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Digital Signal ProcessingAdvanced Methods

Nicolas Dobigeon

University of ToulouseIRIT/INP-ENSEEIHT

http://www.enseeiht.fr/[email protected]

Nicolas Dobigeon Digital Signal Processing - Advanced Methods 1 / 205

http://www.enseeiht.fr/~dobigeon






Prerequisites

Representation tools

Complex variable and z transform

Laplace transform

Digital Signal Processing - Basics (TNS1)

Discrete Fourier Transform (DFT/TFD)

properties fast algorithm

Digital filtering

linear filtering (time-invariant, causal) FIR and IIR filters direct synthesis of FIR filters (windowing) synthesis of IIR filters implementation (direct, canonical, serial and parallel structures)




Bibliography

M. Bellanger, Traitement Numerique du Signal - Theorie et

Pratique , Masson, 1994.

F. Castanie, Traitement Numerique du Signal - Methodes avancees ,

Polycopie ENSEEIHT, 2002. B. Porat, A course in digital signal processing , Wiley, 1997.

J. G. Proakis and D. Manolakis, Digital Signal Processing -

Principles, Algorithms and Applications , Pearson, 1996.

IEEEXplore Digital Library, http://ieeexplore.ieee.org.

www...


http://ieeexplore.ieee.org/

http://ieeexplore.ieee.org/



Examples of applications

signal processing

denoising, compression, detection,...

image processing

edge enhancement, compression, notions of spatial frequency, 2D and 3D processing,...

array processing

notion of wavefront,

digital communications (e.g., mobiles)

notions of multipath, equalization...

⇒ Requires to implement linear filters!




Linear and time-invariant filter (LTI)

Properties

Linearity:

F [ax 1(n) + bx 2(n)] = aF [x 1(n)] + b F [x 2(n)]

= ay 1(n) + by 2(n).

Necessary and sufficient condition:

∃h(n, k ) = F [δ (n − k )] such that y (n) =

+∞k =−∞

x (k )h(n, k )

Time-invariance:

∀n0, y (n) = F [x (n)] ⇔ y (n − n0) = F [x (n − n0)] . (1)

Necessary and sufficient condition:∀n, ∀k , h(n, k ) = h(n − k ).

⇒ Linear and time-invariant filter fully characterized by its impulse responseh(k ) (k ∈ Z) or its transfer function H (z ) = TZ [h(k )]⇒ Filtering identity becomes y (n) =

k x (k )h(n − k ) = h(n) ∗ x (n) (temporal

convolution) or Y (z ) = H (Z )X (Z ) (“frequency” product).Nicolas Dobigeon Digital Signal Processing - Advanced Methods 5 / 205



Linear and time-invariant filter (LTI)

Properties

Linearity:

F [ax 1(n) + bx 2(n)] = aF [x 1(n)] + b F [x 2(n)]

= ay 1(n) + by 2(n).

Necessary and sufficient condition:

∃h(n, k ) = F [δ (n − k )] such that y (n) =

+∞k =−∞

x (k )h(n, k )

Time-invariance:

∀n0, y (n) = F [x (n)] ⇔ y (n − n0) = F [x (n − n0)] . (1)

Necessary and sufficient condition:∀n, ∀k , h(n, k ) = h(n − k ).

⇒ Linear and time-invariant filter fully characterized by its impulse responseh(k ) (k ∈ Z) or its transfer function H (z ) = TZ [h(k )]⇒ Filtering identity becomes y (n) =

k x (k )h(n − k ) = h(n) ∗ x (n) (temporal

convolution) or Y (z ) = H (Z )X (Z ) (“frequency” product).




Outline

OptimizationNon-optimal synthesis of FIR filterNon-optimal synthesis of IIR filterOptimization methodsStabilization of the solutions

Digital effectsSome remindersInput quantizationCoefficient quantizationQuantization in calculations

Non-standard structuresLadder/lattice structuresFraction-based structuresMultirate structures


Optimization



Optimization

Outline

OptimizationNon-optimal synthesis of FIR filter

Direct synthesisSampling in the frequency domain

Non-optimal synthesis of IIR filterDirect synthesis

Pade approximantsOptimization methods

Least-square methodRemez algorithmEigenfilters

Stabilization of the solutions

Digital effects

Non-standard structures


Optimization



Optimization

Non-optimal synthesis of FIR filter

Outline




Pade approximantsOptimization methodsLeast-square methodRemez algorithmEigenfilters


Digital effects



Optimization



Optimization


Some reminders on FIR

Characterized by a finite impulse response:

h(k ) =

b k , k = 0, . . . , M 0, otherwise.

Function transfer:

H (z ) =M

k =0

b k z −k =M

k =0

h(k )z −k

Properties

non-recursive computation

→ stability, weak digital sensivity linear phase possible if symetry of the IR

h(k ) = ±h(M − k ), k = 0, . . . , M , (suivant parite de M )


Optimization



Optimization


Direct synthesis

Direct synthesis (reminder)

Let H I(z )z =e i 2πf denote the ideal frequency transfer function.

Fourier series expansion of H I

e i 2πf

→ RI

H I

e i 2πf

=+∞

k =−∞

c k e i 2πf

with c k = hI(k ) ∝ 1/2−1/2

H I

e i 2πf e −i 2πf df

Windowed truncation of the IR (filter order fixed to M = 2L + 1),

h0(k ) = hI(k )w (k ), k = −L, . . . , L

Optional steps: “Iterations” (order M et window w (·)), Shift of the IR to make it causal

hN(k ) = h0(k − k 0) (with k 0 ≥ L)


Optimization



p


Direct synthesis


Influence of the order (natural window)


Optimization




Direct synthesis


Influence of the order (natural window)


Optimization




Direct synthesis


Influence of the window (order fixed to M = 22)


Optimization




Sampling in the frequency domain


Let a FIR filter whose transfer function H I (z ) is specified by (M + 1) pointsz 0, . . . , z M .

H = {H I (z 0), . . . , H I (z M )} is the set of (strong) constraint.

Let the M-order polynomial approximation

H (z ) =

M n=0

Ln (z ) H I (z n) with Ln (z m) = δ (m − n)

Remark: we have H (z m) = H I (z m) (∀m)

The (unique) Lagrange polynom is

Ln (z ) =M

m=n

1 − z −1z m

1 − z −1n z m


Optimization






H I(z ) can be rewritten

H (z ) =

M n=0

1 − z nz −1

M m=0

Am

1 − z mz −1

with

Am =

H I (z m)n=m

1 − z nz −1m

If the constraints H are specified by equidistant points over the unit circlez n = e i 2π

nM +1 then we have

H (z ) =1 − z M +1

M + 1

M

m=0

H I

e i 2π mM +1

Gain

11 − e i 2π

mM +1 z −1

1st-order cell

Parallel 1st-order cells


Optimization

N i l h i f IIR fil



Non-optimal synthesis of IIR filter

Outline




Pade approximantsOptimization methodsLeast-square methodRemez algorithmEigenfilters


Digital effects



Optimization

N ti l th i f IIR filt




Some reminders on IIR

Characterized by the linear recurrence with constant coefficients:

y (n) = −M

k =1

ak y (n − k ) +M

k =1

b k x (n − k )

where ∃ j ∈ {1, . . . , M } such that a j = 0 (with a0 = 1),

Corresponding function transfer:

H (z ) =

M k =0 b k z −k M k =0 ak z −k

=+∞

k =−∞

h(k )z −k

Properties: efficient ( ≈ selective) but (in)stability defined by the pole positions digital sensivity (error propagations) non-constant group delay


Optimization

Non optimal synthesis of IIR filter




Direct synthesis


Let the ideal transfer function H I(z )z =e i 2πf

Choice of a required feature

Preserving the temporal response


Optimization





Direct synthesis


Let the ideal transfer function H I(z )z =e

i 2πf

Choice of a required feature

Preserving the temporal response Preserving the harmonic response

Computing the analog template thanks to the transformation

f A = f E

π tan

πf N

Synthesis of H A(p ) (p = i 2πf A) ensuring the analog template (choosing

order, prototype...)

