REFLECTION COEFFICIENT ESTIMATION BY

International Workshop on Acoustic Signal Enhancement 2012, 4-6 September 2012, Aachen

REFLECTION COEFFICIENT ESTIMATION BY PSEUDOSPECTRUM MATCHING

D. Markovic´ 1 , C. Hofmann 2 , F. Antonacci 1 , K. Kowalczyk 2 , A. Sarti 1 , W. Kellermann 2 1

Politecnico di Milano

Dipartimento di Elettronica e Informazione Piazza Leonardo da Vinci, 32, 20133 Milano, Italy

ABSTRACT This paper presents a novel model-based method for in-situ estimation of sound reflection coefficients in acoustic enclosures. The method uses a modeling engine for simulation of acoustic propagation and generation of the spatial pseudospectrum using a beamforming technique. The simulated pseudospectrum is matched with the measured pseudospectrum acquired by a real beamforming microphone array in order to estimate reflection coefficients of the walls of the acoustic enclosure. Experimental results confirm that the proposed method allows for an accurate estimation of the reflection coefficients, especially for reflective walls, typical of everyday environments.

2

University of Erlangen-Nuremberg

Multimedia Communications and Signal Processing Cauerstr. 7, 91058 Erlangen, Germany

highly coherent signals. Furthermore, the use of the entire pseudospectrum instead of single DOAs makes the method more robust to noise. The proposed method requires the knowledge or estimation [3] of the environment geometry, sound speed, the source signal and its radiation pattern, and positions of the source and the array. The methodology has been applied in a real environment paneled with walls of different materials and has proven to be more practical with respect to laboratory measurements. The paper is structured as follows: the signal model is described in Sec. 2, the proposed reflection coefficient estimation method is presented in Sec. 3, followed by experimental results shown in Sec. 4 and conclusions in Sec. 5.

Index Terms— Reflection coefficient estimation, beamforming,

2. DATA MODEL

microphone array

1. INTRODUCTION Many audio signal processing algorithms, such as signal enhancement [1] and multichannel upmixing [2], can benefit from the knowledge of parameters characterizing the acoustic environment. Recent research efforts, such as those within the SCENIC project [3], focused primarily on estimating the Directions Of Arrival (DOAs) of early room reflections [4, 5, 6], room volume estimation [7], and the inference of the geometry of an acoustic enclosure [8]. W hen a sound wave propagating through an acoustic space is reflected from a wall, a part of the impinging energy is reflected and another part is absorbed. Such reflective properties of room boundaries are captured by reflection coefficients, which are typically measured using a boundary sample positioned in the so-called impedance tube [9]. However, direct estimation of the reflection coefficient values in a room is highly challenging due to the high sensitivity to noise and the dependency of the reflection coefficient on both frequency and the angle of incidence. In this paper, a novel model-based method to estimate the reflection coefficient of a wall is proposed, which is based on delay and sum beamforming [10]. The presented technique relies on matching the spatial pseudospectrum (defined as the output power of the beamformer for different look directions) generated using the acoustic propagation model with the pseudospectrum obtained from the actually acquired microphone signals. The reflection coefficients and the source signal amplitude can then be estimated by finding optimum model parameters, such that both pseudospectra are closely matched. The proposed algorithm explicitly models the correlations between different acoustic propagation paths, where the reflected signals are modeled as delayed and attenuated versions of the source signal, thereby taking into account the interactions between these

The real-valued reflection coefficient of the material is defined as the amplitude of a reflected wave relative to an incident wave [9]. The assumed acoustic two-dimensional propagation model is depicted in Figure 1, where upon emission of the source signal s(t) with amplitude A through a loudspeaker, the sound wave is specularly reflected from the room boundaries. In general, the amplitude sensed by a microphone for an exemplary reflection path can be written as [9] Aαk (f )gk (θk )l(θl )m(θm ) s(t − τ ), (1) d where αk (f ) denotes the reflection coefficient of the reflector k at frequency f , gk (θk ) models the variation of the reflection coefficient with the angle θk , l(θl ) is the loudspeaker radiation pattern, m(θm ) is the directivity pattern of the microphone, d and τ denote the traveled distance and the corresponding delay, respectively. Note that the signal s(t), environment geometry, gk (θk ), l(θl ) and m(θm ) are assumed to be known, and an acoustic propagation modeling engine [11] is used to obtain θk , θl , θm , d, and τ . Thus we can denote the signal with known attenuation factors as y(t) = gk (θk )l(θl )m(θm )s(t − τ )/d. On the other hand, the unknowns are the frequency-dependent reflection coefficients and the signal amplitude, jointly denoted as Λ(f ) = Aαk (f ). Note that Λ depends on the frequency. In the following we adopt the solution of a subbandbased estimation of Λ(f ). For reasons of simplicity in the notation, however, we omit in the following the subband index. In case of multiple reflections we write the signal acquired by microphone j as xj (t) =

Nr R  

Λri yj,ri (t) + wj (t),

(2)

r=0 i=1

where r is the reflection order. In particular, r = 0 denotes the direct path and r = 1 the first reflection; Nr is the number of paths for reflection order r. As an example, for the two-dimensional model

of a rectangular room, Nr = 1 for r = 0, Nr = 4 for r = 1. Finally, yj,ri (t) is the signal of the acoustic path with indices r and i. Λri is the unknown amplitude caused by room reflections. As an example, Λ01 = A for the direct path, Λ1k = Aαk , k = 1, ..., N1 for the first-order reflection, and Λ2i = Aαk αn , k = n for the second-order reflection. The additive noise wj (t) represents the selfnoise of the microphone, which is supposed to be spatially white and statistically independent from the signal s(t). The early reflections up to the order R are modeled as distinct reflections, whereas the reflection paths with r > R are assumed to have very low energy and to impinge on the array uniformly from all directions, and therefore can be considered as part of the noise.

geometry x1 (t)

xm(t)

s(t)

Estimation procedure

beamforming

estimation step 1

P

xM (t)

path amplitudes

M

noise variance

microphone array log(.) d

estimation step 2

propagation modelling engine + postprocessing

H

m exp(.)

