Can passive phase shifters be implemented without a variable delay element? Revisited

Arghh!! Sorry people scratch the above... or at least the word homogeneous. I guess the distinction I'm looking for is transient and steady state solution. (or something like that.?) The key

Hmm OK. narrow and broad band seem to fit what I'm saying too.

George H.

I've tried the Hilbert Transform for filtering in signal processing, in an IRIG AM signal decoder, but Hilbert turned out to be complicated and too noise sensitive, so I used a simpler method, a synchronous rectifier followed by a moving-average comb filter tuned to cancel all even harmonics of the 1 KHz carrier. This worked quite well.

Joe Gwinn

That's what I thought. This example is getting better and better. Frustrated internal reflection in a close-spaced pair of prisms is also related, although I'd assume that there is a delay that varies with the separation.

It's now clear that while variable delay is a common way to implement phase shifters, it does not follow that this is the only way to get a phase shift. This was the original question.

Joe Gwinn

Yep. That answers the question, and fits my intuition.

By the way, I found a recent and good overall review of the work of Krishnaswamy and his students:

"Non-reciprocal electronics based on temporal modulation", Aravind Nagulu et al, Nature Electronics, Vol.3, May 2020, pages 241-250, . It may be paywalled.

Joe Gwinn

I meant the sign of the voltage waveform. Probably should have called it the amplitude. Anyway, see Phil Hobbs' answer.

Joe Gwinn

Hmm. By that token, a product mixer cannot work in the instant, and yet it does. Hilbert was doing something fancy with analytic signals in Complex Analysis. Some digging seems worthwhile.

Joe Gwinn

In real life, numerical Hilbert transforms are limited to fairly narrow bandwidths, unfortunately.

Cheers

Phil Hobbs

This is a subtle point, I agree. Ideal bilinear multipliers used as mixers produce USB and LSB signals instantaneously, as you say. That moves the focus to how long it takes to separate the two, which is far from instantaneous in general.

Switching mixers and logic-driven phase detectors have an intrinsic delay, because once an output transition has occurred, zero information appears there until the next transition. This distinction can be important in the design of wideband PLLs.

It would be wonderful if you could build wideband phasing SSB upconverters using narrowband phase shift networks in the LO and output ports, but you can't--one of the phase shifters has to be in the baseband port.

Cheers

Phil Hobbs

Yup. Wideband Hilbert transformers are horribly ill-conditioned, because the transfer function has an infinite spike at the origin, and its high-frequency tail also contains infinite energy.

They're okay in reasonably narrow fractional bandwidths.

I use the moving average thing to get rid of ripple in lock-in outputs as well--works great.

Cheers

Phil Hobbs

Yes. In the above case, it was a narrowband signal (1 KHz modulated with 100 bits per second) with quartz-crystal stability, but the numerical differentiation required for the Hilbert transform amplified the noise. IRIG receivers live in a very noisy world.

The max signal swing is something like 6 volts peak to peak, and the minimum is about 0.1 volt pp. The intent is to be able to carry this signal over miles and miles of RG-58 coax cable in desert missile test ranges, the original application circa WW2.

You can use a two-strand barbed wire fence to carry an IRIG-B signal. Until a cow wants to scratch its back.

The test is to see how much PP noise it takes to cause decoder errors.

Yes, same story basically. I got the idea from how DMMs reject power line interference by choosing the integration period to be an exact multiple of the period of the power waveform.

The way I implemented the synchronous rectifier is interesting as well. Originally I implemented a kind of simple PLL, which was a bit complex and fiddly. But it turned out that simply hard-limiting the unfiltered incoming signal and feeding the square wave into one input of a mixer and the original signal to the other input worked well, and was robust. The output went to the comb filter, which yielded the original AM envelope. This went to a synchronous integrate-and-dump filter feeding a threshold detector that turned the AM envelope into a bit stream.

With noisy input, the hard-limited signal looked a bit ratty, but it worked just fine. Those filters did their job well.

Joe Gwinn

Right. Thus, the resort to the filtering approach to SSB generation.

Do you know what problem Hilbert was solving? I don't.

Joe Gwinn

Well, that's FM capture effect for you.

Cheers

Phil Hobbs

From Wiki:

"The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u(t)."

"Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u

• iv admits an extension to the complex upper half-plane satisfying the Cauchy-Riemann equations."

"The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann-Hilbert problem for analytic functions."

A function is analytic if it has a derivative that's the same along any direction in the complex plane, which is where the Cauchy-Riemann equations come from. (Cue sound effect of third-year complex variable calculus being extracted from the archives.) ;)

Cheers

Phil Hobbs