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Coherent-to-Diffuse Power Ratio Estimation for Dereverberation

Andreas Schwarz, Walter Kellermann

arXiv:1502.03784v2cs.SD

TL;DR

Estimating the coherent-to-diffuse power ratio from two-microphone coherence is complicated by model mismatch and estimation variance. The paper formulates and proposes estimators, including unbiased variants that need either target DOA or noise coherence. Its DOA-independent approach improves signal quality across evaluated measures and supports blind dereverberation.

  • Problem

    Direct substitution of estimated coherence into the CDR formula can yield complex values under model mismatch and estimation variance, making that direct estimate infeasible.

    Source: §IV · p. 3

  • Method

    The paper analyzes coherence-based CDR estimators and proposes unbiased variants, with either target DOA or noise coherence sufficient for unbiased estimation.

    Source: §IV · p. 3, §VII · p. 12

  • Results

    The DOA-independent CDR postfilter significantly increased overall signal quality according to all evaluated measures, while CDRprop2 achieved the highest recognition rate across evaluated rooms and highest average PESQ score.

    Source: 2) Results · pp. 10–11

  • Takeaways & Limitations

    The DOA-independent estimator can support effective blind dereverberation without requiring source DOA or other scenario parameters.

    Source: 2) Results · p. 11, §VII · p. 12

    The diffuse-model fit depended on room surfaces: it was good in reflective rooms B and C, while curtains shifted coherence in room A and SR2.

    Source: B. Spatial Properties of Reverberation… · p. 9

Abstract

from arXiv · show

The estimation of the time- and frequency-dependent coherent-to-diffuse power ratio (CDR) from the measured spatial coherence between two omnidirectional microphones is investigated. Known CDR estimators are formulated in a common framework, illustrated using a geometric interpretation in the complex plane, and investigated with respect to bias and robustness towards model errors. Several novel unbiased CDR estimators are proposed, and it is shown that knowledge of either the direction of arrival (DOA) of the target source or the coherence of the noise field is sufficient for unbiased CDR estimation. The validity of the model for the application of CDR estimates to dereverberation is investigated using measured and simulated impulse responses. A CDR-based dereverberation system is presented and evaluated using signal-based quality measures as well as automatic speech recognition accuracy. The results show that the proposed unbiased estimators have a practical advantage over existing estimators, and that the proposed DOA-independent estimator can be used for effective blind dereverberation.

I. INTRODUCTION

The paper develops coherent-to-diffuse power ratio (CDR) estimation from two-microphone coherence for dereverberation, building on earlier coherence-based enhancement and CDR methods. It sets out a signal model and compares estimators that use different available coherence information.

I. INTRODUCTION

  • Earlier work used microphone coherence for direct-sound/reverberation discrimination, coherence-based enhancement –, and explicit CDR estimation,.Related postfilters target uncorrelated or diffuse noise,.Source: §I · p. 1

EDICS: AUD-SIRR, AUD-MAAE, AUD-SEN, AUD-ASAP

  • The contribution studies CDR estimation from two omnidirectional microphones and evaluates CDR-based dereverberation using speech quality measures and ASR accuracy.It proposes unbiased estimators and considers cases where target DOA or noise coherence is unknown.Source: EDICS: AUD-SIRR, AUD-MAAE, AUD-SEN,… · p. 1

II. SIGNAL MODEL

  • The signal model records desired and undesired components with two omnidirectional microphones and represents signals in the short-time Fourier transform domain.The signals are assumed short-time stationary; equal component auto-power spectra across microphones may be affected by early reflections.Source: §II · pp. 1–2
  • The model assumes time-invariant component coherences and mutually orthogonal signal and noise, relating mixed-field coherence to their coherences and SNR.The relationship applies to any signal and noise coherence functions.Source: §II · p. 2
  • For a fully coherent desired signal and diffuse noise, the SNR is termed CDR, and the mixed coherence lies on a complex-plane line between the noise and signal coherences.The line parameter is equivalent to diffuseness.Source: §II · p. 2

III. COHERENCE MODELS FOR DEREVERBERATION

The coherence models represent the desired signal as a plane wave and reverberation as an isotropic field. The paper describes when diffuse reverberation is plausible and considers a two-dimensional isotropic alternative.

