1、Professor A G Constantinides,1,A Prediction Problem,Problem: Given a sample set of a stationary processesto predict the value of the process some time into the future asThe function may be linear or non-linear. We concentrate only on linear prediction functions,Professor A G Constantinides,2,A Predi
2、ction Problem,Linear Prediction dates back to Gauss in the 18th 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
3、 in time is caleed the prediction error.,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 s
4、ignal to follow is the next sample of the given process,Professor A G Constantinides,4,Forward & Backward Prediction,If the prediction is written asThen we have a one-step forward prediction If the prediction is written asThen we have a one-step backward prediction,Professor A G Constantinides,5,For
5、ward Prediction Problem,The forward prediction error is thenWrite the prediction equation asAnd as in the Wiener case we minimise the second order norm of the prediction error,Professor A G Constantinides,6,Forward Prediction Problem,Thus the solution accrues fromExpanding we haveDifferentiating wit
6、h resoect to the weight vector we obtain,Professor A G Constantinides,7,Forward Prediction Problem,HoweverAnd henceor,Professor A G Constantinides,8,Forward Prediction Problem,On substituting with the correspending correlation sequences we haveSet this expression to zero for minimisation to yield,Pr
7、ofessor 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 predictorIn this specific case it is clear that we need only know the autocorrelation propertities of the given process to determine t
8、he predictor coefficients,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,Professor A G Constantinides,11,Forward Prediction Filter,The prediction error is then gi
9、ven as This is a FIR filter known as the prediction-error filter,Professor A G Constantinides,12,Backward Prediction Problem,In a similar manner for the backward prediction case we writeAndWhere we assume that the backward predictor filter weights are different from the forward case,Professor A G Co
10、nstantinides,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,Professor A G Constantinides,14,Backward Predict
11、ion Problem,This can be slightly adjusted asOn 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,15,Backward Prediction Filter,Ie backward prediction filter has the same weights as the
12、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
13、A G Constantinides,16,Levinson-Durbin,Solution of the Normal Equations The Durbin algorithm solves the followingWhere the right hand side is a column of as in the normal equations. Assume we have a solution forWhere,Professor A G Constantinides,17,Levinson-Durbin,For the next iteration the normal eq
14、uations can be written asWhere Set,Is the k-order counteridentity,Professor A G Constantinides,18,Levinson-Durbin,Multiply out to yieldNote that Hence Ie the first k elements of are adjusted versions of the previous solution,Professor A G Constantinides,19,Levinson-Durbin,The last element follows fr
15、om the second equation of Ie,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.,Professor A G Constantinides,21,Levinson-Durbin,The Levinson algorithm solves the problem In the same way
16、as for Durbin we keep track of the solutions to the problems,Professor A G Constantinides,22,Levinson-Durbin,Thus assuming , to be known at the k step, we solve at the next step the problem,Professor A G Constantinides,23,Levinson-Durbin,Where Thus,Professor A G Constantinides,24,Lattice Predictors,
17、Return to the lattice case. We writeor,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 algorith
18、m 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,26,Lattice Predictors,This takes the formWhere is to be chosen to m
19、ake of degree (M-1) . From the above we have,Professor A G Constantinides,27,Lattice Predictors,And henceWhereThus 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,28,Lattice Predic
20、tors,This requirement yields The realisation structure is,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
21、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,30,Lattice Predictors,The form of lattice presented is not t
22、he 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 op
23、posite route, ie we contruct the allpass and read off its denominator.,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.,