Coming back in the Z-domain by the bilinear transform

p = 2f E 1 − z −1

1 + z −1


Optimization




Non optimal synthesis of IIR filter

Pade approximants

Pade approximants

We propose to approximate the ideal transfer functionH I(z ) =

+∞k =−∞ h(k )z −k by the the rational approximant

G m,n(z ) = B m(z )

An(z ) =

mk =0 b k z −k

nk =0 ak z −k

, with u 0 = 1

where B n(·) et Am(·) are polynomials of degrees n and m, resp.

G m,n(z ) is then expanded into the serie

G m,n(z ) =+∞

k =−∞

g (k )z −k

and we impose

g (k ) =

h(k ), for k = 0, . . . , n + m;any, for k ≥ n + m + 1


Optimization




p y

Pade approximants

Pade approximants

If we write

G m,n(z ) =

mk =0 b k z −k nk =0 ak z −k

=m+nk =0

h(k )z −k + O

z −(n+m+1)

then, it comes

mk =0

b k z −k

degree m

=2n+m

k =0

c k z −k

degree 2n + m

+ with c k =

inf(k ,n) j =0

a j h(k − j )

Necessarilyc k = 0 ∀k ∈ {m + 1, . . . , 2n + m} ,

We cancel all the terms c k for k = m + 1, . . . , 2n + m,




Optimization




Pade approximants

Pade approximants

Some remakds When the orders n et m are fixed, if it exists1, the couple {An, B m} is unique2 (with a0 = 1),

sometimes denoted by [m/n] (but the fraction B m(z )/An(z ) is not necessarily irreductible ).

The Pade approximants G n,m = B m/An are grouped in the Pade table indexed by n and m.

The matrices H1 and H2 are Toeplitz et lower triangular Toeplitz for which there are someefficient algorithms to perform inversion and product.

The approximant G m,n(z ) and the targeted transfer function H I(z ) have IRs that coincideon their first (n + m + 1) points. Above, nothing is imposed.→ Possible instability of the approximant!

See also:

C. S. Burrus, T. W. Parks, “Time Domain Design of Recursive Digital Filters”, IEEE Trans.Audio Electroacoustics , vol. AU-18, no. 2, June 1970.

1Sufficient condition: det H1 = 0.2

Frobenius, 1881.Nicolas Dobigeon Digital Signal Processing - Advanced Methods 22 / 205

Optimization

Optimization methods



Outline



Non-optimal synthesis of IIR filterDirect synthesisPade approximants

Optimization methodsLeast-square methodRemez algorithmEigenfilters


Digital effects



Optimization




General principle

Let H I(f ) the ideal transfer function.

We introduce a parameter vector θ that fully characterizes the M -order

digital filter H N(f ) (FIR or IIR) , We define the synthesis error e (f ) = H I(f ) − H N(f ),

If the specified constraints are not the same in all the frequency subbands,we introduce a weighting function P (f )

e P (f ) = P (f ) [H I(f ) − H N(f )] ,

and the corresponding “global” error criterion

J e P (θ) = e P (f ),

Then we are looking for θopt that is solution of

θ

opt

= argminθ∈Θ J e P (θ)Remarks:

H optN (f ) may not satisfy the template → we iterate on M , θ depends in the chosen synthesis structure.


Optimization




Direct structure:

H N(z ) = M k =0 b k z −k

M

k =0 ak z −k

, (with a0 = 1)

→ θD = [a1, . . . , aM , b 0, . . . , b M ]T

Serial structure (after grouping conjugate complex roots):

H N(z ) = b 0 M k =1(1 − z k z −1)

M

k =1(1 − p k z −1)

= b 0

K

k =1

H k (z ), (with M

2

≤ K ≤ M )

where H k (z ) = 1+b k ,1z −1+b k ,2 z −2

1+ak ,1z −1+ak ,2z −2 are bi-quadratic structures,

→ θS = {θS,1, . . . , θS,K } with θS,k = [ak ,1, ak ,2, b k ,1, b k ,2]T

Parallel structure (with simple pole):

H N(z ) = c +K

k =1

H k (z ),

where H k (z ) = 1+b k ,1z −1

1+ak ,1z −1+ak ,2z −2 are bi-quadratic structures,

→ θP = {θP,1, . . . , θP,K } with θP,k = [ak ,1, , ak ,2, b k ,1]T


Optimization




General principle

Let Soit H I(f ) the ideal transfer function.




e P (f ) = P (f ) [H I(f ) − H N(f )] ,


J e P (θ) = e P (f ),

Then we are looking at θopt that is solution of

θ

opt




Optimization




General principle

Let Soit H I(f ) the ideal transfer function.




e P (f ) = P (f ) [H I(f ) − H N(f )] ,


J e P (θ) = e P (f ),

Then we are looking at θopt that is solution of

θ

opt



Which norm?


OptimizationOptimization methods

Least square method



Least-square method

2-norm: least-square method

Let the error term defined in N 0 frequency points

e P (f n) = P (f n) [H I(f n) − H N(f n)] , n = 0, . . . , N 0 − 1,

The criterion can be written

J e P (θ) = eP 2

2 =

N 0−1n=0

P 2

(f n) [H I(f n) − H N(f n)]2

,

where eP = [e P (f 0), . . . , e P (f N 0−1)]T ,

assumption: we have a sub-optimal initial solution θ(0) → local

optimization.

From the initial solution θ(0), we are looking for the growth ∆θ such that

∆θ = argmin∆θ∈Θ

J e P

θ

(0) + ∆θ



Least-square method



Least square method


We expand the criterion J e P θ

(0) + ∆θ

to the 1-st order

J e P

θ

(0) + ∆θ

= J e P

θ

(0)

+

N −1 j =0

∂ J e P

∂θ j

∆θ j

and all the gradient coordinates ∇J e P are set to 0:

∂ J e P

∂θk

+N −10=1

∂ 2J e P

∂θk ∂θ j

∆θ j = 0, k = 0, . . . , N − 1

We obtain the linear system

E1Pe + E

1PET

1 ∆θ = 0

with

e = [e (f 0), . . . , e (f N 0−1)]T , E1 =

∂ e (f n)

∂θk

k ,n

et P = diag

P 2(f n)

.



Least-square method



Least square method


The solution is

∆θ = −

E1PET 1

−1

E1Pe.

Remarks:

method that relies on the computation of ∂ e (f n )∂θ

k

,

→ explicit for most of the synthesis structures,

Bellanger3 provides the computation for a N -order FIR filter with theparameter vector θ = [H 1, . . . , H N −1]T where H k = TFD [h(n)],

no constraints on the stability of the solution,→ stability of the solution not ensures!

3

M. Bellanger, Traitement Numerique du Signal , Masson, 1994.Nicolas Dobigeon Digital Signal Processing - Advanced Methods 30 / 205


Least-square method



q


Example of the synthesis of a FIR filter

→ Flexibility in the attenuated bands not exploited!



Least-square method




Example of the synthesis of a FIR filter

→ Flexibility in the attenuated bands not exploited!





Remez algorithm



∞-norm: Remez algorithm

If f 0, . . . , f M are known then θ are computed by matrix inversion:

θ = θopt ⇔ {e (f i ) = δ }i

→ On ne connaıt pas f 0, . . . , f M −1 !

New optimization problem:

{δ, f , θ} = argminδ,f ,θ

P (f i ) [H I(f i ) − H N(f i )] − (−1)i δ i =0,...,M

with f = (f 0, . . . , f M ),

Alternating minimization (iterative algorithm):

f (r ) = argminf

P (f ) H I (f ) − H (r −1)N (f ) − (−1)i δ (r −1)

δ (r ), θ(r )

= argminδ,θ

P

f (r )i

H I

f (r )i

− H N

f

(r )i

− (−1)i δ

i



Remez algorithm




From a sub-otpimal solution θ(0) et δ (0), at iteration r

computing the error e (r −1)

P (f ) = P (f ) H I(f ) − H (r −1)

N (f ),

localizing the extrema of e (r −1)P (f ), i.e., de

f

(r )0 , . . . , f

(r )M

t.q.

e (r −1)

P

f (

r )i

> δ (r −1),

computing θ(r ) and δ (r ) by resolution of the (M + 1) linear equations

P

f (r )i

H I

f (r )i

− H

(r )N

f

(r )i

= ±δ (r )

i.e., system inversion

H I

f

(r )0

...

H I

f

(r )M

=

1 cos 2πf (r )0 . . . cos 2πf (r )

0 (M − 1) (−1)0

P

f (r )0

.

