure is the lowest as the “optimum actual codevector” and choosing the codevector corresponding to the optimum actual codevector (or its codevector index) to represent the harmonic magnitude vector  306 . 
 
      As was necessary in the vector quantization techniques, before any harmonic magnitudes can be quantized, a codebook must be generated. However, some mathematical difficulties can arise in connection with generating the codebook with the GLA if certain distance measures are used. When using GLA, it is possible to choose a distance measure that results in the need to invert a singular matrix during the centroid computation step, thus making the optimum codevectors extremely difficult to calculate.  
      An example of a distance measure that leads to the need to invert a singular matrix is the distance measure that is defined below in equation (32). This distance measure is commonly used because it is very simple and produces good results at a low computational cost. This distance measure is defined according to: 
 
 d ( x   k   , C ( T   k ) y   i )=∥ x   k   −C ( T   k ) y   i   +g   k   1 ∥ 2    (32) 
 
 where the harmonic magnitude vector x k  and the codevector y i  are in the log domain;  1  is a vector whose elements are all ones with dimension N(T) (the “all-one vector”); and g k  is the optimal gain, where the optimal gain is the gain which satisfies the following equation:  
               g   k     =       1     N   ⁡     (     T   k     )         ⁢     (         y   i   T     ⁢       C   ⁡     (     T   k     )       T     ⁢     1   _       -       1   _     ⁢     x   k         )               (   33   )             
 
 and can also be expressed in terms of the difference between the mean of the actual codevector μ C(Tk)yi  and the mean of the harmonic magnitude vector μ xk  according to the following equation: 
 
 g   k   =μc ( T   k ) yi −μ xk    (34) 
 
 Substituting equation (34) into equation (32) yields the following equation: 
 
 d ( x   k   ,C ( T   k ) y   i )=∥( x   k   −μ   x     k   1)−( C ( T   k ) y   i   −μ   C(T     k     )y     i   1)∥ 2 .   (35) 
 
 As indicated by equation (35), the distance measure given in equation (32) leads to a mean-removed VQ equation (equation (35)) in which the means of both the harmonic magnitude vector and the codevector are subtracted out. To compute the centroid, the codevector y i  that minimizes equation (35), the optimum codevector, needs to be determined. Solving for y i  leads to the following equation:  
                 ∑     k   ,       i   k     =   i         ⁢       Ψ   ⁡     (     T   k     )       ⁢     y   i         =         ∑     k   ,       i   k     =   i         ⁢         C   ⁡     (     T   k     )       T     ⁢     x   k         +       g   k     ⁢       C   ⁡     (     T   k     )       T     ⁢     1   _                 (   36   )             
 
 where Ψ(T k ) is defined according to the following equation: 
 
Ψ( T   k )= C ( T   k ) T   C ( T   k )   (37) 
 
 Equation (36) can be represented in a simplified form by the following equation: 
 
Φ i y i =v i    (38) 
 
 where Φ i  is the centroid matrix and is defined according to the following equation:  
               Φ   i     =       ∑     k   ,       i   k     =   i         ⁢     Ψ   ⁡     (     T   k     )                 (   39   )             
 
 and v i  is defined according to the following equation:  
               v   i     =         ∑     k   ,       i   k     =   i         ⁢         C   ⁡     (     T   k     )       T     ⁢     x   k         +       g   k     ⁢       C   ⁡     (     T   k     )       T     ⁢     1   _                 (   40   )             
 
 Therefore, the optimum codevector is calculated as a function of the inverse of the centroid matrix Φ i   −1  according to the following equation: 
 
 y   i =Φ i   −1   v   i    (41) 
 
 Because Φ i  is a diagonal matrix, its inverse Φ i   −1  is relatively easy to find. However, elements of the main diagonal of ∠ i  might contain zeros, in which case, alternative methods must be used to solve for the optimum codevector. 
 
