PhD Thesis Research
My thesis research is in the field of underwater acoustics and signal processing. In particular, this research explores the possibility using a measured acoustic field with some bandwidth to create new fields that have much lower or much higher frequency content. This is made possible by mathematical constructions that I've developed in my research termed autoproducts. Autoproducts are quadratic products of frequencydomain acoustic fields at two different frequencies. In my research, I've explored some the theory behind these autoproduct fields, as well as some of their applications.
First, we'll discuss what autoproducts are, then we'll discuss how they can be used for source localization.
First, we'll discuss what autoproducts are, then we'll discuss how they can be used for source localization.
Autoproducts
We define the frequencydifference autoproduct and frequencysum autoproduct as:
\[ AP_{\Delta}\left(\textbf{r},\Delta\omega,\omega\right)\equiv P\left(\textbf{r},\omega+\frac{1}{2}\Delta\omega\right)P^*\left(\textbf{r},\omega\frac{1}{2}\Delta\omega\right)\sim P\left(\textbf{r},\Delta\omega\right)\]
\[ AP_{\Sigma}\left(\textbf{r},\Delta\omega,\omega\right)\equiv P\left(\textbf{r},\omega+\frac{1}{2}\Delta\omega\right)P\left(\textbf{r},\omega\frac{1}{2}\Delta\omega\right)\sim P\left(\textbf{r},2\omega\right)\]
To see why the definitions in the first equation might imply the approximation in the second equation, consider a planewave acoustic field.
\[ AP_{\Delta}\left(\textbf{r},\Delta\omega,\omega\right)\equiv P\left(\textbf{r},\omega+\frac{1}{2}\Delta\omega\right)P^*\left(\textbf{r},\omega\frac{1}{2}\Delta\omega\right)\sim P\left(\textbf{r},\Delta\omega\right)\]
\[ AP_{\Sigma}\left(\textbf{r},\Delta\omega,\omega\right)\equiv P\left(\textbf{r},\omega+\frac{1}{2}\Delta\omega\right)P\left(\textbf{r},\omega\frac{1}{2}\Delta\omega\right)\sim P\left(\textbf{r},2\omega\right)\]
To see why the definitions in the first equation might imply the approximation in the second equation, consider a planewave acoustic field.
Plugging these plane waves into the autoproduct definitions, we get:
It's easy to see that, for a plane wave, autoproduct fields are exactly the same as their outofband field counterparts at frequencies above and below the original bandwidth. Note that this procedure appears qualitatively similar to adding fields instead of multiplying fields. Added fields will have a 'beat' patterns at the difference frequency, but this is fundamentally different from the frequencydifference autoproduct: the frequencydifference autoproduct is a genuine field (with real and imaginary components), not just a description of the envelope of the field.
Source Localization
Remote Sensing
Suppose you are out in the ocean, and you have an array of underwater microphones, called hydrophones, and you are simply sitting there in the ocean, quietly listening. Usually you'll just record ambient noise, but suppose you hear something (detection), and determine that whatever that something is, it's of interest to you (classification). Then the next step is to find the origin of that sound (localization), and if it's moving, figure out where it's heading (tracking). These four steps: Detection, Classification, Localization, and Tracking, are commonly grouped together into "remote sensing", with an optional fifth step being Identification (determining not just that it's something of interest, but determining what, or who, specifically, made that sound). In my PhD research, I focus primarily on the localization step, though there are other applications in detection and tracking as well
Matched Field Processing
To perform source localization, there are many methods, but the one we'll focus on is called matched field processing, or MFP for short, which was developed in 1976 by Homer Bucker. The objective of MFP is to use pressurevstime signals from each hydrophone as the input, and then output a spatial map of possible source locations. The way MFP works is to compare measured data (recorded from the real environment) with modeled data (simulated from a computational model of the known environment). Basically, a grid of possible source locations is formed, and comparisons are made between the real, measured data and the simulated, modeled data for a hypothetical source at each grid location. A map is formed showing how strong these comparisons (or more precisely, the spatial crosscorrelations) between measured and modeled data are for each location. This map is called an ambiguity surface, and can be thought of as a plot of most (or least) likely source locations. When functioning properly, the modeled data from the true source location will create the largest crosscorrelation with the measured data, resulting in a peak at the true source location in the ambiguity surface. Through this process, source localization is performed.
Environmental Mismatch
Fast forward to the early 90's. Research greatly expanded the capabilities and flexibility of the MFP technique, though it became clear that there was one problem that seemed insurmountable: environmental mismatch. The issue is that the 'known' acoustic environment (which is necessary for the computational model to create the modeled data) is always slightly different than the true acoustic environment. In other words, unless we knew exactly where all the surface waves, bubbles, sea mounts, temperature gradients and salinity gradients were in the ocean when the sound was broadcast, then MFP will inevitably have errors. The naive expectation is that the worse the match, the poorer the source localization. And while that's not wrong, instead of it being a smooth linear relationship between environmental mismatch and localization error, it's actually a rather sharp and abrupt transition from fully functioning to catastrophically failing to localize. To demonstrate this, consider the following setup:
Example MFPConsider a 100 meter by 100 meter region of a free space region (no boundaries, no temperature gradients) of water with sound speed of 1,500 m/s. Suppose you have a hydrophone array, positioned vertically along the left edge, which is very dense (so we can avoid other problems, like aliasing and array sparseness). A source at an unknown location (let's suppose 60m to the right of the array and 25m above the center of the array) broadcasts sound. This sound has frequency content centered at either 100 Hz, 1,000 Hz, or 10,000 Hz. To imitate unknown environmental variations, each sourcetohydrophone ray path gains a random time delay. This time delay is Gaussian random, centered at 0, and has a standard deviation of 0.25 milliseconds (corresponding to about 40 cm of path length variation). The resulting ambiguity surfaces, as a function of source frequency, are shown below on a logarithmic scale, where a perfect match is shown in dark red, and a poor match (50dB below a perfect match) is shown in dark blue. The white circle in these plots indicates the true source location.

