Digital Systems: Hardware Organization and Design

Digital Systems: Hardware Organization and Design
4/22/2017 Speech Processing Analysis and Synthesis of Pole-Zero Speech Models Architecture of a Respresentative 32 Bit Processor

4/22/2017 Introduction Deterministic: Speech Sounds with periodic or impulse sources Stochastic: Speech Sounds with noise sources Goal is to derive vocal tract model of each class of sound source. It will be shown that solution equations for the two classes are similar in structure. Solution approach is referred to as linear prediction analysis. Linear prediction analysis leads to a method of speech synthesis based on the all-pole model. Note that all-pole model is intimately associated with the concatenated lossless tube model of previous chapter (i.e., Chapter 4). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

All-Pole Modeling of Deterministic Signals
Digital Systems: Hardware Organization and Design 4/22/2017 All-Pole Modeling of Deterministic Signals Consider a vocal tract transfer function during voiced source: Ug[n] A … Glottal Model Vocal Track Model Radiation Model s[n]  Speech T=pitch G(z) V(z) R(z) 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 All-Pole Modeling of Deterministic Signals What about the fact that R(z) is a zero model? A single zero function can be expressed as a infinite set of poles. Note: From the above expression one can derive: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 All-Pole Modeling of Deterministic Signals In practice infinite number of poles are approximated with a finite site of poles since ak0 as k∞. H(z) can be considered all-pole representation: representing a zero with large number of poles ⇒ inefficient Estimating zeros directly a more efficient approach (covered later in this chapter). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Model Estimation Goal - Estimate : filter coefficients {a1, a2, …,ap}; for a particular order p, and A, Over a short time span of speech signal (typically 20 ms) for which the signal is considered quasi-stationary. Use linear prediction method: Each speech sample is approximated as a linear combination of past speech samples ⇒ Set of analysis techniques for estimating parameters of the all-pole model. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Model Estimation Consider z-transform of the vocal tract model: Which can be transformed into: In time domain it can be written as: Referred to us as a autoregressive (AR) model. Current Sample Input Past Samples Scaling Factor – Linear Prediction Coefficients 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Model Estimation Method used to predict current sample from linear combination of past samples is called linear prediction analysis. LPC – Quantization of linear prediction coefficients or of a transformed version of these coefficients is called linear prediction coding (Chapter 12). For ug[n]=0 This observation motivates the analysis technique of linear prediction. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Model Estimation: Definitions
Digital Systems: Hardware Organization and Design 4/22/2017 Model Estimation: Definitions A linear predictor of order p is defined by: Estimate of s[n] Estimate of ak z 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Model Estimation: Definitions Prediction error sequence is given as difference of the original sequence and its prediction: Associated prediction error filter is defined as: If {k}={ak} s[n] P[z] e[n]=Aug[n] ˜ A(z) 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Model Estimation: Definitions Note 1: Recovery of s[n]: Aug[n] s[n] 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Model Estimation: Definitions Note 2: If Vocal tract contains finite number of poles and no zeros, Prediction order is correct, then {k}={ak}, and e[n] is an impulse train for voiced speech and for impulse speech e[n] will be just an impulse. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.1 Consider an exponentially decaying impulse response of the form h[n]=anu[n] where u[n] is the unit step. Response to the scaled unit sample A[n] is: Consider the prediction of s[n] using a linear predictor of order p=1. It is a good fit since: Prediction error sequence with 1=a is: The prediction of the signal is exact except at the time origin. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Important question is: how to derive an estimate of the prediction coefficients al, for a particular order p, that would be optimal in some sense. Optimality is measured based on a criteria. An appropriate measure of optimality is mean-squared error (MSE). Goal is to minimize the mean-squared prediction error: E defined as: In reality, a model must be valid over some short-time interval, say M samples on either side of n: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Thus in practice MSE is time-depended and is formed over a finite interval as depicted in previous figure. [n-M,n+M] – prediction error interval. Alternatively: where 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Determine {k} for which En is minimal: Which results in: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Last equation can be rewritten by multiplying through: Define the function: Which gives the following: Referred to as the normal equations given in the matrix form bellow: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization The minimum error for the optimal solution can be derived as follows: Last term in the equation above can be rewritten as: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Thus error can be expressed as: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Remarks: Order (p) of the actual underlying all-pole transfer function is not known. Order can be estimated by observing the fact that a pth order predictor in theory equals that of a (p+1) order predictor. Also predictor coefficients for k>p equal zero (or in practice close to zero and model only noise-random effects). Prediction error en[m] is non-zero only “in the vicinity” of the time n: [n-M,n+M]. In predicating values of the short-time sequence sn[m], p –values outside of the prediction error interval [n-M,n+M] are required. Covariance method – uses values outside the interval to predict values inside the interval Autocorrelation Method – assumes that speech samples are zero outside the interval. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Error Minimization Matrix formulation Projection Theorem: Columns of Sn – basis vectors Error vector en is orthogonal to each basis vector: SnTen=0; where Orthogonality leads to: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Autocorrelation Method
Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method In previous section we have described a general method of linear prediction that uses samples outside the prediction error interval referred to as covariance method. Alternative approach that does not consider samples outside analysis interval, referred to as autocorrelation method, will be presented next. This method is: Suboptimal, however it Leads to an efficient and stable solution to normal equations. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Assumes that the samples outside the time interval [n-M,n+M] are all zero, and Extends the prediction error interval, i.e., the range over which we minimize the mean-squared error to ±∞. Conventions: Short-time interval: [n, n+Nw-1] where Nw=2M+1 (Note: it is not centered around sample n as in previous derivation). Segment is shifted to the left by n samples so that the first nonzero sample falls at m=0. This operation is equivalent to: Shifting of speech sequence s[m] by n-samples to the left and Windowing by Nw -point rectangular window: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Windowed sequence can be expressed as: This operation can be depicted in the figure presented on the right. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Important observations that are consequence of zeroing the signal outside of interval: Prediction error is nonzero only in the interval [0,Nw+p-1] Nw-window length p-the predictor order The prediction error is largest at the left and right ends of the segment. This is due to edge effects caused by the way the prediction is done: from zeros – from the left of the window to zeros – from the right of the window 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method To compensate for edge effects typically tapered window is used (e.g., Hamming). Removes the possibility that the mean-squared error be dominated by end (edge) effects. Data becomes distorted hence biasing estimates: k. Let the mean-squared prediction error be given by: Limits of summation refer to new time origin, and Prediction error outside this interval is zero. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Normal equations take the following form (Exercise 5.1): where 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Due to summation limits depicted in the figure on the right function n[i,k] can be written as: Recognizing that only samples in the interval [i,k+Nw-1] contribute to the sum, and Changing variable m⇒ m-i: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Since the above expression is only function of difference i-k thus we denote it as: Letting =i-k, referred to as correlation “lag”, leads to short-time autocorrelation function: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method rn[]=sn[]*sn[-] Autocorrelation method leads to computation of the short-time sequence sn[m] convolved with itself flipped in time. Autocorrelation function is a measure of the “self-similarity” of the signal at different lags . When rn[] is large then signal samples spaced by  are said to by highly correlated. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Properties of rn[]: For an N-point sequence, rn[] is zero outside the interval [-(N-1),N-1]. rn[] is even function of  rn[0] ≥ rn[] rn[0] – energy of sn[m] ⇒ If sn[m] is a segment of a periodic sequence, then rn[] is periodic-like with the same period: Because sn[m] is short-time, the overlapping data in the correlation decreases as  increases ⇒ Amplitude of rn[] decreases as  increases; With rectangular window the envelope of rn[] decreases linearly. If sn[m] is a random white noise sequence, then rn[] is impulse-like, reflecting self-similarity only within a small neighborhood. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Letting n[i,k] = rn[i-k], normal equation take the form: The expression represents p linear equations with p unknowns, k for 1≤k≤p. Using the normal equation solution, it can be shown that the corresponding minimum mean-squared prediction error is given by: Matrix form representation of normal equations: Rn=rn. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Expanded form: The Rn matrix is Toepliz: Symmetric about the diagonal All elements of the diagonal are equal. Matrix is invertible Implies efficient solution. Rn  rn 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.3 Consider a system with an exponentially decaying impulse response of the form h[n] = anu[n], with u[n] being the unit step function. Estimate a using the autocorrelation method of linear prediction. h[n] A[n] s[n] Z 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.3 Apply N-point rectangular window [0,N-1] at n=0. Compute r0[0] and r0[1]. Using normal equations: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.3 Minimum squared error (from slide 33) is thus (Exercise 5.5): For 1st order predictor, as in this example here, prediction error sequence for the true predictor (i.e., 1 = a) is given by: e[n]=s[n]-as[n-1]=[n] (see example 5.1 presented earlier). Thus the prediction of the signal is exact except at the time origin. This example illustrates that with enough data the autocorrelation method yields a solution close to the true single-pole model for an impulse input. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Limitations of the linear prediction model
Digital Systems: Hardware Organization and Design 4/22/2017 Limitations of the linear prediction model When the underlying measured sequence is the impulse response of an arbitrary all-pole sequence, then autocorrelation methods yields correct result. There are a number of speech sounds that even with an arbitrary long data sequence a true solution can not be obtained. Consider a periodic sequence simulating a steady voiced sound formed by convolving a periodic impulse train p[n] with an all-pole impulse response h[n]. Z-transform of h[n] is given by: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Limitations of the linear prediction model Thus Normal equations of this system are given by (see Exercise 5.7) Where autocorrelation of h[n] is denoted by rh[]=h[]*h[-]. Suppose now that the system is excited with an impulse train of the period P: P … h[n] 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Limitations of the linear prediction model Normal equations associated with s[n] (windowed over multiple pitch periods) for an order p predictor are given by: It can be shown that rn[] is equal to periodically repeated replicas of rh[]: but with decreasing amplitude due to the windowing (Exercise 5.7). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Limitations of the linear prediction model The autocorrelation function rn[] of the windowed signal s[n] can be thought of as “aliased” version of rh[] due to overlap which introduces distortion: When aliasing is minor the two solutions are approximately equal. Accuracy of this approximation decreases as the pitch period decreases (e.g., high pitch) due to increase in overlap of autocorrelation replicas repeated every P samples. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Limitations of the linear prediction model Sources of error: Aliasing increases with high pitched speakers (smaller pitch period P). Signal is not truly periodic. Speech not always all-pole. Autocorrelation is a suboptimal solution. Covariance method capable of giving optimal solution, however, is not guaranteed to converge when underlying signal does not follow an all-pole model. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

