For my Ph.D. thesis work I’ve been working with a lot of linear systems theory. Because of my mathematics background, I typically think of linear systems details from a functional/harmonic analysis perspective, with real numbers, mostly continuous functions, etc. In the engineering world, however, we use discrete versions of these mathematics.

### Motivation

The application which I’m applying linear systems to is a channel design for a channeled polarimeter. Abstractly we may think of channels in the typical communications sense, sets of $\delta$-functions in the Fourier domain which are convolved with the data (carrier channels/frequencies). The Fourier domain data around each channel can then be filtered out and deconvolved to obtain the spatio-temporal domain data for different channels.

In our polarimetric system design, we must temporally sample (via our focal plane array, i.e., a camera). We also are forced to use a window of samples, since we don’t have infinite time or computer memory. These windows must be integer periods of our carrier (or modulation) frequencies, otherwise we get *spectral leakage. *The seminal paper on spectral leakage and mitigating it is by Harris (1978). Basically when performing a discrete Fourier transform (DFT, including FFT), if the window is not an integer multiple of the period, then frequencies from the sharp cutoff become introduced into the DFT.

### The issues

The camera on our polarimeter has a maximum framerate of 30 frames per second (fps). The bandwidth optimal system design (found so far) for this particular type of system has sets of 9 temporal channels (there are also spatial channels, and the channels are actually three dimensional spatio-temporal channels, but for simplicity here we may ignore this complexity), including the DC or 0 channel, corresponding to the temporal modulation dimension. Due to Hermicity, this means there are really 4 unique temporal frequencies other than 0 for this design, and they are evenly spaced. The maximum frequency is limited by the physical rotation speed limitations of a retarder, and the sample rate of the camera (due to Nyquist).

**The above puts constraints on the windows, frequencies, and sampling rates that we can use without spectral leakage.** It is sometimes theoretically desirable (for noise reasons in this polarimetric instrument) to sample at slightly above Nyquist, however in our case spectral leakage does not allow this. Our **window must be an integer number of periods** of all 4 frequencies. Practically this means that I cannot increase the sampling rate to something like 32 fps, because of the *interplay between Nyquist, spectral leakage, and channel frequency spacing*.

Below is a graph of $$ f(t) = \cos 2\pi(3.75t) + \cos2\pi(7.5t) + \cos2\pi(11.25t) + \cos2\pi(15t),$$ the blue dots denote sampling at 30 fps (or once every $\frac{1}{30}s$), and the red squares denote sampling at 32 fps.

When we take the Fourier transform (I use the unitary version here) $$\mathcal{F}\{f(t)\}_{t \to \nu}$$ for the 30fps sampled case we obtain the graph in blue below. This is the correct representation, there is no imaginary part, and the discrete $\delta$-functions are located precisely at the frequencies expected from the corresponding real (non-discrete) Fourier transform results. The red graphs show the *spectral leakage* *issue,* the imaginary part becomes non-zero, and the $\delta$-functions sort of “split” into positive and negative parts. This creates problems when trying to reconstruct data after filtering. Note that we have the leftmost $\delta$-function at twice the magnitude of the others, and a missing $\delta$-function on the right side, this is due to the channel being right at Nyquist.

In the next post I will go over precisely what the mathematical relationship is between spectral leakage, Nyquist, and channel frequency, and the constraints imposed upon us as engineers when using channeled systems.

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