Basic IIR Digital Filter Structures

Transcription

1 Basic IIR Digital Filter Structures The causal IIR igital filters we are concerne with in this course are characterie by a real rational transfer function of or, equivalently by a constant coefficient ifference equation From the ifference equation representation, it can be seen that the realiation of the causal IIR igital filters requires some form of feeback

2 Basic IIR Digital Filter Structures An N-th orer IIR igital transfer function is characterie by N+ unique coefficients, an in general, requires N+ multipliers an N two-input aers for implementation Direct form IIR filters: Filter structures in which the multiplier coefficients are precisely the coefficients of the transfer function

3 3 Direct Form IIR Digital Filter Direct Form IIR Digital Filter Structures Structures Consier for simplicity a 3r-orer IIR filter with a transfer function We can implement H() as a cascae of two filter sections as shown on the next slie p p p p D P H ) ( ) ( ) (

4 4 Direct Form IIR Digital Filter Direct Form IIR Digital Filter Structures Structures where p p p p P X W H ) ( ) ( ) ( ) ( () H ) H ( W () X () Y () D W Y H ) ( ) ( ) ( ) (

5 Direct Form IIR Digital Filter Structures The filter section H () can be seen to be an FIR filter an can be realie as shown below w n] p x[ n] + p x[ n ] + p x[ n ] + p x[ n [ 0 3 3] 5

6 Direct Form IIR Digital Filter Structures 6 The time-omain representation of H () is given by y[ n] w[ n] y[ n ] y[ n ] 3y[ n 3] Realiation of H () follows from the above equation an is shown on the right

7 Direct Form IIR Digital Filter Structures A cascae of the two structures realiing H () an H () leas to the realiation of H() shown below an is known as the irect form I structure 7

8 Direct Form IIR Digital Filter Structures Note: The irect form I structure is noncanonic as it employs 6 elays to realie a 3r-orer transfer function A transpose of the irect form I structure is shown on the right an is calle the irect form I t structure 8

9 Direct Form IIR Digital Filter Structures Various other noncanonic irect form structures can be erive by simple block iagram manipulations as shown below 9

10 Direct Form IIR Digital Filter Structures Observe in the irect form structure shown below, the signal variable at noes an are the same, an hence the two top elays can be share ' 0

11 Direct Form IIR Digital Filter Structures Likewise, the signal variables at noes an ' are the same, permitting the sharing of the mile two elays Following the same argument, the bottom two elays can be share Sharing of all elays reuces the total number of elays to 3 resulting in a canonic realiation shown on the next slie along with its transpose structure

12 Direct Form IIR Digital Filter Structures Direct form realiations of an N-th orer IIR transfer function shoul be evient

13 Cascae Form IIR Digital Filter Structures 3 By expressing the numerator an the enominator polynomials of the transfer function as a prouct of polynomials of lower egree, a igital filter can be realie as a cascae of low-orer filter sections Consier, for example, H() P()/D() expresse as P( ) P ( ) ( ) ( ) ( ) P P3 H D( ) D ( ) D ( ) D ( ) 3

14 Cascae Form IIR Digital Filter Structures Examples of cascae realiations obtaine by ifferent pole-ero pairings are shown below 4

15 Cascae Form IIR Digital Filter Structures Examples of cascae realiations obtaine by ifferent orering of sections are shown below 5

16 Cascae Form IIR Digital Filter Structures There are altogether a total of 36 ifferent cascae realiations of P ( ) P ( ) P ( ) H( ) D ( ) D ( ) D3 ( ) base on pole-ero-pairings an orering Due to finite worlength effects, each such cascae realiation behaves ifferently from others 6

17 Cascae Form IIR Digital Filter Structures Usually, the polynomials are factore into a prouct of st-orer an n-orer polynomials: + β + k k H p β ( ) 0 k + αk + αk In the above, for a first-orer factor α β 0 k k 7

18 Cascae Form IIR Digital Filter Structures Consier the 3r-orer transfer function + β + β + β H( ) p0 + α + α + α One possible realiation is shown below 8

19 Cascae Form IIR Digital Filter Structures Example- Direct form II an cascae form realiations of H( ) are shown on the next slie

20 Cascae Form IIR Digital Filter Structures Direct form II Cascae form 0

21 Parallel Form IIR Digital Filter Structures A partial-fraction expansion of the transfer function in leas to the parallel form I structure Assuming simple poles, the transfer function H() can be expresse as γ 0k ( ) γ 0 + k + αk In the above for a real pole α + γk H + αk k γ k 0

22 Parallel Form IIR Digital Filter Structures A irect partial-fraction expansion of the transfer function in leas to the parallel form II structure Assuming simple poles, the transfer function H() can be expresse as In the above for a real pole α k δ + 0k + δ k H( ) δ0 + αk + αk k δ k 0