The Levinson Recursion of the Autocorrelation method
Digital Systems: Hardware Organization and Design 4/22/2017 The Levinson Recursion of the Autocorrelation method Direct inversion method (Gaussian elimination): requires p3 multiplies and additions. Levinson Recursion (1947): Requires p2 multiplies and additions Links directly to the concatenated lossless tube model (Chapter 4) and thus a mechanism for estimating the vocal tract area function from an all-pole-model estimation. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 The Levinson Recursion of the Autocorrelation method Step 1: for i=1,2,…,p Step 2: Step 3: Step 4: end ki-partial correlation coefficients - PARCOR 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 The Levinson Recursion of the Autocorrelation method It can be shown that on each iteration that the predictor coefficients k, can be written as solely functions of the autocorrelation coefficients (Exercise 5.11). Desired transfer function is given by: Gain A has yet to be determined. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Properties of the Levinson Recursion of the Autocorrelation method
Digital Systems: Hardware Organization and Design 4/22/2017 Properties of the Levinson Recursion of the Autocorrelation method Magnitude of partial correlation coefficients is less than 1: |ki|<1 for all i. Condition under 1 is sufficient for stability; if all |ki|<1 then all roots of A(z) are inside the unit circle. Autocorrelation Method gives a minimum-phase solution even when the actual system is mixed-phase. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.4 Consider the discrete-time model of the complete transfer function from the glottis to the lips derived in Chapter 4 (Equation 4.40), but without zero contributions from the radiation and vocal tract: Suppose we measure a single impulse response denoted by h[n] wich is equal to the inverse z-transform of H(z) and estimate the model with autocorrelation method setting the number of poles of Ĥ(z) correctly; p=2+2Ci, and with prediction error defined over the entire duration of h[n] which yields a solution 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Experimentation Results
Digital Systems: Hardware Organization and Design 4/22/2017 Experimentation Results 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Properties of the Levinson Recursion of the Autocorrelation method Formal explanation: Suppose s[n] follows an all-pole model Prediction error function is defined over all time (i.e., no window truncation effects: and are the Fourier transform phase functions for the minimum- and maximum-phase contributions of S(), respectively. Autocorrelation solution can be expressed as (Exercise 5.14): 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Properties of the Levinson Recursion of the Autocorrelation method Exercise 5.14 Rationalization of the Result: is the minimum-phase contribution due to the vocal tract poles inside the unit circle, and is maximum-phase contribution due to glottal poles outside the unit circle. Resulting estimated frequency response can be expressed as: The phase distortion of synthesized speech can have perceptual consequence since a gradual onset of the glottal flow, and thus of the speech waveform during the open phase of the glottal cycle, is transformed to a “sharp attack” consistent with the energy concentration property of minimum-phase sequences (Chapter 2). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Properties of the Levinson Recursion to Autocorrelation method
Digital Systems: Hardware Organization and Design 4/22/2017 Properties of the Levinson Recursion to Autocorrelation method Reverse Levinson Recursion: How to obtain lower level model from higher ones? Autocorrelation matching: Let rn[] be the autocorrelation of the speech signal s[n+m]w[m] and rh[] the autocorrelation of h[n]=-1{H(z)} then: rn[] = rh[] for ||≤p 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Gain Computation: En – is the average minimum prediction error for the pth-order predictor. If the energy in the all-pole impulse response h[m] equals the energy in the measurement sn[m] ⇒ Squared gain equal to the minimum prediction error. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Autocorrelation Method Relationship to Lossless Tube Model: Recall that for the lossless concatenated tube model, with glottal impedance Zg(z)= ∞ (open circuit), with the transfer function: Recursively obtained from: N-number of tubes and where reflection coefficients rk is a function of cross-sectional areas of successive tubes, i.e., 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Relationship to Lossless Tube Model:
Digital Systems: Hardware Organization and Design 4/22/2017 Relationship to Lossless Tube Model: Levinson Recursion: Can be written in the ℤ domain (see Appendix 5.B) Starting condition is obtained by mapping a00=0 to Two recursions are identical when ri=-ki which then makes Di(z)=Ai(z). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Relationship to Lossless Tube Model:
