Patent Publication Number: US-6708154-B2

Title: Method and apparatus for using formant models in resonance control for speech systems

Description:
RELATED APPLICATIONS 
     This application is a divisional of U.S. patent application Ser. No. 09/389,898 filed on Sep. 3, 1999 U.S. Pat. No. 6,505,152. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to speech recognition and synthesis systems and in particular to speech systems that exploit formants in speech. 
     In human speech, a great deal of information is contained in the first three resonant frequencies or formants of the speech signal. In particular, when a speaker is pronouncing a vowel, the frequencies and bandwidths of the formants indicate which vowel is being spoken. 
     To detect formants, some systems of the prior art utilize the speech signal&#39;s frequency spectrum, where formants appear as peaks. In theory, simply selecting the first three peaks in the spectrum should provide the first three formants. However, due to noise in the speech signal, non-formant peaks can be confused for formant peaks and true formant peaks can be obscured. To account for this, prior art systems qualify each peak by examining the bandwidth of the peak. If the bandwidth is too large, the peak is eliminated as a candidate formant. The lowest three peaks that meet the bandwidth threshold are then selected as the first three formants. 
     Although such systems provided a fair representation of the formant track, they are prone to errors such as discarding true formants, selecting peaks that are not formants, and incorrectly estimating the bandwidth of the formants. These errors are not detected during the formant selection process because prior art systems select formants for one segment of the speech signal at a time without making reference to formants that had been selected for previous segments. 
     To overcome this problem, some systems use heuristic smoothing after all of the formants have been selected. Although such post-decision smoothing removes some discontinuities between the formants, it is less than optimal. 
     In speech synthesis, the quality of the formant track in the synthesized speech depends on the technique used to create the speech. Under a concatenative system, sub-word units are spliced together without regard for their respective formant values. Although this produces sub-word units that sound natural by themselves, the complete speech signal sounds unnatural because of discontinuities in the formant track at sub-word boundaries. Other systems use rules to control how a formant changes over time. Such rule-based synthesizers never exhibit the discontinuities found in concatenative synthesizers, but their simplified model of how the formant track should change over time produces an unnatural sound. 
     SUMMARY OF THE INVENTION 
     The present invention utilizes a formant-based model to improve the creation of formant tracks in synthesized speech. Text is divided into a sequence of formant model states, which are used to retrieve a sequence of stored excitation segments. The states are also provided to a formant path generator, which determines a set of most likely formant paths given the sequence of model states and the formant models for each state. The formant paths are then used to control a series of resonators, which introduce the formants into the sequence of excitation segments. This produces a sequence of speech segments that are later combined to form the synthesized speech signal. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of a general computing environment in which the present invention may be practiced. 
     FIG. 2 is a graph of the magnitude spectrum of a speech signal. 
     FIG. 3 is a graph of the first three formants of a speech signal. 
     FIG. 4 is a block diagram of a formant tracker and formant model trainer of one embodiment of the present invention. 
     FIG. 5 is a block diagram of a speech compression unit of one embodiment of the present invention. 
     FIG. 6A is a graph of the magnitude spectrum of a speech signal. 
     FIG. 6B is a graph of the magnitude spectrum of a speech signal with its formants removed. 
     FIG. 6C is a graph of the magnitude spectrum of a voiced portion of the signal of FIG.  6 B. 
     FIG. 6D is a graph of the magnitude spectrum of an unvoiced portion of the signal of FIG.  6 B. 
     FIG. 7A is a graph of the magnitude spectrum of a voiced portion of a speech signal showing a set of compression triangles. 
     FIG. 7B is a graph of the magnitude spectrum of an unvoiced portion of a speech signal showing a set of compression triangles. 
     FIG. 8 is a block diagram of a system for reconstructing a speech signal under one embodiment of the present invention. 
     FIG. 9 is a block diagram of a speech synthesis system of one embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS 
     FIG.  1  and the related discussion are intended to provide a brief, general description of a suitable computing environment in which the invention may be implemented. Although not required, the invention will be described, at least in part, in the general context of computer-executable instructions, such as program modules, being executed by a personal computer. Generally, program modules include routine programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. 
     With reference to FIG. 1, an exemplary system for implementing the invention includes a general purpose computing device in the form of a conventional personal computer  20 , including a processing unit (CPU)  21 , a system memory  22 , and a system bus  23  that couples various system components including the system memory  22  to the processing unit  21 . The system bus  23  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The system memory  22  includes read only memory (ROM)  24  and random access memory (RAM)  25 . A basic input/output (BIOS)  26 , containing the basic routine that helps to transfer information between elements within the personal computer  20 , such as during start-up, is stored in ROM  24 . The personal computer  20  further includes a hard disk drive  27  for reading from and writing to a hard disk (not shown), a magnetic disk drive  28  for reading from or writing to removable magnetic disk  29 , and an optical disk drive  30  for reading from or writing to a removable optical disk  31  such as a CD ROM or other optical media. The hard disk drive  27 , magnetic disk drive  28 , and optical disk drive  30  are connected to the system bus  23  by a hard disk drive interface  32 , magnetic disk drive interface  33 , and an optical drive interface  34 , respectively. The drives and the associated computer-readable media provide nonvolatile storage of computer readable instructions, data structures, program modules and other data for the personal computer  20 . 
     Although the exemplary environment described herein employs the hard disk, the removable magnetic disk  29  and the removable optical disk  31 , it should be appreciated by those skilled in the art that other types of computer readable media which can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memory (ROM), and the like, may also be used in the exemplary operating environment. 
     A number of program modules may be stored on the hard disk, magnetic disk  29 , optical disk  31 , ROM  24  or RAM  25 , including an operating system  35 , one or more application programs  36 , other program modules  37 , and program data  38 . A user may enter commands and information into the personal computer  20  through local input devices such as a keyboard  40 , pointing device  42  and a microphone  43 . Other input devices (not shown) may include a joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit  21  through a serial port interface  46  that is coupled to the system bus  23 , but may be connected by other interfaces, such as a sound card, a parallel port, a game port or a universal serial bus (USB). A monitor  47  or other type of display device is also connected to the system bus  23  via an interface, such as a video adapter  48 . In addition to the monitor  47 , personal computers may typically include other peripheral output devices, such as a speaker  45  and printers (not shown). 
     The personal computer  20  may operate in a networked environment using logic connections to one or more remote computers, such as a remote computer  49 . The remote computer  49  may be another personal computer, a hand-held device, a server, a router, a network PC, a peer device or other network node, and typically includes many or all of the elements described above relative to the personal computer  20 , although only a memory storage device  50  has been illustrated in FIG.  1 . The logic connections depicted in FIG. 1 include a local area network (LAN)  51  and a wide area network (WAN)  52 . Such networking environments are commonplace in offices, enterprise-wide computer network Intranets, and the Internet. 
     When used in a LAN networking environment, the personal computer  20  is connected to the local area network  51  through a network interface or adapter  53 . When used in a WAN networking environment, the personal computer  20  typically includes a modem  54  or other means for establishing communications over the wide area network  52 , such as the Internet. The modem  54 , which may be internal or external, is connected to the system bus  23  via the serial port interface  46 . In a network environment, program modules depicted relative to the personal computer  20 , or portions thereof, may be stored in the remote memory storage devices. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. For example, a wireless communication link may be established between one or more portions of the network. 
     Under the present invention, a Hidden Markov Model (HMM) is developed for formants found in human speech. The invention has several aspects including formant tracking, training a formant model, using the model to compress speech signals for later use in speech synthesis, and using the model to generate smooth formant tracks during speech synthesis. Each of these aspects is discussed separately below. 
     FORMANT TRACKING 
     FIG. 2 is a graph of the frequency spectrum of a section of human speech. In FIG. 2, frequency is shown along horizontal axis  200  and the magnitude of the frequency components is shown along vertical axis  202 . The graph of FIG. 2 shows that human speech contains resonances or formants, such as first formant  204 , second formant  206 , third formant  208 , and fourth formant  210 . Each formant is described by its center frequency, F, and its bandwidth, B. 
     FIG. 3 is a graph of changes in the center frequencies of the first three formants during a lengthy utterance. In FIG. 3, time is shown along horizontal axis  220  and frequency is shown along vertical axis  222 . Solid line  224  traces changes in the frequency of the first formant, F 1 , solid line  226  traces changes in the frequency of the second formant, F 2 , and solid line  228  traces changes in the frequency of the third formant, F 3 . Although not shown, the bandwidth of each formant also changes during an utterance. 
     One embodiment of the present invention for tracking these changes in the formants is shown in the block diagram of FIG.  4 . In FIG. 4, input speech  280  is generated by a speaker while reading text  282 . Speech  282  is sampled and held by a sample and hold circuit  284 , which in one embodiment, samples training speech  282  across successive overlapping Hanning windows. 
     The sampled values are then passed to a formant tracker  287  that consists of a formant identifier  288 , a group generator  290  and a Viterbi search unit  292 . Formant identifier  288  receives the sampled values and uses the values to identify possible formants. In one embodiment, formant identifier  288  consists of a Linear Predictive Coding (LPC) unit that determines the roots of the LPC predictor polynomial. Each root describes a possible frequency and bandwidth for a formant. In other embodiments, formants are identified as peaks in the LPC-spectrum. Both of these techniques are well known in the art. 
     In the prior art, only those candidate formants with sufficiently small bandwidths were used to select the formants for a sampling window. If a candidate formant&#39;s bandwidth was too large it was discarded at this stage. In contrast, the present invention retains all candidate formants, regardless of their bandwidth. 
     The candidate formants produced by formant identifier  288  are provided to a group generator  290 , which groups the candidate formants based on their frequencies. In particular, group generator  290  forms unique groups of N candidate formants, with the candidates ordered from lowest frequency to highest frequency within each group. Thus, if N=3 and there are seven candidate formants, the group generator will create 35 3-formant groups. 
     In most embodiments, N=3, with the lowest frequency candidate designated as the first formant, the second lowest frequency candidate designated as the second formant, and the highest frequency candidate designated as the third formant. 
     The groups of formant candidates are provided to a Viterbi search unit  292 , which is used to identify the most likely sequence of formant groups based on training text  282  and a formant Hidden Markov Model  296 . Training text  282  is parsed into sub-word units or states by a parser  294  and the states are provided to Viterbi search unit  292 . For example, in embodiments that model phonemes using a left-to-right three-state model, each word is divided into the constituent states of its phonemes and these states are provided to Viterbi search unit  292 . 
     For each state it receives, Viterbi search unit  292  requests a state formant model from Hidden Markov Model  296 , which contains a model for each possible state in a language. In one embodiment, the state model contains a mean frequency, a mean bandwidth, a frequency variance and a bandwidth variance for each formant in the model. Thus, for state, i, the state formant model takes the form of a 
     vector, h i , defined as:                h   i     =     {             μ     i   ,   F1       ,     σ     i   ,   F1       ,     μ     i   ,   B1       ,     σ     i   ,   B1       ,     μ     i   ,   F2       ,     σ     i   ,   F2       ,                 μ     i   ,   B2       ,     σ     i   ,   B2       ,     μ     i   ,   F3       ,     σ     i   ,   F3       ,     μ     i   ,   B3       ,     σ     i   ,   B3               }             EQ   .              1                         
     where μ i,Fx  is the mean frequency of the xth formant,        σ     i   ,   Fx     2                   
     is the variance of the xth formant&#39;s frequency, μ i,Bx  is the mean bandwidth of the xth formant,        σ     i   ,   Bx     2                   
     is the variance of the xth formant&#39;s bandwidth. 
     Under one embodiment, in order to provide better smoothing during formant tracking, the state vector shown in Equation 1 is augmented by providing means and variances that describe the slope of change of a formant over time. With the additional means and variances, Equation 1 becomes:                h   i     =     {             μ     i   ,   F1       ,     σ     i   ,   F1       ,     μ     i   ,   B1       ,     σ     i   ,   B1       ,     μ     i   ,   F2       ,     σ     i   ,   F2       ,                 μ     i   ,   B2       ,     σ     i   ,   B2       ,     μ     i   ,   F3       ,     σ     i   ,   F3       ,     μ     i   ,   B3       ,     σ     i   ,   B3       ,                 δ     i   ,     Δ                 F1         ,     γ     i   ,     Δ                 F1         ,     δ     i   ,     Δ                 B1         ,     γ     i   ,     Δ                 B1         ,     δ     i   ,     Δ                 F2         ,     γ     i   ,     Δ                 F2                       δ     i   ,     Δ                 B2         ,     γ     i   ,     Δ                 B2         ,     δ     i   ,     Δ                 F3         ,     γ     i   ,     Δ                 F3         ,     δ     i   ,     Δ                 B3         ,     γ     i   ,     Δ                 B3                 }             EQ   .              2                         
     where δ i,ΔF1  and γ iΔF1  are the mean and standard deviation of the change in frequency of the first formant, δ i,ΔB1  and γ i,ΔB1  are the mean and standard deviation of the change in bandwidth of the first formant, δ i,ΔF2 , γ i,ΔF2  and δ i,ΔB2 , γ i,ΔB2  are the mean and standard deviation of the change in frequency and change in bandwidth, respectively, of the second formant, and δ i,ΔF3 , γ i,ΔF3  and δ i,ΔB3 , γ i,ΔB3  are the mean and standard deviation of the change in frequency and bandwidth, respectively, of the third formant. 
     To calculate the most likely sequence of observed formant groups, Ĝ, Viterbi search unit  292  calculates a separate probability for each possible sequence of observed groups: 
     
