Signal processing system

[0001] The present invention relates to a signal processing method and apparatus. The invention is particularly relevant to a statistical analysis of signals output by a plurality of sensors in response to signals generated by a plurality of sources. The invention may be used in speech applications and in other applications to process the received signals in order to separate the signals generated by the plurality of sources. The invention can also be used to identify the number of sources that are present.

[0002] There exists a need to be able to process signals output by a plurality of sensors in response to signals generated by a plurality of sources. The sources may, for example, be different users speaking and the sensors may be microphones. Current techniques employ arrays of microphones and an adaptive beam forming technique in order to isolate the speech from one of the speakers. This kind of beam forming system suffers from a number of problems. Firstly, it can only isolate signals from sources that are spatially distinct. It also does not work if the sources are relatively close together since the “beam” which it uses has a finite resolution. It is also necessary to know the directions from which the signals of interest will arrive and also the spacing between the sensors in the sensor array. Further, if N sensors are available, then only N−1 “nulls” can be created within the sensing zone.

[0003] An aim of the present invention is to provide an alternative technique for processing the signals output from a plurality of sensors in response to signals received from a plurality of sources.

[0004] According to one aspect, the present invention provides a signal processing apparatus comprising: one or more receivers for receiving a set of signal values representative of signals generated by a plurality of signal sources; a memory for storing a probability density function for parameters of a respective signal model, each of which is assumed to have generated a respective one of the signals represented by the received signal values; means for applying the received signal values to the probability density function; means for processing the probability density function with those values applied to derive samples of parameter values from the probability density function; and means for analysing some of the derived samples to determine parameter values that are representative of the signals generated by at least one of the sources.

[0028] Embodiments of the present invention can be implemented on computer hardware, but the embodiment to be described is implemented in software which is run in conjunction with processing hardware such as a personal computer, workstation, photocopier, facsimile machine or the like.

[0029] FIG. 1 is a personal computer (PC) 1 which may be programmed to operate an embodiment of the present invention. A keyboard 3, a pointing device 5, two microphones 7-1 and 7-2 and a telephone-line 9 are connected to the PC 1 via an interface 11. A keyboard 3 and pointing device 5 allow the system to be controlled by a user. The microphones 7 convert the acoustic speech signal of one or more users into equivalent electrical signals and supplies them to the PC 1 for processing. An internal modem and speech receiving circuit (not shown) may be connected to the telephone line 9 so that the PC 1 can communicate with, for example, a remote computer or with a remote user.

[0030] The program instructions which make the PC 1 operate in accordance with the present invention may be supplied for use with an existing PC 1 on, for example, a storage device such as a magnetic disc 13, or by downloading the software from the Internet (not shown) via the internal modem and telephone line 9.

[0031] The operation of a speech recognition system which receives signals output from multiple microphones in response to speech signals generated from a plurality of speakers will be described. However, in order to facilitate the understanding of the operation of such a recognition system, a speech recognition system which performs a similar analysis of the signals output from the microphone for the case of a single speaker and single microphone will be described first with reference to FIG. 2 to 9.

Single Speaker Single Microphone

[0032] As shown in FIG. 2, electrical signals representative of the input speech from the microphone 7 are input to a filter 15 which removes unwanted frequencies (in this embodiment frequencies above 8 kHz) within the input signal. The filtered signal is then sampled (at a rate of 16 kHz) and digitised by the analogue to digital converter 17 and the digitised speech samples are then stored in a buffer 19. Sequential blocks (or frames) of speech samples are then passed from the buffer 19 to a statistical analysis unit 21 which performs a statistical analysis of each frame of speech samples in sequence to determine, amongst other things, a set of auto regressive (AR) coefficients representative of the speech within the frame. In this embodiment, the AR coefficients output by the statistical analysis unit 21 are then input, via a coefficient converter 23 to a cepstral based speech recognition unit 25. In this embodiment, therefore, the coefficient converter 23 converts the AR coefficients output by the analysis unit 21 into cepstral coefficients. This can be achieved using the conversion technique described in, for example, “Fundamentals of Speech Recognition” by Rabiner and Juang at pages 115 and 116. The speech recognition unit 25 then compares the cepstral coefficients for successive frames of speech with a set of stored speech models 27, which may be template based or Hidden Markov Model based, to generate a recognition result.

Statistical Analysis Unit—Theory and Overview

[0033] As mentioned above, the statistical analysis unit 21 analyses the speech within successive frames of the input speech signal. In most speech processing systems, the frames are overlapping. However, in this embodiment, the frames of speech are non-overlapping and have a duration of 20 ms which, with the 16 kHz sampling rate of the analogue to digital converter 17, results in a frame size of 320 samples.

[0034] In order to perform the statistical analysis on each of the frames, the analysis unit 21 assumes that there is an underlying process which generated each sample within the frame. The model of this process used in this embodiment is shown in FIG. 3. As shown, the process is modelled by a speech source 31 which generates, at time t=n, a raw speech sample s(n). Since there are physical constraints on the movement of the speech articulators, there is some correlation between neighbouring speech samples. Therefore, in this embodiment, the speech source 31 is modelled by an auto regressive (AR) process. In other words, the statistical analysis unit 21 assumes that a current raw speech sample (s(n)) can be determined from a linear weighted combination of the most recent previous raw speech samples, i.e.:
s(n)=a₁s(n−1)+a₂s(n−2)+ . . . +a_ks(n−k)+e(n)  (1)
where a₁, a₂ . . . a_k are the AR filter coefficients representing the amount of correlation between the speech samples; k is the AR filter model order; and e(n) represents random process noise which is involved in the generation of the raw speech samples. As those skilled in the art of speech processing will appreciate, these AR filter coefficients are the same coefficients that the linear prediction (LP) analysis estimates albeit using a different processing technique.

[0035] As shown in FIG. 3, the raw speech samples s(n) generated by the speech source are input to a channel 33 which models the acoustic environment between the speech source 31 and the output of the analogue to digital converter 17. Ideally, the channel 33 should simply attenuate the speech as it travels from the source 31 to the microphone 7. However, due to reverberation and other distortive effects, the signal (y(n)) output by the analogue to digital converter 17 will depend not only on the current raw speech sample (s(n)) but it will also depend upon previous raw speech samples. Therefore, in this embodiment, the statistical analysis unit 21 models the channel 33 by a moving average (MA) filter, i.e.:
y(n)=h₀s(n)+h₁s(n−1)+h₂s(n−2)+ . . . +h_rs(n−r)+ε(n)  (2)
where y(n) represents the signal sample output by the analogue to digital converter 17 at time t=n; h₀, h₁, h₂ . . . h_r are the channel filter coefficients representing the amount of distortion within the channel 33; r is the channel filter model order; and ε(n) represents a random additive measurement noise component.

[0036] For the current frame of speech being processed, the filter coefficients for both the speech source and the channel are assumed to be constant but unknown. Therefore, considering all N samples (where N=320) in the current frame being processed gives: $\begin{array}{cc}s\left(n\right)=a_{1}s\left(n-1\right)+a_{2}s\left(n-2\right)+\dots +a_{k}s\left(n-k\right)+e\left(n\right)s\left(n-1\right)=a_{1}s\left(n-2\right)+a_{2}s\left(n-3\right)+\dots +a_{k}s\left(n-k-1\right)+e\left(n-1\right)⋮s\left(n-N+1\right)=a_{1}s\left(n-N\right)+a_{2}s\left(n-N-1\right)+\dots +a_{k}s\left(n-k-N+1\right)+e\left(n-N+1\right)& \left(3\right)\end{array}$
which can be written in vector form as:
s(n)=S·a+e(n)  (4)
where $S={\left[\begin{array}{ccccc}s\left(n-1\right)& s\left(n-2\right)& s\left(n-3\right)& \dots & s\left(n-k\right)\\ s\left(n-2\right)& s\left(n-3\right)& s\left(n-4\right)& \dots & s\left(n-k-1\right)\\ s\left(n-3\right)& s\left(n-4\right)& s\left(n-5\right)& \dots & s\left(n-k-2\right)\\ ⋮& & & ⋰& \\ s\left(n-N\right)& s\left(n-N-1\right)& s\left(n-N-2\right)& \dots & s\left(n-k-N+1\right)\end{array}\right]}_{N\times k}$ $\text{and}$ $\underset{\_}{a}={\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\\ ⋮\\ a_k\end{array}\right]}_{k\times 1}\underset{\_}{s}\left(n\right)={\left[\begin{array}{c}s\left(n\right)\\ s\left(n-1\right)\\ s\left(n-2\right)\\ ⋮\\ s\left(n-N+1\right)\end{array}\right]}_{N\times 1}\underset{\_}{e}\left(n\right)={\left[\begin{array}{c}e\left(n\right)\\ e\left(n-1\right)\\ e\left(n-2\right)\\ ⋮\\ e\left(n-N+1\right)\end{array}\right]}_{N\times 1}$

