What has led you from your undergraduate time here at MIT to your graduate research in RLE with the Digital Signal Processing Group?

During my freshman year I considered both physics and electrical engineering as possible majors of study. I thoroughly enjoyed the former in high-school and an acquaintance spoke highly of the latter. Introductory courses in both fields proved interesting and challenging.

The instructors in physics were particularly logical in their solutions of problems that I had struggled to understand, and I aspired to gain this degree of mastery over sensible, coherent thought processes. Physics was displayed as a wonderful expression of our world, made wholly comprehensible through sequential reasoning. Instructors in electrical engineering engaged my attention through the immediate application of new concepts to practical, hands-on projects. The seemingly effortless transition they made between classroom and workshop was a skill I respected and wished to develop in myself. 

Electrical engineering and physics were equally absorbing subjects of study to me. Both required disciplined reasoning and intuition, and so I eventually chose to major in both. My favorite classes announced themselves to me immediately.  These were the signal processing courses – Signals and Systems (6.003), Signals, Systems, and Inference (6.011), and Discrete-Time Signal Processing (6.341) – none of which I found effortless. Although I was adept at manipulating the mathematics of the courses, it was only after having completed all three that I finally gained the intuitive understanding I sought. I could now apply the concepts to ideas beyond homework exercises. That there is pleasure to be found in learning about sampling theory and filter banks is still a wonder to me.

It was very fortunate timing that in my senior year, Professors Al Oppenheim and Randy Davis sought an undergraduate student to assist in a signal processing project. It concerned the detection and measurement of essential tremor through the use of a digitizing pen. The further objective was to discriminate between essential and Parkinsonian tremor, thereby contributing to a more reliable diagnosis of Parkinson’s disease. I was asked to help in the design of the bandpass filter and the digital differentiator necessary for the detection of essential tremor. It was gratifying to now apply my coursework to a practical problem, and our work in collaboration with others was eventually patented. These months led to a continued mentorship by Professor Oppenheim, as I completed a Masters of Engineering thesis in the following year.

What problem are you trying to solve with your current research and what are some possible applications?

My current research draws on my background in both physics and signal processing. Professor Oppenheim encourages his students to think creatively, to find “solutions in search of problems” rather than “problems in search of solutions.” I have found that in an intellectual sense, I thrive in this environment which dismisses rigid boundaries. The necessary regimentation of undergraduate homework assignments has now been replaced by the freedom to contemplate projects that excite me.

This attitude toward research led me to study a topic for my masters thesis – classical receiver operating characteristic (ROC) curves – that is decades old and initially appeared to offer no new areas of exploration. An ROC curve displays the tradeoff between the probabilities of detection and false alarm (also sometimes called the true positive and false positive rates, or the sensitivity and specificity) of a classical binary hypothesis testing system. It was initially developed for use in radar systems during World War II. These curves have since seen expanded use in a broad range of applications including signal detection in noisy environments, screening and diagnostic algorithms in medical tests, and damage detection in airplanes and ships. 

We discovered that while the theoretical results surrounding optimal ROC curves are well-established and ‑understood, it is often the case that in practice, the ROC curves themselves are generated experimentally in a non-optimal way. We derived a condition under which an experimental ROC curve is optimal, even if generated in a non-optimal manner. We also developed a constructive procedure for the generation of such an optimal ROC curve from a non-optimal one, with minimal knowledge of the underlying data. These results were presented at and published in the proceedings of a major international signal processing conference.

I am now immersed in my doctoral studies and over the past year and a half, the direction of my research has steadily emerged: signal processing in the context of quantum mechanics. It began when we became intrigued by the problem of binary quantum state detection, which is closely related to the topic of my masters thesis, but in a quantum setting. I am currently exploring how this work expands into other topics such as quantum state estimation, quantum sensing, entanglement concentration, quantum control theory, classical frame theory, and generalized uncertainty principles.

Our most recent publication relating to this research details results regarding the operating characteristics of quantum binary hypothesis testing systems. Two types of operating characteristics are discussed. The first is the direct analogue of a classical ROC curve, while the second does not have a classical analogue that is commonly used. We further derive a generalization of previous results concerning the correspondence between positive operator valued measures (POVMs), and classical Parseval frames.  POVMs are used in quantum mechanics to model the measurement of a quantum particle.

What are your future plans?

After graduation, I plan to continue my research in the signal processing field, ideally in the same type of creative and collaborative environment that now supports me. It has been said by several mentors that the research culture of the old Bell Laboratories was exactly this. Thus I will use it as a touchstone for any future employment. An area of future research that I am currently eager to explore would include conducting experiments involving quantum particles, as this would greatly improve my practical understanding of the problems I am currently studying. But as I have well-learned from Professor Oppenheim, any beginning starts with the erasure of boundaries around research possibilities.