.

....

.

.

....

1 cos

2πf (r )M

. . . cos

2πf

(r )M

(M − 1)

(−1)M

P

f (r )M

h(r )0

.

.

.

h(r )M −1

δ(r )



Remez algorithm




Example of the synthesis of a FIR filter (iteration 1)



Remez algorithm







Remez algorithm




Comparison 2/ ∞



Remez algorithm

R l i h




Stopping rule: iteration number R ,

error:e R

p (f ) < seuil,

error variation:

e R −1

p (f )

−

e R

p (f )

< seuil,

Advantages: optimal method (!),

constant amplitude fluctuation in the attenuated and non-attenuatedbands,

minimum filter order,

Drawbacks:

Computationally intensive (iterative method, matrix inversion),



Remez algorithm

R l i h




Regarding the synthesis method:

convergence of the algorithm not demonstrated,→ “correct” behavior if “correct” initialization→ non-optimal solution required!

algorithm implemented with Matlab,

extension to the synthesis of a IIR filter proposed by Bellanger

4

.Some more general remarks:

minimization of a ∞-norm sometimes called Chebyshev approximation,

minimizing the error max.: minimax approximation,

general polynomial approximation method

→ for DSP, Parks-McClellan synthesis algorithm5.

4M. Bellanger, Traitement Numerique du Signal , Masson, 1994.5T. W. Parks et al., “Chebyshev approximation for non recursive digital filters with linear

phase”, IEEE. Trans. Circuit Theory, vol. 19, no. 2, March 1972.



Eigenfilters

Ei filt (FIR)



Eigenfilters (FIR)

Let consider a FIR filter with linear phase

H N(f ) =M −1k =0

hk cos (2πfk ) = θT c(f ), with c(f ) = [cos (2πfk )]

k ,

The criterion6 can be written θ = [h0, . . . , hM −1]T

J e (θ) = e (f )22 = f 2

f 1

|H I(f ) − H N(f )|2 df ,

If the ideal filter owns a constant transfer function H I(f 1) ≈ θT c(f 1) in the

subband [f 1, f 2], J e (θ) can be rewritten

J e (θ) = f 2

f 1

θT c(f 1) − θT c(f )2

df = θT Qrif θ

with Qrif =

f 2

f 1

[c (f 1) − c (f )] [c (f 1) − c (f )]T df

6 · 2 is the 2-norm in the functional space C0([f 1, f 2])Nicolas Dobigeon Digital Signal Processing - Advanced Methods 43 / 205


Eigenfilters

Eigenfilters (FIR)



Eigenfilters (FIR)

The synthesis problem consists of looking for

θopt = argmin

θ∈Θ

J e (θ) ⇔ θopt = argmin

θ∈Θ

θT Qrif θ

→ the solution is given by analyzing the Rayleigh ratio R Qrif (·) !

Property: Let A an hermitian matrix and R A (x) = xT AxxT x

then

minθ∈Θ

R A (x) = λmin

et xopt = argminθ∈Θ

R A (x) ⇔ xopt = vmin

where vmin is the eigenvector associated with the smallest eigenvalue λmin of A.



Eigenfilters

Eigenfilters (IIR)



Eigenfilters (IIR)

Let consider the IIR filter of transfer function

H N(z ) =

M −1k =0 b k z −k M −1k =0 ak z −k

,

If x (n) = δ (n) is the input, we want y (n) = hI(n) (ideal IR) at theoutput7:

M −1k =0

b k hI(n − k ) ≈M −1k =0

ak δ (n − k )

The error8 can be written for n = 0, . . . , N 0:

e (n) =

M −1k =0

b k hI(n − k ) −

M −1k =0

ak δ (n − k ),

7S.-C. Pei and J.-J. Shyu, “Design of 1-D and 2-D IIR Eigenfilters”, IEEE Trans. Signal Processing , vol. 42, no. 4, April 1994.

8Warning: e (n) = hI(n) − hN(n) !



Eigenfilters

Eigenfilters (IIR)



Eigenfilters (IIR)

We define the parameter and error vectors:

θ = [a0, . . . , aM −1, b 0, . . . , b M −1]T , e = [e (0), . . . , e (N 0)]T

Using a matrix formulation:

Hθ = e where H = H1 −I

H2 0 ,

with

H1 =

hI(0) 0 . . . 0hI(1) hI(0) 0

.

.

....

. . . 0hI(M − 1) hI(M − 2) . . . hI(0)

,

H2 =

hI(M ) hI(M − 1) . . . hI(1)

hI(M + 1) hI(M ) hI(2)

.

.

.... 0

hI(N 0) hI(N 0 − 1) . . . hI(N 0 − M + 1)

.



Eigenfilters

Eigenfilters (IIR)



Eigenfilters (IIR)

The (weighted) quadratic criterion is:

J e P (θ) = eP

22 =

N 0n=0

P 2(n)e 2(n),

with eP = [e P (0), . . . , e P (N 0)]T et e P (n) = P (n)e (n),

Or eP 22 = e

T

Pe with P = diag P 2

(n) (et e = Hθ), so

J e P (θ) = θ

T Qriiθ,

with Qrii = HPH,

The synthesis problem consists of looking for

θopt = argmin

θ∈Θ

J e P (θ) ⇔ θ

opt = argminθ∈Θ

θT Qriiθ

→ The solution is given by analyzing the Rayleigh ratio R Qrii (·) !



Eigenfilters

Optimization methods: summary



Optimization methods: summaryLeast-square method

θ depends of the synthesis structure,

J e P (θ) = eP 22 = n P

2

(f n) [H I(f n) − H N(f n)]2

,→ optimization by matrix inversion,

Remez algorithm

θ = [h0, . . . , hM −1]T for a FIR with linear phase,

J e P (θ) = e P (f )L∞ = supf P (f ) [H I(f ) − H N(f )] ,→ iterative procedure: optimization by alternate optimization,

Eigenfilters (FIR)

θ = [h0, . . . , hM −1]T pour un FIR a phase lineaire,

J e (θ) = e (f )2L2

=

|H I(f ) − H N(f )|2 df ,

→ optimization by spectral analysis (SVD of Qrif ),Eigenfilters (IIR)

θ = [a0, . . . , aM −1, b 0, . . . , b M − 1]T ,

J e P (θ) = e (f )2

2 =

n P 2(n)

k b k hI(n − k ) − ak δ (n − k )

2,

→ optimization by spectral analysis (SVD of Qrii),


OptimizationStabilization of the solutions

Outline



Outline



Non-optimal synthesis of IIR filterDirect synthesisPade approximants

Optimization methodsLeast-square methodRemez algorithmEigenfilters


Digital effects







S

To ensure the stability of

H N(z ) = V (z )M −1

k =0 (1 − p k z −1),

we should solve the constrained optimization problem

minθ∈Θ

J e P (θ) such that

|p k | < 1

M −1

k =0

→ possible with θ = [p 0, . . . , p M −1]T (structure serie/cascade),→ otherwise?

Alternative: unconstrained optimization, then stabilization.






Let p 0 a unstable 1-st order, i.e., |p 0| > 1,

The transfer function H N(z ) can be factorized:

H N(z ) = 1

1 − p 0z −1H N(z ), with H N(p 0) = 0

Let the all-pass filter

G (z ) = 1

p 0

1 − p 0z −1

1 −

1p ∗0

z −1 , with |G (z )| = 1

We define

H N(z ) = G (z )H N(z ) = 1

p 0 1 − 1

p ∗0 z −1H N(z ),

p 0 is not an (unstable) pole of H N(z ), 1/p ∗0 is a stable pole of H N(z ) car |1/p ∗0 | = 1/|p 0| < 1, and

H N(z )

=

H N(z )

.






Interpretation in the C plan:

p 0 = r 0e i θ0

1p ∗0

= 1r 0

e i θ0

recursively on H N(z ), each pole p k outside C(0, 1) is replaced with 1/p ∗k .

Remark Same procedure on the numerator V (z ) ⇒ filter of minimum phase.


Optimization





Interpretation in the C plan:

p 0 = r 0e i θ0

1p ∗0

= 1r 0

e i θ0

recursively on H N(z ), each pole p k outside C(0, 1) is replaced with 1/p ∗k .

