# Phase conventions for phasors and Fourier transforms

There is a natural isomorphism between the complex plane and the set of sine waves, but this isomorphism is ambiguous up to a rotation and/or flip of the plane. This ambiguity seems to be related to some of the differences in conventions about the Fourier transform and inverse Fourier transform.

In electrical applications, it's conventional that capacitive impedance is negative imaginary, and this requires that the cosine be represented by a point that lies counterclockwise from the one representing the sine. With this handedness, differentiation means multiplication by $$i\omega$$.

The convention I've been teaching is that $$\sin\rightarrow1$$ and $$\cos\rightarrow i$$, which is consistent with this. Call this convention A.

Looking at the form of Euler's equation, it does seem like it would be nicer to have $$\cos\rightarrow1$$. Sticking to the same handedness, we would then have to have $$\sin\rightarrow -i$$. Call this convention B.

Is this standardized? Is there a physicist's convention that is different from an electrical engineer's convention?

The WP article on the Fourier transform has some material that seems relevant, at "The reason for the negative sign convention in the exponent is that in electrical engineering, it is common..." This seems to imply that electrical engineers use convention B, but that other people use some other convention. Are the other people physicists? Some other kind of engineers? Do they use convention A?

Related:

https://math.stackexchange.com/a/2306802/13618

Fourier transform standard practice for physics (I think what I'm calling convention A is not describable in the $$(a,b)$$ parametrization defined in jgerber's answer.)

My response in the linked answer is to clarify the identification of $$e^{i\omega t}$$ as either a positive or negative frequency signal. The text in the Wikipedia article is related to exactly this. The convention that capacitive impedance is negative imaginary fixes this positive/negative frequency convention already. So both your A and B take the same convention for positive/negative frequency.

Regarding the main part of your question it sounds like you are asking about the definition of a phase factor (rather than a frequency scaling factor) in the Fourier transform. Suppose we define the Fourier transform

$$\mathcal{FT}_{a,b,c}[f(t)](\omega) = \sqrt{\frac {|b|}{(2\pi)^{1-a}}}\int_{-\infty}^{+\infty} e^{+i b \omega t} e^{ic} f(t) dt$$

And the inverse Fourier Transform

$$\mathcal{FT}_{a,b,c}^{-1}[\tilde{f}(\omega)](t) = \sqrt{\frac{|b|}{(2\pi)^{1+a}}}\int_{-\infty}^{+\infty} e^{-i b \omega t}e^{-ic} \tilde{f}(\omega) d\omega$$

Let $$\tilde{f}_{a,b,c}(\omega) = \mathcal{FT}_{a,b,c}[f(t)](\omega)$$ $$\check{f}_{a,b,c}(t) = \mathcal{FT}_{a,b,c}^{-1}[\tilde{f}_{a,b,c}(\omega)](t)$$

Here, (compared to my answer in the linked question) I have introduced a phase factor $$e^{ic}$$ into the definition for the Fourier and inverse Fourier transform. As defined we will have that $$\check{f}_{a,b,c}(t) = f(t)$$ for any $$(a,b,c)$$.

What I can tell you is that, as far as I'm aware, everyone (EE, signal processing, math, physics etc.) all take $$c=0$$. This coincides with your convention B. I believe your convention A corresponds to $$c = + \frac{\pi}{2}$$ so that $$e^{ic} = i$$. I've never seen someone use this for their definition of the Fourier transform.