[0037] As will be apparent from the following discussion, it is also convenient to rewrite equation (3) in terms of the random error component (often referred to as the residual) e(n). This gives: $\begin{array}{cc}e\left(n\right)=s\left(n\right)-a_{1}s\left(n-1\right)-a_{2}s\left(n-2\right)-\dots -a_{k}s\left(n-k\right)e\left(n-1\right)=s\left(n-1\right)-a_{1}s\left(n-2\right)-a_{2}s\left(n-3\right)-\dots -a_{k}s\left(n-k-1\right)⋮e\left(n-N+1\right)=s\left(n-N+1\right)-a_{1}s\left(n-N\right)-a_{2}s\left(n-N-1\right)-\dots -a_{k}s\left(n-k-N+1\right)& \left(5\right)\end{array}$
which can be written in vector notation as:
e(n)=Äs(n)  (6)
where $\ddot{A}={\left[\begin{array}{ccccccccccc}1& -a_1& -a_2& -a_3& \dots & -a_k& 0& 0& 0& \cdots & 0\\ 0& 1& -a_1& -a_2& \dots & -a_{k-1}& -a_k& 0& 0& \cdots & 0\\ 0& 0& 1& -a_1& \dots & -a_{k-2}& -a_{k-1}& -a_k& 0& \cdots & 0\\ ⋮& & & & ⋰& & & & & & \\ 0& & & & & & & & & & 1\end{array}\right]}_{N\times N}$

[0038] Similarly, considering the channel model defined by equation (2), with h₀=1 (since this provides a more stable solution), gives: $\begin{array}{cc}q\left(n\right)=h_{1}s\left(n-1\right)+h_{2}s\left(n-2\right)+\dots +h_{r}s\left(n-r\right)+\varepsilon \left(n\right)q\left(n-1\right)\begin{array}{c}\approx h_{1}s\left(n-2\right)+h_{2}s\left(n-3\right)+\dots +\\ h_{r}s\left(n-r-1\right)+\varepsilon \left(n-1\right)\end{array}⋮q\left(n-N+1\right)\begin{array}{c}=h_{1}s\left(n-N\right)+h_{2}s\left(n-N-1\right)+\dots +\\ h_{r}s\left(n-r-N+1\right)+\varepsilon \left(n-N+1\right)\end{array}& \left(7\right)\end{array}$
(where q(n)=y(n)−s(n)) which can be written in vector form as:
q(n)=Y·h+ε(n)  (8)
where $Y={\left[\begin{array}{ccccc}s\left(n-1\right)& s\left(n-2\right)& s\left(n-3\right)& \dots & s\left(n-r\right)\\ s\left(n-2\right)& s\left(n-3\right)& s\left(n-4\right)& \dots & s\left(n-r-1\right)\\ s\left(n-3\right)& s\left(n-4\right)& s\left(n-5\right)& \dots & s\left(n-r-2\right)\\ ⋮& & & ⋰& \\ s\left(n-N\right)& s\left(n-N-1\right)& s\left(n-N-2\right)& \dots & s\left(n-r-N+1\right)\end{array}\right]}_{N\times r}$ $\mathrm{and}$ $\underset{\_}{h}={\left[\begin{array}{c}{h}_1\\ h_2\\ h_3\\ ⋮\\ h_r\end{array}\right]}_{r\times 1}\underset{\_}{q}\left(n\right)={\left[\begin{array}{c}q\left(n\right)\\ q\left(n-1\right)\\ q\left(n-2\right)\\ ⋮\\ q\left(n-N+1\right)\end{array}\right]}_{N\times 1}\underset{\_}{\varepsilon }\left(n\right)={\left[\begin{array}{c}\varepsilon \left(n\right)\\ \varepsilon \left(n-1\right)\\ \varepsilon \left(n-2\right)\\ ⋮\\ \varepsilon \left(n-N+1\right)\end{array}\right]}_{N\times 1}$

[0039] In this embodiment, the analysis unit 21 aims to determine, amongst other things, values for the AR filter coefficients (a) which best represent the observed signal samples (y(n)) in the current frame. It does this by determining the AR filter coefficients (a) that maximise the joint probability density function of the speech model, channel model, raw speech samples and the noise statistics given the observed signal samples output from the analogue to digital converter 17, i.e. by determining: $\begin{array}{cc}\underset{\underset{\_}{a}}{\max }\left\{p\left(\underset{\_}{a},k,\underset{\_}{h},r,{\sigma }_e^2,{\sigma }_{\varepsilon }^2,\underset{\_}{s}\left(n\right)|\underset{\_}{y}\left(n\right)\right)\right\}& \left(9\right)\end{array}$
where σ_e² and σ_ε² represent the process and measurement noise statistics respectively. As those skilled in the art will appreciate, this function defines the probability that a particular speech model, channel model, raw speech samples and noise statistics generated the observed frame of speech samples (y(n)) from the analogue to digital converter. To do this, the statistical analysis unit 21 must determine what this function looks like. This problem can be simplified by rearranging this probability density function using Bayes law to give: $\begin{array}{cc}\frac{\begin{array}{c}p\left(\underset{\_}{y}\left(n\right)❘\underset{\_}{s}\left(n\right),\underset{\_}{h},r,{\sigma }_{\varepsilon }^2\right)p\left(\underset{\_}{s}\left(n\right)❘\underset{\_}{a},k,{\sigma }_e^2\right)\\ p\left(\underset{\_}{a}❘k\right)p\left(\underset{\_}{h}❘r\right)p\left({\sigma }_{\varepsilon }^2\right)p\left({\sigma }_e^2\right)p\left(k\right)p\left(r\right)\end{array}}{p\left(\underset{\_}{y}\left(n\right)\right)}& \left(10\right)\end{array}$

[0040] As those skilled in the art will appreciate, the denominator of equation (10) can be ignored since the probability of the signals from the analogue to digital converter is constant for all choices of model. Therefore, the AR filter coefficients that maximise the function defined by equation (9) will also maximise the numerator of equation (10).

[0041] Each of the terms on the numerator of equation (10) will now be considered in turn.

p(s(n)|a, k, σ_e²)

[0042] This term represents the joint probability density function for generating the vector of raw speech samples (s(n)) during a frame, given the AR filter coefficients (a), the AR filter model order (k) and the process noise statistics (σ_e²). From equation (6) above, this joint probability density function for the raw speech samples can be determined from the joint probability density function for the process noise. In particular p(s(n)|a, k, σ_e²) is given by: $\begin{array}{cc}p\left(\underset{\_}{s}\left(n\right)|\underset{\_}{a},k,{\sigma }_e^2\right)=p\left(\underset{\_}{e}\left(n\right)\right){\frac{\delta \underset{\_}{e}\left(n\right)}{\delta \underset{\_}{s}\left(n\right)}}_{\underset{\_}{e}\left(n\right)=\underset{\_}{s}\left(n\right)-S\underset{\_}{a}}& \left(11\right)\end{array}$
where p(e(n)) is the joint probability density function for the process noise during a frame of the input speech and the second term on the right-hand side is known as the Jacobean of the transformation. In this case, the Jacobean is unity because of the triangular form of the matrix Ä (see equations (6) above).

[0043] In this embodiment, the statistical analysis unit 21 assumes that the process noise associated with the speech source 31 is Gaussian having zero mean and some unknown variance σ_e². The statistical analysis unit 21 also assumes that the process noise at one time point is independent of the process noise at another time point. Therefore, the joint probability density function for the process noise during a frame of the input speech (which defines the probability of any given vector of process noise e(n) occurring) is given by: $\begin{array}{cc}p\left(\underset{\_}{e}\left(n\right)\right)={\left(2{\mathrm{\pi \sigma }}_e^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-{\underset{\_}{e}\left(n\right)}^T\underset{\_}{e}\left(n\right)}{2{\sigma }_e^2}\right]& \left(12\right)\end{array}$

[0044] Therefore, the joint probability density function for a vector of raw speech samples given the AR filter coefficients (a), the AR filter model order (k) and the process noise variance (σ_e²) is given by: $\begin{array}{cc}\begin{array}{c}p\left(\underset{\_}{s}\left(n\right)|\underset{\_}{a},k,{\sigma }_e^2\right)={\left(2{\mathrm{\pi \sigma }}_e^2\right)}^{-\frac{N}{2}}\\ \exp \left[\frac{-1}{2{\sigma }_e^2}\left({\underset{\_}{s}\left(n\right)}^T\underset{\_}{s}\left(n\right)-2{\underset{\_}{a}}^{T}S\underset{\_}{s}\left(n\right)+{\underset{\_}{a}}^T{S}^{T}S\underset{\_}{a}\right)\right]\end{array}& \left(13\right)\end{array}$
p(y(n)|s(n), h, r, σ_ε²)

