Two Domains - John Dunn, Consultant, Ambertec, P.E., P.C.

Bessel functions are solutions of Bessel's differential equation x² d²y/dx² + x dy/dx + (x² - alfa²) y = 0 and they also happen to be descriptive of the sidebands of frequency modulated (FM) radio signals.

If you take a spectrum analyzer and set it up to display an FM signal, you can get some presentations that visually resemble a southwestern desert mesa as seen in silhouette with peaking near the two edges and sort of a soup bowl depression in between.

From Agilent Application Note 150-1:   http://cp.literature.agilent.com/litweb/pdf/5954-9130.pdf

 

I was asked in an interview one day why the spectrum analyzer trace look can look like this.

In the frequency domain explanation, the spectrum analyzer's display can be accounted for in terms of the analyzer's response to the FM signal's sidebands as described by their Bessel functions.

In the time domain explanation however, the answer is that the FM signal dwells near its frequency deviation extremes for longer amounts of time than it spends when sweeping relatively quickly through the center of those deviation extremes and that the spectrum analyzer's amplitude response time has longer in which to rise at those extremes than it does in the middle.

My interviewer got quite angry with me for the time domain description.

The ONLY explanation he would allow or accept was in the frequency domain and I had just better drop the time domain perspective. Now!!

At another juncture, I had supervisor whom I thought of as the Houdini of triacs and I do mean to stress that this man was really good with those things. Using time domain reasoning, he managed to diagnose some really difficult triac control problems in one particular project and then go on to devise some really effective remedial design changes.

However, working under this fellow's auspices on a different project, I had an RF assembly whose design theory needed to be documented. I began that process in terms of the frequency domain and got brought to a screeching halt.

My supervisor got quite angry with me for the frequency domain description.

The ONLY explanation he would allow or accept was in the time domain and I had just better drop the frequency domain perspective. Now!!

It had turned out that these two fellows had diametrically opposing, absolutely unalterable, utterly adamant, immovable object, not-to-be-ever-questioned points of view as to which domain was the only domain that was correct and proper for analytic use in any and every engineering circumstance.

I found their respective attitudes to be utterly astonishing.

Thank goodness they didn't work in the same company.

 

 

 

Comments

[George Storm]

Is it possible that the interviewer had a point (albeit not expressed): the time domain explanation you gave was good (and an accurate indication of the trend) for large deviations; but the picture becomes complex (perhaps too complex?) if the time-domain explanation is to succeed when the deviation is much less than the modulation frequency.
Of course, the duality of time and frequency is unalterable, so you should be able to use either approach; but it's horses for courses, as sometimes the choice of viewpoint can dramatically simplify the full explanation (but perhaps in some cases only for those rare souls for whom Bessel functions are straightforward).

[John Dunn]

It was these two fellows' complete outrage that I wasn't expecting. How **dare** I even suggest an explanation that didn't comply with their respective viewpoints!

[Jonny Cash]

Time and Frequency Domain!

[John Dunn]

Time was time and frequency was frequency and ne'er the twain did meet!