       
           G={g   1   ,g   2   ,g   3   , . . . g   T }  EQ. 3 
       
     
     where T is the total number of states in the utterance under consideration, and g x  is the frequencies and bandwidths for the formants in a group observed for the xth state. The probability for each observed sequence of formant groups, G, given the HMM λ is defined as:                p        (     G      λ     )       =       ∑   q            p        (       G      q     ,   λ     )            p        (     q      λ     )                   EQ   .              4                         
     where p(q|λ) is the probability of a sequence of states q given the HMM λ, p(G|q,λ) is the probability of the sequence of formant groups given the HMM λ and the sequence of states q, and the summation is taken over all possible state sequences: 
     
       
           q={q   1   ,q   2   ,q   3   , . . . q   T }  EQ. 5 
       
     
     In most embodiments, the sequence of states are limited to the sequence, {circumflex over (q)}, created from the segmentation of training text  282  provided by parser  294 . In addition, many embodiments simplify the calculations associated with Equation 4 by replacing the summation with the largest term in the summation. This leads to: 
     
       
           Ĝ=arg   G   max [ln  p ( G|{circumflex over (q)} , λ)]  EQ. 6 
       
     
     At each state i, the HMM vector of Equation 2 can be divided into two mean vectors Θ i  and Δ i , and two covariance matrices Σ i  and Γ i  defined as:                Θ   i     =     {             μ     i   ,   F1       ,     μ     i   ,   F2       ,     μ     i   ,   F3       ,   …              ,     μ     i   ,     FM   /   2         ,                 μ     i   ,   B1       ,     μ     i   ,   B2       ,     μ     i   ,   B3       ,   …              ,     μ     i   ,     BM   /   2                 }             EQ   .              7                 Δ   i     =     {             δ     i   ,     Δ                 F1         ,     δ     i   ,     Δ                 F2         ,     δ     i   ,     Δ                 F3         ,   …              ,     δ     i   ,     Δ                   FM   /   2           ,                 δ     i   ,     Δ                 B1         ,     δ     i   ,     Δ                 B2         ,     δ     i   ,     Δ                 B3         ,   …              ,     δ     i   ,     Δ                   BM   /   2                   }             EQ   .              8                 Σ   i     =     (           σ     i   ,   F1     2         0       0       0       0       0       0       0           0         σ     i   ,   F2     2         0       0       0       0       0       0           0       0       ⋰       0       0       0       0       0           0       0       0         σ     i   ,     FM   /   2       2         0       0       0       0           0       0       0       0         σ     i   ,   B1     2         0       0       0           0       0       0       0       0         σ     i   ,   B2     2         0       0           0       0       0       0       0       0       ⋰       0           0       0       0       0       0       0       0         σ     i   ,     BM   /   2       2           )             EQ   .              9                 Γ   i     =     (           γ     i   ,     Δ                 F1       2         0       0       0       0       0       0       0           0         γ     i   ,     Δ                 F2       2         0       0       0       0       0       0           0       0       ⋰       0       0       0       0       0           0       0       0         γ     i   ,     Δ                   FM   /   2         2         0       0       0       0           0       0       0       0         γ     i   ,     Δ                 B1       2         0       0       0           0       0       0       0       0         γ     i   ,     Δ                 B2       2         0       0           0       0       0       0       0       0       ⋰       0           0       0       0       0       0       0       0         γ     i   ,     Δ                   BM   /   2         2           )             EQ   .              10                         
     where M/2 is the number of formants in each group. Although the covariance matrices are shown as diagonal matrices, more complicated covariance matrices are contemplated within the scope of the present invention. Using these vectors and matrices, the model λ provided by HMM  296  for a language with n possible states becomes: 
     
       
         λ={Θ 1 ,Δ 1 ,Σ 1 ,Γ 1 ,Θ 2 ,Δ 2 ,Σ 2 ,Γ 2 , . . . Θ n , Δ n ,Σ n ,Γ n }  EQ. 11 
       
     
     Combining Equations 7 through 11 with Equation 6, the probability of each individual group sequence is calculated as:                ln                   p        (       G        q   ^       ,   λ     )         =     (               -     TM   2            ln        (     2      π     )         -       1   2            ∑     t   =   1     T          ln             Σ     q   t                  -       1   2            ∑     t   =   2     T          ln             Γ     q   t                                -     1   2              ∑     t   =   1     T              (       g   t     -     Θ     q   t         )     ′            Σ     q   t       -   1            (       g   t     -     Θ     q   t         )                         -     1   2              ∑     t   =   2     T              (       g   t     -     g     t   -   1       -     Δ     q   t         )     ′            Γ     q   t       -   1            (       g   t     -     g     t   -   1       -     Δ     q   t         )                   )             EQ   .              12                         
     where T is the total number of states in the utterance under consideration, M/2 is the number of formants in each group g, g t  is the group observed in the current sampling window t, g t−1  is the group observed in the preceding sampling window t−1, (x)′ denotes the transpose of matrix x, Σ q     t     −1  indicates the inverse of the matrix Σ q     t   , and the subscript q t  indicates the model vector element of state q, which has been parsed as occurring during sampling window t. 
     The probability of Equation 12 is calculated for each possible sequence of groups, G, and the sequence with the maximum probability is selected as the most likely sequence of formant groups. Since each formant group contains multiple formants, the calculation of the probability of a sequence of groups found in Equation 12 simultaneously provides probabilities for multiple non-intersecting formant tracks. For example, where there are three formants in a group, the calculations of Equation 12 simultaneously provided the combined probabilities of a first, second and third formant track. Thus, by using Equation 12 to select the most likely sequence of groups, the present invention inherently selects the most likely formant tracks. 
     In some embodiments, Equation 12 is modified to provide for additional smoothing of the formant tracks. This modification involves allowing Viterbi Search Unit  292  to select formant constituents (i.e. F 1 , F 2 , F 3 , B 1 , B 2 , and B 3 ) that are not actually observed. This modification is based in part on the recognition that due to limitations in the monitoring equipment, the observed formant track is not always the same as the real formant track produced by the speaker. 
     To provide for this modification, a real sequence of formant groups, X, is defined with: 
     
