Microcomputer Systems 2

Microcomputer Systems 2
Digital Systems: Hardware Organization and Design 11/27/2018 Microcomputer Systems 2 Analysis and Synthesis of Pole-Zero Speech Models Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design
11/27/2018 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 predication 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). 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

All-Pole Modeling of Deterministic Signals
Digital Systems: Hardware Organization and Design 11/27/2018 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) 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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 is a more efficient approach (covered later in this chapter). 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. For ug[n]=0 This observation motivates the analysis technique of linear prediction. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Model Estimation: Definitions
Digital Systems: Hardware Organization and Design 11/27/2018 Model Estimation: Definitions A linear predictor of order p is defined by: Estimate of s[n] Estimate of ak z 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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) 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 Model Estimation: Definitions Note 1: Recovery of s[n]: Aug[n] s[n] 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Covariance Method of Linear Prediction

11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 Error Minimization Determine {k} for which En is minimal: Which results in: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 Error Minimization The minimum error for the optimal solution can be derived as follows: Last term in the equation above can be rewritten as: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 Error Minimization Thus error can be expressed as: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Autocorrelation Method of Linear Prediction

Autocorrelation Method
Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 Autocorrelation Method Windowed sequence can be expressed as: This operation can be depicted in the figure presented on the right. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 Autocorrelation Method Normal equations take the following form (Exercise 5.1): where 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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 negihbourhood. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 Autocorrelation Method 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 Example 5.3 Apply N-point rectangular window [0,N-1] at n=0. Compute r0[0] and r0[1]. Using normal equations: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Limitations of the linear prediction model
Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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] 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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). 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

The Levinson Recursion of the Autocorrelation method
Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Properties of the Levinson Recursion of the Autocorrelation method
Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Properties of the Levinson Recursion to Autocorrelation method
Digital Systems: Hardware Organization and Design 11/27/2018 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 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Criterion of “Goodness”
Digital Systems: Hardware Organization and Design 11/27/2018 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) 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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). 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Prediction Error Residuals
Digital Systems: Hardware Organization and Design 11/27/2018 Prediction Error Residuals Autocorrelation method of linear prediction of order 14 Estimation performed over 20 ms Hamming windowed speech segments. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Prediction Error Residuals
Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Synthesis Based on All-pole Modeling Properties:
Digital Systems: Hardware Organization and Design 11/27/2018 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] 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Synthesis Based on All-pole Modeling
Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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. 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 11/27/2018 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: 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

11/27/2018 Synthesis structures 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Alternate Synthesis Structures
Digital Systems: Hardware Organization and Design 11/27/2018 Alternate Synthesis Structures 27 November 2018 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Microcomputer Systems 2

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