Remark Same procedure on the numerator V (z ) ⇒ filter of minimum phase.


Digital effects

Outline



Optimization

Digital effectsSome reminders

Fixed-point representationDigital effects

Input quantizationCoefficient quantizationQuantization in calculations

Constant inputEntrees variables



Digital effects

Some reminders

Outline



Optimization







Digital effects

Some reminders

Fixed-point representation

Some reminders




Every number x is encoded

x q = (±) sign bit

,

integer part b nE , . . . , b 0, b −1, . . . , b nF

fractional part

with b j ∈ {0, 1}

Constraints:

Quantum: ∆q = 2−nF

Dynamic: x maxq =

nE j =−nF

2 j = 2nE+1 − 2−nF

Example : nE = 2, nF = 2

x maxq = 1 × 20 + 1 × 21 + 1 × 22

=7

+ 1 × 2−1 + 1 × 2−2 =3/4

= 8 − 1/4


Digital effects

Some reminders


Some remindersFi d i i




(Un-)biased quantization,

Figure: Un-biased (left) and biased (right) quantizations.


Digital effects

Some reminders


Some remindersFi d i i





Digital effects

Some reminders


Some remindersFi d i t t ti



Digital effects

Some reminders

Digital effects

Some remindersDigital effects



Digital effects

Overflow

If x = x max + ∆q then

x q =

−x max

q , without protection;x maxq , with protection.

Figure: Overflow without (left) and with (right) protection.


Digital effects

Input quantization

Outline



Optimization







Digital effects

Input quantization

Input quantization



ANC ⇒ input noise e x (n) that propagates into the filter

x q(n) ≈ x (n) + e x (n) ⇒ y q(n) ≈ y (n) + e y (n)

Deterministic model (x (n) = constant )

bounded ouptut error:

|e y (n)| =

k

e x (n − k )h(k )

≤

k

|e x (n − k )h(k )|

≤ ∆q

2

k

|h(k )| (if unbiased quantization)


Digital effects

Input quantization

Input quantization



ANC ⇒ input noise e x (n) that propagates into the filter

x q(n) ≈ x (n) + e x (n) ⇒ y q(n) ≈ y (n) + e y (n)

Deterministic model (x (n) = constant )

bounded ouptut error:

|e y (n)| =

k

e x (n − k )h(k )

≤

k

|e x (n − k )h(k )|

≤ ∆q 2

k

|h(k )| (if unbiased quantization)


Digital effects

Input quantization

Input quantization



Random model (x (t ) = constant )

White noise as input:

R e x (τ ) = ∆q 2

12 δ (τ ) ⇔ S e x (f ) =

∆q 2

12

Power spectral density (Wiener-Lee relation):

S e y (f ) = |H (f )|2 ∆q 2

12

Overall power:

P e y =

12

− 12

S e y (f )df = ∆q 2

12

12

− 12

|H (f )|2df

and, from the Parseval identity:

P e y = ∆q 2

12

n

|h(n)|2.


Digital effects

Coefficient quantization

Outline



Optimization







Digital effects





θq = θ + ∆θ ⇒ H q(z ) = B (z )+∆B (z )A(z )+∆A(z )

Sensitivity: moving of the poles (and/or zeros)

We introduce the polynomial:

A(z ) = k

ak z −k = m 1 − p mz −1 We can show that:

∂ p m∂ ak

= p M −k

m

j =m(p j − p m)

→ big displacement if ∃ j = m s.t. p j − p m ≈ 0

consequence (for a stable IIR):if M ↑ ⇒ in the the circle C (0, 1), density of the poles ↑ ⇒ sensivity ↑ !

→ in practice, implementation with 2nd-order cells (∀ structure)


Digital effects


Coefficient quantization2nd-order purely recursive filter



p y

Aq(z ) = 1 + a1,qz −1 + a2,qz −2 = (1 − p 1,qz −1)(1 − p 2,qz −1)

with p i ,q = 12−a1,q ± a21,q − 4a2,q

Location of the stable poles in the z-space

Finite numbers of couples (a1,q, a2,q) such that p i ,q in the unity circle.

Figure: Location of pole and zeros with quantization of quantum∆q = 2−3 (left) and ∆q = 2−4 (right).


Digital effects





Location of the stable poles in the space (a1, a2)

Figure: Location of stable poles and zeros with quantization of quantum∆q = 2−2.


Digital effects





Location of the stable poles in the space (a1, a2)

Figure: Location of stable poles and zeros with quantization of quantum∆q = 2−3.


Digital effects

Quantization in calculations

Outline



Optimization







Digital effects





Figure: Position of the quantizers in a D-N structure.


Digital effects





Figure: Position of the quantizers in a D-N structure.

→ propagated error: w q(n) = Q

−

k Q [ak w q(n − k )] + x (n)


Digital effects

Quantization in calculationsConstant input




w q(n) = Q −k Q [ak w q(n − k )] + x (n)

→ recursive non-linear equation

Several possible behaviors, depending on the initial conditions:

point attractor (fixed point),→ constant input ⇒ constant output

periodic attractor: limit-cycle,→ not satisfying since periodic signal generator!→ oscillation amplitude Aa bounded by

Aa ≤ ∆q

2 k

|h(k )| et Aa ≤ ∆q

2

supf

|H (f )|

(for an un-biased quantizer)

other attractors (strange, periodic-point,...): ...


Digital effects


Quantization in calculationsConstant input: 2nd-order purely recursive filter




Digital effects





Figure: Trajectory of (x 1(n), x 2(n)) with L = 13.from [Ling et al , Int. J. Bifurcation and Chaos , 2003]


Digital effects







Digital effects







Digital effects

Quantization in calculationsEntrees variables

Quantization in calculationsVariable input



Random model of the quantization noise

⇔

with

var [e q,x (t )] = M σ2

e = M ∆q 2

12

var [e q,y (t )] = (M + 1)σ2e = (M + 1) ∆q 2

12


Digital effects





Output quantization noise Bruit:

∆y = e q,x (t ) ∗ h(t ) + e q,y (t )

Spectral density of the output quantization noise:

S ∆y (f ) = |H (f )|2S e q,x (f ) + S e q,y (f )

= M ∆q

2

12|H (f )|2 + M + 1M

Power of the output quantization noise:

σ2∆y =

12

− 12

S ∆y (f )df

= M ∆q 2

6

12

0

|H (f )|2 +

M + 1

M

df

→ Power linearly ↑ with M.




Digital effects





Case of a serial structure (e.g., 2-order DN cells)

S (M )∆y (f ) =

M j =1

S e j (f )|T j (f )|2 =

M j =1

S e j

(f )M

k = j

|H k (f )|2

To minimize the overall quantization noise:

1. Minimization of |T j (f )| = M

k = j |H k (f )|, i.e., supf |H k (f )|

→ judicious matching between poles and zeros of T j (f ), i.e., the couples{Ak (·), B k (·)}

k

2. Optimizing the arrangement of H k (f ): H M (f ) filters M noises

→ ordering the cells such that supf |H 1(f )| ≥ . . . ≥ supf |H M (f )|



Outline



Optimization

Digital effects

Non-standard structuresLadder/lattice structures

Theoretical foundationsApplication to the synthesis of FIR filtersApplication to the synthesis of IIR filters

Fraction-based structuresMultirate structures

Some theoretical notions2-channel filter bankGeneralization to M subbandsAnd the discrete wavelet transform?






Non-standard?...

..., i.e., that are not direct (Direct Form 1),

Y (z ) = B (z )

A(z )X (z )

thus

y (n) = −

k

ak y (n−k )+

k

b k x (n−k )

⇒ (2M + 1) additions/multiplications + 2 lines of delay elements




Non-standard?...i h



..., i.e., that are not

direct (Direct Form 1), canonical (Direct Form 2),

Y (z ) = B (z )

A(z )

X (z ) = X (z )

A(z ) W (Z )

B (z )

thus

y (n) =

k b k w (n − k )w (n) = −k

ak w (n − k ) + x (n)

⇒ (2M + 1) additions/multiplications + 1 line of delay elements




Non-standard?...i th t t



..., i.e., that are not

direct (Direct Form 1), canonical (Direct Form 2),

decomposed : series/cascade or parallel.

