1. Digital Signal Processing II Lecture 3: Filter Realization
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven
homes.esat.kuleuven.ac.be/~moonen/

2. PART-I : Filter Design/Realization
Step-1 : define filter specs
(pass-band, stop-band, optimization criterion,…)
Step-2 : derive optimal transfer funcion
FIR or IIR
Step-3 : filter realization (block scheme/flow graph)
direct form realizations, lattice realizations,…
Step-4 : filter implementation (software/hardware)
accuracy issues, …
question: implemented filter = designed filter ?
Lecture-2
Lecture-3
Lecture-4
Version Lecture-3 Filter Realization

3. Lecture 3 : Filter Realizations
FIR Filter Realizations
IIR Filter Realizations
PS: We will assume real-valued filter coefficients
Version Lecture-3 Filter Realization

4. FIR Filter Realizations
=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
5. Frequency-domain realization: see Part-2
Version Lecture-3 Filter Realization

5. FIR Filter Realizations
1. Direct form
u[k]
u[k-4]
u[k-3]
u[k-2]
u[k-1] x bo + b4 b3 b2 b1
y[k]
Version Lecture-3 Filter Realization

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

7. FIR Filter Realizations
`Retiming’ : results in...
u[k]
u[k-1] x bo + b1
y[k]
u[k-3]
u[k-2] b4 b3 b2
Version Lecture-3 Filter Realization

8. FIR Filter Realizations
`Retiming’ : repeated application results in...
i.e. `transposed direct form’
u[k] x bo +
y[k] b1 b2 b3 b4
different software/hardware, same i/o-behavior
Version Lecture-3 Filter Realization

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
Version Lecture-3 Filter Realization

10. FIR Filter Realizations
Hence starting point is…
u[k]
u[k-1]
u[k-2]
u[k-3]
u[k-4] bo b1 b2 b3 b4 x x x x x b4 b3 b2 b1 bo x x x x x
y[k] + + + +
y[k] ~ + + + +
Version Lecture-3 Filter Realization

11. FIR Filter Realizations
With , this is rewritten as
`sloppy notation’
Version Lecture-3 Filter Realization

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’o +
b’1
b’3
b’2 ko
y[k]
y[k] ~
Version Lecture-3 Filter Realization

13. FIR Filter Realizations
Repeated application results in `lattice form’
u[k] bo x
y[k] + + + + x x x x ko k1 k2 k3
y[k] ~ x x x x + + + +
i.e. different software/hardware, same i/o-behavior
Version Lecture-3 Filter Realization

14. FIR Filter Realizations
Lattice form :
Also known as `LPC Lattice’ (`linear predictive coding lattice’)
Ki’s are so-called `reflection coefficients’
Every set of bi’s corresponds to a set of Ki’s, and vice versa.
Procedure for computing Ki’s from bi’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 bi’s, compute reflection coefficients Ki’s
(following procedure on previous slides). Zeros are stable
if and only if all reflection coefficients statisfy |Ki|<1
Version Lecture-3 Filter Realization

15. FIR Filter Realizations
Lattice form :
Procedure 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).
Lattice form is often applied to `minimum phase’ filters, i.e. filter with only stable zeros (roots of B(z) strictly inside unit circle). Then design procedure never breaks down.
Version Lecture-3 Filter Realization

16. FIR Filter Realizations
4. Lossless lattice :
Derived from combined realization of (possibly scaled) H(z)
with
which is such that
(*)
PS : interpretation ?… (see next slide)
Version Lecture-3 Filter Realization

17. FIR Filter Realizations
PS : Interpretation ?
When evaluated on the unit circle, this 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
Version Lecture-3 Filter Realization

18. FIR Filter Realizations
PS : How is computed ?
Note that if `a’ is a root of R(z), then `1/a’ is also a
root of R(z), hence
Note that ai’s can be selected such that all of them lie inside the
unit circle. Then is a minimum-phase FIR filter.
This is referred to as spectral factorization, =spectral factor.
Version Lecture-3 Filter Realization

19. FIR Filter Realizations
Hence starting point is…
u[k]
u[k-1]
u[k-2]
u[k-3]
u[k-4] bo b1 b2 b3 b4 x x x x x x x x x x
y[k] + + + +
y[k] ~ + + + +
Version Lecture-3 Filter Realization

20. FIR Filter Realizations
From (*) (page 16), it follows that
Hence there exists a theta_0 such that
`sloppy notation’
Version Lecture-3 Filter Realization

21. FIR Filter Realizations
This is equivalent to...
…Now repeat procedure for shaded graph
(=same structure as the one we started from!) (why?)
u[k]
u[k-3]
u[k-2] x
b’o +
b’1
b’2
b’3
y[k] ~
Version Lecture-3 Filter Realization

22. FIR Filter Realizations
Repeated application results in `lossless lattice’
u[k] x
y[k] + x ~ + x + x + x x x x x x x + x + x + x
Version Lecture-3 Filter Realization

