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630F09_hw8 - ENEE 630 Fall 2009 Homework#8 Due on Monday...

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ENEE 630 Fall 2009 Homework #8 Due on Monday, 7 th December Problem 1 In a normalized lattice predictor, the forward and backward prediction errors at the various stage of the predictor are all normalized to have unit variance . Such an operation makes it possible to utilize the full dynamic range of multipliers used in the predictor, the normalized forward and backward prediction errors are defined as follows, respectively: f m ( n ) = f m ( n ) P 1 / 2 m b m ( n ) = b m ( n ) P 1 / 2 m where P m is the average power (or variance) of the forward prediction error f m ( n ) or that of the backward prediction error b m ( n ). Show that the structure of stage m of the normalized lattice predictor is as shown in Figure 1 for real-valued data. Cos w m + + m Csc w m Csc w z -1 f (n) m-1 w = Cos ( ) m m -1 b (n) m f (n) m b (n) m-1 Cos w m Figure 1: P-1 Problem 2 Consider the problem of optimizing stage m of the lattice predictor. The cost function to be used in the optimization is described by J m m ) = αE [ | f m ( n ) | 2 ] + (1 - α ) E [ | b m ( n ) | 2 ] where α
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