# Binaural coherent-to-diffuse-ratio estimation for dereverberation using an ITD model
**Authors**: Chengshi Zheng, Andreas Schwarz, Walter Kellermann, Xiaodong Li
∗
## BINAURAL COHERENT-TO-DIFFUSE-RATIO ESTIMATION FOR DEREVERBERATION USING AN ITD MODEL
Chengshi Zheng ∗† , Andreas Schwarz † , Walter Kellermann † , Xiaodong Li ∗
Communication Acoustics Laboratory Institute of Acoustics, CAS 100190 Beijing, China
† Chair of Multimedia Communications and Signal Processing Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg 91058 Erlangen, Germany
## ABSTRACT
Most previously proposed dual-channel coherent-to-diffuseratio (CDR) estimators are based on a free-field model. When used for binaural signals, e.g., for dereverberation in binaural hearing aids, their performance may degrade due to the influence of the head, even when the direction-of-arrival of the desired speaker is exactly known. In this paper, the head shadowing effect is taken into account for CDR estimation by using a simplified model for the frequency-dependent interaural time difference and a model for the binaural coherence of the diffuse noise field. Evaluation of CDR-based dereverberation with measured binaural impulse responses indicates that the proposed binaural CDR estimators can improve PESQ scores.
Index Terms -Binaural speech dereverberation, interaural time difference, coherent-to-diffuse-ratio
Recently, coherent-to-diffuse-ratio (CDR) estimators have been proposed, which can be seen as an alternative formulation of coherence-based dereverberation approaches [12]. In [6], the assumption was made that binaural signals are time-aligned before calculating the spectral weights of the Wiener filter. In [11], two CDR estimators were proposed, where one requires knowledge on both the direction of arrival (DOA) of the desired speaker and the spatial coherence of the late reverberant speech, and the other does not need the DOA information. In [12], Schwarz and Kellermann proposed improved estimators both for the case of known and unknown DOA, which were shown to lead to improved dereverberation performance (see [12, Table III] for details). To the best of our knowledge, these CDR estimators have not been applied to binaural dereverberation and their performance has not been reported until now.
## 1. INTRODUCTION
Both speech quality and speech intelligibility may dramatically degrade in reverberant and noisy environments. Many different algorithms were proposed to suppress noise and the reverberation during the past decades (see [1-3] and references therein). This paper focuses on binaural speech dereverberation, where the binaural signals are recorded with two microphones located at two human ears.
Previous studies have already shown that it is important to preserve both the interaural time difference (ITD) and the interaural level difference (ILD) cues when applying binaural dereverberation methods for hearing aids [4-9], since, when binaural cues are distorted, localization of sound sources becomes difficult [10]. This condition is ensured by a twochannel postfiltering approach where the same gain is applied to both channels [6]. In [6], Jeub et al. took the shadowing effect of the head into account in the diffuse sound field model. In [8, 9], interaural coherence histograms were mapped to a gain function to suppress the reverberant components in each frequency channel.
This work was supported by the National Science Fund of China (NSFC) under Grants 61201403 and 61302126.
After briefly reviewing CDR estimators for free-field conditions, i.e., for a sound field with no obstructions close to the microphones, we describe models for the ITD and the coherence of diffuse noise under the influence of the head in a binaural scenario, and show that the direction-dependent CDR estimators based on a free-field assumption are not robust under this model. We propose to modify the CDR estimators to use binaural models. Experimental results confirm that the proposed estimators achieve higher PESQ scores than the free-field estimators when applied to coherence-based dereverberation. The proposed binaural CDR estimators have numerous applications, such as binaural hearing aids, robotics, or immersive audio communication systems.
