We propose an approach for simultaneous diarization and separation of meeting data. It consists of a complex Angular Central Gaussian Mixture Model (cACGMM) for speech source separation, and a von-Mises-Fisher Mixture Model (vMFMM) for diarization in a joint statistical framework. Through the integration, both spatial and spectral information are exploited for diarization and separation. We also develop a method for counting the number of active speakers in a segment of a meeting to support block-wise processing. While the total number of speakers in a meeting may be known, it is usually not known on a per-segment level. With the proposed speaker counting, joint diarization and source separation can be done segment-by-segment, and the permutation problem across segments is solved, thus allowing for block-online processing in the future. Experimental results on the LibriCSS meeting corpus show that the integrated approach outperforms a cascaded approach of diarization and speech enhancement in terms of WER, both on a per-segment and on a per-meeting level.

We propose a modified teacher-student training for the extraction of frame-wise speaker embeddings that allows for an effective diarization of meeting scenarios containing partially overlapping speech. To this end, a geodesic distance loss is used that enforces the embeddings computed from regions with two active speakers to lie on the shortest path on a sphere between the points given by the d-vectors of each of the active speakers. Using those frame-wise speaker embeddings in clustering-based diarization outperforms segment-level clustering-based diarization systems such as VBx and Spectral Clustering. By extending our approach to a mixture-model-based diarization, the performance can be further improved, approaching the diarization error rates of diarization systems that use a dedicated overlap detection, and outperforming these systems when also employing an additional overlap detection.

Diarization is a crucial component in meeting transcription systems to ease the challenges of speech enhancement and attribute the transcriptions to the correct speaker.Particularly in the presence of overlapping or noisy speech, these systems have problems reliably assigning the correct speaker labels, leading to a significant amount of speaker confusion errors.We propose to add segment-level speaker reassignment to address this issue.By revisiting, after speech enhancement, the speaker attribution for each segment, speaker confusion errors from the initial diarization stage are significantly reduced.Through experiments across different system configurations and datasets, we further demonstrate the effectiveness and applicability in various domains.Our results show that segment-level speaker reassignment successfully rectifies at least 40% of speaker confusion word errors, highlighting its potential for enhancing diarization accuracy in meeting transcription systems.

Many real-life applications of automatic speech recognition (ASR) require processing of overlapped speech. A common method involves first separating the speech into overlap-free streams on which ASR is performed. Recently, TF-GridNet has shown impressive performance in speech separation in real reverberant conditions. Furthermore, a mixture encoder was proposed that leverages the mixed speech to mitigate the effect of separation artifacts. In this work, we extended the mixture encoder from a static two-speaker scenario to a natural meeting context featuring an arbitrary number of speakers and varying degrees of overlap. We further demonstrate its limits by the integration with separators of varying strength including TF-GridNet. Our experiments result in a new state-of-the-art performance on LibriCSS using a single microphone. They show that TF-GridNet largely closes the gap between previous methods and oracle separation independent of mixture encoding. We further investigate the remaining potential for improvement.

Since diarization and source separation of meeting data are closely related tasks, we here propose an approach to perform the two objectives jointly. It builds upon the target-speaker voice activity detection (TS-VAD) diarization approach, which assumes that initial speaker embeddings are available. We replace the final combined speaker activity estimation network of TS-VAD with a network that produces speaker activity estimates at a time-frequency resolution. Those act as masks for source extraction, either via masking or via beamforming. The technique can be applied both for single-channel and multi-channel input and, in both cases, achieves a new state-of-the-art word error rate (WER) on the LibriCSS meeting data recognition task. We further compute speaker-aware and speaker-agnostic WERs to isolate the contribution of diarization errors to the overall WER performance.

This paper proposes reverberation as supervision (RAS), a novel unsupervised loss function for single-channel reverberant speech separation. Prior methods for unsupervised separation required the synthesis of mixtures of mixtures or assumed the existence of a teacher model, making them difficult to consider as potential methods explaining the emergence of separation abilities in an animal’s auditory system. We assume the availability of two-channel mixtures at training time, and train a neural network to separate the sources given one of the channels as input such that the other channel may be predicted from the separated sources. As the relationship between the room impulse responses (RIRs) of each channel depends on the locations of the sources, which are unknown to the network, the network cannot rely on learning that relationship. Instead, our proposed loss function fits each of the separated sources to the mixture in the target channel via Wiener filtering, and compares the resulting mixture to the ground-truth one. We show that minimizing the scale-invariant signal-to-distortion ratio (SI-SDR) of the predicted right-channel mixture with respect to the ground truth implicitly guides the network towards separating the left-channel sources. On a semi-supervised reverberant speech separation task based on the WHAMR! dataset, using training data where just 5% (resp., 10%) of the mixtures are labeled with associated isolated sources, we achieve 70% (resp., 78%) of the SI-SDR improvement obtained when training with supervision on the full training set, while a model trained only on the labeled data obtains 43% (resp., 45%).

