# Orthogonality

Started by September 28, 2014
```Is there such a thing as a circuit (or more generally, system of
differential equations) where the output frequency is always orthogonal
(anharmonic) to the input?

Example response would be, you have an arbitrary input frequency, and you
want to pick an output frequency that *doesn't* line up with the input.
If you use a fixed oscillator, it'll work for some inputs, but there will
be special frequencies where they start to line up.  Suppose that, as the
input frequency varies, the output gets pushed away from input harmonics,
so that they never line up.  A phase-UNlocked loop, if you will!

The behavior in my mind is a slowly varying oscillator, so if you looked
at the output on the oscilloscope, it would always be a reasonably clean
wave (sine or square), but as you sweep the input, it drifts and hops in
just such a way as to avoid the input.  Though I think cycle-to-cycle
consistency is not a requirement, in which case a possibly simpler
solution could be random noise with the input filtered out of it.

An example use would be, suppose you have a two channel analog
oscilloscope and you want to add horizontal cursors to it.  Easy solution,
just put a square wave on the other channel, and as long as it's not
triggered, it'll blur into two horizontal lines most of the time.  Vary
the "high" and "low" voltages and there's your cursors.  Trouble is, when
the input is one of those lucky frequencies where the trigger lines up,
the illusion breaks down.  This also applies to CHOP mode in the scope
itself.  The actual limitation is evidently more sloppy: if they line up
within persistence of vision (~20Hz BW), you get a flickering, rolling or
locked display.

I have no application in mind, just a Sunday afternoon musing.  It seems
rather useless, but would probably find application in FDMA or something
like that where orthogonality is handy.  Actually, such a function
probably has even deeper theoretical applications from communications to
encryption (e.g., use a discretized version to produce a message digest
which is not just a scrambling of the input, but deterministically
orthogonal in all dimensions) and number theory.  Which means it might fit
into one of those "computable but infinite" or "unknowable" categories...
hmm...

Tim

--
Seven Transistor Labs
Electrical Engineering Consultation
Website: http://seventransistorlabs.com

```
```On Sun, 28 Sep 2014 12:17:17 -0700, Tim Williams
<tiwill@seventransistorlabs.com> wrote:

> ...snip...
>
> I have no application in mind, just a Sunday afternoon musing.  It seems
> rather useless, but would probably find application in FDMA or something
> like that where orthogonality is handy.  Actually, such a function
> probably has even deeper theoretical applications from communications to
> encryption (e.g., use a discretized version to produce a message digest
> which is not just a scrambling of the input, but deterministically
> orthogonal in all dimensions) and number theory.  Which means it might
> fit
> into one of those "computable but infinite" or "unknowable" categories...
> hmm...
>
> Tim
>

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.
```
```On Sun, 28 Sep 2014 14:17:17 -0500, "Tim Williams"
<tiwill@seventransistorlabs.com> wrote:

>Is there such a thing as a circuit (or more generally, system of
>differential equations) where the output frequency is always orthogonal
>(anharmonic) to the input?
>
>Example response would be, you have an arbitrary input frequency, and you
>want to pick an output frequency that *doesn't* line up with the input.
>If you use a fixed oscillator, it'll work for some inputs, but there will
>be special frequencies where they start to line up.  Suppose that, as the
>input frequency varies, the output gets pushed away from input harmonics,
>so that they never line up.  A phase-UNlocked loop, if you will!
>
>The behavior in my mind is a slowly varying oscillator, so if you looked
>at the output on the oscilloscope, it would always be a reasonably clean
>wave (sine or square), but as you sweep the input, it drifts and hops in
>just such a way as to avoid the input.  Though I think cycle-to-cycle
>consistency is not a requirement, in which case a possibly simpler
>solution could be random noise with the input filtered out of it.