## 2. FREE-FIELD SIGNAL MODEL AND CDR ESTIMATION
We model two reverberant and noisy microphone signals x i ( t ) , i = 1 , 2 , as the sum of a desired speech component x i, coh ( t ) and an undesired component x i, diff ( t ) consisting of diffuse reverberation and/or noise:
<!-- formula-not-decoded -->
As in previous studies, we assume both microphones to be omnidirectional and the desired component to be a plane wave in the free (locally unobstructed) field, so that x 2 , coh ( t ) is a time-shifted version of x 1 , coh ( t ) [6, 11, 12]:
<!-- formula-not-decoded -->
where τ 12 is the time difference of arrival (TDOA) of the desired sound between the first and the second microphone. The free-field model for the spatial coherence between the desired speech component at both microphones, x 1 , coh ( t ) and x 2 , coh ( t ) , is given by
<!-- formula-not-decoded -->
If θ = 0 ◦ corresponds to broadside direction, the TDOA in the free field can be expressed as
<!-- formula-not-decoded -->
where d is the distance of the two microphones and c is the speed of sound.
The spatial coherence between the reverberation/noise components x 1 , diff ( t ) and x 2 , diff ( t ) is given by the spatial coherence function of two omnidirectional sensors in a diffuse (spherically isotropic), locally unobstructed sound field:
<!-- formula-not-decoded -->
where f is the frequency in Hz. For the cylindrically isotropic field, the spatial coherence can be given by
<!-- formula-not-decoded -->
Generally, (5) often fits better than (6) in practical applications [12], therefore we use Γ FF diff ( f ) in this paper, although Γ FF 2D -iso ( f ) may be applied analogously.
The CDR at the i -th microphone can be given by
<!-- formula-not-decoded -->
where Φ i, coh ( k, f ) and Φ i, diff ( k, f ) are the short-time power spectra of x i, coh ( t ) and x i, diff ( t ) , respectively, with the frame index k and frequency f (we will omit both k and f in the following for brevity). We further assume that the power spectra are identical at the two microphones for both the desired and undesired component, i.e., Φ coh = Φ 1 , coh = Φ 2 , coh and Φ diff = Φ 1 , diff = Φ 2 , diff , and therefore
<!-- formula-not-decoded -->
Using the models for the coherence of the desired and diffuse signal components given above, and a short-time estimate of the coherence between x 1 ( t ) and x 2 ( t ) , which is in the following denoted as ˆ Γ x ( k, f ) and which may be obtained by recursive averaging, it is possible to estimate the time- and frequency-dependent CDR, as described in detail in [12]. The CDRestimators which are evaluated in this paper are summarized in Table 1.
Table 1 : Summary of CDR estimators evaluated in this paper. Γ coh and Γ diff indicate the model coherence functions used for desired signal and diffuse noise, respectively, ˆ Γ x indicates the estimated coherence of the mixed sound field. {·} extracts the real part of a complex value and ∗ denotes the complex conjugate.
| Estimator | Direction-dependent |
|-----------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| ˜ η Schwarz1 ˜ η Schwarz2 | { Γ ∗ coh ( Γ diff - ˆ Γ x )}/( { Γ ∗ coh ˆ Γ x } - 1 ) ∣ ∣ ∣ Γ ∗ coh ( Γ diff - ˆ Γ x )/( { Γ ∗ coh ˆ Γ x } - 1 )∣ ∣ ∣ Direction-independent |
| Estimator | {( Γ diff - ˆ Γ x )/( ˆ Γ x - exp ( j ˆ Γ x ))} [12, (25)] |
| ˜ η Thiergart2 ˜ η Schwarz3 | |
## 3. BINAURAL SIGNAL MODEL
When the two microphones are placed at the two ears, the ITD is the propagation delay of the desired sound from the left ear to the right ear and the ILD measures the power level difference between the two microphones. Both the ITD and the ILD have already been widely studied, and various models can be found in [13, 14] and references therein. As in [6], the impact of the ILD is neglected in the following, i.e., we maintain the assumption of equal power at both microphones. Based on this assumption, both the CDRs and the postfilter gain functions are the same at the two microphones placed at the two ears.
In this section, we first describe a simplified model for the frequency-dependent ITD and use it to derive a coherence model for the desired signal component. Then, we describe appropriate models for the diffuse sound field coherence which account for the effect of the head. Finally, we describe the application of these models for binaural CDR estimation and compare the robustness of CDR estimators based on the free-field model to the binaural CDR estimators.