      Although VDVQ procedures offer an improvement over the previously mentioned methods with regard to the accuracy with which the harmonic magnitudes are encoded, in addition to the difficulties encountered when using certain distance measures to optimize the codebook, the rounding function included in the determination of the index relationship introduces errors that ultimately degrade the quality of the synthesized speech.  
     BRIEF SUMMARY  
      Improved variable dimension vector quantization-related (“VDVQ-related”) processes have been developed that not only provide improvements in quality over existing VDVQ processes but can be applied to a wider variety of circumstances. More specifically, the improved VDVQ-related processes provide quality improvements in codebook generation and the quantization of harmonic magnitudes, and facilitate codebook generation or optimization for a broad range of distortion measures, including those that would involve inverting a singular matrix using known centroid computation techniques.  
      The improved VDVQ-related processes include, improved methods for extracting an actual codevector from a codevector, improved methods for codebook optimization, improved VDVQ procedures, improved methods for creating an optimum partition, and improved methods for harmonic coding. Additionally, these improved VDVQ-related processes can be implemented in software and various devices, either alone or in any combination. The various improved VDVQ-related devices include variable dimension vector quantization devices, optimum partition creation devices, and codebook optimization devices. The improved VDVQ-related processes can be further implemented into an improved harmonic coder that encodes the original speech signal for transmission or storage.  
      The improved VDVQ-related processes are based on improvements in the way in which actual codevectors are extracted from the codevectors in a codebook and improvements in the way in which codebooks are generated and optimized. In general, the methods for optimizing codebooks include determining the optimum codevectors using the principles of gradient-descent. By using the principles of gradient-descent, the problems associated with inverting singular centroid matrices are avoided, therefore, allowing the codevectors to be optimized for a greater collection of distance measures. In contrast, the improved methods for extracting an actual codevector from a codevector, in general, redefine the index relationship and use interpolation to determine the actual codevector elements when the index relationship produces a non-integer value. By using interpolation to determine the actual codevector elements, greater accuracy is achieved in coding and decoding the harmonic magnitudes of an excitation because the accuracy of the partitions used in creating the codebook is increased, as well as the accuracy with which the harmonic magnitudes are quantized.  
      In order to test the performance of the improved VDVQ related processes, improved VDVQ quantizers having a variety of dimensions and resolutions were created, tested and the results of the testing were compared with those resulting from similar testing of quantizers implementing various known harmonic magnitude modeling and/or quantization techniques. Experimental results comparing the performance of these improved VDVQ quantizers to the performance of the various known quantizers demonstrated that the improved VDVQ quantizers produce the lowest average spectral distortion under the tested conditions. In fact, the improved VDVQ quantizers demonstrated a lower average spectral distortion than quantizers implementing a known constant magnitude approximation without quantization and quantizers implementing a known partial harmonic magnitude technique without quantization. Additionally, the improved VDVQ quantizers outperformed quantizers based on the known HVXC coding standard implementing a known variable to fixed conversion technique, as well as quantizers obeying the basic principles of a known VDVQ procedure, where the improved VDVQ quantizers had a comparable complexity, or only a moderate increase in computation, respectively.  
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      This disclosure may be better understood with reference to the following figures and detailed description. The components in the figures are not necessarily to scale, emphasis being placed upon illustrating the relevant principles. Moreover, like reference numerals in the figures designate corresponding parts throughout the different views.  
       FIG. 1  is flow chart of a harmonic analysis process, according to the prior art;  
       FIG. 2  is a flow chart of a generalized Lloyd algorithm for optimizing a codebook, according to the prior art;  
       FIG. 3  is a flow chart of a variable dimension vector quantization procedure, according to the prior art;  
       FIG. 4  is a flow chart of a method for extracting an actual codevector from a codevector in a codebook, according to the prior art;  
       FIG. 5  is a graph of codevector indices as a function of pitch period, according to the prior art;  
       FIG. 6  is a flow chart of an embodiment of an improved method for extracting an actual codevector from a codevector in a codebook;  
       FIG. 7  is a flow chart of an embodiment of a method for creating an optimum partitioning for a codebook;  
       FIG. 8  is a flow chart of an embodiment of an improved variable dimension vector quantization procedure;  
       FIG. 9  is a flow chart of an embodiment of an improved method for codebook optimization;  
       FIG. 10  is a flow chart of an embodiment of a method for updating current optimum codevectors using gradient-descent;  
       FIG. 11  is a flow chart of an embodiment of an improved method for harmonic coding; (In Box 910 : VDVQ for the present case is only applied to the harmonic magnitudes, the other parameters use other (undefined) quantization methods).  
       FIG. 12A  is a graph of the spectral distortion resulting from the training data set quantized using an improved VDVQ quantizer as a function of quantizer resolution and according to codevector dimension;  
       FIG. 12B  is a graph of the spectral distortion resulting from the testing data set quantized using an improved VDVQ quantizer as a function of quantizer resolution and according to codevector dimension;  
       FIG. 13A  is a graph of the spectral distortion resulting from the training data set quantized using an improved VDVQ quantizer as a function of codevector dimension and according to quantizer dimension;  
       FIG. 13B  is a graph of the spectral distortion resulting from the testing data set quantized using an improved VDVQ quantizer as a function of codevector dimension and according to quantizer dimension;  
       FIG. 14A  is a graph of the difference in spectral distortion (ASD) resulting from the training data set quantized using an improved VDVQ quantizer and the training data set quantized using a known VDVQ quantizer as a function of quantizer resolution and according to codevector dimension;  
       FIG. 14B  is a graph of the difference in spectral distortion (ASD) resulting from the testing data set quantized using an improved VDVQ quantizer and the training data set quantized using a known VDVQ quantizer as a function of quantizer resolution and according to codevector dimension;  
       FIG. 15A  is a graph of the spectral distortion resulting from the training data set quantized using an improved VDVQ quantizer and modeled and/or quantized using various other models and quantizers as a function of quantizer resolution and according to codevector dimension;  
       FIG. 15B  is a graph of the spectral distortion resulting from the testing data set quantized using an improved VDVQ quantizer and modeled and/or quantized using various other models and quantizers as a function of quantizer resolution and according to codevector dimension;  
       FIG. 16  is a block diagram of an improved VDVQ device; and  
       FIG. 17  is a block diagram of an optimized harmonic coder.  
    
