Imaginary Axis

Closed-Loop System Identi ation Framework

Toon van Waters hoot

∗

(ESAT-SCD/SISTA)

SISTA Seminar 19/02/2004

∗

•

A ousti Feedba k Problem

•

Predi tion Error Method for Closed-Loop System Identi ation

•

Time-varying Autoregressive Noise model

•

Simulation Results

speech/

music

G

F

x

(t)

microphone

loudspeaker

electroacoustic

acoustic

feedback path

forward path

y(t)

u(t)

v(t)

H

w(t)

System:

8

<

:

y(t)

=

F

(q)u(t) + v(t)

u(t)

=

G(q)y(t)

v(t)

=

H

(q)w(t)

Closed-Loop Transfer Fun tion:

u(t) =

G(q)

1 − G(q)F (q)

v(t)

•

Nyquist: instability o urs if

∃ ω :



|G(e

iω

)F (e

iω

)|

≥

1

∠G(e

iω

)F (e

iω

)

=

n

_{2π, n ∈ Z.}

•

instability leads to howling

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nyquist Diagram

Real Axis

Imaginary Axis

•

system operation lose to instability leads to ringing and ex essive reverberation (typi ally avoided by hoosing gain margin

GM >

2dB

)

•

onsequen e: restri tion on forward path ampli ation, e.g. when

G(q) = Kq

−d

hoose

K

≤

10

− GM

20

max

ω

|F (e

iω

)|

In lude an ellation lter

F

0

(q)

:

U

(e

jω

)

V

(e

jω

₎

=

G(e

jω

)

1 − G(e

jω

_{)(F (e}

jω

_{) − F}

0

(e

jω

))

F

x(t)

F

0

G

v(t)

H

w(t)

u(t)

y

(t)

ˆ

y

(t)

e(t)

•

to in rease gain margin

→ |F

0

(e

jω

)|

should model peaks of

|F (e

jω

_)|

→

xed an ellation lter

•

to redu e signal distortion

→ F

0

(q)

should be a onsistent estimate of

F

(q)

Estimate

F

(q)

using the Predi tion Error Method

•

dire t approa h (no probe signal required): identify

F

(q)

from

y

(t) = F (q)u(t) + H(q)w(t)

F

x

(t)

G

r

(t)

H

y(t)

v(t)

w(t)

u

(t)

•

indire t approa h (inje tion of probe signal

r(t)

and knowledge of forward path

G(q)

required): with

F

c

(q) =

_{1−G(q)F (q)}

F

(q)

and

H

c

(q) =

_{1−G(q)F (q)}

H

(q)

, identify

F

c

(q)

from

y(t) = F

c

(q)r(t) + H

c

(q)w(t)

and al ulate

F

(q)

as

F

(q) = F

c

(q)(1 + F

c

(q)G(q))

−1

•

joint input-output approa h (inje tion of probe signal

r(t)

is usual but not required): with

F

c

_{(q) =}

F

(q)

1−G(q)F (q)

,

H

c

_{(q) =}

H

(q)

1−G(q)F (q)

and

S(q) =

1

1−G(q)F (q)

, identify

»F

c

(q)

S

(q)

–

from

»y(t)

u(t)

–

=

»F

c

(q)

S

(q)

–

r(t) +

»

H

c

(q)

G(q)H

c

(q)

–

w(t)

and al ulate

F

(q)

as

F

(q) = F

c

(q)(S(q))

−1

Assume FIR model stru ture

F

ˆ

(q) = ˆ

f

0

(t) + ˆ

f

1

(t)q

−1

_{+ . . . + ˆ}

_f

n ˆ

_F

(t)q

−n ˆ

F

and su ient order ase (

n

ˆ

F

= n

F

).

ˆ

_f

LS

(t)

=

arg min

ˆ

_f

k

2

6

6

4

y(t)

y(t − 1)

. . .

y(1)

3

7

7

5

−

2

6

6

4

u(t)

. . .

u(t − n

_F

ˆ

)

u(t − 1)

. . .

u(t − n

_F

ˆ

− 1)

. . . . . . . . .

u(1)

. . .

u(−n

_F

ˆ

+ 1)

3

7

7

5

·

2

6

4

ˆ

f

₀

(t)

. . .

ˆ

f

_{n ˆ}

F

(t)

3

7

5

k

2

=

arg min

ˆ

_f

ky

t×1

− U

t

×(n ˆ

F

+1)

ˆ

_f

_(t)

(n ˆ

_F

+1)×1

k

2

=

(U

T

U

)

−1

U

T

y

=

f

+ (U

T

U)

−1

U

T

v

Bias term in the time domain:

ˆ

_f

If a noise model

H

ˆ

(q)

is in luded in the PEM identi ation, the bias term in the frequen y domain may be expressed as

ˆ

F

(e

jω

) − F (e

jω

) = (H(e

jω

) − ˆ

H

(e

jω

))Φ

wu

Φ

−1

_u

Sin e

Φ

wu

6= 0

due to one-sided orrelation between

u(t)

and

w(t)

:

u(t) =

G(q)H(q)

1 − G(q)F (q)

w(t),

ˆ

F

(q)

will be a biased estimate of

F

(q)

unless

ˆ

H

(q) = H(q),

∀t.

