{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Filter Structures-ver1

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

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 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 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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 23

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online