Without Environmental UncertaintiesWhen the environment is known precisely, MFP performs perfectly. At 100 Hz, the localization spot size is fairly large, but at 10,000 Hz, the localization spot size is very small, indicating a very precise localization. The spot size scales with the wavelength.

With Environmental UncertaintiesWhen the environment is imperfectly known (represented by random time delays) now the ability to localize has a frequency dependence. At 100 Hz, the plot is nearly unaffected. At 1,000 Hz, the plot is just barely able localize still. At 10,000 Hz, MFP has completely failed to localize the source.

AutoproductBased MFP
If the signal has some bandwidth (as any finite time duration signal will), then the autoproduct definitions can be applied. Each of these plots includes environmental mismatch.
Frequency Difference MFP
High frequency signals can be 'downshifted' to much lower frequencies via the frequencydifference autoproduct, and then processed via conventional MFP at these much lower frequencies. Precision in the localization is sacrificed for robustness to environmental mismatch. FrequencySum MFP
With this technique, the effective frequency is shifted up by a factor of two. In this case, robustness to environmental mismatch is sacrificed in exchange for a more precise localization result. 

In typical shallow ocean environments (shallow meaning depths of around 100 m), the acoustic environment is usually only known wellenough to support localization via MFP for source frequencies up to 1,000 Hz. Typically, anything above 1,000 Hz is effectively impossible to localize via MFP. MFP techniques based on the FrequencyDifference Autoproduct present an opportunity to overcome the environmental mismatch problem.
To see this technique applied to real ocean data, or to learn more about the theoretical limitations of autoproducts, data check out my publications page.
To see this technique applied to real ocean data, or to learn more about the theoretical limitations of autoproducts, data check out my publications page.
For more information, give this a read: 

Last updated 3/27/2019