>
>An example use would be, suppose you have a two channel analog
>oscilloscope and you want to add horizontal cursors to it.  Easy solution,
>just put a square wave on the other channel, and as long as it's not
>triggered, it'll blur into two horizontal lines most of the time.  Vary
>the "high" and "low" voltages and there's your cursors.  Trouble is, when
>the input is one of those lucky frequencies where the trigger lines up,
>the illusion breaks down.  This also applies to CHOP mode in the scope
>itself.  The actual limitation is evidently more sloppy: if they line up
>within persistence of vision (~20Hz BW), you get a flickering, rolling or
>locked display.
>
>I have no application in mind, just a Sunday afternoon musing.  It seems
>rather useless, but would probably find application in FDMA or something
>like that where orthogonality is handy.  Actually, such a function
>probably has even deeper theoretical applications from communications to
>encryption (e.g., use a discretized version to produce a message digest
>which is not just a scrambling of the input, but deterministically
>orthogonal in all dimensions) and number theory.  Which means it might fit
>into one of those "computable but infinite" or "unknowable" categories...
>hmm...
>
>Tim

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
--
| James E.Thompson                                 |    mens     |
| Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    |
| San Tan Valley, AZ 85142     Skype: skypeanalog  |             |
| Voice:(480)460-2350  Fax: Available upon request |  Brass Rat  |
| E-mail Icon at http://www.analog-innovations.com |    1962     |

I love to cook with wine.     Sometimes I even put it in the food.
```
```On Sun, 28 Sep 2014 14:17:17 -0500, Tim Williams wrote:

> Is there such a thing as a circuit (or more generally, system of
> differential equations) where the output frequency is always orthogonal
> (anharmonic) to the input?
>
> Example response would be, you have an arbitrary input frequency, and
> you want to pick an output frequency that *doesn't* line up with the
> input. If you use a fixed oscillator, it'll work for some inputs, but
> there will be special frequencies where they start to line up.  Suppose
> that, as the input frequency varies, the output gets pushed away from
> input harmonics, so that they never line up.  A phase-UNlocked loop, if
> you will!
>
> The behavior in my mind is a slowly varying oscillator, so if you looked
> at the output on the oscilloscope, it would always be a reasonably clean
> wave (sine or square), but as you sweep the input, it drifts and hops in
> just such a way as to avoid the input.  Though I think cycle-to-cycle
> consistency is not a requirement, in which case a possibly simpler
> solution could be random noise with the input filtered out of it.
>
> An example use would be, suppose you have a two channel analog
> oscilloscope and you want to add horizontal cursors to it.  Easy
> solution,
> just put a square wave on the other channel, and as long as it's not
> triggered, it'll blur into two horizontal lines most of the time.  Vary
> the "high" and "low" voltages and there's your cursors.  Trouble is,
> when the input is one of those lucky frequencies where the trigger lines
> up, the illusion breaks down.  This also applies to CHOP mode in the
> scope itself.  The actual limitation is evidently more sloppy: if they
> line up within persistence of vision (~20Hz BW), you get a flickering,
> rolling or locked display.
>
> I have no application in mind, just a Sunday afternoon musing.  It seems
> rather useless, but would probably find application in FDMA or something
> like that where orthogonality is handy.  Actually, such a function
> probably has even deeper theoretical applications from communications to
> encryption (e.g., use a discretized version to produce a message digest
> which is not just a scrambling of the input, but deterministically
> orthogonal in all dimensions) and number theory.  Which means it might
> fit into one of those "computable but infinite" or "unknowable"
> categories... hmm...

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.

--
www.wescottdesign.com
```
```On 9/28/2014 3:17 PM, Tim Williams wrote:
> Is there such a thing as a circuit (or more generally, system of
> differential equations) where the output frequency is always orthogonal
> (anharmonic) to the input?

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

Cheers

Phil Hobbs
```
```On 9/28/2014 5:58 PM, Phil Hobbs wrote:
> On 9/28/2014 3:17 PM, Tim Williams wrote:
>> Is there such a thing as a circuit (or more generally, system of
>> differential equations) where the output frequency is always orthogonal
>> (anharmonic) to the input?
