[dbd]

On Dec 24, 9:50*am, "William Sommerwerck"
wrote:
...


If the noise is a stochastic process (am I using the term correctly?), would
you need an "infinite" sample? And if not, what would be the minimum sample
length? (I assume it would be inversely proportional to the lowest frequency
you wanted to measure.)

You cannot achieve perfect reconstruction of the stochastic noise
process with a finite set of samples. If you simply take the DFT of n
real samples you get n/2 independent Fourier coefficients to calculate
n/2 power spectral density (PSD) estimates. If you simply increase n,
you get more estimates at closer frequency spacing with the same high
variance. To reduce the variance, multiple independent sets of (small)
n real samples are used to generate PSD estimates that are averaged
across multiple blocks to reduce variance. The conventional DFT will
produce one of the estimates at DC. This was in the reference I gave
on noise processing.

For sums of sets of stationary tones you can achieve perfect
reconstruction only if all the tones belong to a set of n/2
frequencies evenly spaced on 0 to Fsample/2. With the conventional
DFT, one of the tones will be DC. If these conditions are met, you can
achieve perfect reconstruction anywhere on the region over which the
tones are stationary. These conditions are seldom met in real
instrumentation and with real-world signals

Fortunately it is not necessary to be able to achieve perfect
reconstruction to make useful applications of Fourier analysis of
finite/discrete data sets. Is it equivalent to infinite/continuous
Fourier analysis of infinite/continuous signals? No, it doesn't need
to be.

Dale B. Dalrymple