To: File periodogram_cyclic_correlation.wpd

From: Jon Hagen

Date: May 18, 2004, corrected June 4, 2004

Subject: Transform of Periodogram - Cyclic Correlation Function

The DFT of a sequence of voltages is given by

and the periodogram is defined as

Note, however, that uk is also the DFT of the function obtained by periodically extending Vn.. Rather than giving this function another designation such as VnP , let us simply regard Vn as being periodically extended.

Writing Equation 2 as a double sum, we have

We can make a change of variable by writing m = n-n', giving us

e

which is the DFT of the term in brackets, which is the cyclic convolution of the input sequence. Consider the specific example where the sequence has length 8 and m=3. The sum becomes

V0V3 + V1V4 +V2V5 + V3V6 +V4V7 + V5V8 +V6V9 + V7V10

= V0V3 + V1V4 +V2V5 + V3V6 +V4V7 + V5V0 +V6V1 + V7V2 5)

In the figure below, the rotor is rotated three steps counterclockwise, corresponding to m=3. This rotor/stator indicates the products that occur in the 'circular' convolution.

So the periodogram is the transform of a cyclic correlation function, rather than the standard correlation function. Of course the data sequence can be extended with zeros so that the wrap-around terms become verso.

Alternate Demonstration

Rather than pulling the answer out of a hat, as above, let us rewrite Equation 4, but with the proviso that the index n+m be always within the range 0 to N-1, i.e. within the range of actual data.

For any given m, the most positive value of m+n is N-1. The next n would put n+m outside the range. So, for this value and higher values of n, we replace m by -(N-m). This keeps n+m always within range and is permissible, because it doesn't change the exponential term,

since exp(-j 2 (-[N-m]k/N) = exp(-j 2 mk/N). In this way, as n ranges from 0 to N-1, the products are formed as a cyclic convolution.

Filter Shape of periodogram