## 3.1. ITD and Desired Signal Coherence Model
Previous studies have shown that, unlike the TDOA in the free-field case given by (4), the ITD is highly dependent on the frequency, the azimuth angle, the elevation angle and the distance of the desired speaker from the head [14-16]. Here, we use a simplified ITD model to make it applicable for practical application to binaural dereverberation. We assume that the distance of the desired speaker from the head is larger than 1m, and thus does not have a significant effect on the ITD [15, Fig. 9]. Furthermore, we neglect separate consideration of elevation and azimuth angles, and instead model the ITD as a function of the angle θ , which we define as the angle between the direction of the desired speaker and the forward median plane of the head. According to the head-related spherical coordinate system [15, Fig. 7], θ = 0 and θ = ± π correspond to the forward and the backward median planes of the head, respectively.
Fig. 1 : Comparison of τ 12 and τ lr ( f ) versus the angle of the desired sound for different frequencies.
<details>
<summary>Image 1 Details</summary>
### Visual Description
## Line Graph: Interaural Time Difference (ITD) vs. Angle (θ)
### Overview
The graph depicts the relationship between Interaural Time Difference (ITD) in milliseconds (y-axis) and angle (θ) in degrees (x-axis). Four distinct lines represent different frequency-dependent ITD responses, with the legend clarifying their associations. The x-axis spans 0° to 180°, and the y-axis ranges from 0 to 0.8 ms.
### Components/Axes
- **X-axis (θ[°])**: Angle, labeled in degrees, spanning 0° to 180°.
- **Y-axis (ITD [ms])**: Interaural Time Difference, labeled in milliseconds, ranging from 0 to 0.8 ms.
- **Legend**: Located in the bottom-left corner, associating line styles/colors with frequency-dependent ITD parameters:
- **Dotted purple**: τ₁₂ (no frequency condition).
- **Solid red**: τₗᵣ(f), f < 500 Hz.
- **Dashed black**: τₗᵣ(f), f = 1000 Hz.
- **Dash-dot green**: τₗᵣ(f), f > 2000 Hz.
### Detailed Analysis
1. **Solid Red Line (τₗᵣ(f), f < 500 Hz)**:
- Peaks at ~100° with ITD ≈ 0.75 ms.
- Symmetrical curve, rising to the peak and declining equally on both sides.
- Matches the highest ITD values across the θ range.
2. **Dashed Black Line (τₗᵣ(f), f = 1000 Hz)**:
- Peaks at ~100° with ITD ≈ 0.7 ms.
- Slightly lower amplitude than the red line, maintaining symmetry.
3. **Dash-Dot Green Line (τₗᵣ(f), f > 2000 Hz)**:
- Peaks at ~100° with ITD ≈ 0.65 ms.
- Lower amplitude than the black line, consistent symmetry.
4. **Dotted Purple Line (τ₁₂)**:
- Peaks at ~100° with ITD ≈ 0.5 ms.
- Lowest amplitude among all lines, forming a baseline.
All lines start and end at 0 ms at 0° and 180°, forming symmetrical parabolic curves centered at 100°.
### Key Observations
- **Frequency-Dependent ITD**: Higher frequencies (f > 2000 Hz) exhibit lower peak ITD values compared to lower frequencies (f < 500 Hz).
- **Symmetry**: All curves are symmetric about 100°, suggesting a consistent relationship between θ and ITD.
- **τ₁₂ Baseline**: The dotted purple line (τ₁₂) consistently underlies other lines, indicating a reference or constant parameter.
### Interpretation
The graph demonstrates that ITD varies with both angle (θ) and sound frequency. Lower frequencies (<500 Hz) produce the largest ITD peaks (~0.75 ms), while higher frequencies (>2000 Hz) show reduced ITD (~0.65 ms). The τ₁₂ parameter (dotted purple) acts as a baseline, possibly representing a non-frequency-specific ITD component. This suggests that auditory localization mechanisms may prioritize lower-frequency sounds for precise angular discrimination, as their ITD responses are more pronounced. The symmetry of all curves implies a linear relationship between θ and ITD within the measured range, with deviations likely influenced by physiological or environmental factors not explicitly modeled here.
