1. DSP-CIS Chapter-5: Filter Realization Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/stadius/

2. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 2 PART-II : 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

3. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 3 Chapter-5 : Filter Realizations FIR Filter Realizations IIR Filter Realizations PS: Will always assume real-valued filter coefficients

4. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 4 Q:Why bother about many different realizations for one and the same filter? Chapter-5 : Filter Realizations A: See Chapter-6 !

5. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 5 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 Part-III

6. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 6 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]

7. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 7 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]

8. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 8 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 +

9. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 9 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)

10. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 10 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

11. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 11 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…

12. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 12 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

13. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 13 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] ~

14. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 14 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

15. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 15 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 !

16. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 16 FIR Filter Realizations Procedure (page 12) breaks down if |K i |=1 is encountered. Means at least one root of B(z) lies on or outside the unit circle (cfr Schur-Cohn). Then have to select other realization (direct form, lossless lattice, …) for B(z). Lattice form is often applied to `minimum phase’ filters, i.e. filters with only stable zeros (roots of B(z) strictly inside unit circle). Then design procedure never breaks down. Lattice form not overly relevant at this point, but sets stage for similar derivations that lead to more relevant realizations (read on…)

17. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 17 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)

18. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 18 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

19. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 19 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.

20. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 20 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…

21. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 21 FIR Filter Realizations From (*) (page 17), it follows that (**) Hence there exists a rotation angle θo such that = orthogonal vectors order reduction

22. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 22 FIR Filter Realizations This is equivalent to... Now shaded graph can again be proven(***) to be power complementary system (Intuition? Hint: p.24). Hence can repeat procedure…

23. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 23 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

24. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 24 Lossless lattice : Also known as `paraunitary lattice’ (see also Chapter-8) 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.18, =freq-domain view) FIR Filter Realizations

25. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 25 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.23 !) u[k] explain/derive! x x x y[k] + + x x x x + + x x x x ~ ~ ~ N=3

26. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 26 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

27. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 27 IIR Filter Realizations 1. Direct form Starting point is… u[k] x bo x b4 x b3 x b2 x b1 ++++ y[k] ++++ x -a4 x -a3 x -a2 x -a1

28. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 28 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 b1 ++++ y[k] u[k] ++++ x -a4 x -a3 x -a2 x -a1 direct form B(z)

29. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 29 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 ++++

30. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 30 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 ++++

31. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 31 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 ++++

32. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 32 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] + +

33. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 33 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’ u[k]y[k]

34. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 34 IIR Filter Realizations Cascade realization is not unique: - multiple ways of pairing poles and zeros - multiple ways of ordering sections in cascade ``Pairing procedure’’ : - pairing of little importance in high-precision (e.g. floating point implementation), but important in fixed-point implementation (with short word-lengths) - principle = pair poles and zeros to produce a frequency response for each section that is as flat as possible (i.e. ratio of max. to min. magnitude response close to unity) - obtained by pairing each pole to a zero as close to it as possible - procedure : start with pole pair nearest to the unit circle, and pair this to nearest complex zeros. remove pole-zero pair, and repeat, etc…. u[k] y[k] Skip this part

35. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 35 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 :

36. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 36 IIR Filter Realizations Starting point is direct form realization… now 2 pages of maths… will use ‘state vector’

37. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 37 FIR Filter Realizations order reduction

38. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 38 IIR Filter Realizations If A(z) = stable polynomial (should be!), then the Schur-Cohn test guarantees that |Ko|<1. Hence define

39. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 39 IIR Filter Realizations Then this is equivalent to… ++ xx + xx xxxx ++ + u[k] y[k] ~ b4 x xxx ++ + - x x x + + x +

40. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 40 IIR Filter Realizations Procedure can be repeated (explain), leads to `lattice-ladder form’ y[k] b4 x + b3’ x + b2’’ x + b1’’’ x + x ‘ladder part’

41. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 41 IIR Filter Realizations PS: Similar derivation leads to 2 nd ‘lattice-ladder’ form y[k] b0 x + b1’ x + b2’’ x + b3’’’ x + x ‘ladder part’

42. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 42 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’ (see p.35, =freq-domain view)

43. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 43 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 ~

44. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 44 IIR Filter Realizations 4. Lossless-lattice : Derived from combined realization of (possibly rescaled, as on p.17) with... such that… (**) i.e. and are `power complementary’ (p.18-19)

45. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 45 IIR Filter Realizations Starting point is direct form realization… now 2 pages of maths… will use ‘state vector’

46. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 46 IIR Filter Realizations PS: right-hand side of (o) has modulus <1 (hence |sinθo|≠1, cosθo≠0), which follows from ‘modulus property’ for (**) p.44: (should not try to prove this…)

47. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 47 IIR Filter Realizations Now it is proved that From (**) p.44 it follows that (compare to p.21) This leads to order reduction = ‘orthogonal’ vectors

48. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 48 IIR Filter Realizations Rearranging rows as on p.38, etc.., and repeating the order-reduction process, leads to… x u[k] y[k] ~ x + + x x x x x x x + x +

49. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 49 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’ (see p.44) 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 ! !

50. DSP-CIS / Chapter-5: Filter Realization / Version 2014-2015 p. 50 IIR Filter Realizations PS : Can be generalized to 1-input N-output lossless systems (=combine p.25 & p.48 !) x x x y[k] + + x x x x + + x x x x ~ ~ ~ x x x + x + u[k]