Using a Teacher-Student training approach we developed a speaker embedding extraction system that outputs embeddings at frame rate. Given this high temporal resolution and the fact that the student produces sensible speaker embeddings even for segments with speech overlap, the frame-wise embeddings serve as an appropriate representation of the input speech signal for an end-to-end neural meeting diarization (EEND) system. We show in experiments that this representation helps mitigate a well-known problem of EEND systems: when increasing the number of speakers the diarization performance drop is significantly reduced. We also introduce block-wise processing to be able to diarize arbitrarily long meetings.

We propose a general framework to compute the word error rate (WER) of ASR systems that process recordings containing multiple speakers at their input and that produce multiple output word sequences (MIMO). Such ASR systems are typically required, e.g., for meeting transcription. We provide an efficient implementation based on a dynamic programming search in a multi-dimensional Levenshtein distance tensor under the constraint that a reference utterance must be matched consistently with one hypothesis output. This also results in an efficient implementation of the ORC WER which previously suffered from exponential complexity. We give an overview of commonly used WER definitions for multi-speaker scenarios and show that they are specializations of the above MIMO WER tuned to particular application scenarios. We conclude with a discussion of the pros and cons of the various WER definitions and a recommendation when to use which.

We introduce a monaural neural speaker embeddings extractor that computes an embedding for each speaker present in a speech mixture.To allow for supervised training, a teacherstudent approach is employed: the teacher computes the target embeddings from each speaker's utterance before the utterances are added to form the mixture, and the student embedding extractor is then tasked to reproduce those embeddings from the speech mixture at its input.The system much more reliably verifies the presence or absence of a given speaker in a mixture than a conventional speaker embedding extractor, and even exhibits comparable performance to a multi-channel approach that exploits spatial information for embedding extraction.Further, it is shown that a speaker embedding computed from a mixture can be used to check for the presence of that speaker in another mixture.

Continuous Speech Separation (CSS) has been proposed to address speech overlaps during the analysis of realistic meeting-like conversations by eliminating any overlaps before further processing. CSS separates a recording of arbitrarily many speakers into a small number of overlap-free output channels, where each output channel may contain speech of multiple speakers. Often, a separation model is trained with Utterance-level Permutation Invariant Training (uPIT), which exclusively maps a speaker to an output channel, and applied in a sliding window approach called stitching. Recently, we introduced an alternative training scheme called Graph-PIT that teaches the separator to produce a speaker-shared output channel format without stitching. It can handle an arbitrary number of speakers as long as the number of overlapping speakers is never larger than the number of output channels. Models trained in this way are able to perform segment-less CSS, i.e., without stitching, and achieve comparable and often better separation quality than the conventional CSS with uPIT and stitching. In this contribution, we further investigate the Graph-PIT training scheme. We show in extended experiments that Graph-PIT also works in challenging reverberant conditions. We simplify the training schedule for Graph-PIT with the recently proposed Source Aggregated Signal-to-Distortion Ratio (SA-SDR) loss, which eliminates unfavorable properties of the previously used A-SDR loss to enable training with Graph-PIT from scratch. Furthermore, we introduce novel signal-level evaluation metrics for meeting scenarios, namely the source-aggregated scale- and convolution-invariant Signal-to-Distortion Ratio (SA-SI-SDR and SA-CI-SDR), which are generalizations of the commonly used SDR-based metrics for the CSS case.

Many state-of-the-art neural network-based source separation systems use the averaged Signal-to-Distortion Ratio (SDR) as a training objective function. The basic SDR is, however, undefined if the network reconstructs the reference signal perfectly or if the reference signal contains silence, e.g., when a two-output separator processes a single-speaker recording. Many modifications to the plain SDR have been proposed that trade-off between making the loss more robust and distorting its value. We propose to switch from a mean over the SDRs of each individual output channel to a global SDR over all output channels at the same time, which we call source-aggregated SDR (SA-SDR). This makes the loss robust against silence and perfect reconstruction as long as at least one reference signal is not silent. We experimentally show that our proposed SA-SDR is more stable and preferable over other well-known modifications when processing meeting-style data that typically contains many silent or single-speaker regions.

