630F09_HW8sol

630F09_HW8sol - ENEE 630 Homework#8 Solution to 1 For real...

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Unformatted text preview: ENEE 630 Homework #8 Solution to 1 For real valued data, f m ( n ) = f m- 1 ( n ) + Γ m b m- 1 ( n − 1) b m ( n ) = b m- 1 ( n ) + Γ m f m- 1 ( n ) Now, we define f m ( n ) = f m ( n ) P 1 / 2 m b m ( n ) = b m ( n ) P 1 / 2 m It follows that P 1 / 2 m f m ( n ) = P 1 / 2 m- 1 f m- 1 ( n ) + Γ m P 1 / 2 m- 1 b m- 1 ( n − 1) P 1 / 2 m b m ( n ) = P 1 / 2 m- 1 b m- 1 ( n − 1) + Γ m P 1 / 2 m- 1 f m- 1 ( n ) Or, f m ( n ) = ( P m- 1 P m ) 1 / 2 f m- 1 ( n ) + Γ m ( P m- 1 P m ) 1 / 2 b m- 1 ( n − 1) b m ( n ) = ( P m- 1 P m ) 1 / 2 b m- 1 ( n − 1) + Γ m ( P m- 1 P m ) 1 / 2 f m- 1 ( n ) Recall that P m = P m- 1 (1 − Γ 2 m ). Thus, ( P m- 1 P m ) 2 = 1 (1 − Γ 2 m ) 1 / 2 Let Γ m = cos w m , then ( P m- 1 P m ) 1 / 2 = csc w m . Thus, f m ( n ) = csc w m ( f m- 1 ( n ) + (cos w m ) b m- 1 ( n − 1)) b m ( n ) = csc w m ( b m- 1 ( n − 1) + (cos w m ) f m- 1 ( n )) See Figure 1 Solution to 2 (a) The index of performance is J m = αE [ | f m ( n ) | 2 ] + (1 − α ) E...
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630F09_HW8sol - ENEE 630 Homework#8 Solution to 1 For real...

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