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Iir Filter

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IIR Filter structures
ELG6163 Miodrag Bolic

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Copyright © 2005, S. K. Mitra

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


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Implementation
Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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]

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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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 )
Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

Cascade Form IIR Digital Filter Structures

Direct form II 22

Cascade form

Copyright © 2005, S. K. Mitra

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
Copyright © 2005, S. K. Mitra

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
Copyright © 2005, S. K. Mitra

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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
Copyright © 2005, S. K. Mitra

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

Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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
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Copyright © 2005, S. K. Mitra

Comparison of the complexity of different IIR filters

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Copyright © 2005, S. K. Mitra

Estimation of area for ASIC implementation

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Copyright © 2005, S. K. Mitra

Estimation of number of processors

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Copyright © 2005, S. K. Mitra

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Copyright © 2005, S. K. Mitra

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

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Copyright © 2005, S. K. Mitra

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