Filter Structures-ver1

Filter Structures-ver1 - Structures of Digital Filters The...

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Unformatted text preview: Structures of Digital Filters The difference equation of a general infinite impulse response (IIR) digital filter is given by = =- +- = N k k N l l k n x b l n y a n y 1 ] [ ] [ ] [ . The filter consists of two sets of coefficients: feedback coefficients } { l a and feed-forward coefficients } { k b . Applying z-transform to the difference equation, we obtain the transfer function (system function) as =- =-- = = N l l l N k k k z a z b z X z Y z H 1 1 ) ( ) ( ) ( We can factor the numerator polynomial and denominator polynomial of the system function and express it in terms of poles } { l p and zeros } { k c =- =--- = N l l N k k z p z c b z H 1 1 1 1 ) 1 ( ) 1 ( ) ( When = l a for l= 1 ,, N , the system is an FIR filter with all the poles of the system located at the origin of the z-plane. In the following, we will study different structures of the implementation (realization) of the system. For a given transfer function, we can find many realizations. 1 Three Basic Elements in Realization (1) Adder (2) Multiplier (3) Unit delay Structures of FIR filters (1) Direct form The transfer function of FIR filter is ] [ ] [ ] 1 [ ] [ ] [ ) ( 1 M h z n h z h z h z n h z H M n M n n--- =- + + + + + = = The difference equation is: ] [ ] [ ] [ ] [ ] 1 [ ] 1 [ ] [ ] [ ] [ M n x M h m n x m h n x h n x h n y- + +- + +- + = For 2 nd order transfer function 2 ] [ 1 n x x 2 [ n ] x 1 [ n ]+ x 2 [ n ] + ] [ 1 n x ] [ 2 n x ] [ ] [ ] [ 2 1 n x n x n y + = ] [ n x ] [ ] [ n x n y = z- 1 ] [ n x ] 1 [ ] [- = n x n y z- 1 z- 1 ] 2 [ ] 2 [ ] 1 [ ] 1 [ ] [ ] [ ] [- +- + = n x h n x h n x h n y The direct form realization of the 2 nd order FIR filter is shown as follows. For a M-th order FIR filter, the direct form realization is shown as follows Computational Complexity: (M+1) multiplications and M additions (2) Transposed form ) ( ] [ ) ( ] 1 [ ) ( ] 1 [ ) ( ] [ ) ( ) ] [ ( ) ( ) 1 ( 1 z X M h z z X M h z z X h z z X h z X z n h z Y M M M n n---- =- +- + + + = = ))...) ( ] [ ) ( ] 1 [ ( ) ( ] 2 [ ( (... ) ( ] [ 1 1 1 1 z X M h z z X M h z z X M h z z z X h---- +- +- + + = For M =3, Y ( z ) can be expressed as ) ( ] 3 [ ) ( ] 2 [ ) ( ] 1 [ ) ( ] [ ) ( 3 2 1 z X h z z X h z z X h z z X h z Y--- + + + = ))) ( ] 3 [ ) ( ] 2 [ ( ) ( ] 1 [ ( ) ( ] [ 1 1 1 z X h z z X h z z X h z z X h--- + + + = 3 ] [ n y ] 2 [ h ] [ h ] 1 [ h ] [ M h 1- z 1- z 1- z ] [ n x x [ n- x [ n-2] x [ n- M+ 1 ] x [ n- M ] For 2 nd order FIR filter, the transposed structure is ] [ ] 2 [ ] [ n x h n x = ] [ ] 1 [ ] 1 [ ] [ 1 n x h n x n x +- = ] [ ] [ ] 1 [ ] [ ] [ 1 2 n x h n x n x n y +- = = ] [ ] [ ] 1 [ ] 1 [ ] 2 [ n x h n x h n x +- +- = ] [ ] [ ] 1 [ ] 1 [ ] 2 [ ] 2 [ n x h n x h n x h +- +- = original difference eqn The transposed structure of M-th order FIR filter is The signal flow of the transposed structure is the reversed version of the direct form....
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This note was uploaded on 01/11/2011 for the course EE 4015 taught by Professor Shuhungleung during the Fall '10 term at City University of Hong Kong.

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Filter Structures-ver1 - Structures of Digital Filters The...

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