Orthogonality

A potential application would be to use it with two analog multipliers whoxe outputs are summed to produce a SSB either USB or LSB signal.

You know the cos baseband modulation, and the other is the sin and 90 phase shifted baseband modulation summed together to produce SSB.

There are analog 'ladder' networks which will do 90 degree phase shift over a bandwidth range. More like stacked bridges and impossible to fathom staring at them, but LTspice gives you an answer quickly.

How about take Trigger Output and use it to toggle a flop? Output of flop selects alternately two different DC inputs to apply to second channel... forming your horizontal cursors. ...Jim Thompson

Just a semantic nit, but orthogonality isn't the property you're looking for -- you're thinking of harmonically related.

Any two sine waves that are not identical in frequency are, very strictly speaking, orthogonal. You seem to be looking to generate a frequency that doesn't match the fundamental frequency of your input signal, or any of its harmonics.

Sure, to some specified accuracy over some range of conditions. A 90 degree phase shifter is one example.

Cheers

Phil Hobbs

Or, just to be a smartass, a zero-ohm resistor to ground. ;)

Cheers

Phil Hobbs

There is such a thing in theory, the Hilbert Transform.

While one cannot implement this exactly in analog hardware, one can make a pretty good approximation. Look into the "phasing method" of generating single-sideband modulation. There is a vast literature.

The other place of possible relevance that the Hilbert Transform comes up is in the generation of "analytic signals" in the complex exponential domain.

Joe Gwinn

The usual way to deal with this is with a pseudorandom noise source, at lea st where optimizing human perceptual issues is the goal.

The problem of generating a signal that is guaranteed to avoid an objection able beatnote with another signal comes down to finding one whose LCM with regard to the first signal is outside the bandwidth of your system, and who se harmonics don't approach each other too closely either. "Orthogonal" ju st refers to vectors that are 90 degrees apart -- sin(x) and cos(x) phasors being the usual examples.

-- john, KE5FX

In some gas-laser systems, each pulse causes a portion of the Doppler-shifted gas particles (gain medium) to become de-excited, so that the SUBSEQUENT pulse is always at a different center frequency (as long as the pulses are timed close together).

And, I suppose it's possible to apply a multiply-accumulate block to detect correlation, and trigger a random-number offset to a VCO if any correlation exceeds its threshold. It's easy to make a 90-degree phase shift this way, which is orthogonal (but not anharmonic), using an XOR phase detector without randomness...

Correlation, though, in the time-series sense, is indeterminate until time has elapsed, so you ought not to expect a feedback scheme to be able to prevent it rather than get burned, then jump away...

Yeah, I should've said correlation. Although harmonics aren't correlated to the fundamental either.. same sort of thing...

Supposing there's an anti-PLL block, which outputs a constant frequency which is uncorrelated to the input (or any harmonics), assuming its behavior is stateful (like a PLL, tracking where it needs to be -- or rather, where it doesn't need to be), you could trap its output between harmonics, then sweep the input through something like a Shepard tone until its output gets stuck at the lowest or highest adjustable range.

It would then have to pop through to the next open space, or anywhere really. At that moment, if it moves in a continuous fashion (as would be typical of a PLL, but not necessarily of a DSP approach), it's not necessary that it become highly correlated over a sufficiently long period -- maybe it can pop through in 1ms, but your correlation period spec is 20ms or something, so you don't mind. In the oscilloscope example, this would be, maybe you see one "frame" worth of glitchiness, but... it's over before you notice. Or in the comms case, maybe you drop a packet, but it's TCP so it's okay. Or something.

Tim

Yeah, there's probably quite a number of trivial or semi-trivial cases where it does something a whole hell of a lot easier than the most general case.

Perhaps the next level up, closer to generality but still not really intended, would be a white noise source which is synchronously notch filtered by the input and its harmonics, in the opposite way that a PLL is a bandpass filter.

A possible specification to add might be, the input and output need to be narrowband signals. That helps express them as variable frequencies rather than arbitrary signals. But also raises the question of bandwidth versus operating frequency, and whether the input and output overlap, or what. It would be very easy to have an input at a frequency of 1-2 (harmonics at >3) (arbitrary units) and an output at 2.5 (fixed, variable or noisy, doesn't matter) which never correlate, at least if both input and output harmonics are not included. Including harmonics, you'd have to consider as many, and as many differences, as bandwidth allows.

Oh, another application: radio, either in synthesis (e.g., dealing with the spurs and aliases from a DDS), or mixing (whistlers at various frequencies and harmonics), or so on.

Tim

Isn't 90 degrees phase shift the definition of orthogonality?

for a sine wave input could do a "dvide-by-pi" PLL annd then mix in the original.

Spectral hole burning works better in solid state systems, because the collisional de-excitation doesn't broaden the natural line width so much. Some systems have as many as 2000 resolvable wavelengths, iirc, which allows recording and replaying a fairly complicated pulse shape.

(An acquaintance of mine, Kelvin Wagner at CU, has done some really interesting stuff with that.)

Cheers

Phil Hobbs

You can't do it with linear constant-coefficient ODEs except by way of phase shift, because of the exponential-in / exponential-out property.

An SSB mixer would be another simple example.

Cheers

Phil Hobbs

Nope. "Orthogonal" means that the overlap integral is zero. Signals at two different frequencies will always be orthogonal, for instance.

Cheers

Phil Hobbs

How about this. It does assume A/D, D/A, and a microprocessor of some kind.

1. Have a table of prime numbers handy.
2. Measure input frequency.
3. Add whatever fudge factor you think you need (10%?) to the measured frequency.
4. Output the next higher prime frequency.
5. ???

1. Profit!

Matt Roberds