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6

Recall that the columns of $\mathbf{W}$ can be thought of as "basis" vectors (or elements of a dictionary - the building blocks of any signal) and elements in each column in $\mathbf{H}$ give the corresponding weights (that vary over time). This allows us to decompose the spectrogram based on not just frequency components but also temporal onset information; ...

6

You are making a wrong assumption on the process. In ICA, the number of mixtures must be at least as many as the number of components. The paper you cite does in fact, acknowledge this: These observed signals from the red, green and blue color sensors are denoted by $x_1(t)$, $x_2(t)$ and $x_3(t)$ respectively, which are amplitudes of the recorded signals ...

5

You might also want to consider Principal Component Analysis (PCA) or an extension of it known as Independent Subspace Analysis which is PCA followed by ICA. These techniques work very well for extracting pitch stationary signals from a single observation signal. I'm an audio specialist but have discussed biomedical signals with colleagues in the past and ...

5

The common practice in the source separation community is to report 3 metrics defined in Vincent et al's paper, called SDR (signal to distortion ratio), SIR (signal to interferences ratio), and SAR (signal to artifacts ratio). The reason for this is that in source separation applications, there can be two types of "noise" in the reconstructed signal: noise ...

5

The opposite of blind-signal separation is not "non-blind signal separation". The thing is, depending on what you know on the signals you will use different algorithms. For example, if you know that there is a low frequency signal mixed with a higher frequency one, you can use low/high-pass filtering to get the two signals. There's no real name for what ...

3

This is a source separation problem, which, in general is quite difficult to solve. One way to attack it is to use non-negative matrix factorization (NMF) with short-time Fourier analysis. Please see this question and a sample script here. The overarching idea is to try to isolate your component sounds A, B, C,... using a time-frequency decomposition (or, ...

3

This can be approached as a general blind source separation problem with an excitation-filter model for both the snare and the synthesizer. For the snare the excitation would be an impulse train, and the filter would have the snare sample as its impulse response. You can try if Flexible Audio Source Separation Toolbox (FASST) finds good enough models to ...

3

So, assuming the linear superposition you suggest, the sound you want to exctract lives in a periodic subspace of your signal space. This subspace is uniquely characterised by the exact spacing between the sound instances, which I assume you know in the following. If you don't know it, you can find it from calculating the autocorrelation sequence of your ...

3

First of all note that there is a certain ambiguity in the problem formulation because the approximation of $x(t)$ has to be divided into the part represented by $x^{\prime}(t)$ and the part represented by $f(y(t))$, and this division is not unique. The approach I chose is to start with a parametrization of $f(y(t))$, and approximate the given $x(t)$ as well ...

3

You have an incorrect approach here. Instead of trying to design small range filters, you should just do frequency-domain analysis. Say you have a signal $x$ consisting of $N$ samples, sampled at the rate of $f_s$ Hz. For simplicity, let's take an example with $N = 1000$ and $f_s$ = 1000 Hz. The Nyquist frequency is half of the sampling frequency, which is $... 3 What if they did all use the same carrier? What you're describing is a single-frequency network (SFN). These are in common use for things that are not stupid FM broadcast. The whole truth is that reception from different senders in an SFN just look like heavy multipath propagation, where the sent signal just takes multiple paths of different length to the ... 2 Let's call your signals$x_1$and$x_2$. The recorded signal is$y$: $$y(n) = (x_1*h_1)(n) + (x_2*h_2)(n)$$ where$n$is the sample (time) index, and$h_1$and$h_2$are the room impulse responses from the respective speakers to the mic. Assume you want to retrieve$x_2$from$y$with the knowledge of$x_1$. You can apply echo cancellation methods to ... 2 Let's sum your problem. Assuming the transfer function of the whole signal transmission chain from the source 1 to the recorded signal (DAC, speaker, room, microphone, ADC...) is$h_1$(and$h_2$from source 2 to microphone), then the signals you have complete access too are:$(h_1 \star x_1)(n) + (h_2 \star x_2)(n)$(the resulting signal)$x_1(n)$(one of ... 2 Choosing points in the horizontal plane with proper spacing seems like the right approach. Aneochic HRTFs are probably the easiest but typically lead to "inside the head" localization. Depending on your application this may or may not be a good thing. If you want externalization it's best to use binaural room impulse responses measured in a space that's ... 2 You do a QR decomposition of$X^T$. Then the system reads as $$Y=AR^TQ^T.$$ Now multiply with$Q$from the right, $$YQ=AR^T,$$ which already should give you hints on the reliability of the eventual solution, since ideally, the block of$YQ$right of the leading$m\times n$block should be zero. If that is true, then$$A=[YQ]_{[1:m,1:n]}[R^T]_{[1:n,1:n]}^{-1}$...

