A titan of CSP, the figure looming over every word of this Blog, the driving force behind so many valuable mathematical tools, Professor William A. Gardner, has died.
Bill Gardner passed away on July 24, about two years after his beloved wife Nancy. In this post, I will chronicle some of my memories of Bill, with various nostalgic images and quotes, some of my interactions with Bill, and above all I will explain his influence on me, my career, and on the CSP Blog.
I believe there would be no CSP Blog without Bill Gardner.
For those readers not familiar with Gardner, he was my Master’s and Doctoral advisor from 1987 to 1992. He is the creator and expositor of modern cyclostationary-signal theory and cyclostationary signal processing. That is why, here on the CSP Blog, the reference The Literature [R1] has always been, and will always be, his seminal book on cyclic spectral analysis called Statistical Signal Processing: A Nonprobabilistic Theory.
I believe he came to regret, a bit, the subtitle of [R1] because the theory he created is not nonprobabilistic; it is nonstochastic. I intend to explain the distinction between probabilistic signal theories that are either stochastic or nonstochastic, which I’ve as yet only feinted at here and there in the dusty corners of the CSP Blog, at some point before I, too, must be consigned to oblivion.
Bill and I had a falling out in 1998 but we managed to reconcile a couple years ago and I tried to help him with some of his final writing projects. Though, even as he was failing, I still had a hard time keeping up with him.
Early Days
I graduated from UC Berkeley in 1986 with a BS in EE and no desire to put university life and studying behind me, so I applied to a couple graduate schools. Got into them, and chose UC Davis. I met Bill when I took a graduate class on spectral analysis. Eventually he asked if I was interested in working with him on my MS thesis, and I said yes. Why did I say yes? Because his math was beautiful and challenging. It really was that simple and I was really that innocent.
And I was challenged. I floundered around, trying to read some material he gave me on cycle detectors. Got confused about what was the detector, what was the input, and what was the output. He told me to go back and read again. So I did.
It all worked out pretty well for my MS on cycle detection (My Papers [12]). I used a VAX/VMS mainframe (we had one called verbena and another called phlox) to program, in Pascal, Monte Carlo simulations involving PSK signals, PSK interferers, and additive white Gaussian noise.
The Doctoral Grind
After the cycle detector projects, I started helping out with bearing estimation through time-difference-of-arrival (TDOA) estimation. This morphed into a study of the two-sensor low-SNR likelihood function. When expanded as a series and approximated due to the low SNR, you can find a bunch of different quadratic processors that can simultaneously detect a signal and provide a TDOA estimate. Did a lot of work on that topic (My Papers [4]) with Bill.
But I didn’t really like TDOA estimation that much. And I didn’t like the idea of writing a doctoral dissertation on TDOA estimation. I was hoping for something newer, something more exciting, something that seemed more like significant new knowledge. Because that’s how I think of the PhD: Development of significant, novel, and correct new knowledge. Something non-trivial and true that no one else has ever created or discovered.
That led, of course, to the theory of higher-order cyclostationarity, and in particular the cumulant theory of cyclostationary time series (My Papers [5,6]), pure and impure sine waves, the spectral moments, and the cyclic polyspectra. Instead of the modern practice of chunking the dissertation by writing and publishing a paper, then calling it a dissertation chapter, and repeating that three times, I slaved over the full draft, withholding it from Bill until I thought it was good.
I was elated to see that remark in the upper right corner of the title page!
But of course it wasn’t all that good, really, and there were a lot of editorial suggestions and corrections and typo identifications in subsequent pages. I’ll talk more about his influence on my writing below.
I made it through, though, and we were both happy with what we had wrought.
Writing Well and Good Writing
I didn’t get or seek out much instruction on writing as a high-school student (I was in general checked out except for math classes). I read a lot growing up, and reading a lot makes you a better writer. But you still have to practice writing to actually get good at it.
I think what happens for most people is they think they can write, and most of the time that belief goes unchallenged, but at some point they get feedback from somebody who does not go gentle with them. This is what is needed. The horror of seeing your pages covered in red ink or pencil markings is what is needed. The … horror. Understanding and skill come only with practice and struggle. There is no large language model route around that truth. You cannot get good at basketball or writing by asking someone–or something–else to do it for you while you sit and watch.