As(t)

l( θl )

d1

θk

Known:

Re fl αk ( ecto f )g rk k (θ k)

θl

g (θ ) l( θl ) m(θm ) y (t) = k k s(t-τ) d Unknown: Λ = A αk(f)

θk

t

d2

1f

reflection signal coefficients amplitude

Fig. 2. Algorithm overview.

θm A αk(f)gk(θk) l( θl ) m(θm ) s(t-τ) d d d + = 1 d2 t

m(θm )

From (3), writing explicitly the unknown amplitudes of different paths yields

Fig. 1. Propagation model. H

P (θ) = a (

The M microphone signals of the array are described by X = [x1 (t), ..., xM (t)]T =

R

r=0

Yri = [y1,ri (t), ..., yM,ri (t)]T ,

Nr

i=1

2 ˆ r1 i1 r2 i2 )a + σw Λ r1 i1 Λ r2 i2 R ,

r1 =0 i1 =1 r2 =0 i2 =1

Λri Yri + W,

2 σw

W = [w1 (t), ..., wM (t)]T . The spatial pseudospectrum obtained using a delay-and-sum beamformer [12] using M microphones for direction θ can be written as ˆ P (θ) = aH (θ)Ra(θ),

r1 r2 R N R N    

(3)

where θ is the look direction of the pseudospectrum, and a(θ) is the steering vector. In particular, we can write that a(θ) =

1, ej2πfs (θ) , ..., ej(M −1)2πfs (θ) , where fs (θ) denotes the spatial frequency, which depends on the angle of arrival θ, temporal ˆ frequency f , sound speed c, and the array geometry. Finally  , Ris the estimate of the observation covariance matrix R = E XXH . √1 M

3. ESTIMATION ALGORITHM Under the assumptions made above, the only unknowns in (2) are the propagation path amplitudes Λri . The estimation of the reflection coefficients is performed in a two-step procedure, as illustrated in Figure 2. First we perform the estimation of Λri from the observed pseudospectrum. After this step we proceed to the estimation of the reflection coefficients αk and of the amplitude A using the estimates of Λri . 3.1. Step 1 The inputs of the estimation algorithm are the values of the pseuˆ = dospectrum for all the possible Directions Of Arrival (DOAs), P T [P (θ1 ), P (θ2 ), ...] . These data are matched with the components of the modeled pseudospectrum in order to estimate the components Λri . A detailed description follows.

(4) Rr1 i1 r2 i2 = E[Yr1 i1 YrH2 i2 ], is the (unknown) noise power/variance. Echoes coming

where from different paths consist in delayed and attenuated replica of the signal s(t) and are highly coherent. It is therefore necessary to take their correlations into account, as expressed by (4). By T denoting Pr1 i1 r2 i2 = [Pr1 i1 r2 i2 (θ1 ), ..., Pr1 i1 r2 i2 (θN )] with H ˆ Pr1 i1 r2 i2 (θ) = a Rr1 i1 r2 i2 a, we can write

P=

r1 r2 R N R N    

2 Λr1 i1 Λr2 i2 Pr1 i1 r2 i2 + σw 1.

(5)

r1 =0 i1 =1 r2 =0 i2 =1

Note that in (5) there are contributions Pr1 i1 r2 i2 that undergo reflections from the same walls but in different order. All these paths will exhibit the same amplitude and therefore we can group them. After this grouping we obtain ˆ ≈ P

N 

2 Pi qi + 1σw = Mq,

(6)

i=1

where N is the number of pseudospectrum components after group2 T ing, M = [P1 , ..., Pi , ..., PN , 1], q = [q1 , ..., qi , ..., qN , σw ] and qi denotes the weight of the corresponding pseudospectrum component Pi . As an example, qi = A2 for the autocorrelation of the direct path, qi = A2 αk2 for the autocorrelation of the first-order reflection, qi = A2 αk for the crosscorrelation between the direct path and the first-order reflection, and qi = A2 αk αn for the crossˆ correlation between two first-order reflections. In (6), the vector P represents the measurements acquired by the microphone array and Pi denotes the modeled components of the pseudospectrum, as illustrated in Figure 3. Equation (6) is a linear system with observations P, model M and vector of unknowns q. It can be solved by minimizing the sum of squared differences between the observations and the model, i.e.

ˆ = arg min{(P − Mq)T (P − Mq)}. q q

Note that the number of variables q in (6) is much higher than 2 and α for the number of independent unknowns, which are A, σw each reflective surface. As it is, therefore, the system in (6) is illconditioned. Different approaches can be used in order to address this problem.

corresponding to di = log qˆi = log (A2 α1 α2 ) is [2, 1, 1, 0, ..., 0]. The least-squares solution to (8) is given by

ˆ ˆ = (HT H)−1 HT d. m

(9)

ˆ α ˆ Finally, the estimates [A, ˆ1, α ˆ 2 , ...]T are obtained as exp(m).