## Help: DFT,FFT Sampling Rates

### Help: DFT,FFT Sampling Rates

I would like to use DFT,FFT or some other technique to derive frequency
domain data of a signal that is sampled at different, but known rates.

An example: my signal is sampled first at T1 and then at T2, T3, and
finally at T4. This pattern continually repeats and the time interval
between T1 and T2, T2 and T3, T3 and T4, T4 and T1 is different for the 4
cases but the time interval is known.

My first thought was to collect all the T1 samples and peform the FFT on
them and effect the same for the T2,T3 and T4 cases.  But this approach
cuts my effective sampling rates by 4 and I really can't live with that.

I guess what I'm looking for is a way to adjust the T2,T3 and T4 cases
relative to the T1 time interval and perform an FFT as if all data was
sampled at the same rate.  Any ideas on this?

Scott Lougheed

### Help: DFT,FFT Sampling Rates

> I would like to use DFT,FFT or some other technique to derive frequency
> domain data of a signal that is sampled at different, but known rates.

> An example: my signal is sampled first at T1 and then at T2, T3, and
> finally at T4. This pattern continually repeats and the time interval
> between T1 and T2, T2 and T3, T3 and T4, T4 and T1 is different for the 4
> cases but the time interval is known.

There is an algorithmm described in

H.G. Feichtinger & A K. Gr\"{o}chenig ``Theory and Practice of
Irregular Sampling'' in {\sl Wavelets: Mathematics and Applications},
Edited by Benedetto and Frazier, CRC Press, 1994.

which is really just the repeated application of FFTs and lowpass
filtering which allows you to do better than just looking at the T1
samples.  It is a bit computationally intensive, but not excessively
so (only a few FFTs are needed).

I seem to recall they had matlab code to do their algorithm.

Ciao,

Peter K.

--
Peter J. Kootsookos                Phone: +353 1 671 8177
MV Technology Ltd                  Fax:   +353 1 671 8470
Unit 24, IDA Enterprise Centre     #include <disclaimer.h>

### Help: DFT,FFT Sampling Rates

> I would like to use DFT,FFT or some other technique to derive frequency
> domain data of a signal that is sampled at different, but known rates.

> An example: my signal is sampled first at T1 and then at T2, T3, and
> finally at T4. This pattern continually repeats and the time interval
> between T1 and T2, T2 and T3, T3 and T4, T4 and T1 is different for the 4
> cases but the time interval is known.

> My first thought was to collect all the T1 samples and peform the FFT on
> them and effect the same for the T2,T3 and T4 cases.  But this approach
> cuts my effective sampling rates by 4 and I really can't live with that.

> I guess what I'm looking for is a way to adjust the T2,T3 and T4 cases
> relative to the T1 time interval and perform an FFT as if all data was
> sampled at the same rate.  Any ideas on this?

> Scott Lougheed

If you have enough CPU time, you could compute the DFT instead of the
FFT.  The FFT assumes uniform sampling.  Take all of your time samples
at all rates (call it t(i)) and your data (call it x[t(i)]) and then
for whatever frequencies you want the spectrum, f(k), compute:

spectrum(k) = sum_over_i { x[t(i)] * exp(-j*2*pi*f(k)*t(i)) }

You probably want to normalize the result by dividing by the number of
samples in t(i).  This is the general idea.

-Mark-

Given time domain data sampled with a repeating, identifiable sequence of
unequal time intervals, is a technique available for computing the FFT or
DFT of this data?

Thanks