Coefficient linear predictive

GHJCOSE 1D H GH 013001.FID. Note the baseline artifacts introduced by the truncated FID. In the Linear Prediction (LP) dialog box make sure that the Execute Backward LP option is enabled and the Execute Forward LP option disabled. Set LP backward to Point to 124. Following the rules given above vary the residual parameters First Point used for LP (recommended 196), Last Point used for for LP (recommended 2047) and Number of Coefficients (recommended 128 or larger). Carefully inspect the resulting spectra with respect to spectral resolution and signal shapes and compare it with the spectrum obtained without LP. 

In a free solution, the electrophoretic mobility (i.e., peiec, the particle velocity per unit applied electric field) is a function of the net charge, the hydrodynamic drag on a molecule, and the properties of the solutions (viscosity present ions—their concentration and mobility). It can be expressed as the ratio of its electric charge Z (Z = q-e, with e the charge if an electron and q the valance) to its electrophoretic friction coefficient. Different predictive models have been demonstrated involving the size, flexibility, and permeability of the molecules or particles. Henry s theoretical model of pdcc for colloids (Henry, 1931) can be combined with the Debye-Hiickel theory predicting a linear relation between mobility and the charge Z ... 

Denotes mth linear prediction coefficient Magnitude signal in the frequency domain nth moment of the lineshape... 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

Linear predictions are commonly performed on instruments of one manufacturer with 8 coefficients while another manufacturer recommends that anywhere from 16 to 32 coefficients be used for its systems. It is important that the correct number of coefficients be employed. Too few will fail to make accurate predictions, and the resulting spectra will, at best, look as if no LP has been performed and, at worst, have even poorer resolution. In contrast, too many coefficients will result in artifacts along the V] axis that resemble noise, and the calculations will take an inordinate amount of time or cause the system to shut down. 

Note that this is the standard FIR form as discussed in Chapter 3, but the output is an estimate of the next sample in time. The task of linear prediction is to select the vector of predictor coefficients ... 

One commonly used method of computing the linear prediction coefficients uses the autocorrelation function, defined as ... 

That is, the Nth order polynomial in can be determined from the N-1 order pol5momial in z z and the reflection coefficient r. If we therefore start at V = 0 and set o(z) = 1 we can calculate the value for i, and then iteratively for all A (z) up to N. This is considerably quicker than carrying out the matrix multiplications explicitly. The real value in this however, will be shown in Chapter 12, when we consider the relationship between the all-pole tube model and the technique of linear prediction. 

The goal of linear prediction is to find the set of coefficients ok which generate the smallest error e[n], over the course of the frame of speech. We will now examine two closely related techniques for finding the Uk coefficients based on error minimisation. We can sate equation 12.16 in terms of the error for a single sample ... 

These steps are repeated until i = p where we have a pol5momial and hence set of predictor coefficients of the required order. We have just seen how the minimisation of error over a window can be used to estimate the linear prediction coefficients. As these are in fact the filter coefficients that define the transfer function of the vocal tract, we can use these in a number of ways to generate other usefiil representations. 

As with any polynomial, these can be factorised to find the roots, but when we do so in this case, we find that because the system is lossless then all the roots lie on Ihe unit circle. If we solve for P(z) and Q z) we find that the roots are interleaved on Ihe unit circle. In essence, the LSF representation converts the linear prediction parameters fi om a set of P/2 pairs of complex conjugate poles which contain frequency and bandwidth information, into a set ofP/2 pairs of frequencies, one of which represents the closed termination and the other the open termination. Note that while the LSF tube is lossless, the loss information in the original LP model is still present. In the process of generating A z) back from P(z) and Q z), the extra 1 reflection coefficient is removed and hence the loss and the bandwidths can be fully recovered. 

A very popular representation in speech recognition is the mel-frequency cepstral coefficient or MFCC. This is one of the few popular represenations lhat does not use linear prediction. This is formed by first performing a DFT on a frame of speech, then performing a filter bank analysis (see Section 12.2) in which the frequency bin locations are defined to lie on the mel-scale. This is set up to give say 20-30 coefficients. These are then transformed to the cepstral domain by the discrete cosine transform (we use this rather than the DFT as we only require the real part to be calculated) ... 

Currently, mel-scale cepstral coeflicients, and perceptual linear prediction coefficients transformed into cepstral coefficients, are popular choices for the above reasons. Specifically they are ehosen because they are robust to noise, can be modelled with diagonal covariance, and with the aid of the perceptual scaling are more discriminative than would otherwise be. From a speech synthesis point of view, these points are worth making, not because the same requirements exist for synthesis, but rather to make the reader aware that the reason MFCCs and PLPs are so often used in ASR systems is for the above reasons, and not because they are intrinsically better in any general purpose sort of way. This also helps explain why there are so many speech representations in the first place each has strengths in certain areas, and will be used as the application demands. In fact, as we shall see in Chapter 16, the application requirements which make, say, MFCCs so suitable for speeeh recognition are almost entirely absent for our purposes. We shall leave a discussion as to what representations really are suited for speech synthesis purposes until Chapter 16. 

Linear prediction performs source/filter separation by assuming an HR system represents the filter. This allows the filter coefficients to be found by a process of minimising the error predicted from the HR filter. 

The direct linear prediction, HR, coefficients are inherently unrobust with regard to quantisation and interpolation. A set of derived representations, including reflection coefficients, log area ratios and line spectral frequencies avoid these problems. 

The source signal, ealled the residual, can be found by inverse filtering the speech signal with the linear prediction coefficients. 

Recall that Equation 13.18 is exactly the same as the linear prediction Equation 12.16, where = fli, 02,..., Op are the predictor coefficients and x[n] is the error signal e n. This shows that the result of linear prediction gives us the same type of transfer function as the serial formant synthesiser, and hence LP can produce exactly the same range of frequency responses as the serial formant S5mthesiser. The significance is of course that we can derive the linear prediction coefficients automatically fi om speech and don t have to make manual or perform potentially errorful automatic formant analysis. This is not however a solution to the formant estimation problem itself reversing the set of Equations 13.14 to 13.18 is not trivial, meaning that while we can accurately estimate the all-pole transfer function for arbitrary speech, we can t necessarily decompose this into individual formants. 

Standard cepstral analysis can be used for a number of purposes, for example FO extraction and spectral envelope determination. One of the main reasons that cepstral coefficients are used for spectral representations is that they are robust and well suited to statistical analysis because the coefficients are to a large extent statistically independent. In synthesis however, measuring the spectral envelope accurately is a critical to good quality and many teclmiques have been proposed for more accurate spectral estimation than classic linear prediction or cepstral analysis. 

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