</details>
Kuhn has shown that the ITD is frequency-dependent [16], which can be approximately summarized as
<!-- formula-not-decoded -->
and, for f ≥ f H = 2000 ,
<!-- formula-not-decoded -->
where (9) and (10) are identical to [16, (7) and (12)], respectively. However, for the middle frequency range, there is not an explicit expression. We propose to use a linear interpolation to model the ITD in the middle frequency range, which agrees well with the measurement results [16] and is given by
<!-- formula-not-decoded -->
where f = f Mid ∈ [500 2000] Hz.
Compared to τ 12 , the ITD τ lr ( f ) is not only a function of the DOA but also of the frequency. τ 12 and τ lr ( f ) versus θ are plotted in Fig. 1 for different frequencies. Fig. 1 shows that the difference between | τ lr ( f ) | and | τ 12 | is largest for f ≤ 500 Hz. For f > 2000 Hz, τ lr ( f ) is close to τ 12 when | θ | or | π ± θ | is smaller than π/ 4 , while | τ lr ( f ) | is much larger than | τ 12 | for | θ | close to π/ 2 .
Without the shadowing effect of the head, the free-field coherence model of the desired signal is given by (3). Based on the frequency-dependent ITD model which accounts for the head effect, we can now define the coherence of the desired component for the binaural case as:
<!-- formula-not-decoded -->
## 3.2. Diffuse Noise Coherence Model
The shadowing effect of the head also has an impact on the spatial coherence of the two microphone signals in a diffuse
Fig. 2 : CDR estimation error of the free-field estimator, ∆ , versus f and θ for (a) the input CDR η in = -20 dB, (b) the input CDR η in = 20 dB.
<details>
<summary>Image 2 Details</summary>
### Visual Description
## Heatmaps: Frequency vs. Angle Analysis
### Overview
The image contains two heatmaps (a) and (b) comparing frequency (y-axis, Hz) and angle θ (x-axis, degrees). Both use color gradients to represent measured values, with distinct color scales and spatial distributions.
---
### Components/Axes
- **X-axis (θ[°])**: Angle in degrees, ranging from 0° to 80°.
- **Y-axis (Frequency [Hz])**: Frequency in Hz, ranging from 0 Hz to 8000 Hz.
- **Heatmap (a)**:
- Color scale: 0 (blue) to 15 (red).
- Legend: Right-aligned, vertical gradient from blue to red.
- **Heatmap (b)**:
- Color scale: -20 (blue) to 0 (red).
- Legend: Right-aligned, vertical gradient from blue to red.
---
### Detailed Analysis
#### Heatmap (a)
- **Structure**: Horizontal bands of color, indicating uniform values across angles for specific frequency ranges.
- **Key Trends**:
- **0–2000 Hz**: Predominantly blue (values ~0–5), suggesting low measured values.
- **2000–4000 Hz**: Yellow-orange bands (~5–10), indicating moderate values.
- **4000–8000 Hz**: Red bands (~10–15), representing the highest values.
- **Spatial Grounding**: Red bands are concentrated at the top (high frequencies), while blue dominates the lower frequencies.
#### Heatmap (b)
- **Structure**: Diagonal gradient from red (top-left) to blue (bottom-right), with a sharp boundary at ~4000 Hz.
- **Key Trends**:
- **0–4000 Hz**: Red-to-yellow gradient (~0 to -10), indicating higher values (closer to 0).
- **4000–8000 Hz**: Blue-to-cyan gradient (~-10 to -20), showing a sharp decline in values.
- **Spatial Grounding**: Red regions dominate the lower-left (low frequencies), while blue regions occupy the upper-right (high frequencies).
---
### Key Observations
1. **Heatmap (a)** exhibits **frequency-dependent uniformity**, with distinct horizontal bands suggesting consistent values across angles for specific frequency ranges.
2. **Heatmap (b)** shows a **frequency-dependent gradient**, with a critical transition at ~4000 Hz where values drop sharply.
3. **Color Consistency**: Red in both heatmaps represents the highest values (15 in (a), 0 in (b)), while blue represents the lowest (0 in (a), -20 in (b)).
