Digital Signal Processing II Lecture 3: Filter Realization

1
Digital Signal Processing IILecture 3 Filter
Realization

• Marc Moonen
• Dept. E.E./ESAT, K.U.Leuven
• marc.moonen_at_esat.kuleuven.be
• www.esat.kuleuven.be/scd/

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)
  
• finite word-length issues,
• question implemented filter designed
  filter ?

Lecture-2
Lecture-3
Lecture-4
3
Lecture 3 Filter Realizations

• FIR Filter Realizations
• IIR Filter Realizations
• PS We will assume real-valued filter
  coefficients

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 Part-2

5
FIR Filter Realizations

• 1. Direct form

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

uk
uk-4
uk-3
uk-2
uk-1
b4
b3
b2
b1
x
x
x
x
yk



7
FIR Filter Realizations

• Retiming results in...
  

uk
uk-1
uk-3
uk-2
b1
b4
b3
b2
x
x
x
x
yk



(different software/hardware, same i/o-behavior)
8
FIR Filter Realizations

• Retiming repeated application results in...
  
• i.e. transposed direct form

(different software/hardware, same i/o-behavior)
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

10
FIR Filter Realizations

• Hence starting point is

uk
uk-1
uk-2
uk-3
uk-4
b3
b2
b1
b4
bo
x
x
x
x
x
b4
b1
b2
bo
b3
x
x
x
x
x





yk



11
FIR Filter Realizations

• With
  this can be rewritten as
• 
  
  (ps find fix for case bo0)

sloppy notation
12
FIR Filter Realizations

• This is equivalent to...
• Now repeat procedure
  for shaded graph
• (same
  structure as the one we started from)

uk
uk-3
uk-2
uk-2
b3
b2
b1
bo
x
x
x
x
bo
b1
b2
b3
x
x
x
x






13
FIR Filter Realizations

• Repeated application results in lattice form

explain
uk
bo
ko
k1
k2
k3
yk
( different software/hardware, same i/o-behavior)
14
FIR Filter Realizations

• Lattice form
• Also known as LPC Lattice (linear predictive
  coding lattice)
• Kis are so-called reflection coefficients
• Every set of bis corresponds to a set of
  Kis, and vice versa.
• Procedure for computing Kis from bis
  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 bis, compute reflection
  coefficients Kis
• (following procedure on previous slides).
  Zeros are (proven to be)
• stable if and only if all reflection
  coefficients statisfy Kilt1 !

15
FIR Filter Realizations

• Procedure (page 11) breaks down if Ki1 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. filter 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)
  

16
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)

17
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)

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 can factorize R(z)
  as
• Note that ais 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.
  

19
FIR Filter Realizations

• Starting point is

uk
uk-1
uk-2
uk-3
uk-4
x
x
x
x
x
x
x
x
x
x








yk
20
FIR Filter Realizations

• From () (page 16), it follows that (prove it)
  
• Hence there exists a theta_0 such that

sloppy notation
21
FIR Filter Realizations

• This is equivalent to...
• Now shaded
  graph can again be proven to be power
• 
  complementary system (Why ? Intuition? Hint page
  23).
• Hence can repeat
  procedure

22
FIR Filter Realizations

• Repeated application results in lossless
  lattice

explain
uk
yk
23
FIR Filter Realizations

• Lossless lattice
• also known as paraunitary lattice (see also
  Lecture 6)
• 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.
• Overall system is realized as cascade of
  lossless sections, hence is itself also
  lossless (see also next slides)

24
FIR Filter Realizations

• State-space description
• Example 2nd-order system
• - R is realization matrix
  (A-B-C-D-matrix) for 1-input/2-output system
• - R is orthogonal ! Orthogonality implies
  losslessness (see next slide)

verify!
!!
25
FIR Filter Realizations

• Orthogonality of R implies losslessness
  
• assume initial internal state is x10,
  x20 (2nd order example)
• input sequence is u0,u1,u2,,uL
• corresponding output sequences are
  y0,y1, and y0,y1,...
• then orthogonality implies that
• energy in initial states inputs
  energy in final states outputs
• Equivalent transfer function based property
  (Linfinity) is
• 
  i.e. H(z) is embedded in 1-input/2-output
• 
  lossless system (see page 17)

26
FIR Filter Realizations

• PS Relevance of lattice realizations
  robustness
• example
• o original transfer function
• transfer function after 8-bit
• truncation of lattice filter
• parameters
• - transfer function after 8-bit
• truncation of direct-form
• coefficients (bis)
• 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

27
FIR Filter Realizations

• PS can be generalized to 1-input M-output
  lossless systems
• (will be used in part II on filter banks)
  (compare to
  page 22 !)

