MAXIMUM A POSTERIORI ESTIMATION OF ROOM IMPULSE RESPONSES. Dinei Florencio and Zhengyou Zhang. Microsoft Research, Redmond, WA, USA

Transcription

1 MAXIMUM A POSTERIORI ESTIMATION OF ROOM IMPULSE RESPONSES Dinei Florencio and Zengyou Zang Microsoft Researc, Redmond, WA, USA ABSTRACT Estimating room impulse responses RIRs) as a number of applications, including personalized audio, analyzing and improving acoustic beavior of concert alls, listening room compensation, sound source localization, and many oters. RIRs ave been estimated in essentially te same fasion for te last 50 years: Compute te cross correlation between a signal played at point A, and te signal received at point B. Best results are obtained wen te signal played is wite noise, or a maximum lengt sequence. No prior knowledge is exploited in computing te RIR, wic is simply assumed to be te cross correlation between played and received signals. In contrast, researc in adaptive RIR estimation a.k.a. adaptive Acoustic Eco Cancellation) as made uge progress by among oter tings) incorporating models for te RIR. In tis paper we propose a new RIR estimation tecnique, based on a maximum a posteriori formulation. More specifically, we estimate te room reverberation time, as well as te room noise level, and use tose as priors for te RIR estimation. Comparison wit ground trut sows an average improvement of 12 db compared to traditional metods. Index Terms RIR, room acoustics, room impulse response, AEC, eco cancellation. 1. INTRODUCTION Wen a sound is played witin an enclosure e.g., a room), it propagates in very complex ways. It reflects off walls and oter surfaces, it diffracts, it is absorbed by air and by tese surfaces, and does all tat in a frequency dependent manner [1, 2]. In general, tis is furter complicated by te directionality of te sound source as well as te directionality of te capturing device e.g., a micropone or te uman ear) [3]. Tus, typically, even if te exact geometry of a room is known, modelling te transfer function between two points in a room is extremely ard, except for crude approximations. Knowledge of tat transfer function is, owever, extremely useful in a number of scenarios. Common application include acoustic eco cancellation AEC) [4, 5, 6, 7], personalized audio [8, 9], analysing and improving acoustic beaviour of concert alls [10, 11], listening room compensation [12, 13, 14], sound source localization [15, 16, 17], and many oters [18, 19, 20, 19, 21]. Due to te aforementioned difficulty in modeling acoustic beavior, room impulse responses RIR) are typically experimentally measured directly in te real environment [22]. More specifically, a sound source and a micropone are placed at eac end-point, a sound is played and recorded, and tat recorded signal is used to compute an estimate of) te RIR between tose two points 1. Furtermore, 1 In applications like AEC, te pat of interest in is between te loudif we need estimates of te RIR in points oter tan te measured ones, RIR interpolation may be needed, wic is, in itself, anoter callenging problem[23]. In tis paper we introduce a tecnique tat makes use of prior knowledge of acoustic beavior to improve te quality of te RIR estimates. Our maximum a posteriori MAP) estimate incorporates a simple model for ambient noise and an exponential decaying model for reverberation. Yet, results are quite appealing: an average 12dB improvement over traditional RIR estimation. A closely related problem is tat of adaptive RIR estimation typically referred as adaptive AEC). In adaptive AEC, one assumes te RIR is canging. Tat is muc more callenging problem, and one as to balance te adaptation parameter according to te dynamics of te system, as well as te ambient noise. State of te art adaptive AEC use models of te RIR to control step size and related parameters [24, 25, 26, 27, 28]. Wile tey greatly improve adaptation, for a fixed system as in our problem), tey underperform a simple matced filter cross correlation), wic as us) uses te wole available signal at once. In a way, our paper brings te RIR estimation in line wit te last 50 years of progress in adaptive AEC. Te remainder of tis paper is organized as follows: Section 2 introduces our notation and explains te traditional RIR estimation metod. Section 3 presents our MAP formulation, and Section 4 presents and discusses te experimental results. Finally, Section 5 presents our main conclusions and future directions. 2. TRADITIONAL RIR ESTIMATION To estimate te room impulse response RIR) between a loudspeaker and a micropone, a signal is played troug te loudspeaker and recorded at te micropone. Witout loss of generality, let us assume te first K samples of an RIR are a reasonable approximation of te RIR for te intended application. Let us denote te training signal as te N 1 vector x, and te corresponding signal received at te micropone as y. To simplify notation and witout loss of generality), we assume x as unit variance. Neglecting boundary effects to simplify te notation, we model te signal y as: y = X + a 1) were is te K 1 vector corresponding to te room impulse response, X is te N K matrix composed of K time-sifted versions and x, and a is te ambient noise, wic we assume to be independent from x. speaker and a micropone, and bot are under te control of te system. In oter scenarios, loudspeakers and/or micropones may ave to be placed at one of bot te desired end points.