Factorization w.r.t. poles/zeros

H (z ) = G

k

H k (z )

Partial fraction decomposition

H (z ) = C + k

H k (z )






Non-standard?...

..., i.e., that are not direct (Direct Form 1),

canonical (Direct Form 2),

decomposed : series/cascade or parallel.

Motivations

weaker sensitivity face to digital errors/effects,same objective as in the cascade structure: matching the poles/zeros10

algorithm-architecture matching (AAM),

more “physical” interpretation of the parameters θ.

10M. Bellanger, Traitement Numerique du Signal , Masson, 1994.



Ladder/lattice structures

Outline



Optimization

Digital effects




Some theoretical notions2-channel filter bank

Generalization to M subbandsAnd the discrete wavelet transform?

Nicolas Dobigeon Digital Signal Processing Advanced Methods 86 / 205


Ladder/lattice structuresTheoretical foundations

Some ideas drawn from linear prediction

Let P m(z ) denote the z -polynomial defined by (m = 1, . . . , M )



P m(z ) =

mk =0

p m,k z −k with p m,0 = 1.

We introduce the associated reciprocal polynomial

P m(z ) = z −mP m 1

z =

mk =0

p m,m−k z −k

Example (m = 3):

P 3(z ) = p 3,0 + p 3,1z −1 + p 3,2z −2 + p 3,3z −3

⇔

P 3(z ) = p 3,3 + p 3,2z −1 + p 3,1z −2 + p 3,0z −3





W i t d th i d t b t 2 l i l



We introduce the inner product between 2 polynomials:

F (z ), G (z ) = 1

2π j

C(0,1)

R M (z )F (z )G †(z )z −1dz

where

G (z ) = k g k z −k ⇔ G †(z ) = k g ∗k z k (= G (1/z ) dans le cas de

coefficients reels) C(0, 1) est le cercle unite

R M (z ) =

P M (z ) P M

z −1

−1

It can be also written

F (z ) , G (z ) = 12π

π−π

R M e j θ F e j θG ∗ e j θ d θ





Let {P0(z) PM (z)} and P0(z) PM (z) denote 2 families of



Let {P 0(z ), . . . , P M (z )} and

P 0(z ), . . . , P M (z )

denote 2 families of

reciprocal polynomial generated as follows P m(z ) = P m−1(z ) + k mz −1P m−1(z )

P m(z ) = k mP m−1(z ) + z −1P m−1(z )

with m = 1, . . . , M and P 0(z ) = P 0(z ) = 1.

A matrix formulation of this forward recursive scheme is P m(z )

P m(z )

=

1 k mz −1

k m z −1

P m−1(z )

P m−1(z )

or, equivalently, (|k m| = 1), the backward recursive scheme is

P m−1(z )P m−1(z )

= 1

1 − k 2m

1 −k m−k mz z

P m(z )P m(z )





Let {P0(z), . . . , PM (z)} and P0(z), . . . , PM (z) denote 2 families of



Let {P 0(z ), . . . , P M (z )} and

P 0(z ), . . . , P M (z )

denote 2 families of

reciprocal polynomial generated as follows P m(z ) = P m−1(z ) + k mz −1P m−1(z )

P m(z ) = k mP m−1(z ) + z −1P m−1(z )

with m = 1, . . . , M and P 0(z ) = P 0(z ) = 1.

A matrix formulation of this forward recursive scheme is P m(z )

P m(z )

=

1 k mz −1

k m z −1

P m−1(z )

P m−1(z )

or, equivalently, (|k m| = 1), the backward recursive scheme is

P m−1(z )P m−1(z )

= 1

1 − k 2m

1 −k m−k mz z

P m(z )P m(z )







Then the following properties hold11,12:



the polynomial series {P 0(z ), . . . , P M (z )} and P 0(z ), . . . , P M (z ) areorthogonal to the canonical basis, i.e.,

P m(z ), z −k

= 0, k = 1, . . . , m

P m(z ), z −k

= 0, k = 1, . . . , m

the polynomial series

P 0(z ), . . . , P M (z )

is an orthogonal basis, i.e.,

P j (z ), P k (z )

= α j δ ( j − k ) =

α j , si j = k ;0, otherwise.

11A. H. Gray, Jr. and J. D. Markel, “Digital Lattice and Ladder Filter Synthesis,” IEEE Trans.Audio. Electroacoust., vol. 21, no. 6, pp. 491-500, Dec. 1973.

12A. H. Gray, Jr., and J. D. Markel, “On Autocorrelation Equations as Applied to SpeechAnalysis,” IEEE Trans. Audio. Electroacoust., vol. 21, no. 2, pp. 69-79, April 1973.

Ni l s D big n Digit l Sign l P ssing Ad n d M th ds 90 / 205




Moreover,



Moreover,

If |k m| = 1 (m = 1, . . . , M ), the series {k 0, . . . , k M } is fully defined by theseries {p M ,0, . . . , p M ,M }

⇒ Any polynomial P M (z ) is uniquely represented by {p M ,0, . . . , p M ,M }ou {k 1, . . . , k M } !

The roots z 1, . . . , z M of P M (z ) are in the unit circle under the following

necessary and sufficient condition:∀m ∈ {1, . . . , M } , |z m| < 1 ⇔ ∀m ∈ {1, . . . , M } , |k m| < 1

Computation of the k m: Levinson-Durbin recursion

p m−1, j = 1

1−k

2m

(p m, j − k mp m,m− j ) , ∀ j ∈ {1, . . . , m − 1}

k m = p m,m, ∀m ∈ {1, . . . , M }

Ni l D bi Di it l Si l P i Ad d M th d 91 / 205


Ladder/lattice structuresApplication to the synthesis of FIR filters

Application to the synthesis of FIR filters

Let the FIR filter to be implemented with transfer function H (z ) = P M (z ) and˜ ˜ ˜



the reciprocal transfer function H (z ) = P M (z ). The outputs Y M (z ) and Y M (z )of filters with input X (z ) are

Y M (z ) = P M (z )X (z )

Y M (z ) = P M (z )X (z )

The forward recursion is written Y M (z ) =

P M −1(z ) + k M z −1P M −1(z )

X (z )

Y M (z ) =

k M P M −1(z ) + z −1P M −1(z )

X (z )

thus

Y M (z ) = Y M −1 + k M z −1Y M −1(z )

Y M (z ) = k M Y M −1 + z −1Y M −1(z )

with Y M −1(z ) = P M −1(z )X (z ) and Y M −1(z ) = P M −1(z )X (z ).

Ni l D bi Di it l Si l P i Ad d M th d 92 /

205


Ladder/lattice structuresApplication to the synthesis of FIR filters


Let the FIR filter to be implemented with transfer function H (z ) = P M (z ) and˜ ˜ ˜



the reciprocal transfer function H (z ) = P M (z ). The outputs Y M (z ) and Y M (z )of filters with input X (z ) are

Y M (z ) = P M (z )X (z )

Y M (z ) = P M (z )X (z )

The forward recursion is written Y M (z ) =

P M −1(z ) + k M z −1P M −1(z )

X (z )

Y M (z ) =

k M P M −1(z ) + z −1P M −1(z )

X (z )

thus

Y M (z ) = Y M −1 + k M z −1Y M −1(z )

Y M (z ) = k M Y M −1 + z −1Y M −1(z )

with Y M −1(z ) = P M −1(z )X (z ) and Y M −1(z ) = P M −1(z )X (z ).










Application to the synthesis of IIR filters

Application to the synthesis of IIR filtersPurely recursive filters



Let the IIR filter of transfer function H (z ) = 1AM (z )

to be implemented.

We generate the polynomial orthogonal basis

A0(z ), . . . , AM (z )

and the

initial forward recurstion system

Am(z ) = Am−1(z ) + k mz −1

Am−1(z )Am(z ) = k mAm−1(z ) + z −1Am−1(z )

is rewritten as Am−1(z ) = Am(z ) − k mz −1Am−1(z )

Am(z ) = k mAm−1(z ) + z −1Am−1(z )





Application to the synthesis of IIR filtersPurely recursive filters








Let the IIR filter of transfer function H (z ) = P M (z )AM (z )

to be implemented.



