### PDF form

Let be the magnitude-squared of the DFT bins of a size- DFT. Then, . Let have mean . Then, the distribution of the zero and Nyquist frequency bins () follows the Chi-squared distribution with 1 degree of freedom scaled by ,

DFT bins 1 through are complex so have the Chi-squared distribution with 2 degrees of freedom scaled by , or equivalently the exponential distribution with mean :

The joint PDF of is obtained from the product of the above bin densities. Interestingly, maximizing over will also maximize the PDF of the input data to the DFT. Let be the length- input to the DFT, then

 (5.6)

where are the DFT bins. Therefore, we can replace with :

 (5.7)

It is mathematically simpler to maximize (5.7) instead of .

Although (5.7) is a function of , we need to evaluate it for a particular value of the parameters and . In Section 5.2.5, we showed how to compute from the AR parameters.

Baggenstoss 2017-05-19