2

Depending on what kind of application is that and what kind of microphones (dynamic, condenser, home-made electret ones) solutions can be different. Are you going to process everything in real-time or to make your analysis offline? If your microphones doesn't require Phantom power and can be plugged into mic-in, then you can buy few of cheapest USB sound ...

2

If I understand correctly what you are trying to achieve is usually called Acoustic Echo Cancellation. In many cases, so-called adaptive filtering algorithms, such as the (normalized) least-mean-squares ([N]LMS) algorithm, can be used effectively. In this context, I can really recommend the book Adaptive Signal Processing by Widrow and Stearns. I'm sure you'...

2

For me, if all you want to deal with is the possible 'debris in the optics,' why not get a reference background image? This will help you to easily identify which are 'noise' caused by your optics. Assume you can get a perfect background, namely everything remains the same except for your cells ( maybe you can get this by taking another image after removing ...

2

You could try calculating Shannon entropy of the spectrum. Normalize the Fourier transform $f(x)$ of your signal so that $\int_{-\infty}^\infty |f(x)|^2\, dx = 1$ and calculate Shannon entropy as $- \int_{-\infty}^\infty |f(x)|^2 \log |f(x)|^2\, dx$. You can clamp $|f(x)|^2 \log |f(x)|^2$ to zero for very small values of $f(x)$ if its logarithm blows up and ...

2

The short answer to your question is yes and the method is called beamforming which you alluded to in your question by mentioning multiple aerials, and knowing the propagation time from two distinct transmission locations broadcasting on the same frequency to each receiving aerial is key to beamforming. The number of required aerials depends on the ...

1

I will sketch an idea how to use Sparse Represenattion (Dictionary Learning) for BSS. Let's say we have $\mathcal{S} = \mathcal{S}_{1} \cup \mathcal{S}_{2}$ where $\mathcal{S}_{1} = \left\{ {x}_{i} \right\}_{i = 1}^{m}$ and $\mathcal{S}_{2} = \left\{ {y}_{i} \right\}_{i = 1}^{n}$. So if we assume different sources for $\mathcal{S}_{1}$ and $\... 1 Will averaging S1 over these 60 acquisitions and S2 over 4 (Figure 1) and then subtracting the averaged signals (S2_avg - S1_avg) give me a better representation of response to Stimulus B or should both the signals be averaged across equal acquisitions (Figure 2). Ideally, both signals should be averaged over the same amount of repetitions and the ideal ... 1 You should try "Harmonic Percussive Source Separation" ... quickly testable on your WAVs using librosa's librosa.decompose.hpss(spectrogram): Median-filtering harmonic percussive source separation (HPSS). Decomposes an input spectrogram S = H + P where H contains the harmonic components, and P contains the percussive components. Here are some ... 1 The key point of blind signal separation is the separation of a mixture of signals when there is mathematically insufficient knowledge to do so. On the contrary if you, mathematically, have sufficient information to distinguish two or more signals from a given mixture of them, then you can, as mathematically expected, perfectly separate them using well known ... 1 What i know in implementations of BSS (e.g jade) is that they first normalise the input data (to variance = 1) and then order the data according to scale. So this is unique and will produce consistent output ordering for the same input data (will add references shortly). Have in mind however that this is just a consistent heuristic (any consistent heuristic ... 1 A great free toolbox is the Flexible Audio Source Separation. Their website contains some demo examples in Python, Matlab, and C++ code. Another toolbox is the Toolbox for Blind Source Separation. However, this toolbox is not user friendly and you need to do some extra programming in C, Matlab or Python. 1 You should look into a generalization of ICA called Independent Subspace Analysis (ISA). Practically it is Principal Component Analysis (PCA) followed by ICA. There are threads on dsp.stackexchange that discuss this as well as github repos (and here) that provide various implementations. 1 It really depends on what you mean by a channel. Normally a channel in a source separation scenario would the number of sensors detecting your unknown/partially known signal sources. In the audio domain sensors are referred to as microphones. If this viewpoint is taken single channel separation and single microphone separation are equivalent. However ... 1 The sound produced using the lowest pitched strings of large stringed music instruments often includes slightly inharmonic overtones (due to finite string thickness and stiffness), which renders the waveform of the note pitch not quite, but "almost periodic". You can record these sounds from the bottom octaves of most (physical, stringed) pianos, and bass ... 1 What you are looking for is exactly Independent Component Analysis, (ICA). The setup for ICA is exactly, given a matrix$Y$, from$Y = AX$, find both$A$and$X\$.

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