Digital Systems: Hardware Organization and Design 4/22/2017 Relationship to Lossless Tube Model: Since the boundary condition was not included in the lossless tube model, V(z) represents the ratio between an ideal volume velocity at the glottis and at the lips: Speech pressure measurement at the lips output, however, has embedded within it the glottal shape G(z), as well as radiation at the lips R(z). Recall that for the voiced case (with no vocal tract zeros): The presence of glottal shape, i.e., G(z), thus introduces poles that are not part of vocal tract. The net effect of glottal shape is typically 6dB/octave fall-off (see slide 94 of the presentation Acoustic of Speech Production) to the spectral tilt of V(z), The influence of the glottal flow shape and radiation load can be approximately removed with a pre-emphasis of 6dB/octave spectral rise. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Example 5.5 In the following figure two examples that show good matches to measured vocal tract area functions for the vowels /a/ and /i/ derived from estimates of the partial correlation coefficients. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Frequency Domain Interpretation
Digital Systems: Hardware Organization and Design 4/22/2017 Frequency Domain Interpretation Consider an all-pole model of speech production: Where A() is given by: Define Q() as the difference of the log-magnitude of measured and modeled spectra: Recall: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Frequency Domain Interpretation
Digital Systems: Hardware Organization and Design 4/22/2017 Frequency Domain Interpretation Thus we can write Q() as: Thus as e[n] is minimized ⇒ E() is minimized, which in turn ⇒ Q() minimized ⇒ spectral difference between actual measured speech and modeled spectrum is minimized. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Linear Prediction Analysis of Stochastic Speech Sounds
Digital Systems: Hardware Organization and Design 4/22/2017 Linear Prediction Analysis of Stochastic Speech Sounds Linear Prediction analysis was motivated with observation that for a single impulse or periodic impulse train input to an all-pole vocal tract model, the prediction error is zero “most of the time”. Such analysis appears not to be applicable to speech sounds with fricative or aspirated sources modeled as a stochastic (or random) process. However, autocorrelation method of linear prediction can be formulated for the stochastic case where a white noise input takes on the role of the single impulse. The solution to a stochastic optimization problem - analogous to the minimization of mean-squared error function En, leads to normal equations which are the stochastic counterparts to our earlier solution. Derivation and interpretation of this stochastic optimization problem is left as an exercise. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Criterion of “Goodness”
Digital Systems: Hardware Organization and Design 4/22/2017 Criterion of “Goodness” How well does linear predication describe the speech signal in time and in frequency? Time Domain Suppose: Underlying speech model is all-pole model of order p, and Autocorrelation method is used in the estimation of the coefficients of the predictor polynomial P(z). If predictor coefficients are estimated exactly then the prediction error: Is perfect impulse train for voiced speech A single impulse for a plosive A white noise for noisy (stochastic) speech. Speech measurement Prediction error s[n] e[n] A(z)=1-P(z) 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Time Domain Autocorrelation method of linear prediction analysis does not yield such idealized outputs when the measurement s[n] is inverse filtered by the estimated system function A(z) (method limitation): Even when the vocal tract response follow an all-pole model, true solution can not be obtained, since the obtained solution approached to the true solution in the limit when infinite amount of data is available. In a typical waveform segment, the actual vocal tract impulse response is not all-pole for variety of reasons: Presence of zeros due to: The radiation load, Nasalization, Back vocal cavity during frication and plosives. Glottal flow shape – even when adequately modeled, is not minimum phase (see example 5.6). 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Prediction Error Residuals
Digital Systems: Hardware Organization and Design 4/22/2017 Prediction Error Residuals Autocorrelation method of linear prediction of order 14 Estimation performed over 20 ms Hamming windowed speech segments. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Prediction Error Residuals