[0045] This term represents the joint probability density function for generating the vector of speech samples (y(n)) output from the analogue to digital converter 17, given the vector of raw speech samples (s(n)), the channel filter coefficients (h), the channel filter model order (r) and the measurement noise statistics (σ_ε²)

[0046] From equation (8), this joint probability density function can be determined from the joint probability density function for the process noise. In particular, p(y(n)|s(n), h, r, σ_ε²) is given by: $\begin{array}{cc}p\left(\underset{\_}{y}\left(n\right)|\underset{\_}{s}\left(n\right),\underset{\_}{h},r,{\sigma }_{\varepsilon }^2\right)=p\left(\underset{\_}{\varepsilon }\left(n\right)\right)\frac{\delta \underset{\_}{\varepsilon }\left(n\right)}{\delta \underset{\_}{y}\left(n\right)}\begin{array}{c}\\ \underset{\_}{\varepsilon }\left(n\right)=\underset{\_}{q}\left(n\right)-Y\underset{\_}{h}\end{array}& \left(14\right)\end{array}$
where p(ε(n)) is the joint probability density function for the measurement noise during a frame of the input speech and the second term on the right hand side is the Jacobean of the transformation which again has a value of one.

[0047] In this embodiment, the statistical analysis unit 21 assumes that the measurement noise is Gaussian having zero mean and some unknown variance σ_ε². It also assumes that the measurement noise at one time point is independent of the measurement noise at another time point. Therefore, the joint probability density function for the measurement noise in a frame of the input speech will have the same form as the process noise defined in equation (12). Therefore, the joint probability density function for a vector of speech samples (y(n)) output from the analogue to digital converter 17, given the channel filter coefficients (h), the channel filter model order (r), the measurement noise statistics (σ_ε²) and the raw speech samples (s(n)) will have the following form: $\begin{array}{cc}p\left(\underset{\_}{y}\left(n\right)|\underset{\_}{s}\left(n\right),\underset{\_}{h},r,{\sigma }_{\varepsilon }^2\right)={\left(2{\mathrm{\pi \sigma }}_{\varepsilon }^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-1}{2{\sigma }_{\varepsilon }^2}\left({\underset{\_}{q}\left(n\right)}^T\underset{\_}{q}\left(n\right)-2{\underset{\_}{h}}^{T}Y\underset{\_}{q}\left(n\right)+{\underset{\_}{h}}^T{Y}^{T}Y\underset{\_}{h}\right)\right]& \left(15\right)\end{array}$

[0048] As those skilled in the art will appreciate, although this joint probability density function for the vector of speech samples (y(n)) is in terms of the variable q(n), this does not matter since q(n) is a function of y(n) and s(n), and s(n) is a given variable (ie known) for this probability density function.

p(a|k)

[0049] This term defines the prior probability density function for the AR filter coefficients (a) and it allows the statistical analysis unit 21 to introduce knowledge about what values it expects these coefficients will take. In this embodiment, the statistical analysis unit 21 models this prior probability density function by a Gaussian having an unknown variance (σ_a²) and mean vector (μ_a) i.e.: $\begin{array}{cc}p\left(\underset{\_}{a}|k,{\sigma }_a^2,{\underset{\_}{\mu }}_a\right)={\left(2{\mathrm{\pi \sigma }}_a^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-{\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}^T\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}{2{\sigma }_a^2}\right]& \left(16\right)\end{array}$

[0050] By introducing the new variables σ_a² and μ_a, the prior density functions (p(σ_a²) and P(μ_a) ) for these variables must be added to the numerator of equation (10) above. Initially, for the first frame of speech being processed the mean vector (μ_a) can be set to zero and for the second and subsequent frames of speech being processed, it can be set to the mean vector obtained during the processing of the previous frame. In this case, p(μ_a) is just a Dirac delta function located at the current value of μ_a and can therefore be ignored.

[0051] With regard to the prior probability density function for the variance of the AR filter coefficients, the statistical analysis unit 21 could set this equal to some constant to imply that all variances are equally probable. However, this term can be used to introduce knowledge about what the variance of the AR filter coefficients is expected to be. In this embodiment, since variances are always positive, the statistical analysis unit 21 models this variance prior probability density function by an Inverse Gamma function having parameters α_a and β_a, i.e.: $\begin{array}{cc}p\left({\sigma }_a^2|{\alpha }_a,{\beta }_a\right)=\frac{{\left({\sigma }_a^2\right)}^{-\left({\alpha }_a+1\right)}}{{\beta }_a\Gamma \left({\alpha }_a\right)}\exp \left[\frac{-1}{{\sigma }_a^2{\beta }_a}\right]& \left(17\right)\end{array}$

[0052] At the beginning of the speech being processed, the statistical analysis unit 21 will not have much knowledge about the variance of the AR filter coefficients. Therefore, initially, the statistical analysis unit 21 sets the variance σ_a² and the α and β parameters of the Inverse Gamma function to ensure that this probability density function is fairly flat and therefore non-informative. However, after the first frame of speech has been processed, these parameters can be set more accurately during the processing of the next frame of speech by using the parameter values calculated during the processing of the previous frame of speech.

p(h|r)

[0053] This term represents the prior probability density function for the channel model coefficients (h) and it allows the statistical analysis unit 21 to introduce knowledge about what values it expects these coefficients to take. As with the prior probability density function for the AR filter coefficients, in this embodiment, this probability density function is modelled by a Gaussian having an unknown variance (σ_h²) and mean vector (μ_h), i.e.: $\begin{array}{cc}p\left(\underset{\_}{h}|r,{\sigma }_h^2,{\underset{\_}{\mu }}_h\right)={\left(2{\mathrm{\pi \sigma }}_h^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-{\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}^T\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}{2{\sigma }_h^2}\right]& \left(18\right)\end{array}$

[0054] Again, by introducing these new variables, the prior density functions (p(σ_h) and p(μ_h)) must be added to the numerator of equation (10). Again, the mean vector can initially be set to zero and after the first frame of speech has been processed and for all subsequent frames of speech being processed, the mean vector can be set to equal the mean vector obtained during the processing of the previous frame. Therefore, p(μ_h) is also just a Dirac delta function located at the current value of μ_h and can be ignored.

[0055] With regard to the prior probability density function for the variance of the channel filter coefficients, again, in this embodiment, this is modelled by an Inverse Gamma function having parameters α_h and β_h. Again, the variance (σ_h²) and the α and β parameters of the Inverse Gamma function can be chosen initially so that these densities are non-informative so that they will have little effect on the subsequent processing of the initial frame.

p(σ_e²) and p(σ_ε²)

[0056] These terms are the prior probability density functions for the process and measurement noise variances and again, these allow the statistical analysis unit 21 to introduce knowledge about what values it expects these noise variances will take. As with the other variances, in this embodiment, the statistical analysis unit 21 models these by an Inverse Gamma function having parameters α_e, β_e and α_ε, β_ε respectively. Again, these variances and these Gamma function parameters can be set initially so that they are non-informative and will not appreciably affect the subsequent calculations for the initial frame.

p(k) and p(r)

[0057] These terms are the prior probability density functions for the AR filter model order (k) and the channel model order (r) respectively. In this embodiment, these are modelled by a uniform distribution up to some maximum order. In this way, there is no prior bias on the number of coefficients in the models except that they can not exceed these predefined maximums. In this embodiment, the maximum AR filter model order (k) is thirty and the maximum channel model order (r) is one hundred and fifty.

