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For a more intuitive explanation, this paper on the advantages of complex wavelets switched on the lightbulb for me why signals are complex in the first place:

For a more intuitive explanation, this paper on the advantages of complex wavelets switched on the lightbulb for me why signals are complex in the first place: http://people.math.sc.edu/blanco/IMI/DTCWT0.pdf http://people.math.sc.edu/blanco/IMI/DTCWT0.pdf

aaaaaaaaaab 8y ago
The most intuitive explanation is simply that in the complex domain sinusoids can be represented as exponentials, and a phase shift becomes a simple multiplication by a constant. Thus sinusoidal signals can be manipulated according to a fairly simple algebra.
whatshisface 8y ago
I'm not sure how much of relationship there would be between the convenience of complex numbers in signal analysis and the appearance of complex numbers in the Schrodinger equation.
- kgwgk 8y ago
  It can be seen as a matter of convenience in Schroedinger’s equation too. Other representations of QM exist. For example, in the phase space representation we have a real function of positions and momentums. In the wave function representation we have a complex function of positions: the (square of the modulus of the) amplitude is related to the position and the (derivative of the) phase is related to the momentum. In the Scroedinger’s formalism we can also go from the position representation to the momentum representation via Fourier transform.
  - whatshisface 8y ago
    While true, the convenience in the Schrodinger equation is due to a symmetry in nature which the algebra of complex numbers mirrors. The pattern of a symmetry in nature corresponding to a convenient algebraic object used to represent it happens in many other places including Pauli spin matrices. In signal processing complex numbers are useful for more purely mathematical reasons.
    - jacobolus 8y ago
      In most parts of physics there are literally things spinning. In signal processing we can take any periodic-in-time signal and pretend it is a function defined on a circle parametrized by angle measure.
      - whatshisface 8y ago
        The complex precession in the Schrodinger equation doesn't involve angular momentum.
        jacobolus 8y ago
        Depends on who you ask. http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf
    - feanaro 8y ago
      > In signal processing complex numbers are useful for more purely mathematical reasons. Yes, but as always you can ask yourself: is this merely a coincidence (if there is such a thing as one) or is this reflecting some deeper relation between nature and mathematics? After all, a "signal" is quite a general and abstract notion.

aaaaaaaaaab 8y ago

The most intuitive explanation is simply that in the complex domain sinusoids can be represented as exponentials, and a phase shift becomes a simple multiplication by a constant. Thus sinusoidal signals can be manipulated according to a fairly simple algebra.

whatshisface 8y ago

I'm not sure how much of relationship there would be between the convenience of complex numbers in signal analysis and the appearance of complex numbers in the Schrodinger equation.

kgwgk 8y ago
It can be seen as a matter of convenience in Schroedinger’s equation too. Other representations of QM exist. For example, in the phase space representation we have a real function of positions and momentums. In the wave function representation we have a complex function of positions: the (square of the modulus of the) amplitude is related to the position and the (derivative of the) phase is related to the momentum. In the Scroedinger’s formalism we can also go from the position representation to the momentum representation via Fourier transform.
- whatshisface 8y ago
  While true, the convenience in the Schrodinger equation is due to a symmetry in nature which the algebra of complex numbers mirrors. The pattern of a symmetry in nature corresponding to a convenient algebraic object used to represent it happens in many other places including Pauli spin matrices. In signal processing complex numbers are useful for more purely mathematical reasons.
  - jacobolus 8y ago
    In most parts of physics there are literally things spinning. In signal processing we can take any periodic-in-time signal and pretend it is a function defined on a circle parametrized by angle measure.
    - whatshisface 8y ago
      The complex precession in the Schrodinger equation doesn't involve angular momentum.
      - jacobolus 8y ago
        Depends on who you ask. http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf
  - feanaro 8y ago
    > In signal processing complex numbers are useful for more purely mathematical reasons. Yes, but as always you can ask yourself: is this merely a coincidence (if there is such a thing as one) or is this reflecting some deeper relation between nature and mathematics? After all, a "signal" is quite a general and abstract notion.