    
     DETAILED DESCRIPTION  
      Improved variable dimension vector quantization-related (“VDVQ-related”) processes have been developed that not only provide improvements in quality over existing VDVQ processes but can be applied to a wider variety of circumstances. More specifically, the improved VDVQ-related processes provide quality improvements in codebook generation and the quantization of harmonic magnitudes, and facilitate codebook generation or optimization for a broad range of distortion measures, including those that would involve inverting a singular matrix using known centroid computation techniques.  
      The improved VDVQ-related processes include, improved methods for extracting an actual codevector from a codevector, improved methods for codebook optimization, improved VDVQ procedures, improved methods for creating an optimum partition, and improved methods for harmonic coding. Additionally, these improved VDVQ-related processes have been implemented in software and various devices to create improved VDVQ-related devices that include actual codevector extraction devices, improved VDVQ devices, and codebook optimization devices.  
      The improved VDVQ-related processes are based on improvements in the way in which actual codevectors are extracted from the codevectors in a codebook and improvements in the way in which codebooks are generated and optimized. In general, the methods for optimizing codebooks include determining the optimum codevectors using the principles of gradient-descent. By using the principles of gradient-descent, the problems associated with inverting singular centroid matrices are avoided, therefore, allowing the codevectors to be optimized for a greater collection of distance measures. In contrast, the improved methods for extracting an actual codevector from a codevector, in general, redefine the index relationship and use interpolation to determine the actual codevector elements when the index relationship produces a non-integer value. By using interpolation to determine the actual codevector elements, greater accuracy is achieved in coding and decoding the harmonic magnitudes of an excitation because the accuracy of the partitions used in creating the codebook is increased, as well as the accuracy with which the harmonic magnitudes are quantized.  
      An improved method for extracting an actual codevector from a codevector in a codebook is shown in  FIG. 6 . This method  320  generally includes: calculating a codevector index according to an interpolation index relationship  362 ; determining whether the codevector index is an integer  364 ; where if the codevector index is an integer, defining the index relationship according to the known index relationship  366 ; and calculating the actual codevector according to the known index relationship  384 ; where if the codevector index is not an integer, defining the index relationship according to an interpolation index relationship  368  and calculating the actual codevector by interpolating the corresponding codevector elements.  
      Calculating a codevector index according to an interpolation index relationship  362  includes determining a value for index(T,j) as a function of the pitch period T and the codevector dimension N v  according to the following equation:  
                   index   ⁡     (     T   ,   j     )       =       2   ⁢     (       N   v     -   1     )       T       ;     j   =   1       ,   …   ⁢           ,   N           (   42   )             
 
 The interpolation index relationship of equation (42) differs from the known index relationship of equation (30) in that the interpolation index relationship does not define the values for the codevector index index(T,j) by rounding off. 
 
      It is then determined in step  364  whether the codevector index as determined by equation (42) is an integer. This determination may be made by determining whether the following equation is satisfied: 
 
┌index( T,j )┐=└index( T,j )┘  (43) 
 
 where ┌x┐ is a ceiling function that returns the smallest integer that is larger than x; └x┘ is a floor function that returns the largest integer that is smaller than x. ┌index(T,j)┐ is a first rounded index and is equal to the value obtained in equation (42) rounded up to the next highest integer; and └index(T,j)┘ is a second rounded index and is equal to the value obtained in equation (42) rounded down to the next lowest integer. If the first rounded index equals the second rounded index, the codevector index as defined by equation (42) must be an integer. 
 
      If it is determined in step  364  that the codevector index as determined by the interpolation codevector relationship is an integer, the index relationship is defined according to a known index relationship  366 , such as is given in equation (30) and the actual codevector u i  is calculated by determining each codevector element u i,j  according to equation (29) where the codevector index index(T,j) is determined according to the known index relationship of equation (30) in step  384 .  
      However, if it is determined in step  364  that the codevector index is not an integer, the index relationship index(T,j) is defined according to the interpolation index relationship of equation (42)  368 . The actual codevector u i  is then determined in step  382  by determining the actual codevector elements u i,j  according to an interpolation of codevector elements. The interpolation may involve any number of codevector elements, each of which is weighted using a weighting function. For example, if the interpolation is between two codevector elements, the interpolation is an interpolation of a first adjacent codevector element y i , ┌index(T,j)┐ and a second adjacent codevector element y i ,└index(T,j)┘ according to the following equation. 
 
 u   i,j =(index( T,j )−└index( T,j )┘) y   i,┌index(   T,j )┐+(┌index( T,j )┐−index( T,j )) y   i ,└index(T,j)┘  (44) 
 
 wherein the weighting function assigned to the first adjacent codevector element is index(T,j)−└index(T,j)┘ and the weighting function assigned to the second adjacent codevector element is ┌index(T,j)┐−index(T,j). 
 
