# FTS Noise Propagation

Posted 2021-03-15

Noise propagates differently through Fourier transform spectrographs (FTS) than traditional dispersive spectrographs. This can be seen as a result of an FTS piling up the light from a whole band into each exposure. This has benefits in terms of read noise, but disadvantages (or at least trade offs) in terms of photon noise.

Normally in dispersive spectrographs, the photon noise is independent of the global spectral content -- the noise in each channel equals the square root of the total counts in that channel. This means that bright spectral regions have quite a lot of photon noise, but that dark regions have very little.

In contrast, the propagation of photon noise through an FTS is more complicated. To first order, an FTS sees the sum of all photons across the band in each exposure. After the FFT, we can then approximate the noise in each channel as the square root of the mean number of counts across all channels. Now, the photon noise is not dependent on the local number of counts in that channel, but on the global number of counts across the band.

This actually has a benefit for bright emission lines -- if the rest of the band is dark, they will be detected with much less photon noise than in a dispersive spectrograph. Absorption features, on the other hand, suffer. They get effectively "filled in" by light from elsewhere in the band.

The details of how photon noise propagates through an FTS is more subtle. An FFT is equivalent to a least squares solution, breaking down a signal into a linear combination of sinusoids. This means there is a covariance matrix that depends on the spectral content of the signal.

We can investigate the noise propagation using some quick Monte Carlo simulations. Here is a made up spectrum that varies smoothly and has a big absorption feature in the middle and a bright emission feature just to the right.

For the first case, let's compare an idealized FTS and dispersive spectrograph. We'll have a zero read-noise detector, and perfect modulation efficiency for the FTS. Here is the output spectra they would record with $1\sigma$ confidence intervals:

The photon noise is overall the same (if they see the same number of photons in total, of course the total uncertainty on that count is the same). The FTS has more noise in the absorption feature, whereas the dispersive spectrograph has more noise in the emission feature.

Here is what the noise plotted by itself looks like. As you can see, the noise in the dispersive case always follows the square root of the counts in that channel, but the FTS noise is roughly uniform across the band. There are a few regions where it is less that anticipated, due to the aforementioned covariance between the Fourier coefficients.

Now, let's see what happens when we add in read noise. We'll add just $5e^-$:

In this case, the FTS pulls ahead in many locations since during each exposure, the read noise is smaller compared to the measured signal of $\approx 1/2$ the whole band. There is still a small penalty at the absorption feature, however.:

If we now also add a somewhat pessimistic modulation efficiency to the FTS of 65%, we see that the noise grows across the band:

Of course, in the real world there are many more factors that these simulations could include, but this should give some intuition at least for how noise propagation is different for an FTS.

The Julia code used to generate these plots is available here.

-WT