Hi Bob,

Glad that was of use to you, as for your questions…

Are these the windows that are called Hamming, Hanning, etc?

Yes, those are some of the many different types of windows used to control the data sample set.

I am also familiar with Euler’s equation that provides the relationship to derive how the time domain can be converted into an infinite series of sine waves.

Euler was THE most prolific Mathematician to ever live. Here I am referring more to the work he did to discover and reveal (through series expansions of trig functions) how a complex exponential could represent a complex number on a Argand plane. I think you are referring to how any waveform can be expressed as a summation of basis orthoganal functions which is a Fourier series expansion (e.g. via an analysis function such as the Fourier Transform).

I believe you are saying that you would use overlapping bins in the freq domain to create an average over say 10 seconds (or 10 minutes, as the case may be).

I am saying that to compute the FFT to measure frequency (and avoid spectral leakage), take each group of consecutive N samples, and multiply them with the chosen N-length window then take the FFT of the result. Take the magnitude sqrt(real^2 + imag^2) of each output and plot it. You can also plot the squared mag to display the power spectrum if you want that.

Now, for the average… If your signal is stable in frequency, then you could benefit from a longer time average (say to reduce noise variations). So, you can add up M complex FFT outputs (bin to bin) and divide sum for each bin by M. Then plot the magnitude and you will see the noise variations reduce, revealing the signal. There are other signal processing techniques that can improve your situation beyond a simple average (e.g. coherent integration, correlation, cyclostationary searches, non-parametric Eigen decomposition of the correlation matirx of the input etc.) These can get quite sophisticated as you might imagine.

I am surprised to hear that you use overlapping windows. I presume that using non-overlapping windows you would get a step function whereas a moving average (across more than one bin) smooths the summations over time.

The method I’m describing is called the Welch periodogram - which is a power spectrum estimation. A 50% overlapped window is often used to multiply the input signal. Because the signal data is being reduced on the end of the windows, you back up by 50% (N/2) and window again and FFT. This attempt to keep the average power unaffected by the characteristics of the window.

Here is my confusion - the sharp boundaries (a kind of quantization artifact) occurs in the frequency domain and will likewise occur in the time domain. To visualize this I would need a 3D plot with the x axis measuring doppler speed (ie f), the y axis would be the summed or averaged magnitudes over the time interval and the z-axis would be position as my drift scan crosses the galactic plane. This would be for each declination I choose and the time interval as the galactic plane rotates into the antenna aperture, this is effectively the RA.

This would also tell me that if I do not obtain a clear galactic rotation curve, if I continue past 24 hours, I would see changes in sidereal time, consistent with signals of galactic origin.

The quantization artifact also occurs in the time domain as one would find jumps in the power from one time bin to the next.

Well, I apologize that I’m not following your description of the signal you are processing and I’m not familiar with the kind of science you are performing. I’m gathering that you have a senor of some kind and that you are looking to see things in the frequency domain. Does the data originate in the time or freq domain? Do you use the FFT to get the freq domain info?

By the way, you say there is a quantization artifact that occurs in the freq domain. I must point out that time sampled data has no concept of frequency quantization, only amplitude. I can represent an infinite number of frequencies as I like - practically speaking, up to the floating point precision of the computations used to create my phase change increments - all within the Nyquist range of course (Fs/2 if Real, or Fs if complex - since there are two sample streams per calculation).

Well, I’m again not sure I’ve been much help.

Cheers,

John