23 Parallel Form IIR Digital Filter Structures The two basic parallel realiations of a 3rorer IIR transfer function are shown below 3 Parallel form I Parallel form II

24 Parallel Form IIR Digital Filter Structures Example - A partial-fraction expansion of H( ) in yiels H( )

25 Parallel Form IIR Digital Filter Structures The corresponing parallel form I realiation is shown below 5

26 Parallel Form IIR Digital Filter Structures 6 Likewise, a partial-fraction expansion of H() in yiels H( ) The corresponing parallel form II realiation is shown on the right

27 Realiation Using MATLAB The cascae form requires the factoriation of the transfer function which can be evelope using the M-file psos The statement sos psos(,p,k) generates a matrix sos containing the coefficients of each n-orer section of the equivalent transfer function H() etermine from its pole-ero form 7

28 Realiation Using MATLAB 8 sos is an L 6 matrix of the form sos p p p 0 0 0L p p p L p p p L 0 0 0L L L whose i-th row contains the coefficients{ p il } an { il }, of the the numerator an enominator polynomials of the i-th norer section

29 9 Realiation Using MATLAB Realiation Using MATLAB L enotes the number of sections The form of the overall transfer function is given by Program 6_ can be use to factorie an FIR an an IIR transfer function L i i i i i i i L i i p p p H H 0 0 ) ( ) (

30 Realiation Using MATLAB Note: An FIR transfer function can be treate as an IIR transfer function with a constant numerator of unity an a enominator which is the polynomial escribing the FIR transfer function 30

31 Realiation Using MATLAB Parallel forms I an II can be evelope using the functions resiue an resiue, respectively Program 6_ uses these two functions 3

32 Realiation of Allpass Filters 3 An M-th orer real-coefficient allpass transfer function A M () is characterie by M unique coefficients as here the numerator is the mirror-image polynomial of the enominator A irect form realiation of A M () requires M multipliers Objective - Develop realiations of A M () requiring only M multipliers

33 Realiation Using Multiplier Extraction Approach Now, an arbitrary allpass transfer function can be expresse as a prouct of n-orer an/or st-orer allpass transfer functions We consier first the minimum multiplier realiation of a st-orer an a n-orer allpass transfer functions 33

34 First-Orer Allpass Structures Consier first the st-orer allpass transfer function given by + A ( ) + We shall realie the above transfer function in the form a structure containing a single multiplier as shown below X Y 34 Y X

35 First-Orer Allpass Structures 35 We express the transfer function A ( ) Y/ in terms of the transfer parameters of the two-pair as tt t ( tt tt) A ( ) t + t t A comparison of the above with A ( ) + yiels t, t, + tt tt X

36 First-Orer Allpass Structures 36 Substituting t an t in tt tt we get t t There are 4 possible solutions to the above equation: Type A: Type B: t t, t, t, t, t, t +, t

37 First-Orer Allpass Structures Type A t : Type B : t, t, t t t, t, t, t, t + We now evelop the two-pair structure for the Type A allpass transfer function 37

38 First-Orer Allpass Structures From the transfer parameters of this allpass we arrive at the input-output relations: Y X X Y X + ( ) X Y + X A realiation of the above two-pair is sketche below 38

39 First-Orer Allpass Structures X By constraining the, Y terminal-pair with the multiplier, we arrive at the Type A allpass filter structure shown below Type A 39

40 First-Orer Allpass Structures In a similar fashion, the other three singlemultiplier first-orer allpass filter structures can be evelope as shown below Type B Type A t 40 Type B t

41 Secon-Orer Allpass Structures A n-orer allpass transfer function is characterie by unique coefficients Hence, it can be realie using only multipliers Type allpass transfer function: A ( )

42 Type Allpass Structures 4

43 Type 3 Allpass Structures Type 3 allpass transfer function: + + A 3( )

44 Type 3 Allpass Structures 44

45 Realiation Using Multiplier Extraction Approach Example - Realie A ( ) ( ( 0. 4 A 3-multiplier cascae realiation of the above allpass transfer function is shown below + 3 )( )( ) ) 45

46 Realiation Using Two-Pair Extraction Approach The stability test algorithm escribe earlier in the course also leas to an elegant realiation of an Mth-orer allpass transfer function The algorithm is base on the evelopment of a series of ( m ) th-orer allpass transfer functions A m () from an mth-orer allpass transfer function A m () for m M, M,..., 46

47 Realiation Using Two-Pair Extraction Approach 47 Let A ( m) m + m + m + + ( ) ( m ) m + m m We use the recursion Am ( ) km A m m ( ) [ ], km Am ( ) where k m Am ( ) m It has been shown earlier that A M () is stable if an only if k for m M, M,..., < m... + m m M, M,...,

