Isolation Of Individual Tracks From Polyphonic Music Nick Starr

Motivation • Retrieving individual source signals from a mixed signal is a goal with broad applications • "Cocktail Party Problem" • Medical Imaging • Separating individual musical instruments

Methodology 1. Map the input signal to the frequency domain via a Fourier Transform 2. Perform Single Value Decomposition (SVD) on the transposed matrix of amplitudes from the transform. 3. Run Independent Component Analysis (ICA) on the frequency components. 4. Classify the components calculated from (3). 5. Map the separated components back to the time domain with an Inverse Fourier Transform

Mathematical Background • Fourier Transform: Mathematically, the Fourier Transform from a function f(t) on the time domain to a function F(v) on the frequency domain (v being a nu, ideally) is F(v) = Integral[f(t)*Exp[-2*pi*i*t*v], t, -infinity, infinity] (where i is the imaginary unit, Sqrt[-i]). Intuitively, F(v) measures how much of the frequency v is present in the original signal. The inverse Fourier Transform, getting f(t) from F(v), is defined similarly: f(t) = Integral[F(v)*Exp[2*pi*i*t*v], v, -infinity, infinity]. (I'll prettify all this notation later) . • Singular Value Decomposition (SVD): For some matrix X, we have: X^T = U*D*V^T. D is a diagonal matrix of singular values in decreasing order, U = (u1, u2, u3 ..... um) where the ui are the eigenvectors of XX^T, and V = (v1, v2, v3, .... vn) where the vi are the eigenvectors of X^TX.  

Mathematical Background (cont) • Independent Component Analysis (ICA): ICA solves the problem of being given a vector x of observation signals, another vector s of source signals, and a matrix A of mixing constants, such that s = Ax. The goal of ICA is to determine A's inverse, so that A^-1s = A^-1Ax = x. (todo: explain HOW it does it)

Diagram of methodology (not quite our algorithm)

Technical Details • We will be using the C programming language to implement our program. • We will be using the FFTW library for the calculation of Fourier Transforms.

Results/Conclusion TBA