---
### Interpretation
- **Heatmap (a)** likely represents a parameter (e.g., signal strength, gain) that remains constant across angles but varies with frequency. The horizontal bands suggest discrete frequency bins with stable measurements.
- **Heatmap (b)** indicates a **frequency-dependent attenuation or loss**, where values decrease significantly above 4000 Hz. The diagonal boundary may represent a cutoff frequency or a material property change (e.g., dielectric loss tangent).
- **Technical Implications**:
- Heatmap (a) could model idealized system behavior (e.g., antenna gain), while (b) might reflect real-world losses (e.g., cable attenuation).
- The sharp transition in (b) at 4000 Hz suggests a physical threshold, such as the onset of dispersion or material resonance.
---
### Limitations
- Exact numerical values are inferred from color scales; precise measurements require calibration data.
- No explicit units or context (e.g., dB, linear scale) are provided for the color gradients.
</details>
sound field. Both theoretical results and experimental results can be found in [17, 18]. Here we use the analytic representation of the binaural correlation function proposed by Lindevald and Benade [17], given by
<!-- formula-not-decoded -->
where α = 2 . 2 and β = 0 . 5 .
The binaural CDR estimators are now obtained by inserting the binaural coherence models Γ Binaural coh and Γ Binaural diff into the estimators given in Table 1. This extension makes the direction-dependent CDR estimators suitable for binaural dereverberation. The corresponding estimators are denoted as ˜ η Binaural · in the following, where · represents the name of the technique that is being used.
## 3.3. Robustness of the Free-Field Estimators in the Binaural Scenario
This part evaluates the robustness of the direction-dependent CDR estimators using the free-field model against the shadowing effect of the head. For the limited space of this paper, only ˜ η Binaural Schwarz2 is chosen to compare with ˜ η FF Schwarz2 , since a previous study [12] has already shown that ˜ η FF Schwarz2 has the best performance among the direction-dependent CDR estimators in Table 1 (see [12, Table III] for details). For the comparison, we generate values of the mixture coherence ˆ Γ x for a certain input CDR η in and different angles and frequencies according to the binaural coherence models defined above, and insert these coherence values into the free-field estimator. We then define the estimation error of the free-field CDR
Table 2 : PESQ scores averaged over all angles for CDR estimators in Table 1, using free-field ( ˜ η FF · ) or binaural coherence models ( ˜ η Binaural · ).
| AIR | Unprocessed | Direction-dependent | Direction-dependent | Direction-dependent | Direction-dependent | Direction-independent | Direction-independent | Direction-independent | Direction-independent |
|----------|---------------|-----------------------|-----------------------|-----------------------|-----------------------|-------------------------|-------------------------|-------------------------|-------------------------|
| Distance | Left/Right | ˜ η FF Schwarz1 | ˜ η Binaural Schwarz1 | ˜ η FF Schwarz2 | ˜ η Binaural Schwarz2 | ˜ η FF Thiergart2 | ˜ η Binaural Thiergart2 | ˜ η FF Schwarz3 | ˜ η Binaural Schwarz3 |
| 1m | 2.24/2.25 | 2.40/2.41 | 2.65/2.68 | 2.57/2.59 | 2.69/2.71 | 2.66/2.67 | 2.64/2.65 | 2.65/2.67 | 2.64/2.65 |
| 2m | 1.88/1.90 | 2.00/2.00 | 2.12/2.13 | 2.10/2.10 | 2.17/2.18 | 2.16/2.17 | 2.15/2.15 | 2.16/2.17 | 2.15/2.16 |
| 3m | 1.77/1.77 | 1.85/1.84 | 1.91/1.90 | 1.92/1.91 | 1.97/1.96 | 1.95/1.95 | 1.95/1.95 | 1.96/1.96 | 1.95/1.95 |
estimator compared to the true CDR η in as
<!-- formula-not-decoded -->
Fig. 2 plots ∆ versus f and θ for the true input CDR η in = -20 dB (a) and η in = 20 dB (b). Only θ ∈ [0 π/ 2] is considered due to the symmetry of the scenario. Fig. 2 shows that the CDR is somewhat overestimated for low input CDR, while for high input CDR, the CDR is seriously underestimated for angles larger than 45 ◦ . The influence of the head on the coherence, especially the one of the desired speech component Γ Binaural coh ( f ) , is significant enough to deteriorate the performance of the free-field CDR estimator considerably.
