Low-rank Modeling in Room Acoustics

When considering the acoustics of a room as a linear time-invariant system, the room impulse response (RIR) describes the response of the room to an audio signal from a given source position to a given receiver position. If the RIR is perfectly known, the response of the room to any acoustic signal can be predicted. In practice, however, there are several difficulties. For a reverberant room, a finite impulse response (FIR) representation of the RIR can be on the order of 100 000 samples. Considering the position dependence of the RIR, the complete description of a room requires the RIRs for a multitude of closely spaced source and receiver positions. This is problematic from both a storage and a processing point of view, as the convolution with long FIR filters, e.g., for the purpose of auralization, is computationally demanding, particularly under the strict latency constraints often found in acoustic signal processing.

In this thesis we consider the concept of low rank, for the purpose of compact modeling of, and fast low-latency convolution with, RIRs. The idea to employ low-rank modeling in room acoustics developed from the observation that the RIR can be well modeled as a sum of decaying sinusoids. It has previously been shown that if such a signal is reshaped into a matrix or tensor, it will have a rank corresponding to twice the number of components of the sum. Due to modeling approximations and measurement noise, the reshaped RIRs considered in this thesis will not be low rank in a strict sense, but rather that when considering a low-rank approximation of the reshaped RIR, the error will be comparatively small.

The thesis begins with a thorough derivation of the sum-of-decaying-sinusoids model and its low-rank properties, along with a confirmation that the model is relevant for real-life RIRs. Next, the low-rank ideas are employed for the purpose of estimation of RIRs from noisy input-output signals. Subsequently, it is shown how the low-rank structure can be utilized for fast low-latency convolution of RIRs with audio signals. Next, it is demonstrated how the compression of RIRs by low-rank approximation preserves many important qualities of the RIR, such as reverberation time and center time, and how these methods perform well also with respect to objective signal-based measures. An algorithm for low-latency convolution of RIRs reshaped into tensors is proposed. Finally, it is shown how low rank can be exploited for the joint compression of multiple RIRs in a room. An algorithm for the simultaneous multi-channel convolution of these compressed RIRs is proposed.

The main contribution of this thesis is providing a novel view on, and modeling tools for, RIRs. It is demonstrated how the low-rank framework is useful for RIR approximation, RIR estimation, fast low-latency convolution with audio, the joint compression of all the RIRs of a room, and how low-rank modeling preserves many important RIR qualities.

In this thesis it is shown that the estimation of RIRs from noisy input-output signals could be improved by considering a prior of the matricization of the RIR being low rank. It is also shown that the joint compression of multiple matricized RIRs was a significant improvement as compared to compressing the matricized RIRs one by one. However, it is also demonstrated that the reshaping of the RIR to a higher-order tensor, as opposed to a 2-D matrix, yielded superior results in every aspect, given a fixed compression rate. In light of this, it is believed that of all the exciting areas of possible future research, the estimation of RIRs with a prior of the tensorization of the RIR being low rank, and the joint compression of multiple tensorized RIRs, hold the most potential.