2 Te traditional way of estimating te RIR is by playing a wite training sequence x, and correlating it wit te received signal y [29]. More specifically, by computing: ĥ = X T y. 2) Te reasoning for tis approximation can be observed by inserting te expression for y from 1) into 2), i.e.: ĥ = X T X + a) 3) = X T X) + X T a. 4) Mismatc db) db SNR 40dB SNR By taking te expected value of 4), we get: E{ĥ} = Rxx + rxa 5) were R xx = E{X T X} i.e., te K-lag autocorrelation matrix of x), and r xa = E{X T a} i.e., te K-lag cross-correlation vector between x and a). Note tat, if te training sequence is wite, tan R xx = I, and, since te noise is assumed independent of te signal r xa = 0. Tus, X T y is an unbiased estimate of te RIR. Furtermore, as te lengt of te training sequence increases, te estimate becomes increasingly close to te true, converging to te correct value as N. For finite training sequences, owever, tese are just approximations. Indeed, we can rewrite 4) as: ) ĥ = + X T X I + X T a 6) Tus, we see tat tere are two noise terms affecting te traditional way of estimating ĥ, bot related to te fact tat x is finite. Te first one reflects te non-diagonal nature of a finite signal covariance matrix, wile te second term originate from te random correlations between two independent) finite signals. Te first can be alleviated by making te training sequence longer, and te second can be alleviated by making te loudspeaker signal stronger, or te sequence longer. Note tat te non-diagonal elements of te covariance matrix can also be reduced by using a special kind of wite noise, called maximum lengt sequences MLS) [30, 31, 32]. Tese special pseudo-random sequences, do improve te quality of te estimates. However, tey sound like wite noise, and cannot be modified muc. As suc, tey are not appropriate for most applications were users will be present during te measurements including our target application of initial AEC filter estimation). Figure 1 sows te ĥ estimation error as a function of training sequence duration, for ambient noise levels of 0 db and 40 db SNR plotted as 10 log 10 i ĥ[i] [i])2 / i [i])2 )). Te quality of estimates vary widely wit room caracteristics, distance, etc. Te data in Figure 1 was obtained using te parameters tat are closest to our target application in AEC, and are described in more detail in Section 4. As expected, te quality of te estimate increases by rougly 3dB for every doubling of te training sequence lengt. However, to get to reasonable estimates, reasonably long sequences ave to be used. For example, at 40dB SNR level, a sequence wit 262k samples, yields only a 23dB approximation of te true RIR. In oter words, we need 16 seconds of a training sequence to get a just reasonable 23 db approximation. In many cases, owever, we cannot afford too long sequences. Wile for acoustic analysis of a concert alls, it migt be reasonable Lengt of training sequence x10 5 samples) Fig. 1. Mismatc between true and conventional RIR estimation as a functions of training sequence lengt. Results correspond to a room wit T 60 = 300ms, a micropone at 2.2m from te loudspeaker, and F s = 16KHz. Estimated filter is also 300ms long. to play several minute-long training sequences, tis is certainly not te case wen te RIR is being used as initialization for a acoustic eco cancellation in a communication system. 3. MAP ESTIMATION We now propose a maximum a posteriori MAP) formulation to estimate. Under te MAP formulation, we ave: MAP = arg max {f y)} 7) = arg max {f)fy )/fy)} 8) = arg max {f)fy )} 9) were f ) denotes te probability density function. Note tat te probability fy ) can be computed by estimating te te probability of te noise a being y X. We assume tat te ambient noise a, and te RIR are multivariate Gaussian random variables, i.e., a N 0, Σ a), and N 0, Σ ), were Σ a and Σ are te covariance matrices for a and, respectively. Note tat, wile a Gaussian is probably not te best model for, it does capture its key statistical caracteristics, and it makes te matematical treatment muc simpler tan if we assume a more complex model. Tus, based on te Gaussian assumption, we ave: f) = 1 2π)k detσ ) exp 1 2 T Σ 1 a ), 10) and 1 fy ) = 2π)k detσ exp 1 ) a) 2 y X)T Σ 1 y X). 11) Inserting 10) and 11) in 9), taking te log, and disregarding te constants and te log itself since we only care about te at te maximum, but not te maximum value per se), we get:

