Blind estimation of the direct-to-reverberant ratio using a beta distribution fit to binaural coherence Open Access

Knowledge of the direct-to-reverberant ratio (DRR) between an acoustic source and a listening position or receiver can be useful in a number of acoustic and audio applications, such as dereverberation,¹ source distance estimation,² and automatic speech recognition.³ The DRR is also known to be a perceptual cue for source distance and the sense of reverberance.⁴ Therefore, appropriate DRR values also are important to promote externalization of virtual sources in augmented-reality environments.⁵ 

Given a room impulse response (RIR), the DRR is defined as, “At a given location, the ratio of the sound pressure level of a direct sound from a directional source to the reverberant sound pressure level simultaneously incident to the same location.”⁶ Mathematically, it is computed from a discrete-time RIR, h(n), as follows:

where n_d is the sample index of the peak of the direct sound arrival and n₀ is the number of samples corresponding to a small temporal window, typically covering a range of 1.0 to 2.5 ms.⁷ When h(n) is known, computation of the DRR is straightforward from Eq. (1). However, when the RIR is not available, the DRR can be estimated blindly using acoustic sources of opportunity.

Blind estimation of DRR can be separated into a variety of categories, e.g., single-channel⁸ and multichannel approaches,⁹ traditional signal-processing¹⁰ and machine-learning¹¹ approaches, and algorithms exploiting various signal features including spectral standard deviation,¹² modulation energy,¹³ and coherence.¹ Eaton et al. provide a recent summary of DRR estimation methods in the context of the ACE Challenge for blind estimation of room-acoustic parameters.¹⁴ 

In this paper, we present a DRR estimation algorithm that exploits the statistics of binaural magnitude-squared coherence (MSC). Binaural MSC values, aggregated over time and frequency, are fit with a beta distribution, and an estimation model is developed from the relationship between the shape parameters of the distribution and the DRR. The performance of the estimator is evaluated both numerically as well as perceptually, the latter with respect to published just-noticeable differences for DRR.

Our DRR estimation relies on the magnitude-squared coherence between two signals collected with binaural microphones.¹⁵ We assume a single sound source at azimuth and elevation angles of (0°,0°) relative to the listener. We compute the time- and frequency-dependent MSC, $|ΓYLYR(k,n)|2$⁠, using a short-time discrete Fourier transform (STFT), $YL/R(k,n)$⁠, of the left and right input signals. Specifically, following Zohourian and Martin,¹⁰ we compute

from the temporally smoothed cross-spectrum and power spectra

for $m,m′∈{L,R}$⁠, where k is the frequency index and n is the STFT time-frame index. $Φ̂YmYm′(k,0)$ is computed with only the second term in Eq. (3). Example time/frequency matrices of MSC, computed using a 10-s speech sample convolved with three binaural room impulse responses (BRIRs) with increasing DRR, are shown in Figs. 1(a)–1(c). The same data, collapsed over time and frequency, are shown in Figs. 1(d)–1(f) in histogram format. While there is not a 1:1 mapping between the MSC matrices and the histograms, we exploit the fact that these matrices, when computed from binaural speech samples with similar DRR values, collapse to similar histogram shapes.

The natural restriction of MSC values to the range [0, 1], as well as the shapes of the MSC histograms in Figs. 1(d)–1(f), suggest that the collapsed MSC values can be described with a beta distribution, the probability density function (PDF) of which is defined on the interval [0, 1] with a pair of shape parameters a and b,

where B(a, b) is the beta function which serves as a normalization constant. Different combinations of the shape parameters allow for skewed PDFs with concentration on either end of the [0, 1] range, as well as a uniform distribution, all of which can approximate the distributions of MSC in different environments with varying DRRs. A scaled beta PDF, fit to the MSC data, is shown with a dashed line in each of Figs. 1(d)–1(f).

Given the relationship between the DRR and the shape of the best-fit beta distribution to MSC values collapsed over time and frequency, we explored various mappings from combinations of the shape parameters a and b to DRR. One logical candidate is the skewness,

since it is precisely the asymmetry of the distribution that we want to exploit. Figure 2(a) depicts DRR as a function of beta skewness for 700 binaural signals (see Sec. 3 for details on these signals). Somewhat surprisingly, though, the simpler mapping from b to DRR led to a more accurate estimator.¹⁶ The relationship between b and DRR is shown in Fig. 2(b) where the dashed line represents a two-term power series fit to the data (⁠$p0bp1+p2$⁠) with $p0=−58.17,$$p1=0.22$⁠, and $p2=60.39$⁠. Using training data to learn this fit, we can collect a new sample of binaural data, compute the MSC between the channels over time and frequency, compute the shape parameter b from a beta PDF fit to the MSC distribution, and estimate the DRR for that new sample.

BRIRs used to evaluate the DRR prediction model were collected from a variety of publicly available datasets, as well as from internal measurements, as indicated in Table 1. The complete set comprised 70 BRIRs, all measured with the source near 0° azimuth and elevation, spanning a DRR range from −15.1 to 17.8 dB. For each BRIR, DRR values were computed for each channel (ear) according to Eq. (1), and averaged to provide a single value. Monaural, anechoic speech samples from the ACE Challenge dataset¹⁴ were merged into a continuous audio stream which was sliced into 10-s segments, and these segments were convolved with the BRIRs to provide input data for the algorithm. Ten randomly chosen speech samples were used with each BRIR for a total of 700 examples. All BRIRs and speech segments were sampled at 48 kHz.

