Magnitude Squared DFT of real data
This building block considers the DFT of real data followed by the
computation of the magnitudesquared of the bins.
Like the previous example, this belongs to a general class of chisquared
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 magnitudesquared,
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 realvalued so
have the Chisquared distribution with 1 degree of freedom
scaled by , which we denote by :
DFT bins 1 through are complex
so have the Chisquared
distribution with 2 degrees of freedom scaled by ,
which we denote by :
The complete PDF
is

(4.1) 
The Jfunction,
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.
Resynthesis of from using UMS (Section 3.3)
is accomplished first by taking the squareroot
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
20170519