Recent speaker diarization studies showed that integration of end-to-end neural diarization (EEND) and clustering-based diarization is a promising approach for achieving state-of-the-art performance on various tasks.Such an approach first divides an observed signal into fixed-length segments, then performs segment-level local diarization based on an EEND module, and merges the segment-level results via clustering to form a final global diarization result.The segmentation is done to limit the number of speakers in each segment since the current EEND cannot handle a large number of speakers.In this paper, we argue that such an approach involving the segmentation has several issues; for example, it inevitably faces a dilemma that larger segment sizes increase both the context available for enhancing the performance and the number of speakers for the local EEND module to handle.To resolve such a problem, this paper proposes a novel framework that performs diarization without segmentation.However, it can still handle challenging data containing many speakers and a significant amount of overlapping speech.The proposed method can take an entire meeting for inference and perform utterance-by-utterance diarization that clusters utterance activities in terms of speakers.To this end, we leverage a neural network training scheme called Graph-PIT proposed recently for neural source separation.Experiments with simulated active-meeting-like data and CALLHOME data show the superiority of the proposed approach over the conventional methods.

Far-field multi-speaker automatic speech recognition (ASR) has drawn increasing attention in recent years. Most existing methods feature a signal processing frontend and an ASR backend. In realistic scenarios, these modules are usually trained separately or progressively, which suffers from either inter-module mismatch or a complicated training process. In this paper, we propose an end-to-end multi-channel model that jointly optimizes the speech enhancement (including speech dereverberation, denoising, and separation) frontend and the ASR backend as a single system. To the best of our knowledge, this is the first work that proposes to optimize dereverberation, beamforming, and multi-speaker ASR in a fully end-to-end manner. The frontend module consists of a weighted prediction error (WPE) based submodule for dereverberation and a neural beamformer for denoising and speech separation. For the backend, we adopt a widely used end-to-end (E2E) ASR architecture. It is worth noting that the entire model is differentiable and can be optimized in a fully end-to-end manner using only the ASR criterion, without the need of parallel signal-level labels. We evaluate the proposed model on several multi-speaker benchmark datasets, and experimental results show that the fully E2E ASR model can achieve competitive performance on both noisy and reverberant conditions, with over 30% relative word error rate (WER) reduction over the single-channel baseline systems.

Time-domain training criteria have proven to be very effective for the separation of single-channel non-reverberant speech mixtures. Likewise, mask-based beamforming has shown impressive performance in multi-channel reverberant speech enhancement and source separation. Here, we propose to combine neural network supported multi-channel source separation with a time-domain training objective function. For the objective we propose to use a convolutive transfer function invariant Signal-to-Distortion Ratio (CI-SDR) based loss. While this is a well-known evaluation metric (BSS Eval), it has not been used as a training objective before. To show the effectiveness, we demonstrate the performance on LibriSpeech based reverberant mixtures. On this task, the proposed system approaches the error rate obtained on single-source non-reverberant input, i.e., LibriSpeech test clean, with a difference of only 1.2 percentage points, thus outperforming a conventional permutation invariant training based system and alternative objectives like Scale Invariant Signal-to-Distortion Ratio by a large margin.

Recently, the end-to-end approach has been successfully applied to multi-speaker speech separation and recognition in both single-channel and multichannel conditions. However, severe performance degradation is still observed in the reverberant and noisy scenarios, and there is still a large performance gap between anechoic and reverberant conditions. In this work, we focus on the multichannel multi-speaker reverberant condition, and propose to extend our previous framework for end-to-end dereverberation, beamforming, and speech recognition with improved numerical stability and advanced frontend subnetworks including voice activity detection like masks. The techniques significantly stabilize the end-to-end training process. The experiments on the spatialized wsj1-2mix corpus show that the proposed system achieves about 35% WER relative reduction compared to our conventional multi-channel E2E ASR system, and also obtains decent speech dereverberation and separation performance (SDR=12.5 dB) in the reverberant multi-speaker condition while trained only with the ASR criterion.