Digital Systems: Hardware Organization and Design 4/22/2017 Prediction Error Residuals Reconstructing residuals form an entire utterance typically one hears in the prediction error: Not a noisy buzz – as expected from idealized residual, but rather Roughly the speech itself ⇒ Some of the vocal tract spectrum is passing through the inverse filter. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Frequency Domain Behavior of linear prediction analysis can be studied alternatively in frequency domain: How well the spectrum derived form linear prediction analysis matches the spectrum of a sequence that follows: An all-pole model, and Not an all-pole model. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Frequency Domain-Voiced Speech
Digital Systems: Hardware Organization and Design 4/22/2017 Frequency Domain-Voiced Speech Recall for voiced speech s[n]: with Fourier transform Ug(). Vocal tract impulse response with all-pole frequency response H(). Windowed speech sn[n] is: Fourier transform of windowed speech sn[n] is: Where: W() - is the window transform o=2/P - is the fundamental frequency 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Frequency Domain-Unvoiced Speech
Digital Systems: Hardware Organization and Design 4/22/2017 Frequency Domain-Unvoiced Speech Recall for unvoiced speech (stochastic sounds): Linear prediction analysis attempts to estimate |H()| - spectral envelope of the harmonic spectrum S(). Spectral envelope Periodogram of noise 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Schematics of Spectra for Periodic and Stochastic Speech Sounds
Digital Systems: Hardware Organization and Design 4/22/2017 Schematics of Spectra for Periodic and Stochastic Speech Sounds 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Properties: For large p |H()| matches the Fourier transform magnitude of the windowed signal |S()|. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Properties: Spectral peeks are better matched than spectral valleys 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Properties: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Synthesis Based on All-pole Modeling Properties:
Digital Systems: Hardware Organization and Design 4/22/2017 Synthesis Based on All-pole Modeling Properties: Now able to synthesize the waveform from model parameters estimated using linear prediction analysis: Synthesized signal: so[n] e[n] A(z)=1-P(z) Au[n] s[n] 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Synthesis Based on All-pole Modeling
Digital Systems: Hardware Organization and Design 4/22/2017 Synthesis Based on All-pole Modeling Important Parameters to Consider: Window Duration – 20-30 [ms] to give a satisfactory time-frequency tradeoff (Exercise 5.20). Duration can be adaptively varied to account for different time-frequency resolution requirement based on: Pitch Voicing state Phoneme class. Frame Interval – Typical rate at which to perform analysis is 10 [ms]. Model Order – There are three components to be considered: Vocal tract: On average “resonant density” of one resonance per 1000 Hz. Order of the system: #poles=2 x #resonances (e.g., for 5000 Hz bandwidth signal 2x5=10 poles) Glottal flow: 2-pole maximum-phase model Radiation at lips: 1 zero inside the unit circle ⇒ 4 poles provide adequate representation. Total of 16 poles Remarks: Magnitude of speech frequency is preserved – frequency phase response is not preserved. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Synthesis Based on All-pole Modeling Voiced/Unvoiced State and Pitch Estimation: Currently no discrimination is done between for example plosive and fricative unvoiced speech sound categories. Pitch is estimated during voiced regions of speech only. However, Pitch estimation algorithms typically estimate pitch as well as perform voiced/unvoiced classification. A degree of voicing may be desired in more complex analysis and synthesis methods: Voicing and turbulence occurs simultaneously Voiced fricatives Breathy vowels. 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 4/22/2017 Synthesis Based on All-pole Modeling Synthesis Structures: Determine excitation for each frame Generate excitation for each frame by: Concatenating an impulse train during voiced signal (spacing determined by the time-varying pitch contour) White noise during unvoiced signal. Compute Gain Directly by measuring frame energy Using Autocorrelation method Voiced Speech: Magnitude of impulse is square root of signal energy. Unvoiced Speech: Noise variance = signal variance. Update filter values on each frame. Overlap and add signal at consecutive frames: 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

4/22/2017 Synthesis structures 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Alternate Synthesis Structures
Digital Systems: Hardware Organization and Design 4/22/2017 Alternate Synthesis Structures 22 April 2017 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design

Similar presentations

Presentation on theme: "Digital Systems: Hardware Organization and Design"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Digital Systems: Hardware Organization and Design

Similar presentations

Presentation on theme: "Digital Systems: Hardware Organization and Design"— Presentation transcript:

Similar presentations

About project

Feedback