       
           X={x   1   ,x   2   ,x   3   , . . . x   T }  EQ. 13 
       
     
     where x i  is the real formant group (also referred to as the real formant vector) at state i. This changes Equation 12 so that it becomes:                ln                   p        (       X        q   ^       ,   λ     )         =     (               -     TM   2            ln        (     2      π     )         -       1   2            ∑     t   =   1     T          ln             Σ     q   t                  -       1   2            ∑     t   =   2     T          ln             Γ     q   t                                -     1   2              ∑     t   =   1     T              (       x   t     -     Θ     q   t         )     ′            Σ     q   t       -   1            (       x   t     -     Θ     q   t         )                         -     1   2              ∑     t   =   2     T              (       x   t     -     x     t   -   1       -     Δ     q   t         )     ′            Γ     q   t       -   1            (       x   t     -     x     t   -   1       -     Δ     q   t         )                   )             EQ   .              14                         
     where Equation 14 is now used to find the most probable sequence of real formant groups, {circumflex over (X)}. 
     With this modification to Equation 12, an additional smoothing term may be added to account for the difference between the real formants and the observed formants. Specifically, if X is the real set of formant tracks, which is hidden, and Ĝ is the most probable observed formant tracks selected above, the joint probability of both X and Ĝ given the Hidden Markov Model λ is defined as:                p        (       G   ^     ,     X   |   λ       )       =         p        (         G   ^     |   X     ,   λ     )            p        (       G   ^     |   λ     )         =       p        (     X   |   λ     )              ∏     t   =   1     T                     p        (       g   t     |     x   t       )                     EQ   .              15                         
     where p(Ĝ|X,λ) is the probability of the most likely observed formant tracks given the real formant tracks and the HMM, p(X|λ) is the probability of the real formant tracks given the HMM, and p(g t |x t ) is the probability of the most likely observed group of formant values at state t given the real group of formant values at state t. In Equation 15 it is assumed that p(G|X,λ) does not depend on λ, and that the probability of a group of most likely observed formants in state t, g t , only depends on the group of actual formants at state t, x t . 
     The probability of a group of most likely observed formant values at state t given the group of real formant values at state t, p(g t |x t ), can be approximated by a Gaussian density function:                p        (       g   t          x   t       )       =       1         (     2      π     )       M   /   2              ∏     j   =   1     M          υ        [   j   ]                exp        {       -     1   2              ∑     j   =   1     M              (       g        [   j   ]       -     x        [   j   ]         )     2         υ   2          [   j   ]             }               EQ   .              16                         
     where M is the number of formant constituents in each group, g[j] represents the jth observed formant constituent (i.e. F 1 , F 2 , F 3 , B 1 , B 2 , or B 3 ) within the group, x[j] represents the jth real formant constituent within the group, and υ 2 [j] is the variance of the jth real formant constituent within the group. In one embodiment, υ[j] of the formant frequency values in group t (F 1   t , F 2   t , or F 3   t ) is set equal to the observed bandwidth for the respective formant frequency value. In these embodiments, υ[j] of the formant bandwidth values was set to the formant bandwidth. 
     Using the far right-hand side of Equation 15, it can be seen that the smoothing equation of Equation 16 can be added to Equation 14 to produce a formant tracking equation that considers unobserved groups of formants. In particular this combination produces:                ln                   p        (       X        q   ^       ,   λ     )         =     (               -     TM   2            ln        (     2      π     )         -       1   2            ∑     t   =   1     T          ln             Σ     q   t                  -       1   2            ∑     t   =   2     T          ln             Γ     q   t                                -     1   2              ∑     t   =   1     T              (       x   t     -     Θ     q   t         )     ′            Σ     q   t       -   1            (       x   t     -     Θ     q   t         )                         -     1   2              ∑     t   =   2     T              (       x   t     -     x     t   -   1       -     Δ     q   t         )     ′            Γ     q   t       -   1            (       x   t     -     x     t   -   1       -     Δ     q   t         )                         -     1   2              ∑     t   =   1     T              (       g   t     -     x   t       )     ′            Ψ   t     -   1            (       g   t     -     x   t       )                   )             EQ   .              17                         
     where Ψ t  is a covariance matrix containing the covariance values υ 2 [j] for the formant constituents of group t. In one embodiment, Ψ t  is a diagonal matrix of the form:                Ψ   i     =     (           υ     i   ,   F1     2         0       0       0       0       0       0       0           0         υ     i   ,   F2     2         0       0       0       0       0       0           0       0       ⋰       0       0       0       0       0           0       0       0         υ     i   ,     F                   M   2         2         0       0       0       0           0       0       0       0         υ     i   ,   B1     2         0       0       0           0       0       0       0       0         υ     i   ,   B2     2         0       0           0       0       0       0       0       0       ⋰       0           0       0       0       0       0       0       0         υ     i   ,     B                   M   2         2           )             EQ   .              18                         
     If Σ q     t    and Γ q     t    are also diagonal matrices, the matrix functions within the last three summations of Equation 17 produces terms of the form:                  ∑     t              =              1     T                (       x   t                -                Θ     q   t         )                  ′                       Σ     q   t       -   1       (                  x   t                -                Θ     q   t         )         =     {                   (       F1   1                -                μ     1   ,              F1         )     2       σ     1   ,              F1     2                  +                    (       F1   2                -                μ     2   ,              F1         )     2       σ     2   ,              F1     2                  +              …              +                    (       F1   T                -                μ     T   ,              F1         )     2       σ     T   ,              F1     2                  +                                  (       F2   1                -                μ     1   ,              F2         )     2       σ     1   ,              F2     2                  +                    (       F2   2                -                μ     2   ,              F2         )     2       σ     2   ,              F2     2                  +              …              +                    (       F2   T                -                μ     T   ,              F2         )     2       σ     T   ,              F2     2                  +              …                 +         (       B3   1                -                μ     1   ,              B3         )     2       σ     1   ,              B3     2                    +                    (       B3   2                -                μ     2   ,              B3         )     2       σ     2   ,              B3     2                  +              …              +                    (       B3   T                -                μ     T   ,              B3         )     2       σ     T   ,              B3     2               }             EQ              .              19                   ∑     t              =              2     T              (       x   t                -                x     t              -              1                  -                Δ     q   t         )     ′                       Γ     q   t       -   1       (                  x   t                -                x     t              -              1                  -                Δ     q   t         )         =       {                 (       F1   2                -                F1   1                -                δ     1   ,              F1         )     2       γ     1   ,              F1     2                  +              …              +                    (       F1   T                -                F1     T              -              1                  -                δ     T   ,              F1         )     2       γ     T   ,              F1     2                  +              …                     (       F2   2                -                F2   1                -                δ     1   ,              F2         )     2       γ     1   ,              F2     2                  +              …              +                    (       F2   T                -                F2     T              -              1                  -                δ     T   ,              F2         )     2       γ     T   ,              F2     2                  +              …                 +         (       B3   2                -                B3   1                -                δ     1   ,              B3         )     2       γ     1   ,              B3     2                    +              …              +                    (       B3   T                -                B3     T              -              1                  -                δ     T   ,              B3         )     2       γ     T   ,              B3     2               }                   and             EQ              .              20                     1   2                       ∑     t              =              1     T              (       g   t                -                x   t       )     ′                       Ψ   t     -   1            (       g   t                -                x   t       )             =     {                   (       g     1   ,              F1                  -                F1   1       )     2       υ     1   ,              F1     2                  +                    (       g     2   ,              F1                  -                F1   2       )     2       υ     2   ,              F1     2                  +              …              +                    (       g     T   ,              F1                  -                F1   T       )     2       υ     T   ,              F1     2                  +                                  (       g     1   ,              F2                  -                F2   1       )     2       υ     1   ,              F2     2                  +                    (       g     2   ,              F2                  -                F2   2       )     2       υ     2   ,              F2     2                  +              …              +                    (       g     T   ,              F2                  -                F2   T       )     2       υ     T   ,              F2     2                  +              …                 +         (       g     1   ,              B3                  -                B3   1       )     2       υ     1   ,              B3     2                    +                    (       g     2   ,              B3                  -                B3   2       )     2       υ     2   ,              B3     2                  +              …              +                    (       g     T   ,              B3                  -                B3   T       )     2       υ     T   ,              B3     2               }                          EQ              .              21                         
     where the subscript notations in Equations 19 through 21 can be understood by generalizing the following small set of examples: F 2   1  is the frequency of the second formant of the first state, F 2   2  is the frequency of the second formant of the second state, B 3   1  is the bandwidth of the third formant of the first state, μ 2,F1  is the Hidden Markov Model mean frequency for the first formant in the second state, σ T,B3   2  is the HMM variance for the bandwidth of the third formant in the last state T, δ 1,F2  is the HMM mean change in the frequency of the second formant of the first state, γ 3,F2   2  is the HMM variance for the frequency of the second formant for the third state, g 2,B3  is the observed value for the third formant&#39;s bandwidth in the second state, and ν 2,F1   2  is the variance for the observed frequency of the first formant in the second state. 
     Since the sequence of formant groups that maximizes Equation 17 is not limited to observed groups of formants, this sequence can be determined by finding the partial derivatives of Equation 17 for each sequence of formant constituents. 
     To find the sequence of formant vectors that maximizes equation 17, each constituent (F 1 , F 2 , F 3 , . . . , B 1 , B 2 , B 3 , . . . ) is considered separately. Thus, a sequence of first formant frequency values, F 1 , is determined, then a sequence of second formant frequency values, F 2 , is determined and so on ending with a sequence of formant bandwidth values for the last formant. Note that the order in which the constituents are selected is arbitrary and the sequence of formant bandwidth values for the last formant may be calculated first. 
     For each constituent (F 1 , F 2 , F 3 , B 1 , B 2 , or B 3 ), the sequence of values that maximizes Equation 17 is determined by determining the partial derivatives of Equation 17 with reference to the constituent in each state. Thus, if the sequence of first formant frequencies, F 1 , is being determined, the partial derivative of Equation 17 is calculated for each F 1   i  across all states, i, of the input speech signal. In other words, the following partial derivatives are taken:                  δ     δ                   F1   1              f        (     EQ   .              17     )         ,       δ     δ                   F1   2              f        (     EQ   .              17     )         ,   …              ,       δ     δ                   F1   T              f        (     EQ   .              17     )                 EQ   .              22                         
     where δ of Equation 22 refers only to the partial derivative of f(EQ. 17) and is not to be confused with the mean of the change in frequency or bandwidth found in the Hidden Markov Model above. 
     Each partial derivative associated with a constituent is then set equal to zero. This produces a set of linear equations for each constituent. For example, the linear equation for the partial derivative with reference to the first formant frequency of the second state, F 1   2 , is:                        δ     δ                   F1   2              f        (     EQ   .              17     )         =                    -     1     γ   q2   2              F1   1       +     (       1     υ   2   2       +     1     σ   q2   2       +     1     γ   q2   2       +     1     γ   q3   2         )                                  F1   2     -       1     γ   q2   2            F1   3       -       g     2   ,   F1         υ   2   2       -       μ   q2       σ   q2   2       -       δ   q2       γ   q2   2       +                                  δ   q3       γ   q3       =   0                   EQ   .              23                         
     where g 2,F1  represents the most likely observed value for the first formant at the second state. 
     The linear equations for a constituent such as F 1  can be solved simultaneously using a matrix notation of the form: 
     
