Литература / Advanced Digital Signal Processing and Noise Reduction (Saeed V. Vaseghi) / 08 - Linear prediction models

Forward prediction

x(m – P) to x(m – 1) are used to predict x(m)

m

x(m) to x(m–P+1) are used to predict x(m–P)

Backward prediction

Figure 8.8 Illustration of forward and backward predictors.

Similarly, as shown in Figure 8.8, we can define a backward predictor, that predicts a sample x(m−P) from P future samples x(m−P+1), . . ., x(m) as

k =1

The backward prediction error is defined as the difference between the actual sample and its predicted value:

b(m) = x(m − P) − xˆ(m − P)

From Equation (8.24), a signal generated by a backward predictor is given by

k =1

The coefficients of the least square error backward predictor, obtained in a similar method to that of the forward predictor in Section 8.1.1, are given by

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Similarly, we can obtain an order-update recursive equation for the backward prediction error by multiplying both sides of Equation (8.33) by the input vector [x(m–i), x(m–i+1), . . . , x(m)] as

Equations (8.49) and (8.50) are interrelated and may be implemented by a lattice network as shown in Figure 8.8. Minimisation of the squared forward prediction error of Equation (8.49) over N samples yields

N−1

∑e(i−1) (m)b(i−1) (m −1)

(b)

Figure 8.9 Configuration of (a) a lattice predictor and (b) the inverse lattice predictor.

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Note that a similar relation for ki can be obtained through minimisation of the squared backward prediction error of Equation (8.50) over N samples. The reflection coefficients are also known as the normalised partial correlation (PARCOR) coefficients.

8.2.4 Alternative Formulations of Least Square Error Prediction

The methods described above for derivation of the predictor coefficients are based on minimisation of either the forward or the backward prediction error. In this section, we consider alternative methods based on the minimisation of the sum of the forward and backward prediction errors.

Burg's Method Burg’s method is based on minimisation of the sum of the forward and backward squared prediction errors. The squared error function is defined as

m=0

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Simultaneous Minimisation of the Backward and Forward Prediction Errors From Equation (8.28) we have that the backward predictor coefficient vector is the reversed version of the forward predictor coefficient vector. Hence a predictor of order P can be obtained through simultaneous minimisation of the sum of the squared backward and forward prediction errors defined by the following equation:

= (x − Xa)T (x − Xa)+(x B − X Ba)T (xB − X Ba)

(8.55)

where X and x are the signal matrix and vector defined by Equations (8.12) and (8.13), and similarly XB and xB are the signal matrix and vector for the backward predictor. Using an approach similar to that used in derivation of Equation (8.16), the minimisation of the mean squared error function of Equation (8.54) yields

Note that for an ergodic signal as the signal length N increases Equation (8.56) converges to the so-called normal Equation (8.10).

8.2.5 Predictor Model Order Selection

One procedure for the determination of the correct model order is to increment the model order, and monitor the differential change in the error power, until the change levels off. The incremental change in error power with the increasing model order from i–1 to i is defined as

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Figure 8.10 Illustration of the decrease in the normalised mean squared prediction error with the increasing predictor length for a speech signal.

Figure 8.10 illustrates the decrease in the normalised mean square prediction error with the increasing predictor length for a speech signal. The order P beyond which the decrease in the error power ∆E(P) becomes less than a threshold is taken as the model order.

In linear prediction two coefficients are required for modelling each spectral peak of the signal spectrum. For example, the modelling of a signal with K dominant resonances in the spectrum needs P=2K coefficients. Hence a procedure for model selection is to examine the power spectrum of the signal process, and to set the model order to twice the number of significant spectral peaks in the spectrum.

When the model order is less than the correct order, the signal is undermodelled. In this case the prediction error is not well decorrelated and will be more than the optimal minimum. A further consequence of undermodelling is a decrease in the spectral resolution of the model: adjacent spectral peaks of the signal could be merged and appear as a single spectral peak when the model order is too small. When the model order is larger than the correct order, the signal is over-modelled. An over-modelled problem can result in an ill-conditioned matrix equation, unreliable numerical solutions and the appearance of spurious spectral peaks in the model.