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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 feedforward coefficients } { k b . Applying ztransform 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 zplane. 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 β ] [ ] [ n x n y β = β z 1 ] [ n x ] 1 [ ] [ = n x n y z 1 β 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 Mth 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 [ n2] 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 Mth order FIR filter is The signal flow of the transposed structure is the reversed version of the direct form....
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Signal Processing, Finite impulse response, Infinite impulse response, Oppenheim, direct form, R W Schafer

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