Fourier Analysis of G0 DAQ

Regular and Irregular Spin Flip

As a quick-and-dirty test/illustration, W. Des Ramsay generated a 2-second long helicity signal from 64 x 1/30 s states and did an FFT.  He did this for both regular spin flip and for spin chosen randomly for each 1/30 s state.

The plots below only show the amplitude, not the phase, of the transform.

 spin1.eps.gz   Regular and random spin sequences with their fourier transforms. (Gnu-zipped postscript, 16.6 kb)
 spin2.eps.gz   Same thing with a different seed in the random number generator. (Gnu-zipped postscript, 16.5 kb)
 

  • The FFT of the regular sequence is just odd multiples of 15 Hz, as you expect for a 15 Hz square wave.

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  • The FFT of the random spin sequence contains all frequencies except  even multiples of 15 Hz.

    • The fourier transforms will be the frequency-domain acceptance of a synchronous detection system operating with the same switching pattern.

    Different Run Times

    The FFT algorithm assumes that the waveform shown in the time window repeats forever.  This is, for example, why the square wave above  shows up as sharp peaks exactly at odd multiples of 15 Hz.  To see the effect of different run times, I simulated synchronous data acquisition by multiplying sine and cosine functions by the switching function (regular or random) and integrating for different lengths of time.  I normalized so that a magnitude of 1.0 means the DAQ sees all of that particular frequency.

     3reg.eps.gz   Frequency domain acceptance of DAQ operating at 1/30 second per state with a regular spin flip sequence and no deadtime between states. The DAQ acts as a filter with a response like the components of a 15Hz square wave (all of the fundamental, 1/3 of the 3rd harmonic, 1/5 of the 5th harmonic, etc.).  At short runtimes, other frequencies get through as well, but as runtime increases, the acceptance peaks shapen up and eventully all that remains is the square wave spectrum.

     3ran.eps.gz   Same as above, but with random spin sequence.  DAQ still blind to even harmonics of 15Hz, but accepting a band of other frequencies, with the amount accepted averaging to zero for very long runtimes.

     peak45hz.eps.gz    Deails of the regular spin flip response.  Response to sine and cosine waveforms as their frequency is scanned from 30Hz to 60Hz.  I assumed a reqular spin sequence of 1/30 second per state, starting with +1, and taking data for 10 states (1/3 s). Note that the magnitude peak is centered at 45 Hz and the zeroes are spaced at 3Hz intervals on either side of this.  3Hz is (1/runtime).

    (It's only at short runtimes that the reqular spin flip DAQ sees much of the cosine component.  At long runtimes the sine component dominates).

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    Send additions and corrections to
    W. Des Ramsay.