Dependent component analysis

For simplicity, assume all individual sets of signals are the same size, k, and total N sets. Building off the basic equations of BSS (seen below) instead of independent source signals, one has independent sets of signals, s(t) = ({s₁(t),...,s_k(t)},...,{s_kN-k+1(t)...,s_kN(t)})^T, which are mixed by coefficients A=[a_ij]εR^mxkN that produce a set of mixed signals, x(t)=(x₁(t),...,x_m(t))^T. The signals can be multidimensional.

${\displaystyle x(t)=A*s(t)}$

The following equation BSS separates the set of mixed signals, x(t), by finding and using coefficients, B=[B_ij]εR^kNxm, to separate and get the set of approximation of the original signals, y(t)=({y₁(t),...,y_k(t)},...,{y_kN-k+1(t)...,y_kN(t)})^T.[1]

${\displaystyle y(t)=B*x(t)}$

Sub-Band Decomposition ICA (SDICA) is based on the fact that wideband source signals are dependent, but that other subbands are independent. It uses an adaptive filter by choosing subbands using a minimum of mutual information (MI) to separate mixed signals. After finding subband signals, ICA can be used to reconstruct, based on subband signals, by using ICA. Below is a formula to find MI based on entropy, where H is entropy.[2]

${\displaystyle {\widehat {I_{H}}}(y)=\sum _{n=1}^{N}{\widehat {H}}(y_{n})-{\widehat {H}}(y)}$

${\displaystyle {\widehat {H}}(y_{n})=-{\frac {1}{T}}\sum _{t=1}^{T}log{\widehat {P}}_{yn}(y_{n}(t))}$

${\displaystyle {\widehat {H}}(y)=-{\frac {1}{T}}\sum _{t=1}^{T}log{\widehat {P}}_{y}(y_{n}(t))}$