0.3 Blind source separation

Background

When separating mixtures of instantaneously mixed signals, independent component analysis works very well, but this ideal situation is rarely found in the real-world. In practical applications, the environment distorts audio signals by adding echoes, reflections, and ambient noise. Additionally independent component analysis, in its purest form, assumes that source signals do not have any propagation delay, which is an assumption that cannot be applied in this case. Recording sources from two microphones placed in different locations will inevitably introduce propagation delays, so the blind source separation method used must also consider this issue.

To solve problems detailed above, the blind source separation problem will be redefined in the time-frequency domain. By taking the short-time Fourier transform (STFT) of the audio inputs, we can represent the inputs as the following.

Xi(ω,t) refers to the input observed at the i th microphone observed at time t . The T symbol refers to the transpose, so in this case, the input sources are represented along the rows. For our application, we will assume that the number of observed inputs and the number of separated sources are both equal to the constant M . We can, now, formulate our blind source separation problem as the following.

In this case, s(ω,t) , refers to the vector of source signals observed at frequency ω and at time t . The second term, n(ω,t) , represents any type of noise or distortions that may be present in the observed signals (noise, reflections, etc.). The blind source separation method attempts to solve for the mixing matrix, A(ω) , which can be represented as the following.

Here, Hi,j(ω) , refers to the transfer function of the j th source to the i th microphone. Additionally, τi,j , refers to the propagation delay from the j th source to the i th microphone.

The system, for which our problem is defined on, is now redefined from an instantaneous mixture of signals to a convolutional mixture in the time-domain of the following form.

This formulation allows for a solution which considers the distortion, n(ω,t) , added by the environment and considers the inherent propagation delay that is inherently present in our application.

Since the new formulation of the blind source separation problem is a convolutional system in the time-domain, we will solve the separation problem in the time-frequency domain formulation shown in Figure 2. Given our approach outlined above, we can solve for the complete frequency-domain separation filter shown below.

B(ω) is the separation filter of the system. Applying this filter will separate the spectrum of the observed signal into the spectra of the source signals observed at frequency, ω . As stated above, by reformulating the problem into the frequency domain, the convolutional mixture becomes transformed into an instantaneous mixture. As such, independent component analysis can be applied to separate the source signals at each frequency. However, in doing so, we introduce other problems, inherent to the independent component analysis method, that the other filters attempt to resolve.