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# Iir Filter

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Submitted By alay1174
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IIR Filter structures
ELG6163 Miodrag Bolic

1

Objective
• Stuctures
– Direct form – Transposed direct form – Lattice-ladder form – Parallel realization – Cascade realization – Bi-quad coupled realization – State space realization

2

Implementation

Basic IIR Digital Filter Structures
• The causal IIR digital filters we are concerned with in this course are characterized by a real rational transfer function of z −1 or, equivalently by a constant coefficient difference equation • From the difference equation representation, it can be seen that the realization of the causal IIR digital filters requires some form of feedback
3

Basic IIR Digital Filter Structures
• An N-th order IIR digital transfer function is characterized by 2N+1 unique coefficients, and in general, requires 2N+1 multipliers and 2N two-input adders for implementation • Direct form IIR filters: Filter structures in which the multiplier coefficients are precisely the coefficients of the transfer function
4

Direct Form IIR Digital Filter Structures
• Consider for simplicity a 3rd-order IIR filter with a transfer function
−1 −2 −3

P( z ) p0 + p1z + p2 z + p3 z = H ( z) = D( z ) 1 + d1z −1 + d 2 z − 2 + d3 z −3

• We can implement H(z) as a cascade of two filter sections as shown on the next slide
5

Direct Form IIR Digital Filter Structures
X (z )

H1 (z )

W (z )

H 2 (z )

Y (z )

where
W ( z) −1 −2 −3 H1( z ) = = P( z ) = p0 + p1z + p2 z + p3 z X ( z) Y ( z) 1 1 H 2 ( z) = = = W ( z ) D( z ) 1 + d1z −1 + d 2 z − 2 + d3 z −3
6

Direct Form IIR Digital Filter Structures
• The filter section H1(z ) can be seen to be an FIR filter and can be realized as shown below w[n] = p0 x[n] + p1x[ n − 1] + p2 x[n − 2] + p3 x[n − 3]

7

Direct Form IIR Digital Filter Structures
• The time-domain representation of H 2 (z ) is given by y[n] = w[n] − d1 y[n − 1] − d 2 y[n − 2] − d3 y[n − 3] Realization of H 2 (z ) follows from the above equation and is shown on the right
8

Direct Form IIR Digital Filter Structures
• A cascade of the two structures realizing H1(z ) and H 2 (z ) leads to the realization of H (z ) shown below and is known as the direct form I structure

9

Direct Form IIR Digital Filter Structures
• Note: The direct form I structure is noncanonic as it employs 6 delays to realize a 3rd-order transfer function • A transpose of the direct form I structure is shown on the right and is called the direct form It structure
10

Direct Form IIR Digital Filter Structures
• Various other noncanonic direct form structures can be derived by simple block diagram manipulations as shown below

11

Direct Form IIR Digital Filter Structures
• Observe in the direct form structure shown below, the signal variable at nodes 1 and 1' are the same, and hence the two top delays can be shared

12

Direct Form IIR Digital Filter Structures
• Likewise, the signal variables at nodes 2 and 2' are the same, permitting the sharing of the middle two delays • Following the same argument, the bottom two delays can be shared • Sharing of all delays reduces the total number of delays to 3 resulting in a canonic realization shown on the next slide along with its transpose structure
13

Direct Form IIR Digital Filter Structures

Direct Form II

Direct Form II t

• Direct form realizations of an N-th order IIR transfer function should be evident
14

Cascade Form IIR Digital Filter Structures
• By expressing the numerator and the denominator polynomials of the transfer function as a product of polynomials of lower degree, a digital filter can be realized as a cascade of low-order filter sections • Consider, for example, H(z) = P(z)/D(z) expressed as P ( z ) P ( z ) P2 ( z ) P3 ( z ) H ( z) = = 1 D ( z ) D1 ( z ) D2 ( z ) D3 ( z )

15

Cascade Form IIR Digital Filter Structures
• Examples of cascade realizations obtained by different pole-zero pairings are shown below

16

Cascade Form IIR Digital Filter Structures
• Examples of cascade realizations obtained by different ordering of sections are shown below

