Lecture 15 Structures for Discrete

Transcript Lecture 15 Structures for Discrete

Structures for Discrete-Time Systems Quote of the Day Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.

Example • Block diagram representation of y

 

 a 1 y



n  1



 a 2 y



n  2



 b 0 x

 

Block Diagram Representation • LTI systems with rational system function can be represented as constant coefficient difference equation • The implementation of difference equations requires delayed values of the – input – output – intermediate results • The requirement of delayed elements implies need for storage • We also need means of – addition – multiplication Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 3

Direct Form I • General form of difference equation k N   0 ˆ k y



n  k



 k M   0 b k x



n  k



• Alternative equivalent form y

 

 k N   1 a k y



n  k



 k M   0 b k x



n  k



Direct Form I • Transfer function can be written as M H   • Direct Form I Represents z  1 k    0 N k b k   1 z a k  k z  k H

 

 

   k M   0 b k z  k   X     1  k N   1 1 a k z  k     V 351M Digital Signal Processing y

 

 k M   0 b k x



n  k



k N   1 a k y



n  k  5

Alternative Representation • Replace order of cascade LTI systems H

 

 H 1 H 2 W

 

 H 2 Y  H 1    k M   0 b k z  k       1  1 k N   1 a k z  k           1  1 k N   1 a k z  k   k M   0 b k z  k   W     X w

 

 y

 

 k N   1 a k w



n  k k M   0 b k w



n  k



Alternative Block Diagram • We can change the order of the cascade systems w

 

 y

 

 k N   1 a k w



n  k k M   0 b k w



n  k



Direct Form II • No need to store the same data twice in previous system • So we can collapse the delay elements into one chain • This is called Direct Form II or the Canonical Form • Theoretically no difference between Direct Form I and II • Implementation wise – Less memory in Direct II – Difference when using finite-precision arithmetic Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 8

Signal Flow Graph Representation • Similar to block diagram representation – Notational differences • A network of directed branches connected at nodes • Example representation of a difference equation Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 9

Example • Representation of Direct Form II with signal flow graphs w 1 w 2 w 3 w 4

       

    y  aw 4 w 1

     

b 0 w 2 w 3 w 2



 

 1



 b 1 w 4 w 1 y

   

  aw 1 b 0 w 1



n 

 

1 

  

b 1 w 1



n  1



Determination of System Function from Flow Graph W 1 W 2 W 3 W 4             Y  W 4    W 1    X W 2 W 3 W 2       z    1 X   W 4 w 1 w 2 w 3 w 4

       

    y  w 4  w 1 w 2 w 3



 

 x n

   

n  1



w 2  w 4 W 2 W 4 Y     X  z  1 X 1   1  z

  

1  1   W 2 1 

 

 z   1 W 4 1  



H   h

 

  Y X      n   1 u



n z  1    1  1



Basic Structures for IIR Systems: Cascade Form • General form for cascade implementation H

 

 A M 1  k N  1 1 k   1   1 1   f k z c k  1 z  1 k   k  2 1    1 1 1   g k z d k  1 z  1  1  1   g  k z  1  d  k z  1  • More practical form in 2 nd order systems H    k M 1   1 b 0 k 1   a b 1 k 1 k z z  1  1   b 2 k a 2 k z z  2  2 Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 14

Example H     1  1   1   1 z  0 .

75 z  1 0 .

5 z  2 z  1  1   1  z  2   1 0 .

125 z  1  z  1  0 .

25 z   2  1    1 • Cascade of Direct Form I subsections   1  0 .

5 z z  1  1   1 1   z  1  0 .

Basic Structures for IIR Systems: Parallel Form • Represent system function using partial fraction expansion H    k N P   0 C k z  k  k N P   1 1  A k c k z  1  k N P   1  1  B k d k  z 1  1   e k 1  z  1 d  k  z  1  • Or by pairing the real poles H    k N P   0 C k z  k  k N S   1 1  e 0 k  a 1 k z  1 e 1 k z  1  a 2 k z  2 Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 16

• Partial Fraction Expansion Example H    1  1  2 z 0 .

75 z  1  1   0 z .

 2 125 z  2  8   1  18 0 .

5 z  1   1  25 0 .

25 z  1  • Combine poles to get H    8  1   7 0 .

75 z  1  8  z 0  1 .

Transposed Forms • Linear signal flow graph property: – Transposing doesn’t change the input-output relation • Transposing: – Reverse directions of all branches – Interchange input and output nodes • Example: H    1  1 az  1 – Reverse directions of branches and interchange input and output Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 18

Example Transpose • Both have the same system function or difference equation y

 

 a 1 y



n  1



 a 2 y



n  2



 b 0 x  b 1 x



n  1



 b 2 x



n  2



Basic Structures for FIR Systems: Direct Form • Special cases of IIR direct form structures • Transpose of direct form I gives direct form II • Both forms are equal for FIR systems • Tapped delay line Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 20

Basic Structures for FIR Systems: Cascade Form • Obtained by factoring the polynomial system function H    n M   0 h

 

Structures for Linear-Phase FIR Systems • Causal FIR system with generalized linear phase are symmetric: h h

 

M M   n n

   

  h n

 

n   0,1,..., M (type I or III) 0,1,..., M (type II or IV) • Symmetry means we can half the number of multiplications • Example: For even M and type I or type III systems: y  k M   0 h

  

n  k



 M k / 2   0  1 h

  

n  k

 

M / 2

 

n  M / 2



 k  M M  / 2 h

  

n  1  k



  M k / 2   0  1 h

  

n  k

 

M / 2

 

n  M / 2



 M k / 2   0  1 h



M  k

 

n  M  k



M k / 2   0  1 h

  

n  k

 

n  M  k



  h



M / 2

 

n  M / 2