>
> 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

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net
```
```"tim" <tim@seemywebsite.com> wrote in message
news:8fOdnSj5ZMI64bXJnZ2dnUVZ5smdnZ2d@giganews.com...
> 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.

Yeah, not quite the right word use.  Also, any phase shift is orthogonal
(in a basis sense), whatever the frequency.

Tim

--
Seven Transistor Labs
Electrical Engineering Consultation
Website: http://seventransistorlabs.com

```
```In article <m09mrt\$cu1\$1@dont-email.me>, Tim Williams
<tiwill@seventransistorlabs.com> wrote:

> Is there such a thing as a circuit (or more generally, system of
> differential equations) where the output frequency is always orthogonal
> (anharmonic) to the input?
>
> Example response would be, you have an arbitrary input frequency, and you
> want to pick an output frequency that *doesn't* line up with the input.
> If you use a fixed oscillator, it'll work for some inputs, but there will
> be special frequencies where they start to line up.  Suppose that, as the
> input frequency varies, the output gets pushed away from input harmonics,
> so that they never line up.  A phase-UNlocked loop, if you will!
>
> The behavior in my mind is a slowly varying oscillator, so if you looked
> at the output on the oscilloscope, it would always be a reasonably clean
> wave (sine or square), but as you sweep the input, it drifts and hops in
> just such a way as to avoid the input.  Though I think cycle-to-cycle
> consistency is not a requirement, in which case a possibly simpler
> solution could be random noise with the input filtered out of it.
>
> An example use would be, suppose you have a two channel analog
> oscilloscope and you want to add horizontal cursors to it.  Easy solution,
> just put a square wave on the other channel, and as long as it's not
> triggered, it'll blur into two horizontal lines most of the time.  Vary
> the "high" and "low" voltages and there's your cursors.  Trouble is, when
> the input is one of those lucky frequencies where the trigger lines up,
> the illusion breaks down.  This also applies to CHOP mode in the scope
> itself.  The actual limitation is evidently more sloppy: if they line up
> within persistence of vision (~20Hz BW), you get a flickering, rolling or
> locked display.
>
> I have no application in mind, just a Sunday afternoon musing.  It seems
> rather useless, but would probably find application in FDMA or something
> like that where orthogonality is handy.  Actually, such a function
> probably has even deeper theoretical applications from communications to
> encryption (e.g., use a discretized version to produce a message digest
> which is not just a scrambling of the input, but deterministically
> orthogonal in all dimensions) and number theory.  Which means it might fit
> into one of those "computable but infinite" or "unknowable" categories...
> hmm...

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

.<http://en.wikipedia.org/wiki/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
```
```On Sunday, September 28, 2014 12:17:17 PM UTC-7, Tim Williams wrote:

> An example use would be, suppose you have a two channel=20
> analog oscilloscope and you want to add horizontal cursors to it.  Easy=
=20
> solution, just put a square wave on the other channel, and as long as=20
> it's not triggered, it'll blur into two horizontal lines most of the time=
. =20
> Vary the "high" and "low" voltages and there's your cursors.  Trouble is,=
=20
> when the input is one of those lucky frequencies where the trigger lines=
=20
> up, the illusion breaks down.  This also applies to CHOP mode in the scop=
e=20
> itself.  The actual limitation is evidently more sloppy: if they line up=
=20
> within persistence of vision (~20Hz BW), you get a flickering, rolling or=
=20
> locked display.
>=20

The usual way to deal with this is with a pseudorandom noise source, at lea=
st where optimizing human perceptual issues is the goal. =20

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. =20

-- john, KE5FX
```
```On Sunday, September 28, 2014 12:17:17 PM UTC-7, Tim Williams wrote:
> Is there such a thing as a circuit (or more generally, system of
> differential equations) where the output frequency is always orthogonal
> (anharmonic) to the input?

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...
```