## 4. EVALUATION
This section evaluates the application of the CDR estimators in Table 1 with the free-field and binaural coherence models to the problem of dereverberation. We use the Aachen Impulse Response (AIR) database [19], which consists of binaural RIRs measured by a dummy head with azimuth angles from -90 ◦ to 90 ◦ with 15 ◦ increments and source-head distances from 1 m to 3 m with 1 m increments.
Ten clean speech samples (five female and five male speakers) are taken from the TIMIT database [20]. The reverberant speech samples are generated by convolving the clean speech with the 'stairway' RIRs from the AIR database. We use the same filterbank, postfilter gain function and parameters as in [12, (29)], with the CDR estimators in Table 1. Knowledge of the true DOA is assumed for computation of the desired signal coherence models. The gain function is applied to the two microphone signals separately. PESQ is chosen as evaluation measure since it was found to be highly correlated with speech quality for the evaluation of noise and reverberation suppression methods [21, 22]. Here, we give raw MOS scores obtained by wideband PESQ. The PESQ scores of the two microphone signals and those of the processed signals are given separately. Note that the average PESQ scores for both ears are very similar, due to the symmetry of the scenario. The experimental results for the different distances are presented in Table 2. From these results, we can make the following observations:
- (1) Using the ITD and binaural diffuse coherence model can improve all of the direction-dependent CDR estimators.
- (2) The direction-independent CDR estimators, which do not rely on a model of the desired signal coherence, are robust
Fig. 3 : PESQ scores versus DOA for the 1 m distance case: (a) direction-dependent CDR estimators; (b) direction-independent CDR estimators. Left represents the unprocessed signal recorded by the microphone located at the left ear.
<details>
<summary>Image 3 Details</summary>
### Visual Description
## Line Chart: PESQ Scores vs. θ (Degrees)
### Overview
Two line charts (a) and (b) depict PESQ scores across three conditions (Left, η_FF, Binaural) as a function of θ (degrees). Both charts share identical axes and legend placement but differ in line trends.
### Components/Axes
- **X-axis**: θ [°], ranging from -50° to 50° in increments of 10°.
- **Y-axis**: PESQ Scores, ranging from 1.8 to 3.0 in increments of 0.2.
- **Legend**: Located at the bottom-left corner of each chart.
- **Blue solid line**: Left
- **Pink dotted line**: η_FF (Schwarz2 in (a), Schwarz3 in (b))
- **Black dashed line**: Binaural
### Detailed Analysis
#### Chart (a)
- **Left (Blue)**:
- Starts at ~2.5 at -50°, peaks at ~2.5 at -30°, then declines to ~2.0 at 50°.
- Notable dip to ~2.1 at 0°.
- **η_FF (Pink)**:
- Starts at ~2.7 at -50°, peaks at ~2.7 at -10°, then declines to ~2.2 at 50°.
- Sharp drop to ~2.0 at 0°.
- **Binaural (Black)**:
- Starts at ~2.8 at -50°, peaks at ~2.8 at -20°, then declines to ~2.4 at 50°.
- Slight rebound to ~2.5 at 0°.
#### Chart (b)
- **Left (Blue)**:
- Starts at ~2.4 at -50°, peaks at ~2.4 at -25°, then declines to ~2.0 at 50°.
- Sharp drop to ~2.1 at 0°.
- **η_FF (Pink)**:
- Starts at ~2.8 at -50°, peaks at ~2.8 at -15°, then declines to ~2.3 at 50°.
- Sharp drop to ~2.1 at 0°.
- **Binaural (Black)**:
- Starts at ~2.8 at -50°, peaks at ~2.8 at -20°, then declines to ~2.4 at 50°.