       
         BX=c  EQ. 24 
       
     
     where B and c are matrices formed by the partial derivatives and X is a matrix containing the constituent&#39;s values at each state. The size of B and c depends on the number of states, T, in the speech signal being analyzed. As a simple example of the types of values in B, c, and X, a small utterance of T= 3  states would produce matrices of:              B   =     (             1     υ   1   2       +     1     σ   q1   2       +     1     γ   q2   2               -     1     γ   q2   2             0             -     1     γ   q2   2                 1     υ   2   2       +     1     σ   q2   2       +     1     γ   q2   2       +     1     γ   q3   2               -     1     γ   q3   2                 0         -     1     γ   q3   2                 1     υ   3   2       +     1     σ   q3   2       +     1     γ   q3   2               )             EQ   .              25               c   =     (         g   1       υ   1   2       +       μ   q1       σ   q1   2       -         δ   q2       γ   q2   2                         g   2       υ   2   2         +       μ   q2       σ   q2   2       +       δ   q2       γ   q2   2       -         δ   q3       γ   q3   2                         g   3       υ   3   2         +       μ   q3       σ   q3   2       +       δ   q3       γ   q3   2         )             EQ   .              26               X   =     (           F1   1               F1   2               F1   3           )             EQ   .              27                         
     Note that B is a tridiagonal matrix where all of the values are zero except those in the main diagonal and its two adjacent diagonals. This remains true regardless of the number of states in the output speech signal. The fact that B is a tridiagonal matrix is helpful under many embodiments of the invention because there are well known algorithms that can be used to invert matrix B much more efficiently than a standard matrix. 
     To solve for the sequence of values for a constituent (F 1 , F 2 , F 3 , B 1 , B 2 , or B 3 ), the inverse of B is multiplied by c. This produces the sequence of values that has a maximum probability. 
     This process is then repeated for each constituent to produce a single most likely sequence of values for each formant constituent in the utterance being analyzed. 
     TRAINING A FORMANT MODEL 
     The formant tracking system described above can be used alone or as part of a system for training a formant model. Note that in the discussion above it was assumed that there was a formant Hidden Markov Model defined for each state. However, when training the formant Model for the first time, this is not true. To overcome this problem, the present invention provides an initial simplistic Hidden Markov Model. In one embodiment, the values for this initial HMM are chosen based on average formant values across all possible states in a language. In one particular embodiment, each state, i, has the same initial vector values of: 
     
       
         μ i,F1 =500  Hz   EQ. 28 
       
     
     
       
         μ i,F2 =1500  Hz   EQ. 29 
       
     
     
       
         μ i,F3 =2500  Hz   EQ. 30 
       
     
     
       
         σ i,F1 =σ i,F2 =σ i,F3 =500  Hz   EQ. 31 
       
     
     
       
         μ i,B1 =μ i,B2 =μ i,B3 =100  Hz    EQ. 32 
       
     
     
       
         σ i,B1 =σ i,B2 =σ i,B3 =100  Hz    EQ. 33 
       
     
     
