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DSP-CIS Chapter-5: Filter Realization Marc Moonen Dept. E.E./ESAT, KU Leuven

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DSP-CIS / Chapter-5: Filter Realization / Version p. 2 Filter Design/Realization Step-1 : define filter specs (pass-band, stop-band, optimization criterion,…) Step-2 : derive optimal transfer function FIR or IIR design Step-3 : filter realization (block scheme/flow graph) direct form realizations, lattice realizations,… Step-4 : filter implementation (software/hardware) finite word-length issues, … question: implemented filter = designed filter ? ‘You can’t always get what you want’ -Jagger/Richards (?) Chapter-4 Chapter-5 Chapter-6

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DSP-CIS / Chapter-5: Filter Realization / Version p. 3 Chapter-5 : Filter Realizations Overview: FIR Filter Realizations IIR Filter Realizations PS: Will always assume real-valued filter coefficients Q: Why bother about many different realizations for one and the same filter? A: See Chapter-6

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DSP-CIS / Chapter-5: Filter Realization / Version p. 4 FIR Filter Realizations FIR Filter Realization =Construct (realize) LTI system (with delay elements, adders and multipliers), such that I/O behavior is given by.. Several possibilities exist… 1. Direct form 2. Transposed direct form 3. Lattice (LPC lattice) 4. Lossless lattice PS: Frequency-domain realization: see Chapter-7

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DSP-CIS / Chapter-5: Filter Realization / Version p. 5 FIR Filter Realizations 1. Direct form u[k]u[k-4]u[k-3]u[k-2]u[k-1] x bo + x b4 x b3 + x b2 + x b1 + y[k]

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DSP-CIS / Chapter-5: Filter Realization / Version p. 6 FIR Filter Realizations 2. Transposed direct form Starting point is direct form : `Retiming’ = select subgraph (shaded) remove one delay element on all inbound arrows add one delay element on all outbound arrows u[k]u[k-4]u[k-3]u[k-2]u[k-1] x bo + x b4 x b3 + x b2 + x b1 + y[k]

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DSP-CIS / Chapter-5: Filter Realization / Version p. 7 FIR Filter Realizations `Retiming’ : results in... (=different software/hardware, same i/o-behavior) u[k]u[k-1] x bo + x b1 + y[k] u[k-4]u[k-3] x b4 x b3 + x b2 +

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DSP-CIS / Chapter-5: Filter Realization / Version p. 8 FIR Filter Realizations ` Retiming’ : repeated application results in... i.e. `transposed direct form’ u[k] x bo + y[k] x b1 + x b2 + x b3 + x b4 (=different software/hardware (`pipeline delays’), same i/o-behavior)

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DSP-CIS / Chapter-5: Filter Realization / Version p. 9 FIR Filter Realizations 3. Lattice form Derived from combined realization of with `flipped’ version of H(z) Reversed (real-valued) coefficient vector results in... i.e. - same magnitude response - different phase response

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DSP-CIS / Chapter-5: Filter Realization / Version p. 10 FIR Filter Realizations Starting point is direct form realization… b4 u[k]u[k-1]u[k-2] x b3 + x b2 + x + x + b1 u[k-3] x b1 + b2 x + x b4 + y[k] bo x + u[k-4] x bo b3 x y[k] ~ Now 1 page of maths…

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DSP-CIS / Chapter-5: Filter Realization / Version p. 11 FIR Filter Realizations With this can be rewritten as order reduction PS: if |ko|=1, then transformation matrix is rank-deficientPS: find fix for case bo=0

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DSP-CIS / Chapter-5: Filter Realization / Version p. 12 FIR Filter Realizations This is equivalent to... …Now repeat procedure for shaded graph (=same structure as the one we started from) u[k]u[k-3]u[k-2] x b’3b’3 + x b’2b’2 + x + x + b’ob’o x b’1b’1 + b’1b’1 x + b’ob’o x b’2b’2 x b’3b’3 y[k] + + x x ko y[k] ~

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DSP-CIS / Chapter-5: Filter Realization / Version p. 13 FIR Filter Realizations Repeated application results in `lattice form’ u[k] y[k] + + x x ko + + x x k1 + + x x k2 + + x x k3 x bo y[k] ~ (= different software/hardware, same i/o-behavior) explain

