My question is as follows. I know that classically speaking, if you have some time varying signal X(t) its power spectral density S_X(f) will be symmetric around f=0 as the signal is real. This is why your spectrum analyzer only shows the positive part, the negative part is redundant.

But if one has a complex signal (such as often encountered in quantum mechanics) the PSD can be asymmetric around 0. This for example leads to the asymmetry between absorption and emission in atoms: atoms have spontaneous emission (emission in the absence of photons in the environment) but not spontaneous absorption.

What I am interested in is if there is a way to physically make such a complex signal with asymmetric PSD. I would specifically think about a voltage signal, ideally generated with some arbitrary waveform generator. Maybe this can be done by using two channels that output the signal with some phase difference? I've tried looking for some source material on this but I have not been successful.

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    \$\begingroup\$ I believe it's too theoretical for this SE. Try Physics.SE \$\endgroup\$
    – Eugene Sh.
    Oct 3, 2016 at 18:10
  • \$\begingroup\$ Obviously, voltages are real signals. To represent a complex signal, you need a pair of real signals, e.g. a pair of voltages. You declare one to be the real part, the other to be the imaginary part, and you're done. How is that useful I wouldn't know, but there you go. Of course you don't even need a complex signal to have an asymmetric PSD, if you drop the constraint of it being asymmetric around 0Hz. It's trivial to generate a signal asymmetric about any carrier frequency other than 0Hz. These are way more useful :) \$\endgroup\$ Oct 3, 2016 at 19:41
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    \$\begingroup\$ @EugeneSh. Or SignalProcessing.SE \$\endgroup\$
    – Andreas
    Oct 3, 2016 at 19:42
  • \$\begingroup\$ @kubaober I see what you mean, which is why I was thinking of the two channel approach. It's not obvious how this would lead to the desired asymmetry though. And yes indeed, other carriers work out just fine! I'm thinking about it in the context of physics experiments, with absorbers and emitters that should be affected in a different way. \$\endgroup\$
    – user129412
    Oct 3, 2016 at 19:57
  • \$\begingroup\$ 1. Draw up an asymmetric real spectrum. 2. Take an IFFT. 3. Output the time-domain signal on two D/A channels. I don't really know what other answer you need... \$\endgroup\$ Oct 3, 2016 at 20:16

1 Answer 1


First think about this picture: -

enter image description here

This is Euler's equation and is all about single rotating vector (\$e^{ix}\$) that moves in and out of both real and imaginery planes. Next consider what a sine or cosine is: -

enter image description here

In other words each are composed of a positive rotating vector and a negative rotating vector. However, as a sum of rotating vectors they only occupy a fixed position - the two rotating parts that make either are contra rotating and this makes sine or cosine fixed and free from rotation.

Here is what they look like: -

enter image description here

Cos(x) exists on the real axis and iSin(x) exists on the imaginery axis.

I think what you are asking for is a method of creating an imaginery voltage like the path taken by the red line: -

enter image description here

  • \$\begingroup\$ Yes, that seems like a key insight into the situation. Is something like that possible? \$\endgroup\$
    – user129412
    Oct 3, 2016 at 19:54
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    \$\begingroup\$ @user129412 No unfortunately not but that's just my opinion - it's a bit like expecting a normal spinning wheel also be able to rotate on an axis tangential to the axle that it normally spins on! \$\endgroup\$
    – Andy aka
    Oct 3, 2016 at 20:16

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