       
         δ i,ΔF1 =δ i,ΔF2 =δ i,F3 =δ i,ΔB1 =δ i,ΔB2 =δ i,ΔB3 =0  Hz   EQ. 34 
       
     
      γ i,ΔF1 =γ i,ΔF2 =γ i,ΔF3 =γ i,ΔB1 =γ i,ΔB2 =γ i,ΔB3 =100  Hz   EQ. 35 
     Using these initial values, a training speech signal is processed by Viterbi search unit  292 , to produce an initial set of most likely formants for each state of the training signal. This initial set of formants includes a frequency and bandwidth for each formant. The formant values in this initial set are stored in a storage unit  298 , which is later accessed by a model building unit  300 . 
     Model building unit  300  collects the formants associated with each occurrence of a state in the speech signal and combines these formants to generate a distribution of formants for the state. For example, if a state appeared five times in the speech signal, model building unit  300  would combine the formants from the five appearances of the state to form a distribution for each formant. In one embodiment, this distribution is characterized as a Gaussian distribution, which is described by its mean and variance. 
     For any one formant in a state, several distributions are determined. In one particular embodiment, four distributions are created for each formant in each state. Specifically, distributions are calculated for the formant&#39;s frequency, bandwidth, change in frequency, and change in bandwidth. Thus, model building unit  300  determines the mean and variance of the frequency, bandwidth, change in frequency and change in bandwidth for each formant in each possible state in the language. 
     The formant Hidden Markov Model calculated by model building unit  300  is then designated as the new Hidden Markov Model  296 . Training speech  280  is then sampled again and the most likely sequence of formant groups is re-calculated using the new HMM. This process of determining a most likely sequence of formant groups and generating a new Hidden Markov Model is repeated until the formant Hidden Markov Model does not change significantly between iterations. In some embodiments, it has been found that three iterations are sufficient. 
     COMPRESSING SPEECH SIGNALS 
     In many applications, such as audio delivery over the Internet, it is advantageous to compress speech signals so that they are accurately represented by as few values as possible. One aspect of the present invention is to use the formant tracking system described above to generate small representations of speech. 
     FIG. 5 is a block diagram of one embodiment of the present invention for compressing speech. In FIG. 5, training speech  350  is generated by a speaker while reading training text  352 . Training speech  350  is sampled and held by a sample and hold circuit  354 . In one embodiment, sample and hold circuit  354  samples training speech  350  across successive overlapping Hanning windows. 
     The set of samples is provided to a formant tracker  362 , which is the same as formant tracker  287  of FIG. 4 . Formant tracker  362  also receives text  352  after it has been segmented into HMM states by a parser  360 . For each state received from parser  360 , formant tracker  362  identifies a set of most likely formants using the techniques described above for formant tracking under the present invention. 
     The frequencies and bandwidths of the identified formants are provided to a filter controller  358 , that also receives the speech samples produced by sample and hold circuit  354 . Filter controller  358  aligns the speech samples of a state with the formants identified for that state by formant tracker  362 . 
     With the samples properly aligned, one sample at a time is passed though a series of filters  364 ,  366 , and  368  that are adjusted by filter controller  358 . Filter controller  358  adjusts these filters based on the frequency and bandwidth of the respective formants identified for this state by formant tracker  362 . In particular, first formant filter  364  is adjusted so that it filters out a set of frequencies centered on the first formant&#39;s frequency and having a bandwidth equal to the first formant&#39;s bandwidth. Similar adjustments are made to second formant filter  366  and third formant filter  368  so that their center frequencies and bandwidths match the respective frequencies and bandwidths of the second and third formants identified for the state by formant tracker  362 . 
     With the three formant filters adjusted, the sample values for the current sampling window are passed through the three filters in series. This causes the first, second and third formants to be filtered out of the current sampling window. The effects of this sampling can be seen in FIGS. 6A and 6B. In FIG. 6A, the magnitude spectrum of a current sampling window for speech signal Y, is shown with the frequency components shown along horizontal axis  430  and the magnitude of each component shown along vertical axis  432 . Four formants,  434 ,  436 ,  438 , and  440  are present in FIG.  6 A and appear as localized peaks. FIG. 6B shows the magnitude spectrum of the excitation signal that is provided at the output of third formant filter  368  of FIG.  5 . Note that in FIG. 6B, first formant  434 , second formant  436  and third formant  438  have been removed but fourth formant  440  is still present. 
     The excitation signal produced at the output of third formant filter  368  is provided to a voiced/unvoiced decomposer  370 , which separates the voiced portion of the excitation signal from the unvoiced portion. In one embodiment, decomposer  370  separates the two signals by identifying the pitch period of the excitation signal. Since voiced portions of the signal are formed from waveforms that repeat at the pitch period, the identified pitch period can be used to determine the shape of the repeating waveform. Specifically, successive sections of the excitation signal that are separated by the pitch period can be averaged together to form the voiced portion of the excitation signal. The unvoiced portion can then be determined by subtracting the voiced portion from the excitation signal. 
     In other embodiments, each frequency component of the excitation signal is tracked over time to provide a time-based signal for each component. Since the voiced portion of the excitation signal is formed by portions of the vocal tract that change slowly over time, the frequency components of the voiced portion should also change slowly over time. Thus, to extract the voiced portion, the time-based signals of each frequency component are low-pass filtered to form smooth traces. The values along the smooth traces then represent the voiced portion&#39;s frequency components over time. By subtracting these values from the frequency components of the excitation signal as a whole, the decomposer extracts the frequency component of the unvoiced component. This filtering technique is discussed in more detail in pending U.S. patent application Ser. No. 09/198,661, filed on Nov. 24, 1998 and entitled METHOD AND APPARATUS FOR SPEECH SYNTHESIS WITH EFFICIENT SPECTRAL SMOOTHING, which is hereby incorporated by reference. 
     FIGS. 6C and 6D show the result of the decomposition performed by decomposer  370  of FIG.  5 . FIG. 6C shows the magnitude spectrum of the voiced portion of the excitation signal and FIG. 6D shows the magnitude spectrum of the unvoiced portion. 
     The magnitude spectrum of the voiced portion of the excitation signal is routed to a compression unit  372  in FIG.  5  and the magnitude spectrum of the unvoiced portion is routed to a compression unit  374 . Compression units  372  and  374  compress the magnitude spectrums of the voiced component and unvoiced component into a smaller set of values. In one embodiment, this compression involves using overlapping triangles to approximate the magnitude spectrum of each portion. FIGS. 7A and 7B show graphs depicting this approximation. In FIG. 7A, magnitude spectrum  460  of the voiced portion is shown as being approximated by ten overlapping triangles,  462 ,  464 ,  466 ,  468 ,  470 ,  472 ,  474 ,  476 ,  478 , and  480 . The location and width of these triangles is the same for each sampling window of the speech signal. Thus, only the peak values need to be recorded to represent the magnitude spectrum of the voiced portion. FIG. 7B shows a similar graph with magnitude spectrum  482  of the unvoiced portion being approximated by four overlapping triangles  484 ,  486 ,  488 , and  490 . Thus, using compression units  372  and  374 , the voiced portion of each sampling window is represented by ten values and the unvoiced portion is represented by four values. 
     The values output by compression units  372  and  374  are placed in a storage unit  376 , which also receives the frequencies and bandwidths of the first three formants produced by formant tracker  362  for this sampling window. Alternatively, these values can be transmitted to a remote location. In one embodiment, the values are transmitted across the Internet. 