- Slight rebound to ~2.5 at 0°.
### Key Observations
1. **Consistent Trends**:
- Binaural consistently achieves the highest PESQ scores across all θ values.
- Left scores are the lowest, with sharper declines as θ increases.
- η_FF scores vary slightly between charts (Schwarz2 vs. Schwarz3) but follow similar patterns.
2. **Peak Positions**:
- Binaural peaks at -20° in both charts.
- η_FF peaks at -10° (a) and -15° (b).
- Left peaks at -30° (a) and -25° (b).
3. **Divergence at θ = 0°**:
- All lines drop sharply at θ = 0°, with η_FF showing the steepest decline.
### Interpretation
- **Performance Implications**:
- Binaural’s higher scores suggest superior performance in the measured metric (e.g., audio quality, signal processing).
- Left’s lower scores and sharper declines indicate reduced effectiveness as θ increases.
- **Condition-Specific Behavior**:
- η_FF (Schwarz2/3) shows intermediate performance, with peaks at moderate negative θ values.
- The divergence in η_FF labels (Schwarz2 vs. Schwarz3) may indicate methodological differences or experimental conditions.
- **θ = 0° Anomaly**:
- The sharp drop at θ = 0° across all conditions suggests a critical threshold or baseline reference point in the data.
### Spatial Grounding
- Legends are consistently placed in the bottom-left corner of both charts.
- Lines are clearly differentiated by color and style (solid, dotted, dashed).
- Axis labels and scales are identical in both charts, ensuring direct comparability.
### Content Details
- **Numerical Approximations**:
- All values are estimated from visual inspection (e.g., ~2.5, ~2.7).
- Uncertainty: ±0.1 due to scale granularity.
- **No Outliers**: All lines follow smooth trends without abrupt deviations.
### Final Notes
The charts highlight the impact of θ on PESQ scores across three conditions. Binaural’s consistent dominance suggests it is the optimal configuration, while Left’s performance degrades more rapidly with increasing θ. The η_FF variations (Schwarz2 vs. Schwarz3) warrant further investigation into their underlying parameters.
</details>
in the binaural case, even when using the free-field diffuse coherence model. This indicates that the choice of the diffuse coherence model is not critical, and the main effect of the head is on the ITD.
- (3) The direction-independent estimators almost reach the performance of the best binaural direction-dependent estimator.
To reveal the mechanism of the better performance of the proposed direction-dependent binaural CDR estimators, the PESQ scores versus θ are plotted in Fig. 3 (due to symmetry, only PESQ scores for the left microphone are shown). As can be seen from this figure, ˜ η Binaural Schwarz2 is much better than ˜ η FF Schwarz2 for | θ | ≥ 45 ◦ . This phenomenon can be explained by the robustness analysis results in Fig. 2, where it was found that the estimation error of ˜ η FF Schwarz2 becomes significant for | θ | > 45 ◦ . However, ˜ η FF Schwarz3 and ˜ η Binaural Schwarz3 nearly have the same performance for all angles, which confirms that the effect of using Γ FF diff ( f ) or Γ Binaural diff ( f ) is not critical for the direction-independent CDR estimators.
The estimators ˜ η Thiergart2 and ˜ η Schwarz3 show similar behavior in this scenario, although the former is biased [12]. This can be explained by the fact that the bias is roughly proportional to the noise coherence and disappears for Γ diff → 0 ; since, for binaural signals, the noise coherence is lower than for the setup investigated in [12], due to the large spacing of the sensors and the shadowing effect of the head, the practical impact of the bias is not significant here.
## 5. CONCLUSIONS
This paper extends previously proposed free-field CDR estimators to binaural dereverberation by using a simplified model for the ITD. Experimental results show that this extension is important for the direction-dependent CDR estimators, where PESQ scores for dereverberation can be significantly improved. It is further shown that the direction-independent CDR estimators, which do not require a model of the desired signal coherence, can achieve similar performance and are robust towards the shadowing effect of the head. Further work could concentrate on studying the impact of the ILD on binaural dereverberation and the theoretical limits of the CDR estimators by using statistical analysis [23].
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