      Alternatively, the actual codevector u i  can be determined in step  382  as a function of a selection matrix C(T) according to equation (26). The selection matrix C(T) is essentially a matrix of all the weighting functions and is defined according to equation (27). The selection matrix elements C j,m   T  are determined according to the following equations: 
 
 c   j,m   T =index( T,j )−└index( T,j )┘; if ┌index( T,j )┐= m    (45a) 
 
c j,m   T =0; otherwise   (45b) 
 
      The improved methods for extracting an actual codevector from a codevector, such as the one shown in  FIG. 6 , can also be implemented in a method for creating an optimum partition. The method for creating an optimum partition uses an interpolation index relationship to produce the optimum partition for a given codebook. An example of a method for creating an optimized partition  600  is shown in  FIG. 7  and includes: defining a codebook  601 ; collecting a training data set  602 ; defining a distortion measure  604 ; and determining the optimum partition by extracting an actual codevector from each codevector in the codebook using an interpolation index relationship  606 .  
      Defining a codebook  601  generally includes, defining a number of codevectors to use as a starting point according to a known method, such as a partition creation and optimization method using a nearest-neighbor search. Collecting a training data set includes defining a set of N t  training vectors that will represent all possible harmonic magnitudes  602  includes defining a number of training vectors x k  associated with a pitch period T k  for k=0 to N t −1, and denoted according to equation (22), where N t  is the size of the training data set. Defining a distortion measure  604  generally includes defining the distortion measure using some distance measure of the distance between a training vector x k  and a codevector y j . One example of such a distance measure is the distance measure defined in equation (32). Therefore, the next step, determining the optimum partition by extracting an actual codevector from each codevector in the codebook using an interpolation index relationship  606 , includes determining the optimum partition using an improved method for extracting an actual codevector to create an actual codevector for each codevector in the codebook and associating each training vector with the codevector corresponding to the actual codevector with which that training vector minimizes the distance measure. The actual codevector with which a training vector minimizes the distance measurement can be found by satisfying equation (23) according to a known method such as the nearest-neighbor search.  
      The improved method for extracting an actual codevector from a codevector, such as the one shown in  FIG. 6 , can be implemented in an improved VDVQ procedure. The improved VDVQ procedure maps harmonic magnitude vector having a variable input vector dimension N(T k ) to the appropriate codevector y i  in a codebook, where the codevector has a codevector dimension N v  and N(T k ) does not necessarily equal N v . An example of an improved VDVQ procedure  500  is shown in  FIG. 8  and includes: extracting an actual codevector from each codevector in a codebook using an interpolation index relationship  502 ; computing the distortion measure between the harmonic magnitude and each actual codevector  504 ; and choosing the codevector corresponding to the optimum actual codevector  506 . Extracting an actual codevector from each codevector in a codebook using an interpolation index relationship  502 , generally includes performing an improved method for extracting an actual codevector from a codevector, such as the one shown in  FIG. 6  and described herein. Step  502  in  FIG. 8 , therefore produces, for each codevector in a codebook, an actual codevector. This actual codevector is a function of a known index relationship when the index, as determined by an interpolation index relationship, is an integer, and is a function of the interpolation index relationship when the index is not an integer.  
      Once an actual codevector is extracted for each codevector, the distortion measure between the harmonic magnitude vector and each actual codevector is computed  504 . The distortion measure is defined as the same distortion measure used to determine the optimum codevectors when the codebook was generated and optimized. Although it can be defined by any distortion measure, the distortion measure can be defined as a distance measure according to equation (31), which is the distance between the actual codevector u i , as determined in step  502 , and the harmonic magnitude. The step of choosing the codevector corresponding to the optimum actual codevector  506  includes designating the actual codevector with which the harmonic magnitude produced the lowest distortion as the “optimum actual codevector” and choosing the codevector corresponding to the optimum actual codevector to represent the harmonic magnitude vector  506 . Alternately, the codevector index of the codevector corresponding to the optimum actual codevector may be chosen to represent the harmonic magnitude.  
      The improved method for extracting an actual codevector from a codevector can also be implemented in an improved method for codebook optimization as shown in  FIG. 9 . This method  800  uses the principle of gradient-descent instead of centroid computation to determine the optimum codevectors and thus avoids the problem of having to invert a singular centroid matrix. Gradient-descent is an iterative method for finding the minimum of function in terms of a variable by determining the partial derivative of the function with respect to the variable, adjusting the variable in a direction negative to the gradient to update the function, and redetermining the partial derivative of the updated function until the partial derivative of the function equals or is acceptably close to zero. The value for the variable that produces the function for which the partial derivative is zero or approaches zero is the value that minimizes the function.  
      The improved method for codebook optimization  800  generally includes: collecting a training data set  802 ; defining a codebook, partition rule and distortion measure  804 ; finding a current optimum codevector for each input vector  806 ; updating the current optimum codevectors using gradient-descent to create new optimum codevectors  808 ; determining whether the optimization criterion has been met  810 ; wherein if the optimization criterion has not been met, updating the codebook with the new optimum codevectors and repeating steps  806 ,  808 ,  810  and  812  until it is determined in step  810  that the optimization criterion has been met; wherein if the optimization criterion has been met, designating the current optimum codevectors as the optimum codevectors.  
      Collecting a training data set  802  generally consists of gathering a number of vectors from the signal source of interest that, in the present case, are a number of harmonic magnitude vectors from some speech signals. Defining a codebook in step  804  generally includes defining a number of codevectors according to any known method. Defining a partition rule in step  804  involves determining the rules by which the harmonic magnitude vectors are to be mapped to the codevectors. This generally includes defining the nearest-neighbor condition as the partition rule. Defining a distortion measure in step  804  includes defining a distance measure, such as the distance measure specified in equation (31).  
      Once the codevectors, partition rule and distortion measure are defined, they are used to find a current optimum codevector for each input vector  806 . Finding a current optimum codevector for each input vector  806  involves finding the nearest codevector for each input vector using an interpolation index relationship by performing the improved VDVQ procedure for each input vector. Performing the improved VDVQ procedure for each input vector includes: extracting an actual codevector from each codevector using an interpolation index relationship; computing the distortion between the harmonic magnitude vector and each actual codevector; and choosing the codevector corresponding to the optimum actual codevector.  
      Once a current optimum codevector is determined for each input vector, these current optimum codevectors are updated using gradient-descent to create new optimum codevectors in step  808 . Updating the current optimum codevectors  808  is shown in more detail in  FIG. 10  and generally includes with regard to each of the current optimum codevectors: determining the partial derivative of the distance measure with respect to each codevector element  852 ; determining the gradient of the distance measure  854 ; and updating the codevector closest to the corresponding input vector in a direction negative to the gradient  856 . Determining the partial derivative of the distance measure with respect to each codevector element  852  includes calculating the partial derivative of the distance measure in terms of each codevector element. If the distance measure is defined according to equation (32) the partial derivative of the distance measure with respect to each codevector element  
         ∂     ∂     y     i   ,   m           ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )           
 