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DSP-CIS / Chapter-5: Filter Realization / Version p. 14 FIR Filter Realizations Also known as `LPC Lattice’ (`linear predictive coding lattice’) K i ’s are so-called `reflection coefficients’ Every set of b i ’s corresponds to a set of K i ’s, and vice versa. Procedure for computing K i ’s from b i ’s corresponds to the well-known `Schur-Cohn’ stability test (from control theory): problem = for a given polynomial B(z), how do we find out if all the zeros of B(z) are stable (i.e. lie inside unit circle) ? solution = from b i ’s, compute reflection coefficients K i ’s (following procedure on previous slides). Zeros are (proved to be) stable if and only if all reflection coefficients statisfy |K i |<1 ! Procedure (page 11) breaks down if |Ki|=1 is encountered. Means at least one root of B(z) lies on or outside the unit circle (cfr Schur-Cohn). Then have to use other realization

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DSP-CIS / Chapter-5: Filter Realization / Version p. 15 FIR Filter Realizations 4. Lossless lattice : Derived from combined realization of H(z) with which is such that (*) PS : interpretation ?… (see next slide) PS : may have to scale H(z) to achieve this (why?) (scaling omitted here)

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DSP-CIS / Chapter-5: Filter Realization / Version p. 16 FIR Filter Realizations PS : Interpretation ? When evaluated on the unit circle, formula (*) is equivalent to (for filters with real-valued coefficients) i.e. and are `power complementary’ (= form a 1-input/2-output `lossless’ system, see also below ) 1

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DSP-CIS / Chapter-5: Filter Realization / Version p. 17 FIR Filter Realizations PS : How is computed ? Note that if z i (and z i *) is a root of R(z), then 1/z i (and 1/z i *) is also a root of R(z). Hence can factorize R(z) as… Note that z i ’s can be selected such that all roots of lie inside the unit circle, i.e. is a minimum-phase FIR filter. This is referred to as spectral factorization, =spectral factor.

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DSP-CIS / Chapter-5: Filter Realization / Version p. 18 FIR Filter Realizations Starting point is direct form realization… u[k]u[k-1]u[k-2] x + x + x + x + u[k-3] x + x + x + y[k] x + u[k-4] x b1b2b3bob4x y[k] ~ ~ Now 1 page of maths…

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DSP-CIS / Chapter-5: Filter Realization / Version p. 19 FIR Filter Realizations From (*) (page 15), it follows that (**) Hence there exists a rotation angle θo such that = orthogonal vectors order reduction

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DSP-CIS / Chapter-5: Filter Realization / Version p. 20 FIR Filter Realizations This is equivalent to... Now shaded graph can again be proven(***) to be power complementary system (intuition?). Hence can repeat procedure…

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DSP-CIS / Chapter-5: Filter Realization / Version p. 21 FIR Filter Realizations Repeated application results in `lossless lattice’ explain + + x x x x y[k] + + x x x x ~ ~ u[k] x x + + x x x x + + x x x x

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DSP-CIS / Chapter-5: Filter Realization / Version p. 22 Lossless lattice : also known as `paraunitary lattice‘ each 2-input/2-output section is based on an orthogonal transformation, which preserves norm/energy/power i.e. forms a 2-input/2-output `lossless’ system (=time-domain view). Overall system is realized as cascade of lossless sections (+delays), hence is itself also `lossless’ (see p.16, =freq-domain view) FIR Filter Realizations

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DSP-CIS / Chapter-5: Filter Realization / Version p. 23 FIR Filter Realizations PS : can be generalized to 1-input N-output lossless systems (will be used in Part III on filter banks) (compare to p.21 !) u[k] explain/derive! x x x y[k] + + x x x x + + x x x x ~ ~ ~ N=3

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DSP-CIS / Chapter-5: Filter Realization / Version p. 24 IIR Filter Realizations Construct LTI system such that I/O behavior is given by.. Several possibilities exist… 1. Direct form 2. Transposed direct form PS: Parallel and cascade realization 3. Lattice-ladder form 4. Lossless lattice

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DSP-CIS / Chapter-5: Filter Realization / Version p. 25 IIR Filter Realizations 1. Direct form Starting point is… u[k] x bo x b4 x b3 x b2 x b y[k] ++++ x -a4 x -a3 x -a2 x -a1

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DSP-CIS / Chapter-5: Filter Realization / Version p. 26 IIR Filter Realizations which is equivalent to... PS : If all a i =0 (i.e. H(z) is FIR), then this reduces to a direct form FIR x bo x b4 x b3 x b2 x b y[k] u[k] ++++ x -a4 x -a3 x -a2 x -a1 direct form B(z)

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DSP-CIS / Chapter-5: Filter Realization / Version p. 27 IIR Filter Realizations 2. Transposed direct form Starting point is… x -a4 x -a3 x -a2 x -a1 y[k] u[k] x bo x b4 x b3 x b2 x b1 ++++