     Note that the phase of both the voiced component and the unvoiced component can be ignored. The present inventors have found that the phase of the voiced component can be adequately approximated by a constant phase across all frequencies without detrimentally affecting the re-creation of the speech signal. It is believed that this approximation is sufficient because most of the significant phase information in a speech signal is contained in the formants. As such, eliminating the phase information in the voiced portion of the excitation signal does not significantly diminish the audio quality of the recreated speech. 
     The phase of the unvoiced component has been found to be mostly random. As such, the phase of the unvoiced component is approximated by a random number generator when the speech is recreated. 
     From the discussion above, it can be seen that the present invention is able to compress each sampling window of speech into twenty values. (Ten values describe the magnitude spectrum of the voiced component, four values describe the magnitude spectrum of the unvoiced component, three values describe the frequencies of the first three formants, and three values describe the bandwidths of the first three formants.) This compression reduces the amount of information that must be stored to recreate a speech signal. 
     FIG. 8 is a block diagram of a system for recreating a speech signal that has been compressed using the embodiment of FIG.  5 . In FIG. 8, the compressed magnitude values of the voiced portion  510  and unvoiced portion  512  are provided to two overlap-and-add circuits  514  and  516 . These circuits recreate approximations of the voiced portion and unvoiced portion, respectively, of the current sampling window. To do this, the circuits sum the overlapping portions of the triangles represented by the compressed voiced values and the compressed unvoiced values. 
     The output of overlap-and-add circuit  516  is provided to a summing circuit  518  that adds in the phase spectrum of the unvoiced portion of the excitation signal. As noted above, the phase spectrum of the unvoiced portion can be approximated by random values. In FIG. 8, these values are provided by a random number generator  520 . 
     The output of overlap and add circuit  518  is provided to a summing circuit  522 , which adds in the phase spectrum of the voiced portion of the excitation signal. As noted above, the phase spectrum of the voiced component can be approximated by a constant value  524 , for all frequencies. 
     After the phase spectrums of the voiced and unvoiced portions have been added to the recreated magnitude spectrums, the recreated voiced and unvoiced portions are summed together by a summing circuit  526 . The output of summing circuit  526  represents the Fourier Transform of a recreated excitation signal. An inverse Fast Fourier Transform  538  is performed on this signal to produce one window of the recreated excitation signal. A succession of these windows is then combined by an overlap-and-add circuit  540  to produce the recreated excitation signal. The excitation signal is then passed through three formant resonators  528 ,  530 , and  532 . 
     Each of the resonators is controlled by a resonator controller  534 , which sets the resonators based on the stored frequencies and bandwidths  536  for the first three formants. Specifically, resonator controller  534  sets resonators  528 ,  530  and  532  so that they resonate at the frequency and bandwidth of the first formant, the second formant and the third formant, respectively. The output of resonator  532  represents the recreated speech signal. 
     SPEECH SYNTHESIS USING A FORMAT HMM 
     Another aspect of the present invention is the synthesis of speech using a formant Hidden Markov Model like the one trained above. FIG. 9 provides a block diagram of one embodiment of such a speech synthesizer under the present invention. 
     In FIG. 9, text  600  that is to be converted into speech is provided to a parser  602  and a semantic identifier  604 . Parser  602  segments the input text into sub-word units and provides these units to a prosody generator  606 . In one embodiment, the sub-word units are states of the formant Hidden Markov Model. 
     Semantic identifier  604  examines the text to determine its linguistic structure. Based on the text&#39;s structure, semantic identifier  604  generates a set of prosody marks that indicate which parts of the text are to be emphasized. These prosody marks are provided to prosody generator  606 , which uses the marks in determining the pitch and cadence for the synthesized speech. 
     To generate the proper pitch and cadence for the synthesized speech, prosody generator  606  controls the rate at which it releases the states it receives from parser  602 . In addition, by repeatedly releasing a single state it receives from parser  602 , prosody generator  606  is able to extend the duration of the sound associated with that state. To extend the duration of a particular sound, prosody generator  606  also has the ability to repeatedly release a single state it receives from parser  602 . To increase the pitch of a phoneme, prosody calculator  606  reduces the time period between successive HMM states at its output. This causes more waveforms to be generated during a period of time, thereby increasing the pitch of the speech signal. 
     Based on the HMM states provided by prosody calculator  606 , component locator  608  locates compressed values for the magnitude spectrums of the voiced and unvoiced portions of the speech signal. These compressed values are stored in a component storage area  610 , which was created during a training speech session that determined the average magnitude spectrums for each HMM state. In one embodiment, these compressed values represent the magnitude of overlapping triangles as discussed above in connection with the re-creation of a speech signal. 
     The compressed magnitude spectrum values for the voiced portion of the speech signal are combined by an overlap-and-add circuit  612 . This produces an estimate of the magnitude spectrum values for the voiced portion of the speech signal. These estimated magnitude values are then combined with a set of constant phase spectrum values  614  by a summing circuit  616 . As discussed above, the same phase value can be used across all frequencies of the voiced portion without significantly impacting the output speech signal. The combination of the magnitude and phase spectrums provides an estimate of the voiced portion of the speech signal. 
     The compressed magnitude spectrum values for the unvoiced component are provided to an overlap-and-add circuit  618 , which combines the triangles represented by the spectrum values to produce an estimate of the unvoiced portion&#39;s magnitude spectrum. This estimate is provided to a summing circuit  620 , which combines the estimated magnitude spectrum with a random phase spectrum that is provided by a random noise generator  622 . As discussed above, random phase values can be used for the phase of the unvoiced portion without impacting the quality of the output speech signal. The combination of the phase and magnitude spectrums provides an estimate of the unvoiced portion of the speech signal. 
     The estimates of the voiced and unvoiced portions of the speech signal are combined by a summing circuit  624  to provide a Fourier Transform estimate of an excitation signal for the speech signal. The Fourier Transform estimate is passed through an inverse Fast Fourier Transform  638  to produce a series of windows representing portions of the excitation signal. The windows are then combined by an overlap-and-add circuit  640  to produce the estimate of the excitation signal. This excitation signal is then passed through a delay unit  626  to align it with a set of formants that are calculated by a formant path generator  628 . 
     In one embodiment, formant path generator  628  calculates a most likely formant track for the first three formants in the speech signal. To do this, one embodiment of formant path generator  628  relies on the HMM states provided by prosody calculator  606  and a formant HMM  630 . The algorithm for generating the most likely formant tracks for a synthesized speech signal is similar to the technique described above for detecting the most likely formant tracks in an input speech signal. 
     Specifically, the formant path generator determines a most likely sequence of formant vectors given the Hidden Markov Model and the sequence of states from prosody calculator  606 . Each sequence of possible formant vectors is defined as: 
     