Automatic transcription of meetings requires handling of overlapped speech, which calls for continuous speech separation (CSS) systems. The uPIT criterion was proposed for utterance-level separation with neural networks and introduces the constraint that the total number of speakers must not exceed the number of output channels. When processing meeting-like data in a segment-wise manner, i.e., by separating overlapping segments independently and stitching adjacent segments to continuous output streams, this constraint has to be fulfilled for any segment. In this contribution, we show that this constraint can be significantly relaxed. We propose a novel graph-based PIT criterion, which casts the assignment of utterances to output channels in a graph coloring problem. It only requires that the number of concurrently active speakers must not exceed the number of output channels. As a consequence, the system can process an arbitrary number of speakers and arbitrarily long segments and thus can handle more diverse scenarios. Further, the stitching algorithm for obtaining a consistent output order in neighboring segments is of less importance and can even be eliminated completely, not the least reducing the computational effort. Experiments on meeting-style WSJ data show improvements in recognition performance over using the uPIT criterion.

We present ESPnet-SE, which is designed for the quick development of speech enhancement and speech separation systems in a single framework, along with the optional downstream speech recognition module. ESPnet-SE is a new project which integrates rich automatic speech recognition related models, resources and systems to support and validate the proposed front-end implementation (i.e. speech enhancement and separation).It is capable of processing both single-channel and multi-channel data, with various functionalities including dereverberation, denoising and source separation. We provide all-in-one recipes including data pre-processing, feature extraction, training and evaluation pipelines for a wide range of benchmark datasets. This paper describes the design of the toolkit, several important functionalities, especially the speech recognition integration, which differentiates ESPnet-SE from other open source toolkits, and experimental results with major benchmark datasets.

Continuous speech separation (CSS) is an arising task in speech separation aiming at separating overlap-free targets from a long, partially-overlapped recording. A straightforward extension of previously proposed sentence-level separation models to this task is to segment the long recording into fixed-length blocks and perform separation on them independently. However, such simple extension does not fully address the cross-block dependencies and the separation performance may not be satisfactory. In this paper, we focus on how the block-level separation performance can be improved by exploring methods to utilize the cross-block information. Based on the recently proposed dual-path RNN (DPRNN) architecture, we investigate how DPRNN can help the block-level separation by the interleaved intra- and inter-block modules. Experiment results show that DPRNN is able to significantly outperform the baseline block-level model in both offline and block-online configurations under certain settings.

We previously proposed an optimal (in the maximum likelihood sense) convolutional beamformer that can perform simultaneous denoising and dereverberation, and showed its superiority over the widely used cascade of a Weighted Prediction Error (WPE) dereverberation filter and a conventional Minimum-Power Distortionless Response (MPDR) beamformer. However, it has not been fully investigated which components in the convolutional beamformer yield such superiority. To this end, this paper presents a new derivation of the convolutional beamformer that allows us to factorize it into a WPE dereverberation filter, and a special type of a (non-convolutional) beamformer, referred to as a weighted MPDR (wM-PDR) beamformer, without loss of optimality. With experiments, we show that the superiority of the convolutional beamformer in fact comes from its wMPDR part.

In recent years time domain speech separation has excelled over frequency domain separation in single channel scenarios and noise-free environments. In this paper we dissect the gains of the time-domain audio separation network (TasNet) approach by gradually replacing components of an utterance-level permutation invariant training (u-PIT) based separation system in the frequency domain until the TasNet system is reached, thus blending components of frequency domain approaches with those of time domain approaches. Some of the intermediate variants achieve comparable signal-to-distortion ratio (SDR) gains to TasNet, but retain the advantage of frequency domain processing: compatibility with classic signal processing tools such as frequency-domain beamforming and the human interpretability of the masks. Furthermore, we show that the scale invariant signal-to-distortion ratio (si-SDR) criterion used as loss function in TasNet is related to a logarithmic mean square error criterion and that it is this criterion which contributes most reliable to the performance advantage of TasNet. Finally, we critically assess which gains in a noise-free single channel environment generalize to more realistic reverberant conditions.

The rising interest in single-channel multi-speaker speech separation sparked development of End-to-End (E2E) approaches to multi-speaker speech recognition. However, up until now, state-of-the-art neural network-based time domain source separation has not yet been combined with E2E speech recognition. We here demonstrate how to combine a separation module based on a Convolutional Time domain Audio Separation Network (Conv-TasNet) with an E2E speech recognizer and how to train such a model jointly by distributing it over multiple GPUs or by approximating truncated back-propagation for the convolutional front-end. To put this work into perspective and illustrate the complexity of the design space, we provide a compact overview of single-channel multi-speaker recognition systems. Our experiments show a word error rate of 11.0% on WSJ0-2mix and indicate that our joint time domain model can yield substantial improvements over cascade DNN-HMM and monolithic E2E frequency domain systems proposed so far.