[0058] Therefore, inserting the relevant equations into the numerator of equation (10) gives the following joint probability density function which is proportional to p(a,k,h,r,σ_a²,σ_h²,σ_e²,σ_ε²,s(n)|y(n)): $\begin{array}{cc}{\left(2{\mathrm{\pi \sigma }}_{\varepsilon }^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-1}{2{\sigma }_{\varepsilon }^2}\left({\underset{\_}{q}\left(n\right)}^T\underset{\_}{q}\left(n\right)-2{\underset{\_}{h}}^{T}Y\underset{\_}{q}\left(n\right)+{\underset{\_}{h}}^T{Y}^{T}Y\underset{\_}{h}\right)\right]\times \begin{array}{c}{\left(2{\mathrm{\pi \sigma }}_e^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-1}{2{\sigma }_e^2}\left({\underset{\_}{s}\left(n\right)}^T\underset{\_}{s}\left(n\right)-2{\underset{\_}{a}}^{T}S\underset{\_}{s}\left(n\right)+{\underset{\_}{a}}^T{S}^{T}S\underset{\_}{a}\right)\right]\times \end{array}\begin{array}{c}{\left(2{\mathrm{\pi \sigma }}_a^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-{\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}^T\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}{2{\sigma }_a^2}\right]\times \end{array}{\left(2{\mathrm{\pi \sigma }}_h^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-{\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}^T\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}{2{\sigma }_h^2}\right]\times \begin{array}{c}\frac{{\left({\sigma }_a^2\right)}^{-\left({\alpha }_a+1\right)}}{{\beta }_a\Gamma \left({\alpha }_a\right)}\exp \left[\frac{-1}{{\sigma }_a^2{\beta }_a}\right]\times \frac{{\left({\sigma }_h^2\right)}^{-\left({\alpha }_h+1\right)}}{{\beta }_h\Gamma \left({\alpha }_h\right)}\exp \left[\frac{-1}{{\sigma }_h^2{\beta }_h}\right]\times \end{array}\frac{{\left({\sigma }_e^2\right)}^{-\left({\alpha }_e+1\right)}}{{\beta }_e\Gamma \left({\alpha }_e\right)}\exp \left[\frac{-1}{{\sigma }_e^2{\beta }_e}\right]\times \frac{{\left({\sigma }_{\varepsilon }^2\right)}^{-\left({\alpha }_{\varepsilon }+1\right)}}{{\beta }_{\varepsilon }\Gamma \left({\alpha }_{\varepsilon }\right)}\exp \left[\frac{-1}{{\sigma }_{\varepsilon }^2{\beta }_{\varepsilon }}\right]& \left(19\right)\end{array}$
Gibbs Sampler

[0059] In order to determine the form of this joint probability density function, the statistical analysis unit 21 “draws samples” from it. In this embodiment, since the joint probability density function to be sampled is a complex multivariate function, a Gibbs sampler is used which breaks down the problem into one of drawing samples from probability density functions of smaller dimensionality. In particular, the Gibbs sampler proceeds by drawing random variates from conditional densities as follows: $\mathrm{first}\mathrm{iteration}$ $\begin{array}{c}p\left(\underset{\_}{a},k|h^0,r^0,{\sigma }_e^{2^0},{\sigma }_{\varepsilon }^{2^0},{\sigma }_a^{2^0},{\sigma }_h^{2^0},{\underset{\_}{s}\left(n\right)}^0,\underset{\_}{y}\left(n\right)\right)\to {\underset{\_}{a}}^1,k^1\\ p\left(\underset{\_}{h},r|{\underset{\_}{a}}^1,k^1,{\sigma }_e^{2^0},{\sigma }_{\varepsilon }^{2^0},{\sigma }_a^{2^0},{\sigma }_h^{2^0},{\underset{\_}{s}\left(n\right)}^0,\underset{\_}{y}\left(n\right)\right)\to {\underset{\_}{h}}^1,k^1\\ p\left({\sigma }_e^2|{\underset{\_}{a}}^1,k^1,{\underset{\_}{h}}^1,r^1,{\sigma }_{\varepsilon }^{2^0},{\sigma }_a^{2^0},{\sigma }_h^{2^0},{\underset{\_}{s}\left(n\right)}^0,\underset{\_}{y}\left(n\right)\right)\to {\sigma }_e^{2^1}\\ ⋮\\ p\left({\sigma }_h^{2^1}|{\underset{\_}{a}}^1,k^1,{\underset{\_}{h}}^1,r^1,{\sigma }_{\varepsilon }^{2^1},{\sigma }_a^{2^1},{\sigma }_h^{2^1},{\underset{\_}{s}\left(n\right)}^0,\underset{\_}{y}\left(n\right)\right)\to {\sigma }_h^{2^1}\end{array}$ $\mathrm{second}\mathrm{iteration}$ $\begin{array}{c}p\left(\underset{\_}{a},k|{\underset{\_}{h}}^1,r^1,{\sigma }_e^{2^1},{\sigma }_{\varepsilon }^{2^1},{\sigma }_h^{2^1},{\underset{\_}{s}\left(n\right)}^1,\underset{\_}{y}\left(n\right)\right)\to {\underset{\_}{a}}^2,k^2\\ p\left(\underset{\_}{h},r|{\underset{\_}{a}}^2,k^2,{\sigma }_e^{2^1},{\sigma }_{\varepsilon }^{2^1},{\sigma }_a^{2^1},{\sigma }_h^{2^1},{s\left(n\right)}^1,y\left(n\right)\right)\to {\underset{\_}{h}}^2,r^2\\ ⋮\end{array}$ $\mathrm{etc}.$
where (h⁰, r⁰, (σ_e²)⁰, (σ_ε²)⁰, (σ_a²)⁰, (σ_h²)⁰, s(n)⁰) are initial values which may be obtained from the results of the statistical analysis of the previous frame of speech, or where there are no previous frames, can be set to appropriate values that will be known to those skilled in the art of speech processing.

[0060] As those skilled in the art will appreciate, these conditional densities are obtained by inserting the current values for the given (or known) variables into the terms of the density function of equation (19). For the conditional density p(a,k| . . . ) this results in: $\begin{array}{cc}p\left(\underset{\_}{a},k|\dots \right)\propto \exp \left[\frac{-1}{2{\sigma }_e^2}\left({s\left(n\right)}^T\underset{\_}{s}\left(n\right)-2{\underset{\_}{a}}^{T}S\underset{\_}{s}\left(n\right)+{\underset{\_}{a}}^T{S}^{T}S\underset{\_}{a}\right)\right]\times \exp \left[\frac{-{\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}^T\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}{2{\sigma }_a^2}\right]& \left(20\right)\end{array}$
which can be simplified to give: $\begin{array}{cc}p\left(\underset{\_}{a},k|\dots \right)\propto \exp \left[\frac{-1}{2}\left(\begin{array}{c}\frac{{\underset{\_}{s}\left(n\right)}^T\underset{\_}{s}\left(n\right)}{{\sigma }_e^2}+\frac{{\underset{\_}{\mu }}_a^T{\underset{\_}{\mu }}_a}{{\sigma }_a^2}-2{\underset{\_}{a}}^T\left[\frac{S\underset{\_}{s}\left(n\right)}{{\sigma }_e^2}+\frac{{\underset{\_}{\mu }}_a}{{\sigma }_a^2}\right]+\\ {\underset{\_}{a}}^T\left[\frac{S^{T}S}{{\sigma }_e^2}+\frac{I}{{\sigma }_a^2}\right]\underset{\_}{a}\end{array}\right)\right]& \left(21\right)\end{array}$
which is in the form of a standard Gaussian distribution having the following covariance matrix: $\begin{array}{cc}\sum _{\underset{\_}{a}}={\left[\frac{S^{T}S}{{\sigma }_e^2}+\frac{I}{{\sigma }_a^2}\right]}^{-1}& \left(22\right)\end{array}$

[0061] The mean value of this Gaussian distribution can be determined by differentiating the exponent of equation (21) with respect to a and determining the value of a which makes the differential of the exponent equal to zero. This yields a mean value of: $\begin{array}{cc}{\widehat{\underset{\_}{\mu }}}_a={\left[\frac{S^{T}S}{{\sigma }_e^2}+\frac{I}{{\sigma }_a^2}\right]}^{-1}\left[\frac{S^T\underset{\_}{s}\left(n\right)}{{\sigma }_e^2}+\frac{{\underset{\_}{\mu }}_a}{{\sigma }_a^2}\right]& \left(23\right)\end{array}$

[0062] A sample can then be drawn from this standard Gaussian distribution to give a^g (where g is the g^th iteration of the Gibbs sampler) with the model order (k^g) being determined by a model order selection routine which will be described later. The drawing of a sample from this Gaussian distribution may be done by using a random number generator which generates a vector of random values which are uniformly distributed and then using a transformation of random variables using the covariance matrix and the mean value given in equations (22) and (23) to generate the sample. In this embodiment, however, a random number generator is used which generates random numbers from a Gaussian distribution having zero mean and a variance of one. This simplifies the transformation process to one of a simple scaling using the covariance matrix given in equation (22) and shifting using the mean value given in equation (23). Since the techniques for drawing samples from Gaussian distributions are well known in the art of statistical analysis, a further description of them will not be given here. A more detailed description and explanation can be found in the book entitled “Numerical Recipes in C”, by W. Press et al, Cambridge University Press, 1992 and in particular at chapter 7.

[0063] As those skilled in the art will appreciate, however, before a sample can be drawn from this Gaussian distribution, estimates of the raw speech samples must be available so that the matrix S and the vector s(n) are known. The way in which these estimates of the raw speech samples are obtained in this embodiment will be described later.