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DSP-CIS / Chapter-5: Filter Realization / Version p. 28 IIR Filter Realizations which is equivalent to... x -a4 x -a3 x -a2 x -a1 y[k] u[k] x bo x b4 x b3 x b2 x b1 ++++

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DSP-CIS / Chapter-5: Filter Realization / Version p. 29 IIR Filter Realizations Transposed direct form is obtained after retiming... PS : If all a i =0 (i.e. H(z) is FIR), then this reduces to a transposed direct form FIR transposed direct form B(z) x -a4 x -a3 x -a2 x -a1 y[k] u[k] x bo x b4 x b3 x b2 x b1 ++++

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DSP-CIS / Chapter-5: Filter Realization / Version p. 30 IIR Filter Realizations PS: Parallel & Cascade Realization Parallel real. obtained from partial fraction decomposition, e.g. for simple poles: –similar for the case of multiple poles –each term realized in, e.g., direct form –transmission zeros are realized iff signals from different sections exactly cancel out. =Problem in finite word-length implementation u[k] y[k] + +

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DSP-CIS / Chapter-5: Filter Realization / Version p. 31 IIR Filter Realizations PS: Parallel & Cascade Realization Cascade realization obtained from pole-zero factorization of H(z) e.g. for L even: –similar for L odd –each section realized in, e.g., direct form –second-order sections are called `bi-quads‘ –non-unique: multiple ways of pairing poles and zero & multiple ways of ordering sections in cascade u[k]y[k]

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DSP-CIS / Chapter-5: Filter Realization / Version p. 32 IIR Filter Realizations 3. Lattice-ladder form Derived from combined realization of with… - numerator polynomial is denominator polynomial with reversed coefficient vector (see also p.10) - hence is an `all-pass’ (=`SISO lossless’) filter :

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DSP-CIS / Chapter-5: Filter Realization / Version p. 33 IIR Filter Realizations Lattice-ladder form (v1) (no derivation) y[k] b4 x + b3’ x + b2’’ x + b1’’’ x + x ‘ladder part’

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DSP-CIS / Chapter-5: Filter Realization / Version p. 34 IIR Filter Realizations Lattice-ladder form (v2) (no derivation) y[k] b0 x + b1’ x + b2’’ x + b3’’’ x + x ‘ladder part’

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DSP-CIS / Chapter-5: Filter Realization / Version p. 35 IIR Filter Realizations K i ’s =sin(theta i ) are `reflection coefficients’ Procedure for computing K i ’s from a i ’s again corresponds to `Schur-Cohn’ stability test Orthogonal transformations correspond to 2-input/2-output `lossless’ sections (=time-domain view). Cascade of lossless sections (+delays) is also `lossless’, i.e. `all-pass‘

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DSP-CIS / Chapter-5: Filter Realization / Version p. 36 IIR Filter Realizations PS : Note that the all-pass part corresponds to A(z) (i.e. L angles θ i correspond to L coeffs a i ) while the ladder part corresponds to B(z). If all a i =0 (i.e. H(z) is FIR), then all θ i =0, hence the all-pass part reduces to a delay line, and the lattice-ladder form reduces to a direct-form FIR. PS : `All-pass’ part (SISO u[k]->y[k]) is known as `Gray-Markel’ structure ~

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DSP-CIS / Chapter-5: Filter Realization / Version p. 37 IIR Filter Realizations 4. Lossless-lattice : Derived from combined realization of (possibly rescaled) with... such that… (**) i.e. and are `power complementary’ (p.16-17)

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DSP-CIS / Chapter-5: Filter Realization / Version p. 38 IIR Filter Realizations Lossless-lattice form (no derivation) x u[k] y[k] ~ x + + x x x x x x x + x +

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DSP-CIS / Chapter-5: Filter Realization / Version p. 39 IIR Filter Realizations Orthogonal transformations correspond to (3-input 3-output) `lossless sections‘ Overall system is realized as cascade of lossless sections (+delays), hence is itself also `lossless‘ PS : If all a i =0 (i.e. H(z) is FIR), then all θ i =0 and then this reduces to FIR lossless lattice PS : If all φ i =0, then this reduces to Gray-Markel structure ! !

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DSP-CIS / Chapter-5: Filter Realization / Version p. 40 IIR Filter Realizations PS : can be generalized to 1-input N-output lossless systems (=combine p.23 & p.38 !) x x x y[k] + + x x x x + + x x x x ~ ~ ~ x x x + x + u[k]

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