 is determined according to the following equation:  
                 ∂     ∂     y     i   ,   m           ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )         =       ∑     j   =   1       N   ⁡     (     T   k     )         ⁢     2   ⁢     (       u     i   ,   j       -     x     k   ,   j       -     g   k       )     ⁢       ∂     u     i   ,   j           ∂     y     i   ,   m                       (   46   )             
 
 where ∂u i,j /∂y i,m  is the partial derivative of an actual codevector element u i,j  with respect to a codevector element y i,m , where u i,j  is determined according to equation (29) if equation (43) is satisfied and according to equation (44) otherwise. Therefore, ∂u i,j /∂y i,m  can be determined according to the following equations:  
                   ∂     u     i   ,   j           ∂     y     i   ,   m           =   1     ;       if   ⁢           ⁢     ⌈     index   ⁡     (     T   ,   j     )       ⌉       =         ⌊     index   ⁡     (     T   ,   j     )       ⌋     ⁢   and   ⁢           ⁢   m     =     index   ⁡     (     T   ,   j     )                   (     47   ⁢   a     )                       ∂     u     i   ,   j           ∂     y     i   ,   m           =       index   ⁡     (     T   ,   j     )       -     ⌊     index   ⁡     (     T   ,   j     )       ⌋         ;     ⁢     
     ⁢         if   ⁢           ⁢     ⌈     index   ⁡     (     T   ,   j     )       ⌉       ≠       ⌊     index   ⁡     (     T   ,   j     )       ⌋     ⁢           ⁢   and   ⁢           ⁢   m       =     ⌈     index   ⁡     (     T   ,   j     )       ⌉               (     47   ⁢   b     )                       ∂     u     i   ,   j           ∂     y     i   ,   m           =       ⌈     index   ⁡     (     T   ,   j     )       ⌉     -     index   ⁡     (     T   ,   j     )           ;     ⁢     
     ⁢         if   ⁢           ⁢     ⌈     index   ⁡     (     T   ,   j     )       ⌉       ≠       ⌊     index   ⁡     (     T   ,   j     )       ⌋     ⁢           ⁢   and   ⁢           ⁢   m       =     ⌈     index   ⁡     (     T   ,   j     )       ⌉               (     47   ⁢   c     )                     ∂     u     i   ,   j           ∂     y     i   ,   m           =   0     ;   otherwise           (     47   ⁢   d     )             
 
 Determining the gradient of the distance measure  854  includes determining the gradient of the distance measure according to the following equation:  
               ∇     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )         =     (         ∂     ∂     y     i   ,   1           ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )         ,       ∂     ∂     y     i   ,   2           ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )         ,   …   ⁢           ,       ∂     ∂     y     i   ,     N   ⁡     (     T   k     )               ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )           )             (   48   )             
 
      Once the gradient of the distance measure ∇d(x k , C(T k )y i ) has been determined, the current closest codevectors are updated in a direction negative to the gradient  856  according to the following equation:  
               y     i   ,   m       ←       y     i   ,   m       -     γ   ⁢     ∂     ∂     y     i   ,   m           ⁢     d   ⁡     (       x   k     ,       C   ⁡     (     T   k     )       ⁢     y   i         )                   (   49   )             
 
 where γ is a step size parameter, a value for which is generally determined prior to performing the method for determining the optimum codevectors  400  and is chosen based on considerations such as desired accuracy, update speed and stability. Additionally, the step size parameter γ can be chosen according to the following equation:  
             γ   =       2   ⁢     N   c         N   t               (   50   )             
 
 where N c  is the number of codevectors and N t  is the number of training vectors. 
 