[0064] A similar analysis for the conditional density p(h,r| . . . ) reveals that it also is a standard Gaussian distribution but having a covariance matrix and mean value given by: $\begin{array}{cc}\sum _{\underset{\_}{h}}={\left[\frac{Y^{T}Y}{{\sigma }_{\varepsilon }^2}+\frac{I}{{\sigma }_h^2}\right]}^{-1}{\widehat{\underset{\_}{\mu }}}_h={\left[\frac{Y^{T}Y}{{\sigma }_{\varepsilon }^2}+\frac{I}{{\sigma }_h^2}\right]}^{-1}\left[\frac{Y^T\underset{\_}{q}\left(n\right)}{{\sigma }_{\varepsilon }^2}+\frac{{\underset{\_}{\mu }}_h}{{\sigma }_h^2}\right]& \left(24\right)\end{array}$
from which a sample for h^g can be drawn in the manner described above, with the channel model order (r^g) being determined using the model order selection routine which will be described later.

[0065] A similar analysis for the conditional density p(σ_e²| . . . ) shows that: $\begin{array}{cc}p\left({\sigma }_e^2|\dots \right)\propto {\left({\sigma }_e^2\right)}^{-\frac{N}{2}}\exp \left[\frac{-E}{2{\sigma }_e^2}\right]\frac{{\left({\sigma }_e^2\right)}^{-\left({\alpha }_e+1\right)}}{{\beta }_e\Gamma \left({\alpha }_e\right)}\exp \left[\frac{-1}{{\sigma }_e^2{\beta }_e}\right]& \left(25\right)\end{array}$
where:
E=s(n)^Ts(n)−2a^TSs(n)+a^TS^TSa
which can be simplified to give: $\begin{array}{cc}p\left({\sigma }_e^2|\dots \right)\propto {\left({\sigma }_e^2\right)}^{-\left[\left(\frac{N}{2}+{\alpha }_e\right)+1\right]}\exp \left[\frac{-1}{{\sigma }_e^2}\left(\frac{E}{2}+\frac{1}{{\beta }_e}\right)\right]& \left(26\right)\end{array}$
which is also an Inverse Gamma distribution having the following parameters: $\begin{array}{cc}{\hat{\alpha }}_e=\frac{N}{2}+{\alpha }_e\mathrm{and}{\hat{\beta }}_e=\frac{2{\beta }_e}{2+{\beta }_e·E}& \left(27\right)\end{array}$

[0066] A sample is then drawn from this Inverse Gamma distribution by firstly generating a random number from a uniform distribution and then performing a transformation of random variables using the alpha and beta parameters given in equation (27), to give (σ_e²)^g.

[0067] A similar analysis for the conditional density p(σ_ε²| . . . ) reveals that it also is an Inverse Gamma distribution having the following parameters: $\begin{array}{cc}{\hat{\alpha }}_{\varepsilon }=\frac{N}{2}+{\alpha }_{\varepsilon }\mathrm{and}{\hat{\beta }}_{\varepsilon }=\frac{2{\beta }_{\varepsilon }}{2+{\beta }_{\varepsilon }·E^{*}}& \left(28\right)\end{array}$
where:
E*=q(n)^Tq(n)−2h^Tq(n)+h^TY^TYh

[0068] A sample is then drawn from this Inverse Gamma distribution in the manner described above to give (σ_ε²)^g.

[0069] A similar analysis for conditional density p(σ_a²| . . . ) reveals that it too is an Inverse Gamma distribution having the following parameters: $\begin{array}{cc}{\hat{\alpha }}_a=\frac{N}{2}+{\alpha }_a\mathrm{and}{\hat{\beta }}_a=\frac{2{\beta }_a}{2+{\beta }_a·{\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}^T\left(\underset{\_}{a}-{\underset{\_}{\mu }}_a\right)}& \left(29\right)\end{array}$

[0070] A sample is then drawn from this Inverse Gamma distribution in the manner described above to give (σ_a²)^g.

[0071] Similarly, the conditional density p(σ_h²| . . . ) is also an Inverse Gamma distribution but having the following parameters: $\begin{array}{cc}{\hat{\alpha }}_h=\frac{N}{2}+{\alpha }_h\mathrm{and}{\hat{\beta }}_h=\frac{2{\beta }_h}{2+{\beta }_h·{\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}^T\left(\underset{\_}{h}-{\underset{\_}{\mu }}_h\right)}& \left(30\right)\end{array}$

[0072] A sample is then drawn from this Inverse Gamma distribution in the manner described above to give (σ_h²)^g.

[0073] As those skilled in the art will appreciate, the Gibbs sampler requires an initial transient period to converge to equilibrium (known as burn-in). Eventually, after L iterations, the sample (a^L, k^L, h^L, r^L, (σ_e²)^L, σ_ε²)^L, (σ_a²)^L, (σ_h²)^L, s(n)^L) is considered to be a sample from the joint probability density function defined in equation (19). In this embodiment, the Gibbs sampler performs approximately one hundred and fifty (150) iterations on each frame of input speech and discards the samples from the first fifty iterations and uses the rest to give a picture (a set of histograms) of what the joint probability density function defined in equation (19) looks like. From these histograms, the set of AR coefficients (a) which best represents the observed speech samples (y(n)) from the analogue to digital converter 17 are determined. The histograms are also used to determine appropriate values for the variances and channel model coefficients (h) which can be used as the initial values for the Gibbs sampler when it processes the next frame of speech.

Model Order Selection

[0074] As mentioned above, during the Gibbs iterations, the model order (k) of the AR filter and the model order (r) of the channel filter are updated using a model order selection routine. In this embodiment, this is performed using a technique derived from “Reversible jump Markov chain Monte Carlo computation”, which is described in the paper entitled “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination” by Peter Green, Biometrika, vol 82, pp 711 to 732, 1995.

[0075] FIG. 4 is a flow chart which illustrates the processing steps performed during this model order selection routine for the AR filter model order (k). As shown, in step s1, a new model order (k₂) is proposed. In this embodiment, the new model order will normally be proposed as k₂=k₁±1, but occasionally it will be proposed as k₂=k₁±2 and very occasionally as k₂=k₁±3 etc. To achieve this, a sample is drawn from a discretised Laplacian density function centred on the current model order (k₁) and with the variance of this Laplacian density function being chosen a priori in accordance with the degree of sampling of the model order space that is required.

[0076] The processing then proceeds to step s3 where a model order variable (MO) is set equal to: $\begin{array}{cc}\mathrm{MO}=\max \left\{\frac{p\left({\underset{\_}{a}}_{\langle 1:k_2\rangle },k_2|\dots \right)}{p\left({\underset{\_}{a}}_{\langle 1:k_1\rangle },k_1|\dots \right)},1\right\}& \left(31\right)\end{array}$
where the ratio term is the ratio of the conditional probability given in equation (21) evaluated for the current AR filter coefficients (a) drawn by the Gibbs sampler for the current model order ((k₁)) and for the proposed new model order (k₂). If k₂>k₁, then the matrix S must first be resized and then a new sample must be drawn from the Gaussian distribution having the mean vector and covariance matrix defined by equations (22) and (23) (determined for the resized matrix S), to provide the AR filter coefficients (a_<1:k2>) for the new model order (k₂). If k₂<k₁ then all that is required is to delete the last (k₁−k₂) samples from the a vector. If the ratio in equation (31) is greater than one, then this implies that the proposed model order (k₂) is better than the current model order whereas if it is less than one then this implies that the current model order is better than the proposed model order. However, since occasionally this will not be the case, rather than deciding whether or not to accept the proposed model order by comparing the model order variable (MO) with a fixed threshold of one, in this embodiment, the model order variable (MO) is compared, in step s5, with a random number which lies between zero and one. If the model order variable (MO) is greater than this random number, then the processing proceeds to step s7 where the model order is set to the proposed model order (k₂) and a count associated with the value of k₂ is incremented. If, on the other hand, the model order variable (MO) is smaller than the random number, then the processing proceeds to step s9 where the current model order is maintained and a count associated with the value of the current model order (k₁) is incremented. The processing then ends.

[0077] This model order selection routine is carried out for both the model order of the AR filter model and for the model order of the channel filter model. This routine may be carried out at each Gibbs iteration. However, this is not essential. Therefore, in this embodiment, this model order updating routine is only carried out every third Gibbs iteration.