       
           X={x   1   ,x   2   ,x   3   , . . . x   T }  EQ. 36 
       
     
     where T is the total number of states in the utterance being constructed, and x i  is the formant vector for the ith state. In Equation 36, each formant vector is defined as: 
     
       
           x   i   ={F   1   ,F   2   i   ,F   3   i   ,B   1   i   ,B   2   i   ,B   3   i }  EQ. 37 
       
     
     where F 1   i , F 2   i , and F 3   i , are the first, second and third formant&#39;s frequencies and B 1   i , B 2   i , and B 3   i , are the first, second and third formant&#39;s bandwidths for the ith state of the speech signal. 
     Ignoring the sequence of states provided by prosody calculator  606  for the moment, the probability for each sequence of formant vectors, X, given a HMM, λ, is defined as:                p        (     X      λ     )       =       ∑   q            p        (       X      q     ,   λ     )            p        (     q      λ     )                   EQ   .              38                         
     where p(q|λ) is the probability of a sequence of states q given the HMM λ, p(X|q,λ) is the probability of the sequence of formant vectors given the HMM λ and the sequence of states q, and the summation is taken over all possible state sequences: 
     
       
           q={q   1   ,q   2   ,q   3   , . . . q   T }  EQ. 39 
       
     
     Although detecting the most likely sequence of states using Equation 38 would in theory provide the most accurate speech signal, in most embodiments, the sequence of states are limited to the sequence, {circumflex over (q)}, created by prosody calculator  606 . In addition, many embodiments simplify the calculations associated with Equation 38 by replacing the summation with the largest term in the summation. This leads to: 
     