      Returning to  FIG. 9 , it is then determined whether an optimization criterion has been met  810 . Determining whether an optimization criterion has been met  810  is performed pursuant to the nature of the optimization criterion used. The optimization criterion may include includes determining whether a specified number of iterations or epochs have been performed, a specified amount of time has passed, the SD has saturated or other optimization criterion has been met. Determining whether the SD has saturated includes determining the SD of the current optimum codevectors and the new optimum codevectors and determining whether the SD has decreased by less than a predetermined difference value from the current optimum codevectors to the new optimum codevectors. Additionally, the optimization criterion (or criteria) may include the gradient reaching or becoming less than a predetermined minimum value. Both the predetermined difference value and the predetermined minimum value are generally determined before the method for determining the optimum codevectors  400  is performed and represent a desired level of accuracy. The predefined difference value and the predefined minimum value are generally chosen in view of considerations such as desired computation speed, accuracy and computational load.  
      If it is determined in step  810  that the optimization criterion has not been met, the codebook is updated  812  by replacing the current optimum codevectors with the new current optimum codevectors so that the new current optimum codevectors become the current optimum codevectors. Thereafter, steps  806 ,  808 , and  810  are reperformed and steps  812 ,  806 ,  808 , and  810  are repeated until it is determined in step  810  that the optimization criterion has been met. When it is determined in step  810  that the optimization criterion has been met, the current optimum codevectors are designated as the optimum codevectors  814 .  
      The improved VDVQ procedure, such as the one shown in  FIG. 8 , can be implemented in an improved method for harmonic coding. An example of an improved method for harmonic coding  900  is shown in  FIG. 11  and includes: determining the LP coefficients  902 ; producing the excitation signal  904 ; determining the pitch period and the harmonic magnitudes  906 ; determining the other parameters  908 ; and quantizing the harmonic magnitudes, pitch period and other parameters  910 .  
      Determining the LP coefficients  902  generally includes performing an LP analysis on each frame of a speech signal that is being coded. Producing the excitation signal  904  generally includes using the LP coefficients to define an analysis filter, which is the inverse of a synthesis filter, and filtering each frame of the speech signal with the inverse filter to produce an excitation signal in frames (each an “excitation signal frame”). Determining the pitch period and the harmonic magnitudes  906  is accomplished by performing harmonic analysis on each excitation signal frame to determine the harmonic magnitudes for that frame. Determining the other parameters  908  generally includes determining parameters such as gain, and those relating to power estimation, the voiced/unvoiced decision and filtering operations for each frame of the speech signal.  
      After the harmonic magnitudes, pitch period and other parameters are determined, they are quantized and encoded into a bit-stream in step  910 . Quantizing the harmonic magnitudes, pitch period and other parameters  910  includes quantizing the pitch period and other parameters using known methods and quantizing the harmonic magnitudes using an improved variable dimension vector quantization procedure, such as is shown in  FIG. 8 . The improved variable dimension vector quantization procedure determines the index for the codevector in a codebook corresponding to the optimum actual codevector for each harmonic magnitude in an excitation frame. These indices, pitch period and other parameters are then encoded into a bit-stream for transmission or storage.  
      In order to test the performance of the improved VDVQ related processes, improved VDVQ quantizers having a variety of dimensions and resolutions were created, tested and the results of the testing were compared with those resulting from similar testing of quantizers implementing various known harmonic magnitude modeling and/or quantization techniques. Experimental results comparing the performance of these improved VDVQ quantizers to the performance of the various known quantizers demonstrated that the improved VDVQ quantizers produce the lowest average SD under the tested conditions. In fact, the improved VDVQ quantizers demonstrated a lower average SD than quantizers implementing a known constant magnitude approximation without quantization (the “known LPC models”) and quantizers implementing a known partial harmonic magnitude technique without quantization (the “known MELP models”). Additionally, the improved VDVQ quantizers outperformed quantizers based on the known HVXC coding standard implementing a known variable to fixed conversion technique (the “known HVXC quantizers”), as well as quantizers obeying the basic principles of a known VDVQ procedure (the “known VDVQ quantizers”). The improvement in quality was achieved at a complexity comparable to that of the known HXVC quantizers and with only a moderate increase in computation when compared to the known VDVQ quantizers.  
      The training data used to design the improved VDVQ quantizers and the known VDVQ quantizers; and the testing data used to test all the quantizers was obtained from the TIMIT database. The training data was obtained from 100 sentences chosen from the TIMIT database that were downsampled to 8 kHz. To obtain the training data, the 100 sentences were windowed to obtain frames of 160 samples/frame. The harmonic magnitudes of these sentences were obtained from the prediction error and had variable dimensions. The prediction error of each frame was determined using LP analysis and then mapped into the frequency domain by windowing the prediction error with a Hamming window and using a 256-sample FFT. An autocorrelation-based pitch period estimation algorithm was designed and used to determine the pitch period. The pitch period was determined to have a range of [20, 147] at steps of 0.25; thus, allowing fractional values for the pitch periods. The harmonic magnitudes were then extracted only from the voiced frames which were determined according to the estimated pitch period. This process yielded approximately 20000 training vectors in total. To obtain the testing data set, a similar procedure was used to extract the testing data from 12 sentences, which yielded approximately 2500 vectors.  
