DSP-CIS%20Chapter-5:%20Filter%20Realization

1
DSP-CISChapter-5 Filter Realization

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

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 cant always get what you want
  -Jagger/Richards (?)

Chapter-4
Chapter-5
Chapter-6
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

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

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
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x
x
x
x
yk



7
FIR Filter Realizations

• Retiming results in...
  

uk
uk-1
uk-4
uk-3
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 (pipeline
delays), 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

• Starting point is direct form realization

uk
uk-1
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bo
x
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x
b4
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bo
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x
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x





yk



Now 1 page of maths
11
FIR Filter Realizations

• With
  this can be rewritten as
• 
  
  

PS if ko1, then transformation matrix
is rank-deficient
PS find fix for case bo0
12
FIR Filter Realizations

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

uk
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bo
x
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bo
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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

• 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 (proved to be)
• stable if and only if all reflection
  coefficients statisfy Kilt1 !
• 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 use other realization

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)

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)

17
FIR Filter Realizations

• PS How is computed ?
• Note that if zi (and zi) is a root of
  R(z), then 1/zi (and 1/zi) is also
• a root of R(z). Hence can factorize R(z)
  as
• Note that zis 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.
  

18
FIR Filter Realizations

• Starting point is direct form realization

uk
uk-1
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uk-3
uk-4
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yk
Now 1 page of maths
19
FIR Filter Realizations

• From () (page 15), it follows that ()
  
• Hence there exists a rotation angle ?o such that

orthogonal vectors
order reduction
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

21
FIR Filter Realizations

• Repeated application results in lossless
  lattice

explain
uk
yk
22
FIR Filter Realizations

• 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)

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 !)

uk
N3
yk
explain/derive!
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

25
IIR Filter Realizations

• 1. Direct form
• Starting point is

uk
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-a3
-a2
-a1
x
x
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26
IIR Filter Realizations

• which is equivalent to...
• PS If all ai0 (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




27
IIR Filter Realizations

• 2. Transposed direct form
• Starting point is

uk








-a4
-a3
-a2
-a1
x
x
x
x
yk
28
IIR Filter Realizations

• which is equivalent to...

uk




-a4
-a3
-a2
-a1
x
x
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yk
29
IIR Filter Realizations

• Transposed direct form is obtained after retiming
  ...
• PS If all ai0 (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
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yk
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

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

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
• 
  
  

33
IIR Filter Realizations

• Lattice-ladder form (v1) (no derivation)

ladder part
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34
IIR Filter Realizations

• Lattice-ladder form (v2) (no derivation)

ladder part
b0
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35
IIR Filter Realizations

• Kis sin(thetai) are reflection coefficients
• Procedure for computing Kis from ais 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

36
IIR Filter Realizations

• PS Note that the all-pass part corresponds to
  A(z) (i.e. L angles ?i correspond to L coeffs
  ai) while the ladder part corresponds to B(z).
  If all ai0 (i.e. H(z) is FIR), then all ?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

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)

38
IIR Filter Realizations

• Lossless-lattice form (no derivation)

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 ai0 (i.e. H(z) is FIR), then all
  ?i0 and then this reduces to FIR lossless
  lattice
• PS If all fi0, then this reduces to
  Gray-Markel structure

!
40
IIR Filter Realizations

• PS can be generalized to 1-input N-output
  lossless systems
• (combine p.23 p.38
  !)

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yk