23. FIR Filter Realizations
Lossless lattice :
also known as `paraunitary lattice form’ (see Lecture 6)
each section is based on an orthogonal transformation, which preserves norm/energy/power
(= forms a 2-input/2-output `lossless’ system)
hence overall structure preserves the input energy,
i.e. is `lossless’…
Version Lecture-3 Filter Realization

24. FIR Filter Realizations
Orthogonality implies `losslessness’ :
- assume initial internal state is x1[0], x2[0] (2nd order example)
input sequence is u[0],u[1],u[2],…,u[L]
corresponding output sequences are y[0],y[1],… and y[0],y[1],...
then orthogonality implies that (up to scaling with alpha on page 22)
`energy’ in initial states + inputs = `energy’ in final states + outputs
- equivalent transfer function based property is…
i.e. SISO transfer function H(z) is `embedded’
in 1-input/2-output lossless system (see page 17) ~
Version Lecture-3 Filter Realization

25. FIR Filter Realizations
PS : Relevance of lattice realizations : robustness
example :
o = original transfer function
+ = transfer function after 8-bit
truncation of lattice filter
parameters (angles theta)
- = transfer function after 8-bit
truncation of direct-form
coefficients (bi’s)
PS : A 2x2 orthogonal transformation (rotation) may be implemented in
hardware based on a so-called `CORDIC’ (`COR’ for co-ordinate
rotation’) architecture, which is more efficient (and more robust) than a multiply-add based implementation
Version Lecture-3 Filter Realization

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
3. Lattice-ladder form
4. Lossless lattice
5. Parallel realization
6. Cascade realization
7. Bi-quad coupled realization
8. State space realization
Version Lecture-3 Filter Realization

27. IIR Filter Realizations
1. Direct form
Starting point is …
u[k] + + + +
-a1
-a2
-a3
-a4 x x x x
Direct form
for B(z) x bo b4 b3 b2 b1
y[k] + + + +
Version Lecture-3 Filter Realization

28. IIR Filter Realizations
which is equivalent to...
PS : If all a_i=0 (hence H(z) is FIR), then this reduces to a
direct form FIR realization. x bo b4 b3 b2 b1 +
y[k]
u[k]
-a4
-a3
-a2
-a1
Version Lecture-3 Filter Realization

29. IIR Filter Realizations
-State space description:
-N delay elements (=minimum number =max of numerator and
denominator polynomial order = max(p,q))
-2N+1 multipliers, 2N adders
u[k] x bo b4 b3 b2 b1 +
y[k]
-a4
-a3
-a2
-a1
x1[k]
x2[k]
x3[k]
x4[k]
Version Lecture-3 Filter Realization

30. IIR Filter Realizations
2. Transposed direct form
Starting point is…
u[k] + x
-a4
-a3
-a2
-a1
y[k] bo b4 b3 b2 b1
Version Lecture-3 Filter Realization

31. IIR Filter Realizations
which is equivalent to...
u[k] x bo b4 b3 b2 b1 + + + +
-a1
-a2
-a3
-a4 x x x x
y[k]
Version Lecture-3 Filter Realization

32. IIR Filter Realizations
Transposed direct form is obtained after retiming ...
PS : If all a_i=0 (hence H(z) is FIR), then this reduces to a
transposed direct form FIR realization.
Transposed
direct form B(z) x
-a4
-a3
-a2
-a1
y[k]
u[k] bo b4 b3 b2 b1 +
Version Lecture-3 Filter Realization

33. IIR Filter Realizations
- State space description
i.e. direct forms state space matrices are `transpose’ of
each other, which justifies the name `transposition’.
u[k] x
-a4
-a3
-a2
-a1
y[k] bo b4 b3 b2 b1 +
x1[k]
x2[k]
x3[k]
x4[k]
Version Lecture-3 Filter Realization

34. IIR Filter Realizations
3. Lattice-ladder form
Derived from combined realization of
with...
- numerator polynomial is denominator polynomial with
reversed coefficient vector (see also page 9)
- hence is an `all-pass’ (=`SISO lossless’) filter :
Version Lecture-3 Filter Realization

35. IIR Filter Realizations
Starting point is…
u[k] + x b1 b2 b0 b3 b4 a3 a2 a1 1 a4
y[k] ~
x[k] -
Version Lecture-3 Filter Realization

36. IIR Filter Realizations
This is equivalent to…
(prove it!) + x
x’[k-1]
u[k] b0
y[k] ~ -
Version Lecture-3 Filter Realization

37. IIR Filter Realizations
Right-hand part has the same structure as what we started
from, hence repeated application leads to `lattice-ladder form’
u[k]
y[k] ~ x + b0
=lossless section
Version Lecture-3 Filter Realization

38. IIR Filter Realizations
Lattice-Ladder form :
Ki’s are `reflection coefficients’
Procedure for computing Ki’s (=sin(theta_i) !!) from ai’s again corresponds to `Schur-Cohn’ stability test (see lecture-2):
all zeros of A(z) are stable (i.e. lie inside unit circle)
iff all reflection coefficients statisfy |Ki|<1 (i=1,…,N-1)
(ps: procedure breaks down if |Ki|=1 is encountered)
Orthogonal transformations correspond to `lossless’ sections
which also explains `all-pass’ (lossless) characteristic of
Version Lecture-3 Filter Realization