       
           {circumflex over (X)}=arg   x   max[ln p ( X|{circumflex over (q)} ,λ)]  EQ. 40 
       
     
     As in the formant tracking discussion above, at each state, i, of the synthesized speech signal, the HMM vector of Equation 2 can be divided into two mean vectors Θ i , and Δ i , and two covariance matrices Σ i  and Γ i  defined as:                Θ   i     =     {             μ     i   ,   F1       ,     μ     i   ,   F2       ,     μ     i   ,   F3       ,   …              ,     μ     i   ,     FM   /   2         ,                 μ     i   ,   B1       ,     μ     i   ,   B2       ,     μ     i   ,   B3       ,   …              ,     μ     i   ,     BM   /   2                 }             EQ   .              41                 Δ   i     =     {             δ     i   ,     Δ                 F1         ,     δ     i   ,     Δ                 F2         ,     δ     i   ,     Δ                 F3         ,   …              ,     δ     i   ,     Δ                   FM   /   2           ,                 δ     i   ,     Δ                 B1         ,     δ     i   ,     Δ                 B2         ,     δ     i   ,     Δ                 B3         ,   …              ,     δ     i   ,     Δ                   BM   /   2                   }             EQ   .              42                 Σ   i     =     (           σ     i   ,   F1     2         0       0       0       0       0       0       0           0         σ     i   ,   F2     2         0       0       0       0       0       0           0       0       ⋰       0       0       0       0       0           0       0       0         σ     i   ,     FM   /   2       2         0       0       0       0           0       0       0       0         σ     i   ,   B1     2         0       0       0           0       0       0       0       0         σ     i   ,   B2     2         0       0           0       0       0       0       0       0       ⋰       0           0       0       0       0       0       0       0         σ     i   ,     BM   /   2       2           )             EQ   .              43                 Γ   i     =     (           γ     i   ,     Δ                 F1       2         0       0       0       0       0       0       0           0         γ     i   ,     Δ                 F2       2         0       0       0       0       0       0           0       0       ⋰       0       0       0       0       0           0       0       0         γ     i   ,     Δ                   FM   /   2         2         0       0       0       0           0       0       0       0         γ     i   ,     Δ                 B1       2         0       0       0           0       0       0       0       0         γ     i   ,     Δ                 B2       2         0       0           0       0       0       0       0       0       ⋰       0           0       0       0       0       0       0       0         γ     i   ,     Δ                   BM   /   2         2           )             EQ   .              44                         
     where M/2 is the number of formants in each group, with M=6 in most embodiments. Although the covariance matrices are shown as diagonal matrices, more complicated covariance matrices are contemplated within the scope of the present invention. Using these vectors and matrices, the model λ provided by formant HMM  630  for a language with n possible states becomes: 
     
       
         λ={Θ 1 ,Δ 1 ,Σ 1 ,Γ 1 ,Θ 2 ,Δ 2 ,Σ 2 ,Γ 2 , . . . Θ n , Δ n ,Σ n ,Γ n }  EQ. 45 
       
     
     Combining Equations 41 through 45 with Equation 40, the probability of each individual sequence of formant vectors is calculated as:                ln                   p        (       X        q   ^       ,   λ     )         =     (               -     TM   2            ln        (     2      π     )         -       1   2            ∑     t   =   1     T          ln             Σ     q   t                  -       1   2            ∑     t   =   2     T          ln             Γ     q   t                                -     1   2              ∑     t   =   1     T              (       x   t     -     Θ     q   t         )     ′            Σ     q   t       -   1            (       x   t     -     Θ     q   t         )                         -     1   2              ∑     t   =   2     T              (       x   t     -     x     t   -   1       -     Δ     q   t         )     ′            Γ     q   t       -   1            (       x   t     -     x     t   -   1       -     Δ     q   t         )                   )             EQ   .              46                         
     where T is the total number of states or output windows in the utterance being synthesized, M/2 is the number of formants in each formant vector x, x t  is the formant vector in the current output window t, x t−1  is the formant vector in the preceding output window t−1, (y)′ denotes the transpose of matrix y, Σ q     i     −1  indicates the inverse of the matrix Σ q     i   , and the subscript q t  indicates the HMM element of state q, which has been assigned to output window t. Note that in many embodiments, the formant tracks are selected on a sentence basis so the number of states T is the number of states in the current sentence being constructed. 
     To find the sequence of formant vectors that maximizes equation 46, the partial derivative technique described above for Equation 17 is applied to Equation 46. This results in linear equations that can be represented by the matrix equation BX=C as discussed further above. Examples of the values in these matrices for a synthesized utterance of three states are:              B   =     (             1     σ   q1   2       +     1     γ   q2   2               -     1     γ   q2   2             0             -     1     γ   q2   2                 1     σ   q2   2       +     1     γ   q2   2       +     1     γ   q3   2               -     1     γ   q3   2                 0         -     1     γ   q3   2                 1     σ   q3   2       +     1     γ   q3   2               )             EQ   .              47               c   =     (               μ   q1       σ   q1   2       -       δ   q2       γ   q2   2                   μ   q2       σ   q2   2       +       δ   q2       γ   q2   2       -       δ   q3       γ   q3   2                   μ   q3       σ   q3   2       +       δ   q3       γ   q3   2               )             EQ   .              48               X   =     (             F1   1                            F1   2               F1   3           )             EQ   .              49                         
     Note that B is once again a tridiagonal matrix where all of the values are zero except those in the main diagonal and its two adjacent diagonals. This remains true regardless of the number of states in the output speech signal. 
     To solve for the sequence of values for a constituent (F 1 , F 2 , F 3 , B 1 , B 2 , or B 3 ), the inverse of B is multiplied by c. This produces the sequence of values that has a maximum probability. 
     This process is then repeated for each constituent to produce a single most likely sequence of values for each formant constituent in the utterance being produced. 
     Once the most likely sequence of values for each formant constituent has been determined by formant path generator  628  of FIG. 9, the path generator adjusts three resonators  632 ,  634  and  636  so that they respectively resonate at the first, second and third formant frequencies for that state. Formant path generator  628  also adjust resonators  632 ,  634 , and  636  so that they resonate with a bandwidth equal to the respective bandwidth of the first, second and third formants of the current state. 
     Once the resonators have been adjusted, the excitation signal is serially passed through each of the resonators. The output of third resonator  636  thereby provides the synthesized speech signal. 
     Although the present invention has been described with reference to particular embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing from the spirit and scope of the invention.