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Towards optimal regularization

by incorporating prior knowledge in an acoustic echo canceller

Toon van Waterschoot, Geert Rombouts and Marc Moonen

Katholieke Universiteit Leuven, ESAT-SCD, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium toon.vanwaterschoot@esat.kuleuven.be

1 Regularization in least squares parameter estimation

Consider the linear estimation problem

y(1) y(2) ...

y (N )

=

u(1) u(0) . . . u(1 − n F ) u(2) u(1) . . . u(2 − n F )

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

u (N ) u(N − 1) . . . u(N − n F )

·

 f 0 f 1 ...

f n

F

 +

v(1) v(2) ...

v (N )

 ,

or in matrix notation

y = Uf + v. (1)

An estimator for f may be obtained by minimizing the least squares (LS) criterion

min ˆ f

V LS (ˆ f ) = min

ˆ f

(y − Uˆ f ) T (y − Uˆ f ) (2) which results in the well-known LS estimator





ˆ f LS = (U T U ) −1 U T y. (3) When the input signal u(t) is not (or hardly) persistently excit- ing, as is often the case when using audio signals, the matrix U T U may be ill-conditioned or even singular. A common so- lution is to apply Tikhonov regularization by adding a scaled identity matrix to U T U :









ˆ f RgLS = (U T U + αI) −1 U T y. (4) This is in fact the solution to a ridge regression problem as for- mulated by the criterion

min ˆ f

V RR (ˆ f ) = min

ˆ f

(y − Uˆ f ) T (y − Uˆ f ) + αkˆ f k 2 2 . (5) We address the question whether it is convenient to replace the scaled identity matrix αI by a non-identity, possibly non- diagonal regularization matrix P:









ˆ f RgLS = (U T U + P) −1 U T y, (6) and if it is, what is the optimal choice for P?

2 Linear mimimum mean square error estimation

Let us derive an expression for the minimum mean square error (MMSE) estimator of f :

min ˆ f

V M M SE (ˆ f ) = min

ˆ f

E (ˆ f − f ) T (ˆ f − f ) (7) under the following assumptions:

• The estimator is a linear function of the data in y:

ˆ f MMSE = Z T y. (8)

• The measurement noise in v is drawn from a stationary white noise process with zero mean and variance σ v 2 :

µ v , Ev = 0, (9)

R v , cov(v) = Evv T = σ v 2 I. (10)

• The true parameter vector f is considered as a random vari- able on which some prior knowledge may be available.

More specifically, let the prior probability density function (PDF) p(f ) be characterized by its first and second order mo- ments:

µ f , Ef, (11)

R f , cov(f) = E(f − Ef)(f − Ef) T . (12) Then the linear MMSE estimator can be obtained as the mean of the posterior PDF p(f |y) after the data have been recorded:

ˆ f MMSE = E(f |y) = µ+(U T R v −1 U +R f −1 ) −1 U T R v −1 (y−Uµ).

We will construct the prior knowledge on f in such a way that µ f = 0. Then, also using the white noise assumption in (10), the expression for ˆf MMSE can be rewritten as





ˆ f MMSE = (U T U + σ v 2 R f −1 ) −1 U T y. (13) From this point of view, applying Tikhonov regularization as in (4) with α = σ v 2 is equivalent to assuming that the true pa- rameter vector f is drawn from a stationary white noise pro- cess. In room acoustics applications however, more informa- tion on the true parameter may be available and an appropri- ate non-identity covariance matrix R f can be constructed.

3 Gathering prior knowledge on room acoustics

Consider an acoustic echo cancellation (AEC) application:

far-end from

far-end to

x (t) y (t)

F ˆ u (t)

d (t)

acoustic echo path

F

v (t)

If an LS estimator with Tikhonov regularization is applied in the AEC problem, the regularization matrix P = αI (with the regularization parameter typically chosen as α = σ v 2 ) looks like this:

0

20

40

60

80

1000

20

40

60

80

100 0

0.5 1 1.5

A room impulse response (RIR) has a very typical form, which may be characterized by three parameters:

• the initial delay, which corresponds to the time needed by the loudspeaker sound wave to reach the microphone through a direct path (i.e. without reflections),

• the direct path attenuation, which determines the peak re- sponse in the RIR, and

• the exponential decay rate, which models the reverberant tail of the RIR.

0 0.02 0.04 0.06 0.08 0.1 0.12

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

t (s)

direct path attenuation

initial delay

exponential decay rate

These three parameters may be calculated from the acoustic setup (distance between loudspeaker and microphone, acous- tic absorption of the walls, room volume, etc.). Hence they can be considered as prior knowledge. If these three param- eters are taken into account, a diagonal regularization matrix may be constructed that looks like this:

0 20

40 60

80

100 0 10 20 30 40 50 60 70 80 90 100

0 20 40 60 80 100 120 140

An idealized case, which is interesting as a reference method, occurs when the true RIR is known. In this case a diagonal reg- ularization matrix may be constructed with the diagonal ele- ments equal to the inverse square values of the true parameter vector coefficients:

0

20

40

60

80

100 0

20

40

60

80

100 0

2000 4000 6000 8000 10000

4 Off-line simulation results

Four different least squares estimators were compared for the AEC application:

• the LS estimator ˆf LS without regularization,

• the regularized LS estimator ˆf RgLS with Tikhonov regular- ization: P = σ v 2 I ,

• the regularized LS estimator ˆf RgLS with regularization based on the three RIR parameters described above: P = σ v 2 R −1 f ,synth , and

• the regularized LS estimator ˆf RgLS with regularization based on the true RIR: P = σ v 2 R −1 f ,true .

For each estimator, 100 simulation runs were performed with a different near-end noise realization.

Two types of loudspeaker signals were used:

• a sum of n F − 1 sinusoids with random frequencies, uni- formly distributed in the interval between DC and the Nyquist frequency, and

• a male speech signal.

Two types of acoustic impulse responses were used:

• a hearing aid impulse response with n F + 1 = 100 coeffi- cients, and

• a room impulse response with n F + 1 = 1000 coefficients.

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

ˆfRgLS

f−fk2 kfk2

ˆfRgLS ˆfRgLS

P= σv2I

ˆfLS

P= σv2R−1f ,synth P= σv2R−1f ,true

F (q) = hearing aid IR u (t) = Σ sin

0.05 0.1 0.15 0.2 0.25 0.3 0.35

f−fk2 kfk2

ˆfLS ˆfRgLS

P= σv2I

ˆfRgLS ˆfRgLS

P= σv2R−1f ,true P= σv2R−1f ,synth

u (t) = speech F (q) = hearing aid IR

0.1 0.12 0.14 0.16 0.18 0.2 0.22

f−fk2 kfk2

ˆfLS ˆfRgLS

P= σv2I

ˆfRgLS

P= σv2R−1f ,synth

ˆfRgLS

P= σv2R−1f ,true

u (t) = Σ sin F (q) = room IR

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

f−fk2 kfk2

ˆfLS ˆfRgLS

P= σv2I

ˆfRgLS

P= σv2R−1f ,synth

ˆfRgLS

P= σv2R−1f ,true

u (t) = speech

F (q) = room IR

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