AGC A Prediction Problem DSP n Problem Given

AGC A Prediction Problem DSP n 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 Professor A G Constantinides© 1

AGC A Prediction Problem DSP n n n 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. Professor A G Constantinides© 2

AGC A Prediction Problem DSP n n 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 Professor A G Constantinides© 3

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

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

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

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

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

AGC Forward Prediction Problem DSP n n 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 Professor A G Constantinides© 9

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

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

AGC Backward Prediction Problem DSP n n n 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 Professor A G Constantinides© 12

AGC Backward Prediction Problem DSP n n Thus on comparing 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 n Professor A G Constantinides© 13

AGC Backward Prediction Problem DSP n n n 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 Professor A G Constantinides© 14

AGC Backward Prediction Filter DSP n n n 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 Professor A G Constantinides© 15

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

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

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

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

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

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

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

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

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

AGC Lattice Predictors DSP n n 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 Professor A G Constantinides© 25

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

AGC Lattice Predictors DSP n And hence n Where n 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. Professor A G Constantinides© 27

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

AGC Lattice Predictors DSP n n n 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 Professor A G Constantinides© 29

AGC Lattice Predictors DSP n n n 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. Professor A G Constantinides© 30

AGC PSD Estimation DSP n n n 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. Professor A G Constantinides© 31