3 { MAP = arg max T Σ 1 } y X) T Σ 1 y X) 12) Taking te derivative of te function being maximized in te rigt side of 12) and making it equal to zero, we get: Σ 1 MAP X T Σ 1 a y X MAP ) = 0 13) Wic can be rewritten as: ) 1 MAP = X T Σ 1 a X + Σ 1 X T Σ 1 a y 14) Tus, to estimate MAP, besides te received signal y, we need estimates of te covariance matrices of te ambient noise and of te taps of te RIR. Tese are, in effect, our priors on te te ambient noise, and on te RIR itself. We discuss our metod for obtaining tem next Estimating te priors Σ a and Σ Te covariance matrix Σ a captures te caracteristics of te ambient noise. Assuming suc noise is stationary, a reasonable estimate can be easily computed during silence periods, e.g., before playing te training sequence x. Tus, in effect it is truly a priori knowledge. As a proof of concept, and a demonstration of te potential of te tecnique, we ave cosen to use a reasonably simple model for Σ. We already simplified te model by assuming te taps of are jointly Gaussian. Wile te result in 14) would be valid for a generic Σ, we furter simplify our metod by assuming te taps are independent, and tat tey decay at an exponential rate wit a time constant proportional to te room reverberation time T 60. Te independence assumption makes Σ a diagonal matrix. Ten, eac element of te diagonal corresponds to te estimate of te variance of tat tap [33, 34]: σ 2 [k] = A.10 6k/FsT 60, 15) were A is a scaling constant to account for te starting sound level. Note tat tis simplified model disregards te fact tat te direct pat does not obey te same exponential decay as te rest of te reverberation, and te sparseness of early reverberation. Te constants A and T 60 are estimated by computing te cross correlation between x and y i.e., te RIR using te traditional metod), and fitting a function in te form of 15) to te later part of te correlation to avoid te direct pat and early reflections). More specifically, we compute ĥ using Eq. 2), and estimate A and T 60 as: {A, T 60} = arg min A,T 60 K k=k s ĥ[k])2 A.10 6k/FsT 60) 2, 16) were k s is te starting point, cosen to elp ignore te direct pat and te early reflections. Our estimation of te covariance matrix of te taps of te RIR, Σ, is based on te actual received signal y. Tus, rigorously speaking, it is not prior knowledge. Te prior is effectively te exponential decay assumption. In oter words, tere are two estimation steps. In te first, we use te received signal to estimate Σ. Ten, on a second step, we estimate MAP using our estimate of Σ as prior. a Because te first step is simply an estimation of two real numbers A and T 60), we assume it is reasonably robust, and don t consider tis as a significant source of error, and focus mostly on te second step. Altoug subtle, we make tat distinction to clarify our use of te MAP term in naming our estimation algoritm. 4. EXPERIMENTAL RESULTS As we mentioned earlier, obtaining true RIRs is callenging, so te standard way of evaluating RIR estimation metods is based on syntetic data. We compare te results of our MAP metod wit te traditional correlation metod, bot using a random training sequence as well as a maximum lengt sequence. We denote tese by TRAD and MLS, respectively, wile refer to our metod as MAP. For syntetic data, we can directly compare te estimated RIR wit te actual one, and precisely measure te estimation error. Table 1 sows te results for ambient SNRs varying from 0 to 30dB, and sequences lengts of 4095, 8191, and Note tat MLS sequences are restricted to certain lengts e.g., 2 N 1), so we evaluate results only at tose lengts, even toug te MAP and TRAD metods do not ave suc restriction. Te simulated room is a rectangular box measuring meters, wit a T 60 of 300ms, and a micropone at 2.2 m from te loudspeaker. Sampling frequency is 16KHz, and te estimated filter lengt is 300ms tus K = 4800). Te true RIR was obtained using te image metod [35]. For TRAD and MAP, a different training sequence is generated eac trial, as a pseudo-random gaussian noise wit te desired lengt. For te MLS te training sequence is computed as in [31]. For all cases, te received signal y was ten computed by Eq. 4), and used as input for te estimation metod. Results are averaged over 100 trials wit different environment wite) noise. For te MAP metod, we compute A and T 60 by using 16) wit k s = 800 and K = 4800, and use a quadratic error parametric fitting based on te minsearc function from Matlab 2013Ra. As it can be seen in Table 1, te MAP estimation is better tan TRAD and