17

Cascade Form IIR Digital Filter Structures
• There are altogether a total of 36 different cascade realizations of P ( z ) P2 ( z ) P2 ( z ) H ( z) = 1
D1 ( z ) D2 ( z ) D3 ( z )

based on pole-zero-pairings and ordering • Due to finite wordlength effects, each such cascade realization behaves differently from others
18

Cascade Form IIR Digital Filter Structures
• Usually, the polynomials are factored into a product of 1st-order and 2nd-order polynomials: ⎛ 1 + β1k z −1 + β 2 k z − 2 ⎞ ⎟ H ( z ) = p0 ∏ ⎜ ⎜ 1 + α z −1 + α z − 2 ⎟ k ⎝ ⎠ 1k 2k • In the above, for a first-order factor α 2k = β 2k = 0
19

Cascade Form IIR Digital Filter Structures
• Consider the 3rd-order transfer function ⎛ 1+ β11z −1 ⎞⎛ 1+ β12 z −1 + β 22 z −2 ⎞ H ( z ) = p0 ⎜ ⎜ ⎟ −1 ⎟⎜ −1 −2 ⎟ ⎝ 1+α11z ⎠⎝ 1 + α12 z + α 22 z ⎠ • One possible realization is shown below

20

Cascade Form IIR Digital Filter Structures
• Example - Direct form II and cascade form realizations of

H ( z) =

0.44 z −1 + 0.362 z − 2 + 0.02 z −3 1+ 0.4 z −1 + 0.18 z − 2 −0.2 z −3 ⎛ 0.44+ 0.362 z −1 + 0.02 z −2 ⎞⎛

=⎜ 1+ 0.8 z −1 + 0.5 z − 2 ⎝ are shown on the next slide

⎞ ⎟⎜ −1 ⎟ ⎠⎝ 1−0.4 z ⎠

z −1

21

Cascade Form IIR Digital Filter Structures

Direct form II 22

Parallel Form IIR Digital Filter Structures
• A partial-fraction expansion of the transfer function in z −1 leads to the parallel form I structure • Assuming simple poles, the transfer function H(z) can be expressed as ⎛ γ 0 k +γ 1k z −1 ⎞ H ( z) = γ 0 + ∑ ⎜ −1 −2 ⎟ k ⎝ 1+α1k z +α 2 k z ⎠ • In the above for a real pole α 2 k = γ 1k = 0

23

Parallel Form IIR Digital Filter Structures
• A direct partial-fraction expansion of the transfer function in z leads to the parallel form II structure • Assuming simple poles, the transfer function H(z) can be expressed as ⎛ δ1k z −1 + δ2 k z −2 ⎞ H ( z ) = δ0 + ∑ ⎜ −1 −2 ⎟ k ⎝ 1+ α1k z + α 2 k z ⎠ • In the above for a real pole α 2 k = δ 2 k = 0

24

Parallel Form IIR Digital Filter Structures
• The two basic parallel realizations of a 3rdorder IIR transfer function are shown below

Parallel form I 25

Parallel form II

Parallel Form IIR Digital Filter Structures
• Example - A partial-fraction expansion of H ( z) = in z −1 yields H ( z ) = − 0.1 +
0.6 1− 0.4 z
−1

0.44 z −1 + 0.362 z − 2 + 0.02 z −3 1+ 0.4 z −1 + 0.18 z − 2 −0.2 z −3 − 0.5 − 0.2 z −1 1+ 0.8 z −1 + 0.5 z − 2

+

26

Parallel Form IIR Digital Filter Structures
• The corresponding parallel form I realization is shown below

27

Parallel Form IIR Digital Filter Structures
• Likewise, a partial-fraction expansion of H(z) in z yields H ( z) =
0.24 z −1 1− 0.4 z
−1

+

0.2 z −1 + 0.25 z − 2 1+ 0.8 z −1 + 0.5 z − 2

• The corresponding parallel form II realization is shown on the right
28

Comparison of the complexity of different IIR filters

29

Estimation of area for ASIC implementation

30

Estimation of number of processors

31

32

Other possibilities for comparisson
• Predicting pipelining improvement using timing metrics. • Predicting retiming improvement

33

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