I am thinking of the typical (sound, electromagnetic or whatever) signal which can be represented in time or frequency domain as a collection of infinite amplitudes for, respectively, each time-point or each frequency, so that one can say that this function (where the input is the time points or the frequencies, as the case may be, and output is their amplitudes) is a vector in a generalized sense (where coefficients are the amplitudes and the basis vectors could be represented by complex exponentials of the relevant time point or frequency, as the case may be).

So this vector, in a generalized sense, has clearly "magnitude" and one could think that it has "direction", in some sense, if we consider that we could play with the weight of the components, raising it here and lowering it there, so that the resulting signal sort of “points to somewhere else”, even if its magnitude remains the same.

Is this a simple theoretical speculation or does it have practical application in physics? Is there any real-life situation where you would need to ascertain or alter the “direction” of a signal for some practical purpose?

(Well, an obvious answer is that through this playing with the weights you generate a different signal and this will always have some practical utility, but I was wondering if, in the scientific literature or in practice, this is an expression that you would typically use in some common situations: let us change the "direction" of the signal to achieve this or that result.)

  • $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$ – David Z Jul 19 '20 at 19:51

Sure. Suppose you have a sine wave and a square wave with the same magnitude (in your vector sense).

Without being able to distinguish "direction", how can you tell them apart?

  • $\begingroup$ Good. What about the other example that you had mentioned in the comment: a sine and a cosine wave? Would you agree that they have same direction but different complex magnitude (phase)? $\endgroup$ – Sierra Jul 19 '20 at 20:25
  • $\begingroup$ @Sierra, I would say they have different direction, but it's your question so you get to define your terms, and if you say they have the same direction by your definitions, then I can't argue with it. $\endgroup$ – The Photon Jul 19 '20 at 20:26
  • $\begingroup$ @Sierra, if you want me to be able to comment on it, you should define with mathematics, how, given a signal $x(t)$ or $X(f)$ are the magnitude and direction calculated. For example, one traditional definition of the magnitude would be $\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}x^2(t) dt$. $\endgroup$ – The Photon Jul 19 '20 at 20:28
  • $\begingroup$ Fine with your definition of magnitude, which I see as the analog of Pythagorean Theorem. The problem with direction is that it is an invariant thing but without a numerical invariant expression, at least that I know. In finite dimensions you would need angles against n-1 axes, cosine of each angle being the ratio between relevant coefficient and magnitude of the vector. $\endgroup$ – Sierra Jul 19 '20 at 21:06

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