M ( )

We generate the polynomial orthogonal basis A0(z ), . . . , AM (z ) and wedecompose P M (z ) on this basis

P M (z ) =M

m=0

ν mAm(z ) ⇒ H (z ) =M

m=0

ν mAm(z )

AM (z ).

where the initial forward recurstion system Am(z ) = Am−1(z ) + k mz −1Am−1(z )

Am(z ) = k mAm−1(z ) + z −1Am−1(z )

is rewritten as Am−1(z ) = Am(z ) − k mz −1Am−1(z )Am(z ) = k mAm−1(z ) + z −1Am−1(z )








In the decomposition on the basis

P M (z ) =M

m=0

ν mAm(z )

the coefficients {ν 0, . . . , ν M } are computed thanks to the the backward

recursion (m = M , . . . , 2) P m−1(z ) = P m(z ) − ν mP m−1(z )ν m−1,m−1 = p m−1,m−1.

where P m(z ) = mk =0 p m,k z −k and ν 0 = p 0,0.



Fraction-based structures

Outline

Optimization



Digital effects









Continued fraction expansion

Let the filter of transfer function



H (z ) =M k =0 b k z k M

k =0 ak z k

to be implemented.We want to write H (z ) as a continued fraction expansion...Several solutions exist13, for example

H (z ) = A0 + 1

B 1z + 1

A1 + 1...

B M z + 1AM

13See the conditions stated by Mitra & Sherwood (1972).





We introduce the elementary sections

U k (z ) = 1B k z +T k (z )

(k = 1, . . . , M )



k + k ( )

T k (z ) = 1Ak +U k +1(z )

, (k = 1, . . . , M − 1) with T M (z ) = 1AM

.





The transfer function

H (z ) = A0 + 1



B 1z + 1A1 + 1

...B M z + 1

AM

can be rewrittenH (z ) = A0 + U 1(z )

with

U 1(z ) = 1B 1z +T 1(z )

,

T 1(z ) = 1A1+U 2(z )

,

U 2(z ) = 1B 2z +T 2(z )

,

. . .

T M (z ) = 1AM

.





This results in the following ladder structure with U 1(z ) = 1B 1z +T 1(z )







This results in the following ladder structure with T 1(z ) = 1A1 +U 2(z )






This results in the following ladder structure with T 2(z ) = 1A2 +U 3(z )







This results in the following ladder structure







Computing the {A0, . . . , AM } and {B 1, . . . , B M }Recursive Euclide algorithm: Euclidean division + remainder inversion.



Exemples

H (z ) = 72z 2 + 78z + 9

24z 2 + 18z 2 + 1

= 3 +

1

z + 12+ 1

3z + 14

H (z ) = 720z 2 + 240z + 12

720z 3 + 600z 2 + 72z + 1

= 1z + 1

2+ 1

3z + 14+ 1

5z + 16



Multirate structures

Outline

Optimization



Digital effects


Theoretical foundations

Application to the synthesis of FIR filtersApplication to the synthesis of IIR filters







Multirate systems?



Monorate system

amplifier (×),

summation (+),

delay (×z −1).




Multirate systems?

M



Monorate system

amplifier (×),

summation (+),

delay (×z −1).

Multirate systemAdditional operations:

downsampling (decimation),

upsampling (stretch).




Some theoretical notions

Down-sampling (decimation)

Let y D (n) denote a down-sampled version of the signal x (n) by a factor M :

y D (n) = x (nM ), n ∈ Z.

I h h d l d i l i d b



In the z -space, the down-sampled signal is represented by

Y D (z ) = 1

M

M −1k =0

X

z 1M ωk

M

where ωM = e j 2π

M .

Proof

We introduce x M (n) = x (n)u M (n) where

u M (n) =

1, if n divisible by M ;0, otherwise.

= 1

M

M −1

k =0

ω−knM

Then Y D (z ) = X M

z

1M

,

We can show that X M (z ) = 1M

M −1k =0 X

ωk

M z

.






Let y D (n) denote a down-sampled version of the signal x (n) by a factor M :

y D (n) = x (nM ), n ∈ Z.

I th th d l d i l i t d b



In the z -space, the down-sampled signal is represented by

Y D (z ) = 1

M

M −1k =0

X

z 1M ωk

M

where ωM = e j 2π

M .

Proof

We introduce x M (n) = x (n)u M (n) where

u M (n) =

1, if n divisible by M ;0, otherwise.

= 1

M

M −1

k =0

ω−knM

Then Y D (z ) = X M

z

1M

,

We can show that X M (z ) = 1M

M −1k =0 X

ωk

M z

.












Up-sampling (stretch)



Similarly, let y E (n) denote a up-sampled version of the signal x (n) by a factor L(interpolation by zero insertion):

y E (n) = x n

L , if n is divisible par L;0, otherwise.








Similarly, let y E (n) denote a up-sampled version of the signal x (n) by a factor L(interpolation by zero insertion):

y E (n) = x n

L , si n est divisible par L;0, otherwise.

In the z -space, the up-sampled signal is represented by

Y E (z ) = X

z L

.





Noble identities14



14P. P. Vaidyanathan, “Multirate digital filters, filter banks, polyphase networks, andapplications: a tutorial,” Proc. of the IEEE , vol. 78, no. 1, pp. 56-93, Jan. 1990.





Polyphase representation15

We decompose the impulse response into two sub-series of samples with oddand even indexes:

{h(n)}n∈Z = {h(2m)}m∈Z ∪ {h(2m + 1)}n∈Z



{h(n)} ∈ = {h(2m)} ∈ ∪ {h(2m + 1)} ∈

The resulting transfer function is

H (z ) =n∈Z

h(n)z −n

= m

h(2m)z −2m E 0(z 2)

+m

h(2m + 1)z −(2m+1) z −1E 1(z 2)

with the two sub-series

E 0(z ) = TZ [e 0(n)] = TZ [h(2m)]

E 1(z ) = TZ [e 1(n)] = TZ [h(2m + 1)]

15M. Bellanger, Traitement Numerique du Signal , Masson, 1994.





Polyphase representation

To generalize, we decompose the IR onto the M (polyphase) sub-series comingfrom down-sampling of order M :

{h(n)} =M −1

{h (Mm + )}



{h(n)}n∈Z =

=0

{h (Mm + )}m∈Z

The resulting transfer function is

H (z ) = n∈Z h(n)z −n

= E 0

z M

+ z −1E 1

z M

+ . . . + z −(M −1)E M −1

z M

=M −1=0

z −E

z M

with the M sub-series ( = 1, . . . , M − 1):

E (z ) = TZ [e (m)] with e (m) = h(Mm + ).






Example M = 4

Let consider a FIR filter of order 11



Let consider a FIR filter of order 11

H (z ) =10

n=0

b k z −n

= E 0(z 4) + z −1E 1(z 4)

+ z −2E 2

(z 4) + z −3E 3

(z 4)

with

E 0(z ) = b 0 + b 4z −1 + b 8z −2

E 1(z ) = b 1 + b 5z −1 + b 9z −2

E 2(z ) = b 2 + b 6z −1

+ b 10z −2

E 3(z ) = b 3 + b 7z −1












Outline

Optimization

Di it l ff t



Digital effects






Generalization to M

subbandsAnd the discrete wavelet transform?





Multirate and filter banks



Filter-banks: analysis








Vocoder LPC (Lecture “Compression parole et musique”, Corinne Mailhes)






Issue no. 1: computational complexity









Issue no. 1: computational complexity⇒ decrease the rate (down-sampling)









Issue no. 1: computational complexity⇒ decrease the rate (down-sampling)

Issue no. 2: reconstruction (⇔ synthesis))



( ))

Filter-banks: analysis/synthesis





Multirate and filter banksApplications

Vocoders16

,Ici, slide 43

http://mailhes.perso.enseeiht.fr/compression/compressionParoleMusiquetotale_mailhes.pdf




Ici, slide 27-28

Decomposition into subbands: lossy coding17,Ici, slide 39

Decomposition into subbands: multi-resolution analysis18

,Ici

Decomposition into subbands: spectral estimation,Bonacci et al., ICASSP 2002.

Bonacci et al., EUSIPCO 2002.

16Lecture “Compression parole et musique”, Corinne Mailhes.17Lecture “Compression”, Corinne Mailhes.18Lecture “Representation et analyse des signaux”, Marie Chabert.