A. Desired Signal

  • The desired signal is modeled as a plane wave with DOA θ, whose coherence has unit magnitude and equals one when the TDOA is zero.TDOA depends on microphone spacing, direction of arrival, and sound speed.Source: A. Desired Signal · p. 2

B. Reverberation as Isotropic Sound Field

  • A diffuse noise field models uncorrelated sources distributed spherically around the receiver, producing a real-valued spatial coherence for two omnidirectional sensors.The model is motivated by many independent sources around the receiver.Source: B. Reverberation as Isotropic Sound… · p. 2
  • Room reverberation can appear approximately diffuse when limited observation windows and source temporal correlation make delayed reflections behave like uncorrelated sources.Larger rooms, highly reflective surfaces, and low source temporal correlation favor this approximation; diffraction and diffuse reflection may also increase diffuseness.Source: B. Reverberation as Isotropic Sound… · pp. 2–3
  • With highly absorbing floors or ceilings, a 2D isotropic field of sources in the microphone plane may model reverberation more accurately than a spherical diffuse field.The paper evaluates room geometry and surface-reflectivity effects using simulated and measured RIRs.Source: B. Reverberation as Isotropic Sound… · p. 3

IV. COHERENT-TO-DIFFUSE POWER RATIO ESTIMATION

The paper estimates CDR from measured short-time coherence rather than directly applying the mixture relationship. It reformulates existing and proposed estimators around coherence models and examines their bias and robustness.

IV. COHERENT-TO-DIFFUSE POWER RATIO ESTIMATION

  • The approach estimates CDR from short-time spatial coherence, then uses the estimate with a suppression technique such as Wiener filtering or spectral subtraction.This contrasts with postfilters formulated directly from auto- and cross-power spectral estimates.Source: §IV · p. 3
  • Directly substituting estimated coherence into the CDR formula is infeasible because model mismatch and estimation variance can produce complex-valued results.CDR and diffuseness are real-valued, so the paper considers estimators that return positive real values.Source: §IV · p. 3
  • The estimator analysis uses the estimated coherence and assumed signal and noise coherence models, distinguishing methods that use both models from those using only one.The paper also proposes an estimator for known signal coherence with unknown noise coherence, then evaluates bias and robustness.Source: §IV · p. 3

A. Interpretation of Estimator Behavior in the Complex Plane

Estimator behavior is judged by bias along the model-consistent coherence line and robustness to deviations from it. For known DOA and noise coherence, several unbiased formulations trade off phase-error robustness and practical dereverberation behavior.

A. Interpretation of Estimator Behavior in the Complex Plane

  • Bias is deviation from the ideal CDR along the model-consistent coherence line, while robustness concerns sensitivity to coherence estimates off that line.Figure 1 plots estimates by estimator across coherence values, with rows varying TDOA and the theoretical mixed-signal line connecting fully coherent and diffuse endpoints.Source: A. Interpretation of Estimator… · p. 4, Figure 1 · p. 4

B. CDR Estimation for Known DOA and Noise Coherence

  • Known-DOA estimators use the target TDOA to phase-align the measured coherence, but rotating only the direct component leaves the estimator biased at nonzero TDOAs.The same rotation affects diffuse coherence, which the basic estimator does not account for.Source: B. CDR Estimation for Known DOA and… · pp. 4–5
  • The Thiergart estimator is unbiased but highly sensitive to phase deviations, with small errors near unit coherence causing large changes in its CDR estimate.The estimate can sharply drop to zero for coherence values close to the unit circle.Source: B. CDR Estimation for Known DOA and… · p. 5
  • Correcting diffuse coherence for the phase rotation removes the bias of the estimator based on (17) while preserving its robustness to phase errors.This estimator is identical to (17) for zero TDOA.Source: B. CDR Estimation for Known DOA and… · p. 5
  • Replacing the real-part and maximum operations in the numerator with a magnitude yields a heuristic unbiased variant reported to improve dereverberation performance.It has a small bias for nonzero TDOAs, but compensating for it has negligible practical effect.Source: B. CDR Estimation for Known DOA and… · p. 5
  • Several unbiased estimators are possible when both signal and noise coherence are known, because unbiasedness constrains behavior only on the model-consistent line.The proposed second variant has practical advantages but is not established as optimal; optimizing it would require a statistical model of coherence-model deviations.Source: B. CDR Estimation for Known DOA and… · p. 5

C. CDR Estimation for Unknown DOA

Estimating CDR without explicit DOA information can use the phase of the measured coherence, but that phase is biased at low CDR. A geometric constraint instead yields a uniquely determined unbiased DOA-independent estimator.

C. CDR Estimation for Unknown DOA

  • Using mixture coherence phase as the direct-signal phase avoids explicit DOA estimation, but is biased for nonzero TDOAs because diffuse coherence dominates at low CDR.Diffuse coherence is real-valued and has zero phase.Source: C. CDR Estimation for Unknown DOA · p. 5
  • The direct-signal coherence can be recovered without DOA by intersecting the line through noise and mixture coherence with the unit circle.This uses the known facts that fully coherent direct sound has magnitude one and the noise coherence lies on the real axis.Source: C. CDR Estimation for Unknown DOA · p. 5
  • Taking the positive quadratic solution gives a uniquely determined unbiased DOA-independent CDR estimator.Its unbiasedness follows from requiring the direct-signal coherence magnitude to equal one.Source: C. CDR Estimation for Unknown DOA · pp. 5–6

D. CDR Estimation for Unknown Noise Coherence

The section derives a CDR estimator that uses known direct-signal coherence without requiring noise coherence, then evaluates its bias and robustness to coherence-model errors. The proposed estimators are unbiased under the evaluated conditions, but their sensitivity to model errors differs.