Bill was going to edit the **** out of anything you gave him. And it helped. A lot. The incipient writer must get over themselves, get over their pride, grapple with the feedback, and try again. He taught me that, dear reader, and we all are the beneficiaries.
Here is a page from the first paper I tried to write. It was for MILCOM 1988.
Over those Master’s and Doctoral years, we wrote many things together, and I grappled with his feedback many times. I got better! And Bill Gardner did not relent, so I kept getting better.
In this situation, small positive editorial notes are precious gems that you cling to in the sea of corrections and challenges.
He gave those too, and I try to do the same with my students (and CSP Blog commenters). Small compliments are quite meaningful when they are rare. The trick is to ensure that the number does not go to zero, else you risk alienation.
Near the end of my doctoral work, Bill was working on a paper with other students and asked me to give a read and make comments. I did, and something clicked that day. In addition to the usual grammar, spelling, and usage edits, I saw the whole paper as one entity and didn’t like what I saw. It was the first time I wanted to reorganize someone else’s writing. I made my suggestions and got not a small compliment, but a big one:
(Chad: You’ve outdone me. I’ll have to start taking writing lessons from you. Thanks for your very careful reading of my papers. Your comments are much appreciated.)
That was a good day.
And here we are, me and you my dear reader: I’m still writing and you’re still reading what I put down. Your attention is much appreciated. We both have benefited from Bill’s teaching of scholarly writing and his dedication to his own high standards.
Introduction to Fine Dining
But it wasn’t all writing, reading, and arithmetic! As I progressed through graduate school, I went to conferences with my groupmates and Bill, we traveled to Silicon Valley to work with various tech companies, and we had get-togethers in Bill’s neighborhood of Napa/Yountville in California. It was during these excursions that I was introduced to Italian fine dining. I didn’t know much about fine dining in those days, but I could definitely cook, having been a line cook and pizza cook in my teens and twenties.
My introduction was Piatti. I discovered carpaccio. And eating crusty bread with olive oil instead of butter. And many other things. I eventually went to better restaurants, for sure, but it was all very exciting and new at the time, and I look back on this introduction fondly. Grazie mille, Professor.
Spreading the Gospel of CSP
In 1992 Bill organized the Workshop on Cyclostationary Signals. It was exciting to have our own conference! It was held in Yountville, CA, and out of it came what is probably Bill’s most-purchased book: Cyclostationarity in Communications and Signal Processing.
I don’t have much left from that workshop except a paper copy of some slides I must have delivered.
A couple years later, Steve Schell and I organized the Second Workshop on Cyclostationary Signals, which was held in Monterey, CA. The turnout was good, and we enjoyed ourselves. What came out of it was a proceedings volume. I did the cover art for that. You can see that it is a very early version of my cyclic-cumulant plots, like the ones in the banner of the CSP Blog.
There never was a third Workshop on Cyclostationary Signals. But at least we have the CSP Blog and cyclostationarity.com to sustain us here in the twenty-first century.
Final Thoughts
What makes a man’s life a rare fossil? One hundred years from now, if someone stumbles on this post, on Gardner’s papers, or on old CS books, what makes that a great find? What distinguishes an ordinary fossil from a rare and precious fossil?
I think one ingredient is novelty. Doing something new, being something new, or thinking something new is a critical factor. Otherwise, the fossil looks like all the other fossils.
But novelty is not enough. Significance is another key ingredient. Doing something new and important. The importance doesn’t have to be contemporaneously recognized. A man can be ahead of his time, as they say, and that will show up in the fossil record.
Add in a flaw, a mar, a small imperfection and you have something interesting, rare, and quite possibly valuable. Perhaps Bill’s flaw–his imperfection–was perfectionism itself.
But, taken all for all, he blazed a trail I’m still following, he held himself to high standards, he was brilliant, and he did it his way. So I think the future will find him a rare fossil indeed.
Requiescat In Pace, Professor Gardner. We will remember you.