Simulation Smoother

[0078] As mentioned above, in order to be able to draw samples using the Gibbs sampler, estimates of the raw speech samples are required to generate s(n), S and Y which are used in the Gibbs calculations. These could be obtained from the conditional probability density function p(s(n)| . . . ). However, this is not done in this embodiment because of the high dimensionality of s(n). Therefore, in this embodiment, a different technique is used to provide the necessary estimates of the raw speech samples. In particular, in this embodiment, a “Simulation Smoother” is used to provide these estimates. This Simulation Smoother was proposed by Piet de Jong in the paper entitled “The Simulation Smoother for Time Series Models”, Biometrika (1995), vol 82, 2, pages 339 to 350. As those skilled in the art will appreciate, the Simulation Smoother is run before the Gibbs Sampler. It is also run again during the Gibbs iterations in order to update the estimates of the raw speech samples. In this embodiment, the Simulation Smoother is run every fourth Gibbs iteration.

[0079] In order to run the Simulation Smoother, the model equations defined above in equations (4) and (6) must be written in “state space” format as follows:
ŝ(n)=÷ŝ(n−1)+ê(n)
y(n)=h^T·ŝ(n−1)+ε(n)  (32)
where $\tilde{A}={\left[\begin{array}{cccccccc}{a}_1& a_2& a_2& \dots & a_k& 0& \dots & 0\\ 1& 0& 0& \dots & 0& 0& \dots & 0\\ 0& 1& 0& \dots & 0& 0& \dots & 0\\ ⋮& & & ⋰& & & & \\ 0& & & & & & 1& 0\end{array}\right]}_{r\times r}$ $\mathrm{and}$ $\widehat{\underset{\_}{s}}\left(n\right)={\left[\begin{array}{c}\hat{s}\left(n\right)\\ \hat{s}\left(n-1\right)\\ \hat{s}\left(n-2\right)\\ ⋮\\ \hat{s}\left(n-r+1\right)\end{array}\right]}_{r\times 1}\widehat{\underset{\_}{e}}\left(n\right)={\left[\begin{array}{c}\hat{e}\left(n\right)\\ 0\\ 0\\ ⋮\\ 0\end{array}\right]}_{r\times 1}$

[0080] With this state space representation, the dimensionality of the raw speech vectors (ŝ(n)) and the process noise vectors (ê(n)) do not need to be N×1 but only have to be as large as the greater of the model orders—k and r. Typically, the channel model order (r) will be larger than the AR filter model order (k). Hence, the vector of raw speech samples (ŝ(n)) and the vector of process noise (ê(n)) only need to be r×1 and hence the dimensionality of the matrix à only needs to be r×r.

[0081] The Simulation Smoother involves two stages—a first stage in which a Kalman filter is run on the speech samples in the current frame and then a second stage in which a “smoothing” filter is run on the speech samples in the current frame using data obtained from the Kalman filter stage. FIG. 5 is a flow chart illustrating the processing steps performed by the Simulation Smoother. As shown, in step s21, the system initialises a time variable t to equal one. During the Kalman filter stage, this time variable is run from t=1 to N in order to process the N speech samples in the current frame being processed in time sequential order. After step s21, the processing then proceeds to step s23, where the following Kalman filter equations are computed for the current speech sample (y(t)) being processed:
w(t)=y(t)−h^Tŝ(t)
d(t)=h^TP(t)h+σ_ε²
k_f(t)=(ÃP(t)h)·d(t)⁻¹
ŝ(t+1)=Ãŝ(t)+k_f(t)·w(t)
L(t)=Ã−k_f(t)·h^T
P(t+1)=ÃP(t)L(t)^T+σ_e²·I  (33)
where the initial vector of raw speech samples (ŝ(1)) includes raw speech samples obtained from the processing of the previous frame (or if there are no previous frames then s(i) is set equal to zero for i<1); P(1) is the variance of ŝ(1) (which can be obtained from the previous frame or initially can be set to σ_e²); h is the current set of channel model coefficients which can be obtained from the processing of the previous frame (or if there are no previous frames then the elements of h can be set to their expected values˜zero); y(t) is the current speech sample of the current frame being processed and I is the identity matrix. The processing then proceeds to step s25 where the scalar values w(t) and d(t) are stored together with the r×r matrix L(t) (or alternatively the Kalman filter gain vector k_f(t) could be stored from which L(t) can be generated). The processing then proceeds to step s27 where the system determines whether or not all the speech samples in the current frame have been processed. If they have not, then the processing proceeds to step s29 where the time variable t is incremented by one so that the next sample in the current frame will be processed in the same way. Once all N samples in the current frame have been processed in this way and the corresponding values stored, the first stage of the Simulation Smoother is complete.

[0082] The processing then proceeds to step s31 where the second stage of the Simulation Smoother is started in which the smoothing filter processes the speech samples in the current frame in reverse sequential order. As shown, in step s31 the system runs the following set of smoothing filter equations on the current speech sample being processed together with the stored Kalman filter variables computed for the current speech sample being processed:
C(t)=σ_e²(I−σ_e²U(t))
η(t)˜N(0,C(t))
V(t)=σ_e²U(t)L(t)
r(t−1)=hd(t)⁻¹w(t)+L(t)^Tr(t)−V(t)^TC(t)⁻¹η(t)
U(t−1)=hd(t)⁻¹h^T+L(t)^TU(t)L(t)+V(t)^TC(t)⁻¹V(t)
{tilde over (e)}(t)=σ_e²r(t)+η(t) where {tilde over (e)}(t)=[{tilde over (e)}(t){tilde over (e)}(t−1){tilde over (e)}(t−2) . . . {tilde over (e)}(t−r+1)]^T
ŝ(t)=Ãŝ(^t−1)+ê(t) where ŝ(t)=[ŝ(t)ŝ(t−1)ŝ(t−2) . . . ŝ(t−r+1)]^T
and ê(t)=[{tilde over (e)}(t) 0 0 . . . 0]^T  (34)
where η(t) is a sample drawn from a Gaussian distribution having zero mean and covariance matrix C(t); the initial vector r(t=N) and the initial matrix U(t=N) are both set to zero; and s(0) is obtained from the processing of the previous frame (or if there are no previous frames can be set equal to zero). The processing then proceeds to step s33 where the estimate of the process noise ({tilde over (e)}(t)) for the current speech sample being processed and the estimate of the raw speech sample (ŝ(t)) for the current speech sample being processed are stored. The processing then proceeds to step s35 where the system determines whether or not all the speech samples in the current frame have been processed. If they have not, then the processing proceeds to step s37 where the time variable t is decremented by one so that the previous sample in the current frame will be processed in the same way. Once all N samples in the current frame have been processed in this way and the corresponding process noise and raw speech samples have been stored, the second stage of the Simulation Smoother is complete and an estimate of s(n) will have been generated.

[0083] As shown in equations (4) and (8), the matrix S and the matrix Y require raw speech samples s(n−N−1) to s(n−N−k+1) and s(n−N−1) to s(n−N−r+1) respectively in addition to those in s(n). These additional raw speech samples can be obtained either from the processing of the previous frame of speech or if there are no previous frames, they can be set to zero. With these estimates of raw speech samples, the Gibbs sampler can be run to draw samples from the above described probability density functions.

Statistical Analysis Unit—Operation

[0084] A description has been given above of the theory underlying the statistical analysis unit 21. A description will now be given with reference to FIGS. 6 to 8 of the operation of the statistical analysis unit 21.

[0085] FIG. 6 is a block diagram illustrating the principal components of the statistical analysis unit 21 of this embodiment. As shown, it comprises the above described Gibbs sampler 41, Simulation Smoother 43 (including the Kalman filter 43-1 and smoothing filter 43-2) and model order selector 45. It also comprises a memory 47 which receives the speech samples of the current frame to be processed, a data analysis unit 49 which processes the data generated by the Gibbs sampler 41 and the model order selector 45 and a controller 50 which controls the operation of the statistical analysis unit 21.

[0086] As shown in FIG. 6, the memory 47 includes a non volatile memory area 47-1 and a working memory area 47-2. The non volatile memory 47-1 is used to store the joint probability density function given in equation (19) above and the equations for the variances and mean values and the equations for the Inverse Gamma parameters given above in equations (22) to (24) and (27) to (30) for the above mentioned conditional probability density functions for use by the Gibbs sampler 41. The non volatile memory 47-1 also stores the Kalman filter equations given above in equation (33) and the smoothing filter equations given above in equation 34 for use by the Simulation Smoother 43.