uk
M3
yk
explain/derive!
28
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
• PS State space realizations

29
IIR Filter Realizations

• 1. Direct form
• Starting point is

uk
-a4
-a3
-a2
-a1
x
x
x
x
yk




30
IIR Filter Realizations

• which is equivalent to...
• PS If all a_i0 (i.e. H(z) is FIR), then this
  reduces to a direct form FIR

uk




-a4
-a3
-a2
-a1
x
x
x
x
Direct form B(z)
yk




31
IIR Filter Realizations

• -State-space description
• -N delay elements (minimum number max of
  numerator and
• 
  denominator polynomial order max(p,q))
• -2N1 multipliers, 2N adders (minimum number)

32
IIR Filter Realizations

• 2. Transposed direct form
• Starting point is

uk








-a4
-a3
-a2
-a1
x
x
x
x
yk
33
IIR Filter Realizations

• which is equivalent to...

uk




-a4
-a3
-a2
-a1
x
x
x
x
yk
34
IIR Filter Realizations

• Transposed direct form is obtained after retiming
  ...
• PS If all a_i0 (i.e. H(z) is FIR), then this
  reduces to a transposed direct form FIR

uk
Transposed direct form B(z)




-a4
-a3
-a2
-a1
x
x
x
x
yk
35
IIR Filter Realizations

• - State-space description
• i.e. direct forms state space matrices are
  transpose of
• each other (which justifies the name
  transposed dir.form).

36
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

37
IIR Filter Realizations

• PS Parallel Cascade Realization
• Cascade realization obtained from
• 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

38
IIR Filter Realizations

• Cascade 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.

uk
yk
39
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
• 
  
  

40
IIR Filter Realizations

• Starting point is

41
IIR Filter Realizations

• This is equivalent to

prove it !
uk
-



x
x
x
xk-1
x
x
x
x
b0
x
x
x
x






yk

42
IIR Filter Realizations

• Right-hand part has the same structure as what we
  started
• from, hence procedure can be repeated.
• Repeated application leads to lattice-ladder
  form

43
IIR Filter Realizations

• Lattice-Ladder form
• Kis are reflection coefficients
• Procedure for computing Kis (sin(theta_i) !)
  from ais again corresponds to Schur-Cohn
  stability test (cfr. supra)
• all zeros of A(z) are stable (i.e. lie
  inside unit circle)
• iff all reflection coefficients statisfy
  Kilt1 (i1,,N-1)
• (ps procedure breaks down if Ki1 is
  encountered)
• Orthogonal transformations correspond to
  2-input/2-output lossless sections
• see next slide

44
IIR Filter Realizations

• State-space description
• Example 2nd-order system
• - R is realization matrix
  (A-B-C-D-matrix) for uk -gt yk (SISO)
• - R is orthogonal (RT.RI), which implies
  losslessness , i.e.
• (cfr. page 39)

verify!
!!

45
IIR Filter Realizations

• 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 part
  corresponds to B(z). If all a_i0 (i.e. H(z) is
  FIR), then all theta_i0, 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 uk-gtyk) is known
  as Gray-Markel structure

46
IIR Filter Realizations

• 4. Lossless-lattice
• Derived from combined realization of
  (possibly rescaled, as on page 16)
• with...
• - numerator polynomial is C(z) is such
  that
• i.e. and are power
  complementary (p.17-18)

47
IIR Filter Realizations

• Lossless-lattice
• similar derivation (but more complicated -hence
  skipped) leads to

48
IIR Filter Realizations

• Orthogonal transformations correspond to (3-input
  3-output) lossless sections
• State-space description with orthogonal
  realization matrix R (try it) implies overall
  lossless characteristic, i.e.
• 
  (cfr. Page 46)
• PS If all a_i0 (i.e. H(z) is FIR), then all
  theta_i0 and then this reduces to FIR lossless
  lattice
• PS If all phi_i0, then this reduces to
  (retimed) Gray-Markel structure

!
!
49
IIR Filter Realizations

• PS can be generalized to 1-input M-output
  lossless systems
• (combine p.27 p.47
  !)

uk
yk
50
IIR Filter Realizations

• State-space Realizations
• 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 (transposed) direct form is NOT a
  state-space realization, cfr. elements (bi-bo.ai)
  in B or C (page 31 35)

51
IIR Filter Realizations

• Example bi-quad coupled realization
• - direct form realization of bi-quad (2nd
  order IIR) may be unsatisfactory,
• e.g. for short word-lengths and poles near
  z1 or z-1 (see lecture-4)
• - alternative realization

state space description
poles
52
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
• State-space realization with orthogonal
  realization matrix is referred to as orthogonal
  filter
• Relevance see lecture-4