Most approaches to multi-talker overlapped speech separation and recognition assume that the number of simultaneously active speakers is given, but in realistic situations, it is typically unknown. To cope with this, we extend an iterative speech extraction system with mechanisms to count the number of sources and combine it with a single-talker speech recognizer to form the first end-to-end multi-talker automatic speech recognition system for an unknown number of active speakers. Our experiments show very promising performance in counting accuracy, source separation and speech recognition on simulated clean mixtures from WSJ0-2mix and WSJ0-3mix. Among others, we set a new state-of-the-art word error rate on the WSJ0-2mix database. Furthermore, our system generalizes well to a larger number of speakers than it ever saw during training, as shown in experiments with the WSJ0-4mix database.

This article proposes methods that can optimize a Convolutional BeamFormer (CBF) for jointly performing denoising, dereverberation, and source separation (DN+DR+SS) in a computationally efficient way. Conventionally, a cascade configuration, composed of a Weighted Prediction Error minimization (WPE) dereverberation filter followed by a Minimum Variance Distortionless Response (MVDR) beamformer, has been used as the state-of-the-art frontend of far-field speech recognition, even though this approach's overall optimality is not guaranteed. In the blind signal processing area, an approach for jointly optimizing dereverberation and source separation (DR+SS) has been proposed; however, it requires huge computing cost, and has not been extended for applications to DN+DR+SS. To overcome the above limitations, this paper develops new approaches for jointly optimizing DN+DR+SS in a computationally much more efficient way. To this end, we first present an objective function to optimize a CBF for performing DN+DR+SS based on maximum likelihood estimation on an assumption that the steering vectors of the target signals are given or can be estimated, e.g., using a neural network. This paper refers to a CBF optimized by this objective function as a weighted Minimum-Power Distortionless Response (wMPDR) CBF. Then, we derive two algorithms for optimizing a wMPDR CBF based on two different ways of factorizing a CBF into WPE filters and beamformers: one based on an extension of the conventional joint optimization approach proposed for DR+SS and another based on a novel technique. Experiments using noisy reverberant sound mixtures show that the proposed optimization approaches greatly improve the performance of the speech enhancement in comparison with the conventional cascade configuration in terms of signal distortion measures and ASR performance. The proposed approaches also greatly reduce the computing cost with improved estimation accuracy in comparison with the conventional joint optimization approach.

Despite the strong modeling power of neural network acoustic models, speech enhancement has been shown to deliver additional word error rate improvements if multi-channel data is available. However, there has been a longstanding debate whether enhancement should also be carried out on the ASR training data. In an extensive experimental evaluation on the acoustically very challenging CHiME-5 dinner party data we show that: (i) cleaning up the training data can lead to substantial error rate reductions, and (ii) enhancement in training is advisable as long as enhancement in test is at least as strong as in training. This approach stands in contrast and delivers larger gains than the common strategy reported in the literature to augment the training database with additional artificially degraded speech. Together with an acoustic model topology consisting of initial CNN layers followed by factorized TDNN layers we achieve with 41.6 % and 43.2 % WER on the DEV and EVAL test sets, respectively, a new single-system state-of-the-art result on the CHiME-5 data. This is a 8 % relative improvement compared to the best word error rate published so far for a speech recognizer without system combination.

In this paper, we present Hitachi and Paderborn University's joint effort for automatic speech recognition (ASR) in a dinner party scenario.The main challenges of ASR systems for dinner party recordings obtained by multiple microphone arrays are (1) heavy speech overlaps, (2) severe noise and reverberation, (3) very natural conversational content, and possibly (4) insufficient training data.As an example of a dinner party scenario, we have chosen the data presented during the CHiME-5 speech recognition challenge, where the baseline ASR had a 73.3% word error rate (WER), and even the best performing system at the CHiME-5 challenge had a 46.1% WER.We extensively investigated a combination of the guided source separation-based speech enhancement technique and an already proposed strong ASR backend and found that a tight combination of these techniques provided substantial accuracy improvements.Our final system achieved WERs of 39.94% and 41.64% for the development and evaluation data, respectively, both of which are the best published results for the dataset.We also investigated with additional training data on the official small data in the CHiME-5 corpus to assess the intrinsic difficulty of this ASR task.