[0087] FIG. 7 is a schematic diagram illustrating the parameter values that are stored in the working memory area (RAM) 47-2. As shown, the RAM includes a store 51 for storing the speech samples y_f (1) to y_f (N) output by the analogue to digital converter 17 for the current frame (f) being processed. As mentioned above, these speech samples are used in both the Gibbs sampler 41 and the Simulation Smoother 43. The RAM 47-2 also includes a store 53 for storing the initial estimates of the model parameters (g=0) and the M samples (g=1 to M) of each parameter drawn from the above described conditional probability density functions by the Gibbs sampler 41 for the current frame being processed. As mentioned above, in this embodiment, M is 100 since the Gibbs sampler 41 performs 150 iterations on each frame of input speech with the first fifty samples being discarded. The RAM 47-2 also includes a store 55 for storing W(t), d(t) and L(t) for t=1 to N which are calculated during the processing of the speech samples in the current frame of speech by the above described Kalman filter 43-1. The RAM 47-2 also includes a store 57 for storing the estimates of the raw speech samples (ŝ_f(t)) and the estimates of the process noise ({tilde over (e)}_f(t)) generated by the smoothing filter 43-2, as discussed above. The RAM 47-2 also includes a store 59 for storing the model order counts which are generated by the model order selector 45 when the model orders for the AR filter model and the channel model are updated.

[0088] FIG. 8 is a flow diagram illustrating the control program used by the controller 50, in this embodiment, to control the processing operations of the statistical analysis unit 21. As shown, in step s41, the controller 50 retrieves the next frame of speech samples to be processed from the buffer 19 and stores them in the memory store 51. The processing then proceeds to step s43 where initial estimates for the channel model, raw speech samples and the process noise and measurement noise statistics are set and stored in the store 53. These initial estimates are either set to be the values obtained during the processing of the previous frame of speech or, where there are no previous frames of speech, are set to their expected values (which may be zero). The processing then proceeds to step s45 where the Simulation Smoother 43 is activated so as to provide an estimate of the raw speech samples in the manner described above. The processing then proceeds to step s47 where one iteration of the Gibbs sampler 41 is run in order to update the channel model, speech model and the process and measurement noise statistics using the raw speech samples obtained in step s45. These updated parameter values are then stored in the memory store 53.

[0089] The processing then proceeds to step s49 where the controller 50 determines whether or not to update the model orders of the AR filter model and the channel model. As mentioned above, in this embodiment, these model orders are updated every third Gibbs iteration. If the model orders are to be updated, then the processing proceeds to step s51 where the model order selector 45 is used to update the model orders of the AR filter model and the channel model in the manner described above. If at step s49 the controller 50 determines that the model orders are not to be updated, then the processing skips step s51 and the processing proceeds to step s53. At step s53, the controller 50 determines whether or not to perform another Gibbs iteration. If another iteration is to be performed, then the processing proceeds to decision block s55 where the controller 50 decides whether or not to update the estimates of the raw speech samples (s(t)). If the raw speech samples are not to be updated, then the processing returns to step s47 where the next Gibbs iteration is run.

[0090] As mentioned above, in this embodiment, the Simulation Smoother 43 is run every fourth Gibbs iteration in order to update the raw speech samples. Therefore, if the controller 50 determines, in step s55 that there has been four Gibbs iterations since the last time the speech samples were updated, then the processing returns to step s45 where the Simulation Smoother is run again to provide new estimates of the raw speech samples (s(t)). Once the controller 50 has determined that the required 150 Gibbs iterations have been performed, the controller 50 causes the processing to proceed to step s57 where the data analysis unit 49 analyses the model order counts generated by the model order selector 45 to determine the model orders for the AR filter model and the channel model which best represents the current frame of speech being processed. The processing then proceeds to step s59 where the data analysis unit 49 analyses the samples drawn from the conditional densities by the Gibbs sampler 41 to determine the AR filter coefficients (a), the channel model coefficients (h), the variances of these coefficients and the process and measurement noise variances which best represent the current frame of speech being processed. The processing then proceeds to step s61 where the controller 50 determines whether or not there is any further speech to be processed. If there is more speech to be processed, then processing returns to step S41 and the above process is repeated for the next frame of speech. Once all the speech has been processed in this way, the processing ends.

Data Analysis Unit

[0091] A more detailed description of the data analysis unit 49 will now be given with reference to FIG. 9. As mentioned above, the data analysis unit 49 initially determines, in step s57, the model orders for both the AR filter model and the channel model which best represents the current frame of speech being processed. It does this using the counts that have been generated by the model order selector 45 when it was run in step s51. These counts are stored in the store 59 of the RAM 47-2. In this embodiment, in determining the best model orders, the data analysis unit 49 identifies the model order having the highest count. FIG. 9a is an exemplary histogram which illustrates the distribution of counts that is generated for the model order (k) of the AR filter model. Therefore, in this example, the data analysis unit 49 would set the best model order of the AR filter model as five. The data analysis unit 49 performs a similar analysis of the counts generated for the model order (r) of the channel model to determine the best model order for the channel model.

[0092] Once the data analysis unit 49 has determined the best model orders (k and r), it then analyses the samples generated by the Gibbs sampler 41 which are stored in the store 53 of the RAM 47-2, in order to determine parameter values that are most representative of those samples. It does this by determining a histogram for each of the parameters from which it determines the most representative parameter value. To generate the histogram, the data analysis unit 49 determines the maximum and minimum sample value which was drawn by the Gibbs sampler and then divides the range of parameter values between this minimum and maximum value into a predetermined number of sub-ranges or bins. The data analysis unit 49 then assigns each of the sample values into the appropriate bins and counts how many samples are allocated to each bin. It then uses these counts to calculate a weighted average of the samples (with the weighting used for each sample depending on the count for the corresponding bin), to determine the most representative parameter value (known as the minimum mean square estimate (MMSE)). FIG. 9b illustrates an example histogram which is generated for the variance (σ_e²) of the process noise, from which the data analysis unit 49 determines that the variance representative of the sample is 0.3149.

[0093] In determining the AR filter coefficients (a_i for i=i to k), the data analysis unit 49 determines and analyses a histogram of the samples for each coefficient independently. FIG. 9c shows an exemplary histogram obtained for the third AR filter coefficient (a₃), from which the data analysis unit 49 determines that the coefficient representative of the samples is −0.4977.

[0094] In this embodiment, the data analysis unit 49 only outputs the AR filter coefficients which are passed to the coefficient convertor 23 shown in FIG. 2. The remaining parameter values determined by the data analysis unit 49 are stored in the RAM 47-2 for use during the processing of the next frame of speech. As mentioned above, the AR filter coefficients output by the statistical analysis unit 21 are input to the coefficient convertor 23 which converts these coefficients into cepstral coefficients which are then compared with stored speech models 27 by the speech recognition unit 25 in order to generate a recognition result.

[0095] As the skilled reader will appreciate, a speech processing technique has been described above which uses statistical analysis techniques to determine sets of AR filter coefficients representative of an input speech signal. The technique is more robust and accurate than prior art techniques which employ maximum likelihood estimators to determine the AR filter coefficients. This is because the statistical analysis of each frame uses knowledge obtained from the processing of the previous frame. In addition, with the analysis performed above, the model order for the AR filter model is not assumed to be constant and can vary from frame to frame. In this way, the optimum number of AR filter coefficients can be used to represent the speech within each frame. As a result, the AR filter coefficients output by the statistical analysis unit 21 will more accurately represent the corresponding input speech. Further still, since the underlying process model that is used separates the speech source from the channel, the AR filter coefficients that are determined will be more representative of the actual speech and will be less likely to include distortive effects of the channel.

[0096] Further still, since variance information is available for each of the parameters, this provides an indication of the confidence of each of the parameter estimates. This is in contrast to maximum likelihood and least square approaches, such as linear prediction analysis, where point estimates of the parameter values are determined.

Multi Speaker Multi Microphone

[0097] A description will now be given of a multi speaker and multi microphone system which uses a similar statistical analysis to separate and model the speech from each speaker. Again, to facilitate understanding, a description will initially be given of a two speaker and two microphone system before generalising to a multi speaker and multi microphone system.

[0098] FIG. 10 is a schematic block diagram illustrating a speech recognition system which employs a statistical analysis unit embodying the present invention. As shown, the system has two microphones 7-1 and 7-2 which convert, in this embodiment, the speech from two speakers (not shown) into equivalent electrical signals which are passed to a respective filter circuit 15-1 and 15-2. In this embodiment, the filters 15 remove frequencies above 8 kHz since the filtered signals are then converted into corresponding digital signals at a sampling rate of 16 kHz by a respective analogue to digital converter 17-1 and 17-2. The digitized speech samples from the analogue to digital converters 17 are then fed into the buffer 19. The statistical analysis unit 21 analyses the speech within successive frames of the input speech signal from the two microphones. In this embodiment, since there are two microphones there are two sequences of frames which are to be processed. In this embodiment, the two frame sequences are processed together so that the frame of speech from microphone 7-1 at time t is processed with the frame of speech received from the microphone 7-2 at time t. Again, in this embodiment, the frames of speech are non-overlapping and have a duration of 20 ms which, with the 16 kHz sampling rate of the analogue to digital converters 17, results in the statistical analysis unit 21 processing blocks of 640 speech samples (corresponding to two frames of 320 samples).

