... | ... | @@ -251,10 +251,35 @@ where $`x`$ is the measured signal. Note that n is the elements in X. In the nex |
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```math
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θ = 0.5 tan^{-1}(-2\sum{x_1x_2}/\sum(x_2^2-x_1^2))
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```
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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:
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The ratio gives information between the ratios of the covariance to the variances of $`x_2`$ and $`x_1`$. Once θ is found, we express $`U^*`$ as:
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```math
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U^* = \begin{bmatrix} cos(θ) & sin(θ) \\ -sin(θ) & cos(θ) \end{bmatrix}
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U^* =
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\begin{bmatrix}
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cos(θ) & sin(θ) \\
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-sin(θ) & cos(θ)
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\end{bmatrix}
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```
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The second action was the stretching ($`\sigma^{-1}`$, see the figure above). This step is very easy to calculate as we already found the first principle direction and its variance. Let's call it $`\sigma_1`$ this time:
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```math
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\sigma_1(θ) = \sum_{n}^{N} {[x_1(n) x_2(n)]\begin{bmatrix} cos(θ) \\ sin(θ) \end{bmatrix}}^2
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```
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The other principle component will be orthogonal:
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```math
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\sigma_2(θ) = \sum_{n}^{N} {[x_1(n) x_2(n)]\begin{bmatrix} cos(θ-\pi/2) \\ sin(θ-\pi/2) \end{bmatrix}}^2
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```
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giving $`\sigma^{-1}`$ as:
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```math
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\sigma^{-1} =
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\begin{bmatrix}
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cos(θ) & 0 \\
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0 & cos(θ)
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\end{bmatrix}
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```
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