Outline

Optimization

Digital effects



http://mailhes.perso.enseeiht.fr/compression/datacompression_mailhes.pdf

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2-channel filter bank

Problem of 2-channel analysis/synthesis

Let consider the general problem of analysis/synthesis X (z ) → X (z ) defined by



From above (up/down-sampling):

Y m(z ) = 1

2 X m z 1/2 + X m −z 1/2Z m(z ) = Y m

z 2






With X m(z ) = H m(z )X (z ) and X (z ) = F 0(z )Z 0(z ) + F 1(z )Z 1(z ), it follows:

X (z ) = 1

2 [H 0(z )F 0(z ) + H 1(z )F 1(z )] X (z )

+ 12

[H 0(−z )F 0(z ) + H 1(−z )F 1(z )] X (−z )



2= T (z )X (z ) + A(z )X (−Z )

where

T (z ) = 12

[H 0(z )F 0(z ) + H 1(z )F 1(z )] is the distorsion function

A(z ) = 12 [H 0(−z )F 0(z ) + H 1(−z )F 1(z )] is the aliasing function







X (z ) = 1

2 [H 0(z )F 0(z ) + H 1(z )F 1(z )] X (z )

+ 12

[H 0(−z )F 0(z ) + H 1(−z )F 1(z )] X (−z )



2= T (z )X (z ) + A(z )X (−Z )

where

T (z ) = 12


A(z ) = 12 [H 0(−z )F 0(z ) + H 1(−z )F 1(z )] is the aliasing function

But the operator R : X (z ) → X (−Z ) is not a LTI filter, thus:

It does not exist {t (n)}n∈Z such that x (n) = t (n) ∗ x (n)...







X (z ) = 1

2 [H 0(z )F 0(z ) + H 1(z )F 1(z )] X (z )

+ 12

[H 0(−z )F 0(z ) + H 1(−z )F 1(z )] X (−z )



2= T (z )X (z ) + A(z )X (−Z )

where

T (z ) = 12


A(z ) = 1

2 [H 0(−z )F 0(z ) + H 1(−z )F 1(z )] is the aliasing function

But the operator R : X (z ) → X (−Z ) is not a LTI filter, thus:

It does not exist {t (n)}n∈Z such that x (n) = t (n) ∗ x (n)...

... except if the anti-aliasing condition is fulfilled:

A(z ) = H 0(−z )F 0(z ) + H 1(−z )F 1(z ) = 0






The anti-aliasing condition is:

F 0(z )

H 1(−z ) = −

F 1(z )

H 0(−z ) = C (z )



At the end of the analysis/synthesis chain

X (z ) = 1

2

H 0(z )H 1(−z )

V (z )

− H 1(z )H 0(−z ) V (−z )

C (z )X (z )

= T (z )X (z )

with T (z ) = 12

[V (z ) − V (−z )] C (z ).A sufficient condition (but not necessary) is C (z ) = 1, then

F 0(z ) = H 1(−z )F 1(z ) = −H 0(−z )







Quadrature Mirror Filters (QMF)







Quadrature Mirror Filters (QMF)Polyphase representation

With H 1(z ) = H 0(−z ):

ˆ 1 2 2



X (z ) = 1

2

H 0(z )2 − H 0(−z )2

X (z )

H 0(z ) is decomposed with respect to its 2 polyphase terms:

H 0(z ) = E 0(z 2) + z −1E 1 z 2 .

The analysis filters are written: H 0(z )H 1(z )

=

1 11 −1

E 0(z 2)z −1E 1 z 2






Problem of analysis/synthesis:








Non-aliasing condition and polyphase decomposition:







On the circle C (0, 1), we write T

e 2i πf

=

T

e 2i πf

e i φ(2πf )

Several possible interesting cases:

If T (z ) is a all-pass, i.e,T

e 2i πf

= d , then



X

e 2i πf = d

X

e 2i πf

−→ Amplitude preserving filters (but phase distorsion).

If T (z ) is a linear phase filter, i.e, φ(2πf ) = α2πf + β , then

arg

X

e 2i πf

= arg

T

e 2i πf

+ arg

X

e 2i πf

= α2πf + β + arg

X

e 2i πf

−→ Phase preserving filters (but amplitude distorsion).






If there is no amplitude distorsion and no phase distorsion

∃L ≥ 1 T ( ) K −L



∃L ≥ 1, T (z ) = Kz

Then, at the ouput

ˆX e

2i πf = K X e

2i πf arg

X

e 2i πf

= arg

X

e 2i πf

− L2πf

−→ Output delayed and amplified/attenuated.−→ Perfect reconstruction filter bank.





Perfect reconstruction filter bank

We introduce the product filter V (z ) = H 0(z )H 1(−z )

X (z ) = 1

2

H 0(z )H 1(−z ) − H 1(z )H 0(−z ) X (z )



2

V (z )

V (−z )

To obtain T (z ) = Kz −L in the analysis/synthesis chain (L odd):

T (z ) = 12 [V (z ) − V (−z )] = Kz −L

⇔ v (2n + 1) = K δ (2n + 1 − L)v (2n) any

⇔V (z ) = Kz −L + n

v (2n)z −2n.





Perfect reconstruction filter bank

3 synthesis methods:

1. We build V (z ) such that K δ( L) if dd



v (n) =

K δ (n − L) if n oddany si n even

then we factorize V (z ) = H 0(z )H 1(z ).

2. We fix H 0(z ) and we solve (matrix computation)

H 0(z )H 1(−z ) − H 0(−z )H 1(z ) = Kz −L

3. We choose {H 0(z ), H 1(z )} in appropriate filter families→ CQF...





Conjugate quadrature filters (CQF)

We choose H 0 and H 1 such that

H

0(z )H

0(z −1) + H

0(−

z )H 0

(−

z −1) = 1 (a)H 1(z )H 1(z −1) + H 1(−z )H 1(−z −1) = 1 (b)H0(z)H1(z−1) + H0(−z)H1(−z−1) = 0 (c)



H 0(z )H 1(z ) + H 0( z )H 1( z ) 0 (c)

Properties Let H 0(z ) a FIR filter that follows (a). Then19

H 0(z ) is of even length (odd order),

A NSC for having H 1(z ) a FIR that follows (b) and (c) is

H 1(z ) = ±z −LH 0(−z −1), L odd

In this case, T (z ) = z

−L

.

19Vetterli, IEEE Trans. SP , 1992.Nicolas Dobigeon Digital Signal Processing - Advanced Methods 147 / 205





“Classical” solution from Smith and Barnwell (for L odd and z = e 2i πf ):

H 1(z ) = −z −L

H 0 (−z ) with H 0 (z ) = H 0 1

z



Then we have

symmetry with respect to the power

H 0 e j 2πf 2

+ H 0 e j (2πf +π)2

= 1.













“Classical” solution from Smith and Barnwell (for L odd and z = e 2i πf ):

H 1(z ) = −z −LH 0 (−z ) with H 0 (z ) = H 0 1

z



Then we have

symmetry with respect to the power

H 0e j 2πf 2 + H 0

e j (2πf +π)2 = 1.

complementariry with respect to the power

H 0 e j 2πf

2

+ H 1 e j 2πf

2

= 1.








Filter bank fully defined by H 0(z ):

h0(n) fixedh1(n) = (−1)nh0(L − n)f0(n) = h0(L − n)



f 0(n) h0(L n)f 1(n) = −(−1)nh0(n)

Remarks

If H 0(z ) is a stable IIR, H 1 (z ) is unstable!→ can not be implemented with IIR...→ implemented with FIR.

If H 0(z ) is a causal filter, H 0

− 1

z

is anti-causal. Thus, there is delay

introduced z −L ! (H 1(z ) becomes causal if L ≥order of the FIR H 0(z )).





Conjugate quadrature filters (CQF)Spectral factorization synthesis

We set P (z ) = H 0(z )H 0(−z ) (half-band filter). Then, it comes:

P (z ) + P (−z ) = 1, i.e., P (z ) = 12

+ k

p (2k + 1)z −(2k +1).



1. Synthesis of a zero-phase half-band filter Q (z ) such that

Q (e

j 2πf

) + Q (e

j (2πf +π)

) = 1. (3)














P (z ) + P (−z ) = 1, i.e., P (z ) = 12

+

k

p (2k + 1)z −(2k +1).




