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I'm having a problem to normalize my output signal (extracted EEG signal) after applying Independent Component Analysis (ICA) that may be as similar as the original source (EEG signal). I need do this work to subtract and determine noises. I simulate it on Matlab. Could you help me solve this problem, please? Thank you a lot.

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  • $\begingroup$ When you say normalize, do you mean multiply by a constant or what? $\endgroup$
    – Royi
    Jun 9, 2023 at 9:50

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From your description, you have an original (real-valued) signal and a processed (real-valued) signal, and now you want to normalize the processed signal by multiplying it with a scaling constant, so that the normalized signal is "as similar as the original signal".

Denote the original (discrete) signal by a row vector $\textbf{x}=[x_0, x_1, ... x_{n-1}]$, the processed signal by a row vector $\textbf{y}=[y_0, y_1, ... y_{n-1}]$, and the scaling constant by $c$, where $c$ is to be determined.

The scaling constant $c$ can be optimized so that the mean-squared error (MSE) between $\textbf{x}$ and $c\textbf{y}$ is minimized. The MSE is

$$\epsilon=\frac{1}{n}\sum_{k=0}^{n-1}(cy_k-x_k)^2,$$

$$\epsilon=\frac{1}{n}(c^2\sum_{k=0}^{n-1}y_k^2-2c\sum_{k=0}^{n-1}x_ky_k+\sum_{k=0}^{n-1}x_k^2).$$

To minimize the MSE, let $\frac{d\epsilon}{dc}=0$. This finally yields an optimum value of $c$,

$$c=\frac{\displaystyle\sum_{k=0}^{n-1}x_ky_k}{\displaystyle\sum_{k=0}^{n-1}y_k^2}=\frac{\textbf{xy}'}{\textbf{yy}'}$$

where $\text{y}'$ is the transpose of $\textbf{y}$.

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