Note that

Φ

wu

Φ

−1

Audio signals are ommonly modelled as Autoregressive (AR) sequen es, whi h leads to a Linear

Predi tion (LP) estimation problem.

However, audio signals are only stationary over short data windows (on average

20ms

for spee h). Hen e for onsistent estimation of the sour e signal model

H

(q)

, a time-varying autoregressive (TVAR) model should be identied:

•

either by tra king the TVAR oe ients using an adaptive algorithm (assuming 'slow' time-variation and short data windows, whi h is not always possible when on urrently estimating

H

(q)

and

F

(q)

),

•

or by basis expansion of

 the TVAR oe ients (e.g. on a Fourier, Legendre, wavelet, ... basis)

 the model stru ture (e.g. on a Kautz, Laguerre basis),

•

or by regarding the TVAR oe ients as sto hasti variables and estimating them using Kalman ltering.

The noise model

H

ˆ

(q)

an be in orporated in the PEM identi ation by preltering the loudspeaker and mi rophone signals

u(t)

resp.

y

(t)

with a (time-varying) lter

L(q) = ˆ

H

−1

(q)

:

ˆ

_f

LS

,L

(t)

=

arg min

ˆ

_f

kL

t

×(t+nL)

y

(t+nL)×1

− L

t

×(t+nL)

U

(t+nL)×(n ˆ

F

+1)

ˆ

_f

_(t)

(n ˆ

_F

+1)×1

k

2

=

(U

T

L

T

LU

)

−1

U

T

L

T

Ly

=

f

+ (U

T

L

T

LU

)

−1

U

T

L

T

Lv

where preltering matrix

L

t

×(t+nL)

is dened as

L

_t

_×(t+nL)

=

2

6

6

4

l

0

(t)

l

1

(t)

. . .

l

_nL

(t)

0

. . .

0

0

l

₀

(t − 1)

. . .

l

_nL−1

(t − 1)

l

_nL

(t − 1)

. . .

0

. . . . . . . . . . . . . . . . . . . . .

0

0

. . .

0

0

. . .

l

_nL

(1)

3

7

7

5

.

If

L(q)

is a onsistent estimate of the true inverse sour e signal model

H

−1

_{(q), ∀t}

, then

Lv

= w

and the bias term

(U

T

_L

T

_LU

G

F

y

(t)

v

(t)

w

(t)

x

(t)

H

L

u

(t)

ˆ

F

ˆ

y

L

(t)

e

L

(t)

L

y

L

(t)

u

L

(t)

•

Simulation parameters:

f

s

= 8kHz

,

F

(q)

a pre-measured room impulse response,

n

F

+ 1 =

1000

,forward path

G(q) = Kq

−1

,

N

= 24000

samples,

GM

= 3dB

,

F

ˆ

(q)

anexponentially windowed RLS adaptive lter with

λ

= 0.9997

•

Performan e measure: normalized bias

δ

(t) = 20 log

10

k

ˆ

_fLS

(t)−

f

k

k

f

k

•

Referen e algorithm: de orrelation of

u(t)

and

y(t)

by SSB-AM frequen y shifting

1

•

Preltering te hniques (sour e signal

v(t)

assumed to be known):  onstant preltering

 pie ewise onstant preltering with

N

win

= 3000

samples

 BEM-TVAR preltering: basis expansion of sour e signal on a Fourier basis with

51

basis fun tions over time windows of

N

win

= 3000

samples

•

Sour e signals:

 stationary AR sequen e of order

12

exhibiting a long-term spee h spe trum  true spee h signal

1

Sour e signal

v(t)

= stationary AR sequen e

0

0.5

1

1.5

2

2.5

x 10

4

10

0

10

1

10

2

t/T

_s

(s)

δ

(dB)

no decorrelation

SSB−AM decorrelation

constant prefiltering

Sour e signal

v

(t)

= true spee h signal

0

0.5

1

1.5

2

2.5

x 10

4

10

0

10

1

10

2

no decorrelation

SSB−AM decorrelation

constant prefiltering

piecewise constant prefiltering

BEM−TVAR prefiltering

•

Consistent PEM identi ation of the a ousti feedba k path

F

(q)

in adaptive feedba k an ellation using the dire t method, an be a hieved by in luding a noise model

H

ˆ

(q)

in the identi ation pro edure.

•

The noise model

H

ˆ

(q)

should be a onsistent estimate of the sour e signal model

H

(q)

, whi h is generally time-varying due to the non-stationary nature of audio signals.

•

Several approa hes to time-varying autoregressive (TVAR) modelling exist. A BEM-TVAR te hnique that expands the TVAR oe ients onto a set of predened basis fun tions was implemented under

the assumption that the sour e signal was known.

•

Under the latter assumption the BEM-TVAR preltering te hnique performs similar to a state-of-the-art referen e algorithm in whi h de orrelation by frequen y shifting is applied. However, the

BEM-TVAR preltering te hnique does not introdu e any signal distortion in the forward path,

whereas the frequen y shifting te hnique does.

•

Con urrent estimation of the a ousti feedba k path

F

(q)

and the inverse sour e signal model