Dear Luis,
Processing v103450400.wav
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I have spent a few happy hours grinding up the .wav file from your www page.
Here are some thoughts on what I discover.
There is no doubt that you have detected Voyager-1. I do not think you could
be absolutely certain from this file only, but the evidence of repeated
features in other recordings is sufficient correlation.
Tools
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I write my own tools for computers. This means that what I am reporting here
is processed by independent means. However, I seek to reproduce your findings
where possible, and am in broad agreement with your results.
Doppler
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I checked the expected doppler profile by my own calculations; I use the
library routines of the software that controls the Bochum telescope. This
gives identical results to Horizons, because the algorithms are based on the
same models. In particular I computed the doppler shift at 1 minute intervals,
and ran a linear fit over the frequency to computer the doppler rate, and to
see if there is any significant non-linearity over 15 minutes (there is not).
The rate determined is fdot = -0.7923 Hz/s.
FFTs etc
--------
Next I subjected the file to spectral processing at various FFT resolutions,
with and without doppler correction.
The baseline processing used 16384 point FFTs, which with the sample rate of
22050 S/s gives a bin bandwidth of 1.35 Hz, and a data time span of 0.74s per
FFT. Thus a 900s record results in ~1200 FFTs. These are averaged to one
composite spectrum. With the above parameters, the display noisefloor (jitter)
is ~0.125 dB.
Receiver Passband
-----------------
The basic receiver spectrum is illustrated in fig01.gif. This is the raw data
with NO doppler correction, processed with 2048 point FFT, 10.8 Hz resolution.
Features noted are a 2 dB slope over much of the passband, and an abrupt cut
off at 7000 Hz (-20 dB at 7050 Hz) which I assume is the effect of an anti-
alias pre-filter in the sound card. Also apparent are various ripples in the
passband of order 0.2 dB.
The slope and the ripples are phenomena of similar magnitude to the signal we
seek, so care is needed in subsequent processing. The gross effects can be
removed by subtracting out a precomputed spectrum based on a running average of
(say) 100 adjacent bins. This will increase the display noisefloor by 10%, but
should clarify the trace considerably.
It is also clear that frequencies below 1 kHz and above 6 kHz can be discarded.
With Doppler Correction
-----------------------
Next we look at the spectrum when each contributing FFT is shifted according to
the precomputed doppler change as discussed earlier. Normally the transform is
loaded with real data Real(t) = V(t), with the imaginary component set to 0 .
To 'shift' the input the transform is loaded with Real(t)=V(t)*COSp(t),
Imag(t)=-V(t)*SINp(t) where p(t) = w*t + 0.5*wd*t*t and w is the angular
frequency at the start of the data being loaded, and wd is the rate of change.
This is illustrated in fig02.gif . The display noise is 0.125 dB rms; for a
detection we need a "line" of (say) 4.5x this or 0.6 dB above the mass. This
is more than half a vertical division, and there is nothing that comes close to
this. So one would conclude that there was no signal.
Signal Spreading?
-----------------
On the other hand, the testing hypothesis assumes that the signal carrier is
confined to one display bin, of bandwidth 1.35 Hz. This may not be the true.
Possible sources of unmodelled frequency shift are:
* Spacecraft frequency drift
* Interstellar scintillation
* GPSDO variation exceeding 1E-10
* Doppler change not quite linear
* RX drift
Suppose these sources exceed 1 bin; then we should do a running average on 3
adjacent bins. fig03.gif illustrates this, and reveals some energy clumping at
1350 Hz and at 5325 Hz.
Expanding the previous basic fig02.gif around 5325 Hz shows that in fact 5
adjacent spectral bins are of similar amplitude, suggesting the signal is
spread over about 5*1.35 = 7 Hz.
Dealing with Spreading
----------------------
So, a running average (correlation) on 5 bins is displayed in fig04.gif which
has been zoomed in to 5000-6000 Hz and expanded vertically. The peak of the
correlation is ~0.26 dB, while the display noisefloor is 0.125/sqrt(5) = 0.056
dB. This makes the detection metric = 0.26/0.056 = 4.7:1 which makes it a
strong candidate for a signal.
A (S+N)/N = 0.26 dB is S/N=0.0617. The bin bandwidth is 1.35 Hz, so the signal
strength Pc/No = 0.0617*1.35*5 = 0.415 Hz, of -3.8 dB [Hz].
Removing RX passband slope
--------------------------
The effect of the non-flat RX passband can be largely removed as discussed
earlier. The trace then looks like fig05.gif As with the previous diagram,
the correlation peak is 0.25 dB which maps to Pc/No = -4.0 dB [Hz]. The
uncertainty in this is easily calculated to be 1.4 dB rms.
Conclusion
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The signal to noise power density of -4 dB [Hz] is slightly larger than the
expected value of -5.9 dB [Hz]. On the other hand, the expected value includes
some uncertainties, as does the signal processing, so this result is about
right.
James Miller G3RUH
Original: 2006 Apr 20 [Thu] 1459 utc
Corrected: 2006 Apr 21 [Fri] 0700 utc