[0099] In order to perform the statistical analysis on the input speech, the analysis unit 21 assumes that there is an underlying process similar to that of the single speaker single microphone system described above. The particular model used in this embodiment is illustrated in FIG. 11. As shown, the process is modelled by two speech sources 31-1 and 31-2 which generate, at time t=n, raw speech samples s¹(n) and s²(n) respectively. Again, in this embodiment, each of the speech sources 31 is modelled by an auto aggressive (AR) process. In other words, there will be a respective equation (1) for each of the sources 31-1 and 31-2, thereby defining two unknown AR filter coefficient vectors a¹ and a², each having a respective model order k¹ and k². These source models will also have a respective process noise component e¹(n) and e²(n).

[0100] As shown in FIG. 11, the model also assumes that the speech generated by each of the sources 31 is received by both microphones 7. There is therefore a respective channel 33-11 to 33-22 between each source 31 and each microphone 7. There is also a respective measurement noise component ε¹(n) and ε²(n) added to the signal received by each microphone. Again, in this embodiment, the statistical analysis unit 21 models each of the channels by a moving average (MA) filter. Therefore, the signal received from microphone 7-1 at time t=n is given by:
y¹(n)=h₁₁₀s¹(n)+h₁₁₁s¹(n−1)+h₁₁₂s¹(n−2)+ . . +h_11r1s¹(n−r₁₁)
+h₂₁₀s²(n)+h₂₁₁s²(n−1)+h₂₁₂s²(n−2)+ . . . +h_21r21s²(n−r₂₁)+ε¹(n)  (35)
where, for example, h₁₁₂ is the channel filter coefficient of the channel between the first source 31-1 and the microphone 7-1 at time t=2; and r₂₁ is the model order of the channel between the second speech source 31-2 and the microphone 7-1. A similar equation will exist to represent the signal received from the other microphone 7-2.

[0101] In this embodiment, the statistical analysis unit 21 aims to determine values for the AR filter coefficients for the two speech sources, which best represent the observed signal samples from the two microphones in the current frame being processed. It does this, by determining the AR filter coefficients for the two speakers (a¹ and a²) that maximise the joint probability density function of the speech models, channel models, raw speech samples and the noise statistics given the observed signal samples output from the two analogue to digital converters 17-1 and 17-2, i.e. by determining: $\begin{array}{cc}\underset{{\underset{\_}{a}}^1,{\underset{\_}{a}}^2}{\max }\left\{\begin{array}{c}p({\underset{\_}{a}}^1,{\underset{\_}{a}}^2,k^1,k^2,{\underset{\_}{h}}_{11},{\underset{\_}{h}}_{12},{\underset{\_}{h}}_{21},{\underset{\_}{h}}_{22},r_{11},r_{12},r_{21},r_{22},\\ {\sigma }_{e_1}^2,{\sigma }_{e_2}^2,{\sigma }_{{\varepsilon }_1}^2,{\sigma }_{{\varepsilon }_2}^2,{\underset{\_}{s}}^1\left(n\right),{\underset{\_}{s}}^2\left(n\right)❘{\underset{\_}{y}}^1\left(n\right),{\underset{\_}{y}}^2\left(n\right))\end{array}\right\}& \left(36\right)\end{array}$

[0102] As those skilled in the art will appreciate, this is almost an identical problem to the single speaker single microphone system described above, although with more parameters. Again, to calculate this, the above probability is rearranged using Bayes law to give an equation similar to that given in equation (10) above. The only difference is that there will be many more joint probability density functions on the numerator. In particular, the joint probability density functions which will need to be considered in this embodiment are:

• • [0103] p(y¹(n)|s¹(n),s²(n),h₁₁,h₂₁,r₁₁,r₂₁,σ_ε1²)
  • [0104] p(y²(n)|s¹(n),s²(n),h₁₂,h₂₂,r₁₂,r₂₂,σ_ε2²)
  • [0105] p(s¹(n)|a¹,k¹,σ_e1²), p(s²(n)|a²,k²,σ_e2²)
  • [0106] p(a¹|k¹,σ_a1²,μ_a1) p(a²|k²,σ_a2²,μ_a2)
  • [0107] p(h₁₁|r₁₁,σ_h11²,μ_h11) p(h₁₂|r₁₂,σ_h12²,μ_h12)
  • [0108] p(h₂₁|r₂₁,σ_h21²,μ_h21) p(h₂₂|r₂₂,σ_h22²,μ_h22)
  • [0109] P(σ_a1²|α_a1,β_a1) P(σ_a2²|α_a2,β_a2) p(σ_e1²) p(σ_e2²)
  • [0110] P(σ_h11²|α_h11,β_h11) P(σ_h12²|α_h12,β_h12) P(σ_h21²|α_h21,β_h21)
  • [0111] P(σ_h22²|α_h22,β_h22) p(k¹) p(k²) p(r₁₁) p(r₁₂) p(r₂₁) p(r₂₂)

[0112] Since the speech sources and the channels are independent of each other, most of these components will be the same as the probability density functions given above for the single speaker single microphone system. This is not the case, however, for the joint probability density functions for the vectors of speech samples (y¹(n) and y²(n)) out from the analogue to digital converters 17, since these signals include components from both the speech sources. The joint probability density function for the speech samples output from analogue to digital converter 17-1 will now be described in more detail.

p(y¹(n)|s¹(n),s²(n),h₁₁,h₂₁,r₁₁,r₂₁,σ_ε1²)

[0113] Considering all the speech samples output from the analogue to digital converter 17-1 in a current frame being processed (and with h₁₁₀ and h₂₁₀ being set equal to one), gives: $\begin{array}{cc}{\underset{\_}{\varepsilon }}^1\left(n\right)={\underset{\_}{q}}^1\left(n\right)-\left[\begin{array}{ccc}Y& ⋮& Y_2\end{array}\right]·\left[\begin{array}{c}{\underset{\_}{h}}_{11}\\ \cdots \\ {\underset{\_}{h}}_{21}\end{array}\right]& \left(37\right)\end{array}$
where ${\underset{\_}{h}}_{11}={\left[\begin{array}{c}{h}_{111}\\ h_{112}\\ h_{113}\\ ⋮\\ h_{11r_{11}}\end{array}\right]}_{r_{11}\times 1}{\underset{\_}{h}}_{21}={\left[\begin{array}{c}{h}_{211}\\ h_{212}\\ h_{213}\\ ⋮\\ h_{21r_{21}}\end{array}\right]}_{r_{21}\times 1}$ ${\underset{\_}{q}}^1\left(n\right)={\left[\begin{array}{c}{q}^1\left(n\right)\\ q^1\left(n-1\right)\\ q^1\left(n-2\right)\\ ⋮\\ q^1\left(n-N+1\right)\end{array}\right]}_{N\times 1}{\underset{\_}{\varepsilon }}^1\left(n\right)={\left[\begin{array}{c}{\varepsilon }^1\left(n\right)\\ {\varepsilon }^1\left(n-1\right)\\ {\varepsilon }^1\left(n-2\right)\\ ⋮\\ {\varepsilon }^1\left(n-N+1\right)\end{array}\right]}_{N\times 1}$ $\mathrm{and}$ $Y_1={\left[\begin{array}{cccc}{s}^1\left(n-1\right)& s^1\left(n-2\right)& \dots & s^1\left(n-r_{11}\right)\\ s^1\left(n-2\right)& s^1\left(n-3\right)& \dots & s^1\left(n-r_{11}-1\right)\\ s^1\left(n-3\right)& s^1\left(n-4\right)& \dots & s^1\left(n-r_{11}-2\right)\\ ⋮& & ⋰& \\ s^1\left(n-N\right)& s^1\left(n-N-1\right)& \dots & s^1\left(n-r_{11}-N+1\right)\end{array}\right]}_{N\times r_{11}}$ $Y_2={\left[\begin{array}{cccc}{s}^2\left(n-1\right)& s^2\left(n-2\right)& \dots & s^2\left(n-r_{21}\right)\\ s^2\left(n-2\right)& s^2\left(n-3\right)& \dots & s^2\left(n-r_{21}-1\right)\\ s^2\left(n-3\right)& s^2\left(n-4\right)& \dots & s^2\left(n-r_{21}-2\right)\\ ⋮& & ⋰& \\ s^2\left(n-N\right)& s^2\left(n-N-1\right)& \dots & s^2\left(n-r_{21}-N+1\right)\end{array}\right]}_{N\times r_{21}}$
and q¹(n)=y¹(n)−s¹(n)−s²(n).