Magnitude Squared DFT of real data

This building block considers the DFT of real data followed by the computation of the magnitude-squared of the bins. Like the previous example, this belongs to a general class of chi-squared feature extractors fitting within the ``Gaussian" row of Table 3.1. The energy statistic in all of these problems is the total energy. In the case of DFT magnitude-squared, the ES is contained implicitly (Parseval's theorem). Let $ N$ be even and

$\displaystyle z_k = \left\vert\sum_{i=1}^{N} x_i \; e^{-j 2 \pi (k-1) (i-1)/N}\right\vert^2,
\;\;\; 1\leq k \leq N/2+1.$

The DFT bins are independent under $ H_0$, but not identically distributed. DFT bins 0 and $ N/2$ are real-valued so $ z_k$ have the Chi-squared distribution with 1 degree of freedom scaled by $ N$, which we denote by $ p_0(z)$:

$\displaystyle p_0(z) = \frac{1}{N\sqrt{2\pi}}\; (z/N)^{-1/2} \; \exp\left\{
-\frac{z}{2N}\right\}.
$

DFT bins 1 through $ N/2-1$ are complex so $ z_k$ have the Chi-squared distribution with 2 degrees of freedom scaled by $ N/2$, which we denote by $ p_1(z)$:

$\displaystyle p_1(z) = \frac{1}{N}\; \exp\left\{-\frac{z}{N}\right\}.
$

The complete PDF $ p({\bf z}\vert H_0)$ is

$\displaystyle \log p({\bf z}\vert H_0) = \log p_0(z_0) + \sum_{k=1}^{N/2} \log p_1(z_k) + \log p_0(z_{N/2}).$ (4.1)

The J-function, $ \log J = \log p({\bf x}\vert H_0) - \log p({\bf z}\vert H_0)$ can be simplified for even $ N$ to

$\displaystyle \log J = \frac{\log z_1 + \log z_{N/2+1}}{2}
+ \frac{N}{2} \log N - \frac{N-2}{2} \log(2\pi) ,$

which interestingly is data independent except with respect to the zero and Nyquist frequency bins. For more information, refer to the software module software/module_dftmsq.m.

Re-synthesis of $ {\bf x}$ from $ {\bf z}$ using UMS (Section 3.3) is accomplished first by taking the square-root of each bin. We then multiply the zero and Nyquist bins ($ z_1$ and $ z_{N/2+1}$ for even $ N$) by 1 or -1, each with probability 1/2. The remaining bins are multiplied by a phase term $ e^{j\phi_k}$, where $ \phi_k$ are independent RV uniformly distributed on the interval $ [0,\;2\pi]$. Finally, the length-$ N$ DFT vector is created by appending the conjugate of the complex bins, then taking the inverse DFT to obtain $ {\bf x}$. For more information, refer to the software module software/module_dftmsq_synth.m.

Baggenstoss 2017-05-19