For each example, the short-time discrete Fourier transform was computed with 32-ms windows, 50% overlap, a Hann window, and FFT size equal to the window size (1536 samples). Frequency bins between 200 Hz and 10 kHz were retained, resulting in 196 560 time/frequency points (315 frequency bins × 624 time frames) for each channel. The MSC between the channels was computed using Eqs. (2) and (3). We originally fit the collapsed MSC data using matlab's^® betafit function to compute the shape parameter b, but found a simple calculation using the mean μ and variance $σ2$ of the data to provide equivalent results,

We then fit a two-term power-series model to a subset of the resulting 700 pairs of DRR and b values, and tested that model with the remaining pairs using a tenfold cross-validation scheme.

Example results from our prediction algorithm are shown in Fig. 3. Figure 3(a) contains the DRR estimates for 700 10-s binaural speech signals generated through the process described in Sec. 3: the model was trained with 630 examples (63 of the 70 BRIRs, each convolved with 10 speech samples) and asked to predict the DRR from 70 signals comprising the remaining 7 BRIRs each convolved with 10 samples of unseen speech. 1000 iterations of this process (train on a randomly selected set of 63 BRIRs, test with the remaining 7) yielded an average root-mean-square error (RMSE) of 2.30 dB (min. RMSE = 2.23 dB, max. RMSE = 2.48 dB, median RMSE = 2.30 dB). The shaded gray area represents the just-noticeable difference (JND) for DRR as reported by Larsen et al.⁴ (see their Fig. 3). The black line represents perfect performance. Prediction performance can be seen to deteriorate slightly at DRRs below approximately −10 dB [as a result of the variance of DRR values around $b≈2.5$ in Fig. 2(b)], however, the results remain well within the JND.

While the performance of the algorithm appears to be, for the most part, independent of the BRIR dataset, four of the BRIRs for which the estimates have high error [cyan data points in Fig. 3(a) near measured DRRs of −15, −6.2, and 2.2 dB] come from the Pori dataset.²⁰ All of these BRIRs were measured in a concert hall, the largest of the rooms within our collection of BRIRs. Strong early reflections from a nearby sidewall may be responsible for skewing the MSC distributions toward higher values and thus causing an overestimation of DRR for some of these cases, but further analysis is necessary to confirm what geometric and/or acoustic characteristics of this room, and the specific measurement positions within it, may be responsible for the reduced estimation performance.

In Fig. 3(b), we compare our results to those from our implementation of the estimator by Zohourian and Martin¹⁰ applied to the same 700 examples. Note the improved performance below 0 dB DRR, where their model tends to over-predict the DRR and the error exceeds the JND for DRRs below approximately −5 dB. This characteristic of over-predicting low DRRs also is evident in their Fig. 4 (albeit when the test samples were speech convolved with simulated BRIRs with the source at 45° azimuth).

Figure 3(c) shows the performance of the model, in terms of RMSE, as a function of speech sample length. Each box represents 10 iterations of predictions with 70 BRIRs, each convolved with 10 samples of speech of the designated length. The red, central mark indicates the median value, and the box extents indicate the 25th and 75th percentiles. The whiskers indicate the full data extents excluding outliers, which are marked with red “+” symbols. These results suggest that speech samples as short as 4 s, and possibly 2 s, may provide sufficient data for successful DRR predictions, although evaluation with respect to the JND should be done to confirm this. One factor that may have influenced the performance of the algorithm, particularly with the shorter speech samples, is the lack of a voice-activity detector (VAD) in the processing pipeline. Without a VAD, the presence of pauses between utterances in the shorter samples may have significantly skewed the MSC distribution and thus degraded the estimates.

This paper presents an approach to estimate the direct-to-reverberant ratio blindly from binaural speech signals based on the distribution of magnitude-squared coherence values, computed over time and frequency, between the two channels. Given short segments of speech captured with binaural microphones, we build a prediction model by fitting each set of MSC values with a beta distribution, and map the shape parameter b from the distributions to the DRR with a two-term power series. Initial validation of the model was carried out with 700 examples comprising 70 measured BRIRs, spanning a DRR range from −15 to +18 dB, each convolved with ten 10-s speech samples. Tenfold cross validation with these examples yielded results with an overall RMSE of 2.3 dB, nearly all of which fell within the just-noticeable difference for DRR (which is a function of the DRR value). Further evaluation suggests that speech samples as short as 2 s may be sufficient for an acceptable level of estimation performance.

We currently are exploring several directions for future work. First, similar to reverberation time (RT), DRR is a frequency-dependent parameter, so band-limited predictions should be considered. Preliminary work on this aspect (not shown) suggests a small increase in the RMSE of the predictions, although this may be mitigated by the fact that the JND for DRR based on narrow-band signals is higher than that for broadband signals.⁴ Second, the performance of this algorithm should be evaluated with varying signal-to-noise ratios, which can be accomplished by adding binaural noise with the appropriate RT and coherence to the training and test examples. Third, this estimator should be tested with live binaural speech signals rather than recorded speech convolved with measured BRIRs. To this end we have implemented the algorithm on the Bela embedded computing platform²³ and currently are planning a data-collection campaign. Fourth, we opted for a model whose input is restricted to acoustic sources at 0° azimuth and elevation to avoid the added complexity of requiring a direction-of-arrival estimate and compensation for interaural differences based on that estimate. However, generalization of our model to arbitrary source locations may be a valuable extension. Fifth, we considered only a small number of mappings from both beta shape parameters a and b to DRR before adopting the model in Sec. 2.3 which utilizes only b. However, there may be a function DRR = f(a, b) that outperforms our current estimator, and we are currently developing a simple machine-learning architecture to learn this function. Finally, there is evidence that DRR and RT are not independent,¹¹ and there is previous work using coherence to estimate RT,²⁴ which suggests that joint coherence-based estimation of these two room-acoustics parameters may be beneficial.