D. CDR Estimation for Unknown Noise Coherence

  • When direct-signal coherence is known, assuming real noise coherence lets the estimator eliminate the unknown noise coherence using the measured and direct-signal coherence.The noise coherence is determined geometrically by the real-axis intersection of the line through the direct-signal and mixed-signal coherence.Source: D. CDR Estimation for Unknown Noise… · p. 6
  • The estimator extends problematic coherence estimates to infinite CDR when the estimated imaginary part exceeds the modeled value in magnitude, and zero CDR when their signs differ.It is unusable when direct-signal coherence has zero phase and becomes increasingly sensitive to estimation errors at small TDOAs.Source: D. CDR Estimation for Unknown Noise… · p. 6

E. Evaluation of Estimator Bias and Robustness

  • The proposed estimators and the DOA-dependent Thiergart estimator are unbiased, whereas the Jeub and DOA-independent Thiergart estimators are biased.The Jeub estimator can over- or underestimate depending on TDOA and frequency; the DOA-independent Thiergart estimator consistently underestimates.Source: E. Evaluation of Estimator Bias and… · p. 6, Figure 2 · p. 6
  • The DOA-independent proposed estimator withstands direct-signal phase error but quickly overestimates CDR when noise coherence is mismodeled; proposed variant 2 appears more tolerant of model errors than variant 1.The DOA-dependent Thiergart variant reacts strongly to small phase deviations at high CDR.Source: E. Evaluation of Estimator Bias and… · pp. 6–7, Figure 3 · p. 6

V. APPLICATION TO SPEECH ENHANCEMENT

The system estimates short-time CDR from spatial coherence and uses it to drive a postfilter for reverberation or diffuse-noise suppression. Its preprocessor reduces transfer-function variation, and a DOA-independent estimator makes the system independent of target DOA.

V. APPLICATION TO SPEECH ENHANCEMENT

  • The preprocessor averages microphone squared magnitudes while retaining one microphone's phase, reducing transfer-function variation from interference between early reflections and the direct path.The output is formed by applying the postfilter gain to this preprocessed signal.Source: §V · p. 7
  • The system estimates CDR from short-time spatial coherence using direct-signal and reverberation coherence models, then applies a spectral-magnitude-subtraction postfilter.The gain includes an oversubtraction factor and gain floor and is applied to the preprocessed signal.Source: §V · p. 7
  • With a DOA-independent CDR estimator, the signal-enhancement system is completely independent of the target signal's DOA.Source: §V · p. 7

VI. EVALUATION

The evaluation examines reverberation's spatial properties and assesses CDR estimation and CDR-based dereverberation using simulated and measured impulse responses. The setup includes measurements in three rooms and additional REVERB challenge recordings.

VI. EVALUATION

  • The evaluation assesses reverberation coherence, CDR-estimation accuracy, and the effect of the proposed CDR-based dereverberation system using simulated and measured RIRs.Source: §VI · p. 7

A. Setup and Parameters

  • Measured RIRs from Rooms A, B, and C span reverberation times of approximately 0.4, 1, and 3.5 seconds, with 40–70 source positions per room and microphones spaced 8 cm apart.The source positions cover distances of 1, 2, and 4 m and angles from −90° to 90°.Source: A. Setup and Parameters · p. 7
  • The REVERB challenge recordings add RIRs from six rooms, two source–microphone distances, and two source angles, using neighboring microphones from an 8-channel circular array.Source: A. Setup and Parameters · p. 7
  • Coherence estimates use recursive averaging with forgetting factor λ = 0.68; processing uses 16 kHz sampling and a DFT-based uniform filterbank.The filterbank uses a 1024-sample window, FFT size 512, and downsampling factor 128.Source: A. Setup and Parameters · pp. 7–8

B. Spatial Properties of Reverberation in Simulated and Measured Rooms

Late-reverberation coherence in real rooms generally falls between diffuse and 2D isotropic models, while surface-reflectivity differences and dominant vertical reflections can produce departures. The simulated image-source model reproduces observed late-reverberation spatial characteristics.