This work examines acoustic beamformers employing neural networks (NNs) for mask prediction as front -end for automatic speech recognition (ASR) systems for practical scenarios like voice-enabled home devices. To test the versatility of the mask predicting network, the system is evaluated with different recording hardware, different microphone array designs, and different acoustic models of the downstream ASR system. Significant gains in recognition accuracy are obtained in all configurations despite the fact that the NN had been trained on mismatched data. Unlike previous work, the NN is trained on a feature level objective, which gives some performance advantage over a mask related criterion. Furthermore, different approaches for realizing online, or adaptive, NN-based beamforming are explored, where the online algorithms still show significant gains compared to the baseline performance.

The weighted prediction error (WPE) algorithm has proven to be a very successful dereverberation method for the REVERB challenge. Likewise, neural network based mask estimation for beamforming demonstrated very good noise suppression in the CHiME 3 and CHiME 4 challenges. Recently, it has been shown that this estimator can also be trained to perform dereverberation and denoising jointly. However, up to now a comparison of a neural beamformer and WPE is still missing, so is an investigation into a combination of the two. Therefore, we here provide an extensive evaluation of both and consequently propose variants to integrate deep neural network based beamforming with WPE. For these integrated variants we identify a consistent word error rate (WER) reduction on two distinct databases. In particular, our study shows that deep learning based beamforming benefits from a model-based dereverberation technique (i.e. WPE) and vice versa. Our key findings are: (a) Neural beamforming yields the lower WERs in comparison to WPE the more channels and noise are present. (b) Integration of WPE and a neural beamformer consistently outperforms all stand-alone systems.

In this paper we show how a neural network for spectral mask estimation for an acoustic beamformer can be optimized by algorithmic differentiation. Using the beamformer output SNR as the objective function to maximize, the gradient is propagated through the beamformer all the way to the neural network which provides the clean speech and noise masks from which the beamformer coefficients are estimated by eigenvalue decomposition. A key theoretical result is the derivative of an eigenvalue problem involving complex-valued eigenvectors. Experimental results on the CHiME-3 challenge database demonstrate the effectiveness of the approach. The tools developed in this paper are a key component for an end-to-end optimization of speech enhancement and speech recognition.

This paper presents an end-to-end training approach for a beamformer-supported multi-channel ASR system. A neural network which estimates masks for a statistically optimum beamformer is jointly trained with a network for acoustic modeling. To update its parameters, we propagate the gradients from the acoustic model all the way through feature extraction and the complex valued beamforming operation. Besides avoiding a mismatch between the front-end and the back-end, this approach also eliminates the need for stereo data, i.e., the parallel availability of clean and noisy versions of the signals. Instead, it can be trained with real noisy multi-channel data only. Also, relying on the signal statistics for beamforming, the approach makes no assumptions on the configuration of the microphone array. We further observe a performance gain through joint training in terms of word error rate in an evaluation of the system on the CHiME 4 dataset.

Multi-channel speech enhancement algorithms rely on a synchronous sampling of the microphone signals. This, however, cannot always be guaranteed, especially if the sensors are distributed in an environment. To avoid performance degradation the sampling rate offset needs to be estimated and compensated for. In this contribution we extend the recently proposed coherence drift based method in two important directions. First, the increasing phase shift in the short-time Fourier transform domain is estimated from the coherence drift in a Matched Filter-like fashion, where intermediate estimates are weighted by their instantaneous SNR. Second, an observed bias is removed by iterating between offset estimation and compensation by resampling a couple of times. The effectiveness of the proposed method is demonstrated by speech recognition results on the output of a beamformer with and without sampling rate offset compensation between the input channels. We compare MVDR and maximum-SNR beamformers in reverberant environments and further show that both benefit from a novel phase normalization, which we also propose in this contribution.

We present an algorithm for clustering complex-valued unit length vectors on the unit hypersphere, which we call complex spherical k-mode clustering, as it can be viewed as a generalization of the spherical k-means algorithm to normalized complex-valued vectors. We show how the proposed algorithm can be derived from the Expectation Maximization algorithm for complex Watson mixture models and prove its applicability in a blind speech separation (BSS) task with real-world room impulse response measurements. It turns out that the proposed spherical k-mode algorithm is on par with other state-of-the-art BSS algorithms in terms of signal-to-inference ratio gains although being far easier to implement and using fewer calculations.