...  @@ 249,9 +249,13 @@ We are ready to explore the math behind the curtains. The first step was to figu 
...  @@ 249,9 +249,13 @@ We are ready to explore the math behind the curtains. The first step was to figu 

where $`x`$ is the measured signal. Note that n is the elements in X. In the next step, we take the derivative with respect to θ and make it equal to zero. The maximum will give us the first principle component, where the minimum will give the second principle component, which are orthogonal. In practice, it does not matter whether we find the min or max with the derivative; they differ by 90 degree here. After couple of calculation steps, we can get the angle θ:


where $`x`$ is the measured signal. Note that n is the elements in X. In the next step, we take the derivative with respect to θ and make it equal to zero. The maximum will give us the first principle component, where the minimum will give the second principle component, which are orthogonal. In practice, it does not matter whether we find the min or max with the derivative; they differ by 90 degree here. After couple of calculation steps, we can get the angle θ:






```math


```math


θ = 1/2 tan^{1}(2\sum{x_1x_2}/\sum(x_2^2x_1^2))


θ = 0.5 tan^{1}(2\sum{x_1x_2}/\sum(x_2^2x_1^2))




```




The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and $`x_1`$. Next, we express $`U^*`$ with θ:








```math




U^* = \begin{bmatrix} cos(θ) & sin(θ) \\ sin(θ) & cos(θ) \end{bmatrix}


```


```


The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and $`x_1`$.












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