What's the ideal FFT window for measuring a group of signals of differing amplitudes but close in frequency?

Question

In my system, I need to measure the amplitudes and phases of four lasers, each of a different, known wavelength, chopped at a 50% duty cycle via TTL at frequencies spaced evenly (or not) around some fundamental frequency f0. The lasers are attenuated by some analyte medium, and the resulting laser power (which is measured by a single detector) tells me what I need to know about the medium.

The signals are recovered by performing I/Q demodulation. The application is similar to the one in this question. The signals' individual frequencies can be controlled by me, but should not be more than 6-7 Hz away from f0.

In single-signal applications of my setup, I've used the Exact Blackmann window, which is known to be good for single-frequency measurements, at least according to National Instruments. My question is, is there a more robust choice for recovering four extremely close signals?

Other details: sample rate is 8192 or 16384 Hz for two seconds, f0 is in the neighborhood of 1230 Hz. Typically I have the frequencies evenly spaced, at f0±6 and f0±2, so they're all 4 Hz apart. The input signal is demodulated and windowed four times, once at each of the frequencies of the lasers. Followup question: could spacing them unevenly help?

Answer 1

As for ideal windows for FFT frequency resolution and dynamic range, many assume that the Gaussian window would be best since it has the minimum time-bandwidth product, and due to this it is the filter most often used in spectrum analyzers. However, this is only true when the time domain extends to infinity. For finite time durations the window that actually has the minimum time-bandwidth product is the Discrete Prolate Spheroidal Sequence (DPSS, or Slepian) window which maximizes the energy concentration of the main lobe. The Kaiser window very closely approaches this optimum and both have an controlling parameter to trade frequency resolution (ability to discern two closely spaced frequencies) and dynamic range (ability to discern two frequency tones of different amplitudes). For further details on the comparison of the two, see this link by Julius Smith: https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html

That said, the signal after IQ demodulation could be windowed and then FFT'd to provide the phase and amplitude (relative to the demodulation signal used) of closely spaced modulation frequencies. For this application I would use a Kaiser window with Beta = 8 which provides 80 dB dynamic range of signals captured over a 2 second duration while being able to resolve two signals that are 4 Hz apart.

Answer 2

FFTs seems not the best way of doing it since you need very long time sequences to get enough frequency resolution and you have to deal with significant spectral leakage. Windowing will reduce spectral leakage put also reduce frequency resolution so it's a game of whack-a-mole.

If you only have a few known frequencies a heterodyne detector should work very well:

1. Build a local oscillator at the desired frequency
2. Multiply your input with the local oscillator (complex or cos & sin)
3. Lowpass filter the output

That will give a complex signal that represents the spectral amplitude over time. The bandwidth of the low pass filter determines the bandwidth of your spectral analysis.

If you don't know the frequency exactly or it can drift, you can wrap a PLL (Phase Locked Loop) around the local oscillator to track your spectral component.

Answer 3

If you are using 4 different lasers, are you free to select them at slightly different colors ($\lambda$)? If you could do this then you can make this a WDM type system. The receivers would have bandpass filters for each color, so you wouldn't be required to do any separation at frequencies near 1230Hz. The color spacing is something like 12.5GHz apart. Otherwise I would look at spread spectrum codes. It is for when the inputs are at exactly the same frequency but you send orthogonal signals so that they can be separated.