48 Realiation Using Two-Pair Extraction Approach 48 If the allpass transfer function A m () is expresse in the form... m A m ' ' ' ( m ) ( m) m+ m ( ) '... ' ( m ) ' ( ) m + m then the coefficients of A m () are simply relate to the coefficients of A m () through ' i m m i i, i m m

49 Realiation Using Two-Pair Extraction Approach To evelop the realiation metho we express A m () in terms of A m (): A m ( ) k m + + km Am ( ) ( ) We realie A m () in the form shown below A m 49 A m () X Y t t t t Y A m () X

50 Realiation Using Two-Pair Extraction Approach The transfer function A m ( ) Y / X of the constraine two-pair can be expresse as A m ( ) ( t t tam Comparing the above with t t t ) A ( ) m ( ) 50 A m ( ) k m + + km Am ( ) ( ) we arrive at the two-pair transfer parameters A m

51 Realiation Using Two-Pair Extraction Approach t km, t km tt tt Substituting t k m an t km in the equation above we get tt ( km) There are a number of solutions for an t t 5

52 t Realiation Using Two-Pair Extraction Approach Some possible solutions are given below: m m km), t t k, t k, t ( + t t km, t km, t, t km km, t km, t km, t km km, t km, t ( km), t k m 5

53 Realiation Using Two-Pair Extraction Approach 53 Consier the solution t km, t km, t ( km), t Corresponing input-output relations are Y kmx ( km) X Y X k m X A irect realiation of the above equations leas to the 3-multiplier two-pair shown on the next slie

54 Realiation Using Two-Pair Extraction Approach The transfer parameters t km, t km, t ( km), t + lea to the 4-multiplier two-pair structure shown below k m 54

55 t Realiation Using Two-Pair Extraction Approach Likewise, the transfer parameters km, t km, t km, t lea to the 4-multiplier two-pair structure shown below km 55

56 Realiation Using Two-Pair Extraction Approach A -multiplier realiation can be erive by manipulating the input-output relations: Y kmx ( km) X Y X k m X Making use of the secon equation, we can rewrite the first equation as Y k Y + m X 56

57 Realiation Using Two-Pair Extraction Approach A irect realiation of Y k Y + Y m X X km X lea to the -multiplier two-pair structure, known as the lattice structure, shown below 57

58 Realiation Using Two-Pair Extraction Approach Consier the two-pair escribe by t k, t k, t ( k ), t + m m Its input-output relations are given by Y kmx + ( km) X Y ( + km) X km X Define V km ( X ) X m k m 58

59 Realiation Using Two-Pair Extraction Approach We can then rewrite the input-output relations as Y V + X an Y X + The corresponing -multiplier realiation is shown below V V 59

60 Realiation Using Two-Pair Extraction Approach An mth-orer allpass transfer function A m () is then realie by constraining any one of the two-pairs evelope earlier by the ( m )th-orer allpass transfer function A m () 60

61 Realiation Using Two-Pair Extraction Approach The process is repeate until the constraining transfer function is A0 () The complete realiation of A M () base on the extraction of the two-pair lattice is shown below 6

62 Realiation Using Two-Pair Extraction Approach 6 It follows from our earlier iscussion that A M () is stable if the magnitues of all multiplier coefficients in the realiation are less than, i.e., k m < for m M, M,..., The cascae lattice allpass filter structure requires M multipliers A realiation with M multipliers is obtaine if instea the single multiplier two-pair is use

63 63 Realiation Using Two-Pair Realiation Using Two-Pair Extraction Approach Extraction Approach Example - Realie ) ( A

64 Realiation Using Two-Pair Extraction Approach We first realie A 3( ) in the form of a lattice two-pair characterie by the multiplier coefficient k an constraine by a n-orer allpass as inicate below A ( ) 64 k3 0.

65 Realiation Using Two-Pair Extraction Approach The allpass transfer function A ( ) is of the form ' ' + + A ( ) + ' Its coefficients are given by ' + ' 0.4 ( 0.)(0.8) 3 3 ( 0.) ' 0.8 ( 0.)(0.4) 3 3 ( 0.)

66 Realiation Using Two-Pair Extraction Approach Next, the allpass A ( ) is realie as a lattice two-pair characterie by the multiplier coefficient k ' an constraine by an allpass A ( ) as inicate below 66 k3 0., k

67 Realiation Using Two-Pair Extraction Approach The allpass transfer function A ( ) is of the form " + A ( ) " + It coefficient is given by ' ' ' ' " ( ' ) + '

68 Realiation Using Two-Pair Extraction Approach Finally, the allpass A ( ) is realie as a lattice two-pair characterie by the multiplier coefficient k " an constraine by an allpass A0 ( ) as inicate below A 3( ) 68 k3 0., k , A ( ) A ( ) k

69 Cascae Lattice Realiation Using MATLAB The M-file polyrc can be use to realie an allpass transfer function in the cascae lattice form To this en Program 6_3 can be employe 69