      Thirty (30) improved VDVQ quantizers were created for comparison with the known quantizers. For each of these 30 improved VDVQ quantizers, a codebook including a plurality of codevectors and a partition was determined. These 30 improved VDVQ quantizers included five (5) groups of quantizers where each group of quantizers has a specific dimension Nv and where within each group of quantizers, each improved quantizer has a different resolution. For the first group of improved VDVQ quantizers, the dimension is N v =41; for the second group of quantizers, the dimension is N v =51; for the third group of quantizers, the dimension is N v =76; for the third group of quantizers, the dimension is N v =101; and for the fifth group of quantizers, the dimension is N v =129. Each of these groups of quantizers included six improved quantizers, each with a different resolution. The first improved VDVQ quantizer in each group had a resolution r=5, the second had a resolution r=6; the third had a resolution r=7; the fourth had a resolution r=8, the fifth had a resolution r=9, and the sixth had a resolution r=10.  
      The codebooks for each of the 30 improved VDVQ quantizers were created using the training data and the improved method for codebook optimization as described herein in connection with  FIG. 9 , with the initial values for the codevectors being the codevectors for the corresponding known VDVQ coders (described subsequently). Therefore, the optimum partition for the codebook was determined using an interpolation index relationship and the optimum codevectors were determined using gradient-descent. The optimization criterion used to determine when to stop the training process was the saturation of the SD for the entire training data set. After each epoch (an epoch is defined as one complete pass of all the training data in the training data set through the training process), the average of the SD with regard to the training data was determined and compared with the average SD of the previous epoch. If the SD had not gotten smaller by at least a predefined amount, the average SD was determined to be in saturation and the training procedure was stopped. Furthermore, the step size parameter was chosen according to equation (50) and the distance measure used to create the partition (and later to quantize the test data) was the distance measure defined in equation (32).  
      Additionally, 30 known VDVQ quantizers were created for comparison with the improved VDVQ quantizers. These 30 known VDVQ quantizers have the same dimensions and resolutions as the improved VDVQ quantizers. The codevectors and partitions for each of the 30 known VDVQ quantizers were created using the training data and the GLA to optimize a randomly created initial codebook. For each known VDVQ quantizer, a total of 10 random initializations were performed where each random initialization was followed by 100 epochs of training (where one epoch consists of a nearest neighbor search followed by centroid computation and where after each epoch it was determined if the average SD of the entire training data set had saturated). The distance measure used to create the partition (and later to quantize the test data) was the distance measure defined in equation (32).  
      Further, six (6) known HVXC quantizers were created. All of the known HVXC quantizers were designed to have a codebook with a codevector dimension of 44, where each of the six known HVXC quantizers had a different resolution (5, 6, 7, 8, 9 and 10 bits, respectively). The codevectors and partitions for each of the known HVXC quantizers were created using the GLA where the GLA optimized initial codevector created by interpolating the training vectors to 44 elements. For each known HVXC quantizer, a total of 10 random initializations were performed where each random initialization was followed by 100 epochs of training. One epoch is a complete pass of all the data in the training data set. In actual training, each vector in the training data set is presented sequentially to the GLA, when all the vectors are passed and the codebook updated, one epoch has passed. The training process is then repeated with the next epoch, where the same training vectors are presented.  
      In the experiments, initially the performance of the 30 improved VDVQ quantizers in terms of SD was determined as a function of both dimension and resolution. The performance of these improved VDVQ quantizers was then compared to the performance of the corresponding VDVQ quantizers (the corresponding known VDVQ quantizer is the known VDVQ quantizer having the same resolution and dimension as the improved VDVQ quantizer to which it corresponds), also in terms of both dimension and resolution. Then, the performance as a function of resolution of the improved VDVQ quantizers with a codevector dimension of 41 was compared to the performance of a known LPC model, a known MELP model, the known HVXC quantizers, and the known VDVQ quantizers having a codebook dimension of 41.  
      The SD of the 30 improved VDVQ quantizers is shown in  FIGS. 12A, 12B ,  13 A and  13 B.  FIG. 12A  shows the SD for all 30 improved VDVQ quantizers as a function of resolution for the training data, and  FIG. 12B  shows the SD for all 30 improved VDVQ quantizers as a function of resolution for the testing data.  FIG. 13A  shows the SD for all 30 improved VDVQ quantizers, grouped according to resolution, as a function of dimension for the training data and  FIG. 13B  shows the SD for all 30 improved VDVQ quantizers, grouped according to resolution, as a function of dimension for the testing data.  
       FIGS. 14A, 14B , show the difference between SD resulting from the improved VDVQ quantizers and the SD resulting from the known VDVQ quantizers (“ΔSD”). In  FIG. 14A , the difference in SD ΔSD is shown for the training data and is grouped according to the dimension of the quantizers from which it was produced and presented as a function of resolution. In  FIG. 14B , the difference in SD, ΔSD is shown for the testing data and is grouped according to the dimension of the coders from which it was produced and presented as a function of resolution. With regard to the training data, the introduction of interpolation among the elements of the codevectors through the use of the interpolation index relationship produces a reduction in the average SD. The amount of this reduction tends to be higher for the lower dimension coders with higher resolution. With regard to the testing data, the introduction of interpolation among the elements of the codevectors through the use of the interpolation index relationship generally produces a reduction in the average SD.  