B. Spatial Properties of Reverberation in Simulated and Measured Rooms

  • Simulated reverberation matches the diffuse model when surface reflectivities are equal and the 2D isotropic model when floor and ceiling are highly absorbing.When walls are more absorbing than floor and ceiling, coherence is significantly higher than diffuse coherence; estimates have visibly lower variance in the larger room.Source: B. Spatial Properties of Reverberation… · p. 8
  • Most REVERB rooms match the diffuse model, but SR2 has higher coherence and MR2 nearly matches the 2D isotropic model, consistent with their surface-reflectivity patterns.Coherence-estimate variance is lower in rooms with longer reverberation times.Source: B. Spatial Properties of Reverberation… · pp. 8–9
  • Measured rooms B and C fit the diffuse model, while room A's coherence resembles the simulated case with partially absorbing walls, consistent with its curtains.The room-level estimates use one microphone pair and therefore have significantly higher variance.Source: B. Spatial Properties of Reverberation… · p. 9
  • In real rooms with microphones in the same horizontal plane, reverberation coherence typically lies between diffuse and 2D isotropic coherence, with increased coherence in some rooms due to dominant vertical reflections.The diffuse model fits most rooms unless surface reflectivities differ substantially; the image-source model can reproduce observed late-reverberation spatial characteristics.Source: B. Spatial Properties of Reverberation… · p. 9

C. CDR Estimation for Reverberant Speech

The section tests whether CDR estimates track early-versus-late energy in real reverberant speech, then compares estimator accuracy and dereverberation evaluation methods. It treats direct sound and early reflections as desired signal and the reverberation tail as undesired signal.

C. CDR Estimation for Reverberant Speech

  • The desired component includes the direct path and reflections arriving within 50 ms; the reverberation tail is undesired, making ELR50 ms the relevant enhancement SNR.Including early reflections is motivated by their reported benefit to speech intelligibility and ASR accuracy,.Source: C. CDR Estimation for Reverberant Speech · p. 9
  • Low-reverberation bins resemble plane-wave coherence, while highly reverberant bins show greater spread and model mismatch that can affect estimator performance.The comparison uses bins with ELR50 ms above 10 dB and below −10 dB around 1 kHz.Source: C. CDR Estimation for Reverberant Speech · p. 9, Figure 8 · p. 10
  • CDRprop2 further reduces estimation error over CDRprop1, while the proposed DOA-independent unbiased estimator also reduces error; CDRprop4 has the second-highest error.The errors are evaluated as diffuseness MSE, with true CDR approximated by ELR50 ms and averaged over time and frequency.Source: C. CDR Estimation for Reverberant Speech · pp. 9–10

D. Dereverberation Performance

1) Measures and Evaluation Method:

  • Comparisons include CDR estimators, Allen and Westermann coherence postfilters, and Lebart’s decay model; TDOA-dependent estimators use ideal TDOA knowledge.The evaluation uses 500 GRID utterances per room and source position across rooms A, B, C, MR1/2, and LR1/2.Source: 1) Measures and Evaluation Method · p. 10

2) Results:

CDRprop2 achieves the strongest reported recognition and average PESQ results, while the proposed unbiased estimators generally improve on biased variants. The DOA-independent approach supports blind dereverberation, although estimator performance and suppression trade-offs vary by method and setting.

2) Results:

  • Both DOA-dependent and DOA-independent unbiased variants slightly outperform their biased counterparts; CDRprop2 further improves on CDRprop1.The further improvement is attributed to behavior on coherence values that deviate from the ideal model.Source: 2) Results · pp. 10–11
  • CDRprop2 achieves the highest recognition rate across evaluated rooms and the highest average PESQ score.Source: 2) Results · p. 10, Table III · p. 11
  • The DOA-independent CDR-based postfilter improves overall signal quality across evaluated measures without requiring source DOA or other scenario parameters.The conclusion identifies this method as a blind dereverberation system.Source: 2) Results · p. 11, §VII · p. 12
  • Allen and Westermann yield less ELR improvement but higher signal-to-distortion ratio; their lower suppression benefits fwSegSNR while ASR and PESQ improve little.Source: 2) Results · p. 11
  • 77.4% mean recognition rate follows reducing Lebart’s suppression parameter to µ = 0.6; at the evaluated setting, its highest ELR accompanies the worst signal-to-distortion ratio.The authors suggest overestimated reverberation, possibly from mismatch between the exponential-decay assumption and early impulse responses.Source: 2) Results · p. 11

VII. CONCLUSION

  • The proposed DOA-independent estimator also permits unbiased CDR estimation without signal DOA knowledge, while the paper reports the diffuse reverberation model is sufficiently accurate for this application.Source: §VII · p. 12

APPENDIX: DEFINITION OF THE ELR

The appendix defines ELR from the early and late components of a room impulse response. It also specifies the threshold convention and how ELR is evaluated after enhancement.

APPENDIX: DEFINITION OF THE ELR

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