In this example, different parameters are varied in an attempt to find smallest signal in doppler_noise_onetenth.wav.
Doppler_noise_onetenth.wav contains 7 signals that are each one minute long. They begin 5 minutes into the file and end at 12 minutes into the file. The strength of the signal is reduced by one-half each minute. The pattern is as follows:
Minutes Signal as % of Peak Noise
5-6 10
6-7 5
7-8 2.5
8-9 1.25
9-10 0.63
10-11 0.31
11-12 0.16
The first step is to make sure that you can see at least the larger signals in this file in spectral view. To do this you must set the resolution correctly. Start with Spectral View, which is found in the top tool bar, under View, and the resolution is found under Options/Settings. There is a tab that says Spectral, and the resolution can be set from there. The resolution must be set to above 512 bands. This setting will work, and the signals will be visible. However, if a resolution of 4096 bands is attainable on your computer, the picture is much easier to work with.
At this point, the picture looks something like this:
The signals can be viewed now as the faint yellow line beginning at 5 min. This nonlinear drifting signal occurs around 1300 Hz, and becomes virtually invisible by 9 min. Zooming in on the signal permits more details to be seen. To zoom to a selection, left click on the left side of the selection, and right click on the other side of the selection, and then click the magnifying glass with the box around it (left-hand, bottom corner). Then zoom vertically by using the magnifying glasses to the far right until the signals are easily viewed. The picture should now look more like this:
The example zooms in from 4:30 min. to 12:30 min., because the last signal goes from 11 min. to 12 min., and the goal is to detect this entire repeating signal.
The next task is to dechirp the signal (using the Transform / Stretch function) until it becomes as horizontal as possible (to permit FFT signal integration). Because the signal drift is exactly the same in each successively weaker signal, the dechirp can be optimized on the visible signals and then applied to the invisible ones. A close inspection reveals that, during about the first 5 seconds of each signal, the drift is strongly nonlinear. Stretching is most effective on the most linear parts of each signal. First, the signal to be stretched must be selected. Left click about 5 seconds into the signal, e.g., 5:05 min. for the first signal, and right click at the end of the signal, e.g., 6 min. The selection should look like this:
The Stretch function can be found under the Transform heading, under the last choice, Time/Pitch. When the stretch dialogue box opens, it looks like this:
These example stretches are done with the gliding stretch option. In order to get the signal as visible as possible via FFT integration, it should appear as horizontal as possible. Any whole number stretch that is done will chirp the signal too far; therefore very small stretches need to be applied. The beginning of the signal is lower in frequency than the end of the signal, so the initial pitch must be raised, and the final pitch must be lowered. Experiment with various large and small numbers. If the initial pitch is raised to 99.72% of its original pitch and the final pitch is lowered to 100.125% of its original pitch, the signal becomes fairly horizontal. After applying the same stretch to all of the signals, the spectrogram looks like this:
If you apply different stretches, keep in mind that when you complete two consecutive stretches on one signal, the program executes the second stretch on the result of the first stretch. Therefore, every time you want to try a different stretch, you must open the original file again and attempt the stretch from the beginning. There is a “Revert to Saved” option under the File menu that is helpful in this task.
Once the signals appear horizontal (check it with the Zoom functions), the Frequency Analysis function can be employed. This function can be found under the “Analyze” option on the top tool bar. With the 5 minute signal selected and the frequency analysis dialogue box open, the screen looks like this:
The signal is distinctly visible at approximately 1300 Hz. The longer the FFT duration, the better the S/N enhancement, so that number should be as large as possible (up to the transit time of the telescope!). Welch (Gaussian) is usually the best choice next to the FFT size. After scanning, the frequency analysis window looks like this:
After extended FFT integration, the signal becomes larger and more sharp. (As an added benefit of dechirping, receiver "birdies" and other interference signals that are constant in frequency are reduced in amplitude when frequency drifts are applied to the raw data.)
This pattern can be seen for all the signals up through the one from 8 minutes to 9 minutes, although the 8-9 minute peak is much smaller, of course:
After attempting to stretch the signals from 9 min. to 10 min. and further, I found that the frequency analysis still did not show any signal.
Experiments in which the splicing frequency and the overlap percentage were varied were conducted to determine whether these parameters could make the peak on the frequency analysis of the visible signals larger, and so possibly make the peak on the invisible signals visible. However, after trying many combinations of splicing frequencies and overlap percentages, no change in the peak size of the visible signals was observed, and the peaks after 9 minutes remained invisible. The overlap and splicing frequencies are found at the bottom middle of the stretch dialogue box. It is possible that the 8-bits of intensity resolution in the file may not be enough to capture the smallest signals (less than 1.25% of the peak noise).
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