# 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 be even and

The DFT bins are independent under , but not identically distributed. DFT bins 0 and are real-valued so have the Chi-squared distribution with 1 degree of freedom scaled by , which we denote by :

DFT bins 1 through are complex so have the Chi-squared distribution with 2 degrees of freedom scaled by , which we denote by :

The complete PDF is

 (4.1)

The J-function, can be simplified for even to

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 from using UMS (Section 3.3) is accomplished first by taking the square-root of each bin. We then multiply the zero and Nyquist bins ( and for even ) by 1 or -1, each with probability 1/2. The remaining bins are multiplied by a phase term , where are independent RV uniformly distributed on the interval . Finally, the length- DFT vector is created by appending the conjugate of the complex bins, then taking the inverse DFT to obtain . For more information, refer to the software module software/module_dftmsq_synth.m.

Baggenstoss 2017-05-19