MLS in all cases. Te gains over TRAD vary from 5.5 to 19.4 db, wit an average improvement of 11.8 db. Compared to MLS, te gains are smaller, varying from 1.3 to 6.8 db, wit an average improvement of 2.8 db. Figure 2 sows an example of te true and estimated RIRs by te different metods. As expected, te TRAD metod is muc noisier, and tat estimation noise does not decrease towards te tail of te RIR. Te estimation based on MLS as significantly less noise tan TRAD. However, altoug ard to see in te picture, tat noise does not decrease significantly towards te tail eiter. Finally, te MAP estimation is clearly closer to te ground trut, and te estimation noise is decreasing wit k. Tis is important, particularly in cases were te lengt of te true impulse response is not known a priori RIR estimates and AEC Wile tis paper focuses mostly on te RIR estimation itself, in tis section we quickly comment on our underlying application, as it as some implications on our metod. AEC is typically done by adaptive systems. Tis is partially due to te need to adapt to environment canges, and partially due to te need for a reasonable start up time. In systems were te direct pat between loudspeaker and micropone is known a priori e.g., speakerpones, etc) tat pat is

4 Lengt = 4095 Lengt = 8191 Lengt = SNR TRAD MLS MAP TRAD MLS MAP TRAD MLS MAP Table 1. Estimation mismatc, in db, between te true RIR and te corresponding estimates using traditional correlation TRAD), using correlation based on a MLS, and using te proposed metod MAP). MAP MLS TRAD TRUE Fig. 2. RIR and corresponding estimations for a room wit T 60 = 300ms, and 20dB SNR. From bottom to top: RIR ground trut, traditional RIR estimate, MLS RIR estimate, and MAP RIR estimate. typically measured in te factory or during design), and used as te starting point for te adaptive filter. In systems were te location of loudspeaker is not controlled e.g., PCs, game consoles, etc), tis initial estimate as to be computed after deployment. Indeed, modern communication systems were te placement of te micropone in relation to te loudspeaker is not constrained, typically play a calibration tone to obtain an initial estimate of te AEC filter. Suc calibration tone may be disguised e.g., te sinusoidal sweep at te startup sound in Skype), or not e.g., te Mozart snippet played during te initial Kinect calibration). Regardless of being disguised or not, wite noise is not a desirable sound to be played at loud volumes. For tat reason, as muc as MLS provide improvements over pseudo-random sequences, tey are very specific and cannot be modified. Wile te results reported for TRAD and MAP are also based on wite noise sequences, tese can be easily modified to accommodate virtually any sequence, by witening te received and reference) signals before computing te RIR estimates. Wile some degradation can be expected, te overall trend is similar. In oter words, wile we include results for using MLS in Table 1, we do not consider tem as a valid alternative to many applications. Furtermore, we point out tat, for cases were MLS are valid, te MAP results can be furter improved by using tem as te training sequence as well. 5. CONCLUSIONS Estimating Room impulse responses find application in a number of areas. Estimation time, owever, is a concern, as it can be too long for many applications. Tis time can be reduced by te use of MLS sequences, but tese imply additional constraints on te sequences. In tis paper we presented a new metod for estimation of room impulse responses. Te metod is based on a MAP formulation, using te observed ambient noise and an exponential reverberation decay as priors. Wile priors ave been used in te past in adaptive AEC, tey are used to control te adaptation, and do not extend naturally to fixed or initial) RIR estimation. Using our metod, improvements of 11.8 db over traditional metods, and of 2.8 db over MLS sequences were acieved. In contrast to MLS, te proposed tecnique can be easily modified to support non-wite training sequences, making it particularly appropriate for applications were users will be present, as for example in initial AEC filter estimation. We are currently using tis MAP estimation to speed up product development[36]. Additionally, we are also investigating extensions to multicannel adaptive AEC[7], implications to micropone arrays [37]and evaluating subjective quality of te results using Crowd- MOS [38, 39]

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