       FIGS. 15A and 15B  show the SD as a function of resolution produced by the known LPC models  950 , the known MELP models  952 ; the known HVXC quantizers  954 , the known VDVQ quantizers with a codevector dimension of 41 956; and the improved VDVQ quantizers with a codevector dimension of 41 958.  FIG. 15A  shows the SD as a function of resolution for the training data and  FIG. 15B  shows the SD as a function of resolution for the testing data. The SD of the improved VDVQ quantizers is significantly lower that that of the known HVXC and known VDVQ quantizers. This difference has particular significance with regard to the known HVXC quantizers because the known HVXC quantizers have a codebook resolution higher than that of the improved VDVQ quantizer.  
      Furthermore, the SD for the improved VDVQ quantizers was significantly lower than the SD of the known LPC model and the known MELP model, particularly at higher resolutions. Because both the known LPC model and the known MELP model did not include quantization, their respective resolutions were infinite and therefore, their respective SDs were constant (for the LPC model the SD was 4.44 dB for the training data and 4.36 dB for the testing data; and for the MELP model the SD was 3.29 dB for the training data and 3.33 dB for the testing data). The SD values shown in  FIGS. 19A and 19B  for the known LPC model and the known MELP model reflect only the distortion inherent in the models and do not reflect any distortion due to quantization. Therefore, these SD values represent the best possible performance for these quantizers in that, if quantization were added, the SD would only degrade.  
      Implementations and embodiments of the improved VDVQ-related processes, including improved methods for extracting an actual codevector from a codevector, methods for creating an optimum partition for a codebook, improved variable dimension vector quantization procedures, improved methods for codebook optimization, methods for updating current optimum codevectors using gradient-descent and improved methods for harmonic coding all include computer readable software code. Such code may be stored on a processor, a memory device or on any other computer readable storage medium. Alternatively, the software code may be encoded in a computer readable electronic or optical signal. The code may be object code or any other code describing or controlling the functionality described herein. The computer readable storage medium may be a magnetic storage disk such as a floppy disk, an optical disk such as a CD-ROM, semiconductor memory or any other physical object storing program code or associated data.  
      Additionally, improved VDVQ-related processes may be implemented in an improved VDVQ-related device  1200 , as shown in  FIG. 16 , alone or in any combination. The improved VDVQ-related device  1200  generally includes an improved VDVQ-related unit  1202  and may also include an interface unit  1204 . The improved VDVQ-related unit  1202  includes a processor  1220  coupled to a memory device  1216 . The memory device  1218  may be any type of fixed or removable digital storage device and (if needed) a device for reading the digital storage device including, floppy disks and floppy drives, CD-ROM disks and drives, optical disks and drives, hard-drives, RAM, ROM and other such devices for storing digital information. The processor  520  may be any type of apparatus used to process digital information. The memory device  518  may store a speech signal, and any or all of the improved VDVQ-related processes, or any combination of the foregoing. Upon the relevant request from the processor  1220  via a processor signal  1222 , the memory communicates the requested information via a memory signal  1224  to the processor  1220 .  
      The interface unit  1204  generally includes an input device  1214  and an output device  1216 . The output device  1216  receives information from the processor  1220  via a second processor signal  1212  and may be any type of visual, manual, audio, electronic or electromagnetic device capable of communicating information from a processor or memory to a person or other processor or memory. Examples of output devices include, but are not limited to, monitors, speakers, liquid crystal displays, networks, buses, and interfaces. The input device  1214  communicates information to the processor via an input signal  1210  and may be any type of visual, manual, mechanical, audio, electronic, or electromagnetic device capable of communicating information from a person or processor or memory to a processor or memory. Examples of input devices include keyboards, microphones, voice recognition systems, trackballs, mice, networks, buses, and interfaces. Alternatively, the input and output devices  1214  and  1216 , respectively, may be included in a single device such as a touch screen, computer, processor or memory coupled to the processor via a network.  
      The improved VDVQ-related processes can be implemented into an improved harmonic coder that encodes the original speech signal for transmission or storage. An example of an improved harmonic coder  1300  is shown in  FIG. 17 . A harmonic coder  1300  generally includes an LPA device  1302 ; an inverse filter  1304 ; another process device  1306 ; a harmonic analysis device  1308 ; and a quantizer  1310 . The LPA device  1302  performs LPA on the input signal s(n) to produce the LP coefficients. These LP coefficients are used to define an inverse filter  1304  that is simply the inverse of the synthesis filter. The inverse filter  1304  filters the input signal s(n) to produce the excitation signal u(n). The excitation signal u(n) is then analyzed by the harmonic analysis device  1308  using harmonic analysis to extract the fundamental frequency ω o  and the harmonic magnitudes x j .  
      The LP coefficients are also input into another process device  1306 . The other process device  1306  uses the LP coefficients to determine other parameters such as, those relating to power estimation, the voiced/unvoiced decision and filtering options. The other parameters, the harmonic magnitudes x j , and the pitch period T, are all input into the quantizer. The quantizer, using an improved method for codebook and partition optimization, uses the harmonic magnitudes x j  and the pitch period T to create the optimum codevectors and the optimum partitions to define a codebook. The quantizer then uses the codebook and an improved VDVQ procedure to quantize the harmonic magnitudes to produce quantized harmonic magnitudes y i . Finally, the quantizer produces a bit-stream containing the quantized harmonic magnitudes y i , the pitch period and the other parameters.  
      Although the methods and apparatuses disclosed herein have been described in terms of specific embodiments and applications, persons skilled in the art can, in light of this teaching, generate additional embodiments without exceeding the scope or departing from the spirit of the claimed invention. For example, the methods, devices and systems can be used in connection with image and audio coding.