39. IIR Filter Realizations
State-space description
EX: 1st-order system: EX: 2nd-order system:
- R is `realization matrix’ (A-B-C-D-matrix) for u[k] -> y[k] (SISO)
- Observe that R is an orthogonal matrix : R^T.R=I
- Orthogonality of R again implies `losslessness’ (see next slide) ~
Version Lecture-3 Filter Realization

40. IIR Filter Realizations
- Orthogonality of R implies `losslessness’ :
assume initial state is x1[0], x2[0] (2nd order example)
input sequence is u[0],u[1],u[2],…,u[L]
corresponding output sequence is y[0],y[1],y[2],…,y[L]
then orthogonality of R implies that
`energy’ in initial states + inputs = `energy’ in final states + outputs
- In SISO-case, losslessness implies all-pass transfer
function: (see also page 35) ~
Version Lecture-3 Filter Realization

41. IIR Filter Realizations
PS : `All-pass’ part (u[k]->y’[k]) is known as `Gray-Markel’ structure
PS : Note that the all-pass part corresponds to A(z) (i.e. N angles theta_i
correspond to N coefficients a_i), while the ladder section corresponds to B(z).
If all a_i=0 (hence H(z) is FIR !), then all theta_i=0, hence the
all-pass part reduces to a delay line, and the lattice-ladder form
reduces to a direct-form FIR realization. x + in
out
Version Lecture-3 Filter Realization

42. IIR Filter Realizations
4. Lossless-lattice :
Derived from combined realization of
with...
- numerator polynomial is C(z) is such that
i.e and are `power complementary’
Version Lecture-3 Filter Realization

43. IIR Filter Realizations
`Lossless-lattice’ :
similar derivation (but more complicated -hence skipped) leads to… x +
u[k]
y[k] ~
=lossless section
Version Lecture-3 Filter Realization

44. IIR Filter Realizations
Orthogonal transformations correspond to `lossless sections
hence overall lossless characteristic, i.e. SISO transfer function H(z) is `embedded’ in 1-input/2-output lossless system
Note that 2N+1 filter coefficients (a_i’s and b_I’s) define 2N+1 rotation angles. If all a_i=0 (hence H(z) is FIR), rotation angles will be such that all feedback paths are cancelled (in a magic way)
Example: (draw it/try it!)
Version Lecture-3 Filter Realization

45. IIR Filter Realizations
state-space description
1st-order system: etc….
R is `realization matrix’ (A-B-C-D-matrix) for u[k] ->y[k], y[k] (1-in/2-out)
Observe that R is an orthogonal matrix : R^T.R=I
Orthogonality of R again implies `losslessness’ ~
Version Lecture-3 Filter Realization

46. IIR Filter Realizations
5. Parallel Realization :
partial fraction decomposition
e.g. for simple poles:
similar for the case of multiple poles
each term realized in, e.g., direct form
same number of delay elements, additions and multiplications (if p=q) as direct forms
transmission zeros are realized iff signals from different sections exactly cancel out.
=Problem in finite word-length implementation
u[k]
y[k] +
Version Lecture-3 Filter Realization

47. IIR Filter Realizations
6. Cascade Realizations :
pole-zero factorization of H(z)
e.g. for N even:
similar for N odd
each section realized in, e.g., direct form
second-order sections are called `bi-quads’
same number of delay elements, additions and multiplications as direct forms
u[k]
y[k]
Version Lecture-3 Filter Realization

48. IIR Filter Realizations
realization is non-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]
Version Lecture-3 Filter Realization

49. IIR Filter Realizations
7. Bi-quad `coupled realization’ :
- direct form realization of bi-quad may be unsatisfactory, e.g. for short
word-lengths and poles near z=1 or z=-1 (see lecture-4)
- alternative realization: + -
y[k]
u[k]
state space description :
poles :
Version Lecture-3 Filter Realization

50. IIR Filter Realizations
8. State-space Realization :
State-space description:
State-space realization = realization such that all the elements of A,B,C,D are the multiplier coefficients in the structure
Example: bi-quad `coupled’ realiz.
Example: (transposed) direct form is NOT a state-space realization, cfr. elements (bi-bo.ai) in B or C
Example: lattice-ladder, lattice ? (not quite, but close enough…see p.55) +
y[k]
u[k]
Version Lecture-3 Filter Realization

51. IIR Filter Realizations
State-space description/realization is non-unique:
any non-singular matrix T transforms A,B,C,D into an alternative state-space description/realization
with
transfer function is
(prove it)
impulse response is
Version Lecture-3 Filter Realization

52. IIR Filter Realizations
PS : orthogonal filters : state-space realization with
orthogonal realization matrix:
Relevance : see lecture-4 (scaling, quantization noise power, limit
cycles, etc…)
PS : lattice filters and lattice-part of lattice-ladder ….
Strictly speaking, these are not state-space realizations (cfr. supra), but orthogonal R is realized as a product of matrices, each of which is again orthogonal, such that useful properties of orthogonal states-space realizations indeed carry over….
Version Lecture-3 Filter Realization