1. AGC DSP AGC DSP Professor A G Constantinides©1 A Prediction Problem Problem: Given a sample set of a stationary processes to predict the value of the process some time into the future as  The function may be linear or non-linear. We concentrate only on linear prediction functions

2. AGC DSP AGC DSP Professor A G Constantinides©2 A Prediction Problem Linear Prediction dates back to Gauss in the 18 th century. Extensively used in DSP theory and applications (spectrum analysis, speech processing, radar, sonar, seismology, mobile telephony, financial systems etc) The difference between the predicted and actual value at a specific point in time is caleed the prediction error.

3. AGC DSP AGC DSP Professor A G Constantinides©3 A Prediction Problem The objective of prediction is: given the data, to select a linear function that minimises the prediction error. The Wiener approach examined earlier may be cast into a predictive form in which the desired signal to follow is the next sample of the given process

4. AGC DSP AGC DSP Professor A G Constantinides©4 Forward & Backward Prediction If the prediction is written as Then we have a one-step forward prediction If the prediction is written as Then we have a one-step backward prediction

5. AGC DSP AGC DSP Professor A G Constantinides©5 Forward Prediction Problem The forward prediction error is then Write the prediction equation as And as in the Wiener case we minimise the second order norm of the prediction error

6. AGC DSP AGC DSP Professor A G Constantinides©6 Forward Prediction Problem Thus the solution accrues from Expanding we have Differentiating with resoect to the weight vector we obtain

7. AGC DSP AGC DSP Professor A G Constantinides©7 Forward Prediction Problem However And hence or

8. AGC DSP AGC DSP Professor A G Constantinides©8 Forward Prediction Problem On substituting with the correspending correlation sequences we have Set this expression to zero for minimisation to yield

9. AGC DSP AGC DSP Professor A G Constantinides©9 Forward Prediction Problem These are the Normal Equations, or Wiener- Hopf, or Yule-Walker equations structured for the one-step forward predictor In this specific case it is clear that we need only know the autocorrelation propertities of the given process to determine the predictor coefficients

10. AGC DSP AGC DSP Professor A G Constantinides©10 Forward Prediction Filter Set And rewrite earlier expression as These equations are sometimes known as the augmented forward prediction normal equations

11. AGC DSP AGC DSP Professor A G Constantinides©11 Forward Prediction Filter The prediction error is then given as This is a FIR filter known as the prediction-error filter

12. AGC DSP AGC DSP Professor A G Constantinides©12 Backward Prediction Problem In a similar manner for the backward prediction case we write And Where we assume that the backward predictor filter weights are different from the forward case

13. AGC DSP AGC DSP Professor A G Constantinides©13 Backward Prediction Problem Thus on comparing the the forward and backward formulations with the Wiener least squares conditions we see that the desirable signal is now Hence the normal equations for the backward case can be written as

14. AGC DSP AGC DSP Professor A G Constantinides©14 Backward Prediction Problem This can be slightly adjusted as On comparing this equation with the corresponding forward case it is seen that the two have the same mathematical form and Or equivalently

15. AGC DSP AGC DSP Professor A G Constantinides©15 Backward Prediction Filter Ie backward prediction filter has the same weights as the forward case but reversed. This result is significant from which many properties of efficient predictors ensue. Observe that the ratio of the backward prediction error filter to the forward prediction error filter is allpass. This yields the lattice predictor structures. More on this later

16. AGC DSP AGC DSP Professor A G Constantinides©16 Levinson-Durbin Solution of the Normal Equations The Durbin algorithm solves the following Where the right hand side is a column of as in the normal equations. Assume we have a solution for Where

17. AGC DSP AGC DSP Professor A G Constantinides©17 Levinson-Durbin For the next iteration the normal equations can be written as Where Set Is the k-order counteridentity

18. AGC DSP AGC DSP Professor A G Constantinides©18 Levinson-Durbin Multiply out to yield Note that Hence Ie the first k elements of are adjusted versions of the previous solution

19. AGC DSP AGC DSP Professor A G Constantinides©19 Levinson-Durbin The last element follows from the second equation of Ie

20. AGC DSP AGC DSP Professor A G Constantinides©20 Levinson-Durbin The parameters are known as the reflection coefficients. These are crucial from the signal processing point of view.

21. AGC DSP AGC DSP Professor A G Constantinides©21 Levinson-Durbin The Levinson algorithm solves the problem In the same way as for Durbin we keep track of the solutions to the problems

22. AGC DSP AGC DSP Professor A G Constantinides©22 Levinson-Durbin Thus assuming, to be known at the k step, we solve at the next step the problem

23. AGC DSP AGC DSP Professor A G Constantinides©23 Levinson-Durbin Where Thus

24. AGC DSP AGC DSP Professor A G Constantinides©24 Lattice Predictors Return to the lattice case. We write or

25. AGC DSP AGC DSP Professor A G Constantinides©25 Lattice Predictors The above transfer function is allpass of order M. It can be thought of as the reflection coeffient of a cascade of lossless transmission lines, or acoustic tubes. In this sense it can furnish a simple algorithm for the estimation of the reflection coefficients. We strat with the observation that the transfer function can be written in terms of another allpass filter embedded in a first order allpass structure

26. AGC DSP AGC DSP Professor A G Constantinides©26 Lattice Predictors This takes the form Where is to be chosen to make of degree (M-1). From the above we have

27. AGC DSP AGC DSP Professor A G Constantinides©27 Lattice Predictors And hence Where Thus for a reduction in the order the constant term in the numerator, which is also equal to the highest term in the denominator, must be zero.

28. AGC DSP AGC DSP Professor A G Constantinides©28 Lattice Predictors This requirement yields The realisation structure is

29. AGC DSP AGC DSP Professor A G Constantinides©29 Lattice Predictors There are many rearrangemnets that can be made of this structure, through the use of Signal Flow Graphs. One such rearrangement would be to reverse the direction of signal flow for the lower path. This would yield the standard Lattice Structure as found in several textbooks (viz. Inverse Lattice) The lattice structure and the above development are intimately related to the Levinson-Durbin Algorithm

30. AGC DSP AGC DSP Professor A G Constantinides©30 Lattice Predictors The form of lattice presented is not the usual approach to the Levinson algorithm in that we have developed the inverse filter. Since the denominator of the allpass is also the denominator of the AR process the procedure can be seen as an AR coefficient to lattice structure mapping. For lattice to AR coefficient mapping we follow the opposite route, ie we contruct the allpass and read off its denominator.

31. AGC DSP AGC DSP Professor A G Constantinides©31 PSD Estimation It is evident that if the PSD of the prediction error is white then the prediction transfer function multiplied by the input PSD yields a constant. Therefore the input PSD is determined. Moreover the inverse prediction filter gives us a means to generate the process as the output from the filter when the input is white noise.