Q (e j 2πf ) + Q (e j (2πf +π)) = 1. (4)

2. We set q (n) = q (n) + δ (n) such that Q (e j 2πf ) = |H 0(e j 2πf )|2 ≥ 0.














P (z ) + P (−z ) = 1, i.e., P (z ) = 12

+

k

p (2k + 1)z −(2k +1).




Q (e j 2πf ) + Q (e j (2πf +π)) = 1. (5)

2. We set q (n) = q (n) + δ (n) such that Q (e j 2πf ) = |H 0(e j 2πf )|2 ≥ 0.

3. We factorize Q (z ) = A

i (1 − ai z −1)(1 − a−1i z −1) (with |ai | ≤ 1)

4. We define the minimum phase spectral factor

H 0(z ) = A1+2

i (1 − ai z −1).






Other method proposed by Daubechies20:

P (z ) = (1 + z −1)k (1 + z )k R (z )

under the constraints

R(z) symetric (R(z−1) R (z))



R (z ) symetric (R (z 1) = R (z )),

R (z ) positive for z = e j 2πf

R (z ) of minimal degree.

Then factorization of P (z ) with zeros inside C (0, 1)→ Family of Daubechies’ minimym phase filters.

Drawbacks of the CQF:

only 1 degree of freedom for the analysis/synthesis: H 0(z ),

no filter bank with linear phase analysis and synthesis filters.

20Daubechies, Comm. Pure Appl. Math., 1998.Nicolas Dobigeon Digital Signal Processing - Advanced Methods 158 / 205




Bi-orthogonal filter bank

We choose H 0 and H 1 such that C (z ) = −z −L and

H 0(z )F 0(z ) + H 0(−z )F 0(−z ) = 2H 1(z ) = −z LF 0(z −1), L impair

F 1(z ) = z −LH 0(−z )



In this case, T (z ) = z L. We set P (z ) = H 0(z )F 0(z ) and it comes:

P (z ) + P (−z ) = 2

Synthesis method:

1. We choose P (z ) (half-band, zero phase, real coefficients)

2. We factorize P (z ) = H 0(z )F 0(z )

→ 2 degrees of freedom.→ linear phase filter bank possible.





Outline

Optimization

Digital effects

N t d d t t






Some theoretical notions2-channel filter bankGeneralization to M subbands

And the discrete wavelet transform?






Generalization to M subbands

Full tree decomposition

(Wave packet decomposition)







Dyadic decomposition

(Dyadic discrete wavelet decomposition)







Optimal subband decomposition







Direct M -channel decomposition

The parts “analysis” and “synthesis” own M filters in parallel.








As previously, the properties of down/up-sampling lead to:

X (z ) = 1

M

M −1m=0

X (ωm

M z )M −1k=0

H k (ωmM z )F k (z )



m=0

k =0

=M −1

m=0

X (ωm

M z ) Am(z )

with Am(z ) = 1M

M −1k =0 H k (ωm

M z )F k (z ).






With matrix notations:

X (z ) = a(z )T x(z ) = 1

M

f (z )T H(z )T x(z )

with A0(z )

A1(z )

1

h(z )T

h(ωmz )T

F 0(z )

F 1(z )



...

AM −1(z ) a(z )

=M

...

h(ωM −1

m z )T

H(z )

...

F M −1(z ) f (z )

and

h(z ) =

H 0(z )...

H M −1(z )

and x(z ) =

X (z )...

X (ωM −1M z )

..






However, the operatorsRm : X (z ) → X (ωm

M z )

are not LTI filters (except for m = 0). The anti-aliasing conditions are:

Am(z ) = 0, pour m = 0

If h th l i filt (i H( )) th bl i t f l i



If we choose the analysis filter (i.e., H(z )), the problem consists of solving

1

M f (z ) = H(z )−1a(z ) with a(z ) =

T (z )0

...0

In the perfect reconstruction problem, T (z ) ∝ z −N .





Direct M -channel decompositionPolyphase representation

The polyphase representation of the synthesis and analysis filters are

H k (z ) = m−1

m=0 z

−m

E k ,m z

M F k (z ) =

m−1m=0 z −mE k ,m

z M

m−1m=0

z −M −1−mR k ,m

z M



We define the polyphase matrices

E

z M

tel que

E

z M

k ,m= E k ,m

z M

R

z M

tel que

R

z M

k ,m= R k ,m

z M

andP(z ) = R (z ) E (z ) .













Vaidyanathan enounced numerous properties21,22.

Necessary and sufficient condition for anti-aliasing P(z ) is pseudo-circulant, i.e.,

P(z ) is Toeplitz,

P k ,0(z ) = z −1P 0,M −k (z ), k = 1, . . . , M − 1P 0(z ) P 1(z ) . . . P M −1(z )

1



P (z ) = [P k ,m(z )]k ,m

=

z −1P M −1(z ) P 0(z ) . . . P M −2(z )

.

.

.

.

.

.

. ..

.

.

.z −1P 1(z ) . . . z −1P M −1(z ) P 0(z )

In this case, the transfer function T (z ) is

T (z ) = z −(M −1)M −1

k =0

z −k P 0,k z M

.

21Vaidyanathan, IEEE Trans. ASSP , 1987.22Vaidyanathan and Mitra, IEEE Trans. ASSP , 1988.







Outline

Optimization

Digital effects





Theoretical foundationsApplication to the synthesis of FIR filters

Application to the synthesis of IIR filtersFraction-based structuresMultirate structures






Case of the CQF filter bank

We remind that C (z ) = 1, i.e.,

F 0(z ) = H 1(−z )F 1(z ) = −H 0(−z )

Orthogonal filter bank ( 2 channels)H 0(z )H 0(z −1) + H 0(−z )H 0(−z −1) = 2 (a)



0( ) 0( ) + 0( ) 0( ) ( )H 1(z )H 1(z −1) + H 1(−z )H 1(−z −1) = 2 (b)

H 0(z )H 1(z −1

) + H 0(−z )H 1(−z −1

) = 0 (c)is equivalent to

h0(n), h0(n − 2k ) = δ (k ) (a’)h1(n), h1(n − 2k ) = δ (k ) (b’)h0(n), h1(n − 2k ) = 0 (c’)

with a(n), b (n) =

n a(n)b (n) (a and b real).






We introducex = [. . . , x (−1), x (0), x (1), . . .]T

and the matrix

H0 =

.

.

....

.

.

....

.

.

....

.

.

....

. . . 0 h0(M − 1) h0(M − 2) h0(M − 3) . . . h(0) 0 0 0 . . .

. . . 0 0 0 h0(M − 1) . . . h(2) h(1) h(0) 0 . . .

. . . . . . . .



..

.

...

.

...

.

...

.

.

then

with h0(n) = h0(M − 1 − n) = f 0(n) (M odd).








What is the operator PH0 = H∗0 H0?

From (a), we have H0H∗0 = I,

→ Orthogonal projection onto the space V 0 H∗0 spanned by thecolumns of H∗

0 , i.e., the rows of H0.

Can we complete the space V0 in L2(Z) x(n), n ∈ Z/ x2 < ∞ ?



Can we complete the space V 0 in L (Z)

x (n), n ∈ Z/ x2 < ∞

?

From (c), we have H0H

∗

1 = 0, → H∗

0 ⊥ H∗1

→ V 0 and W 0 H∗1 are in orthogonal direct sum

Corollary

H

∗

0 H0 + H

∗

1 H1 = I.






InterpretationThe CQF filter bank define a decomposition into sub-spaces of L2(Z).

L2(Z) = V 0 ⊕

⊥W 0



We iterate:

L2(Z) = V 0 ⊕⊥ W 0

V 0 =

V 1

⊕⊥ W 1

V 1 = V 2 ⊕⊥ W 2

. . .

⇔ L2(Z) =

M i =0

W i details

⊕⊥ V M approx.






InterpretationThe CQF filter bank define a decomposition into sub-spaces of L2(Z).

L2(Z) = V 0 ⊕

⊥W 0



We iterate:

L2(Z) = V 0 ⊕⊥ W 0

V 0 =

V 1

⊕⊥ W 1

V 1 = V 2 ⊕⊥ W 2

. . .

⇔ L2(Z) =

M i =0

W i details

⊕⊥ V M approx.





Case of the CQF filter bankDyadic decomposition

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