5 ms·

Yes, I mean 'time and spectral domain'. The idea of the gif is to represent the two domains orthogonal to each other graphically. The signal in the time domain

Yes, I mean 'time and spectral domain'. The idea of the gif is to represent the two domains orthogonal to each other graphically. The signal in the time domain is a complex function (time->(magnitude,phase)) so it can be plotted in 3 dimensions. The same is true for the spectral domain. So really there are only 2 3d plots. But moving from the time to the signal domain can also be thought of as a 90 degree rotation. In the gif you can see two blue planes are orthogonal to each other. But the gif shows only the 2d magnitude plots of the time domain and spectral domain. The cuboid embeds a 3d plot of each domain on each of it's side in order to plot the imaginary values as well. The slicing allows you to select up to two frequencies from the frequency domain and plot the corresponding basis function on the top side of the cuboid. The top side of the cube corresponds to the 3d coordinate system in the gif. In hope this helps to clarify.

duped 5y ago
Is the "time domain" function a complex function of time, and "spectral" domain the F.T. of that function? I ask because it's rather unusual to look at the F.T. of complex functions, since most useful F.T.s only deal with real-valued functions of time and it might be extra confusing as the F.T. is no longer symmetric as a result It may also be nice to use different syntax, like `x(t)` and X(\omega)` since "s" can be a bit overloaded in the context of Fourier theory
- quietbritishjim 5y ago
  On the contrary, it's most normal to consider the time domain function [1] as a complex valued function, even if it happens to take only real values. And the reason for that is exactly for symmetry with its transform! The transform is certainly complex valued, thanks to the e^ix\xi factor (unless you take the sine and cosine transforms separately, but of course those are just the real and imaginary components). For example, you can see the first definition on Wikipedia [2] starts with a function f:R->C. Edit: Oh, by "symmetric" you meant the Fourier transform being symmetric about 0. I thought you meant the natural symmetry of the forms of f and \hat{f}. Still, it seems natural to me to treat the two functions similarly, including use of complex values. But I'm a (very) pure mathematician, at least in this area. [1] Or whatever function you want to take the transform of, it doesn't have to be the time domain and is in fact often spatial. This is probably why OP used the phrase "signal domain". [2] https://en.m.wikipedia.org/wiki/Fourier_transform https://en.m.wikipedia.org/wiki/Fourier_transform

duped 5y ago

Is the "time domain" function a complex function of time, and "spectral" domain the F.T. of that function? I ask because it's rather unusual to look at the F.T. of complex functions, since most useful F.T.s only deal with real-valued functions of time and it might be extra confusing as the F.T. is no longer symmetric as a result It may also be nice to use different syntax, like `x(t)` and X(\omega)` since "s" can be a bit overloaded in the context of Fourier theory

quietbritishjim 5y ago
On the contrary, it's most normal to consider the time domain function [1] as a complex valued function, even if it happens to take only real values. And the reason for that is exactly for symmetry with its transform! The transform is certainly complex valued, thanks to the e^ix\xi factor (unless you take the sine and cosine transforms separately, but of course those are just the real and imaginary components). For example, you can see the first definition on Wikipedia [2] starts with a function f:R->C. Edit: Oh, by "symmetric" you meant the Fourier transform being symmetric about 0. I thought you meant the natural symmetry of the forms of f and \hat{f}. Still, it seems natural to me to treat the two functions similarly, including use of complex values. But I'm a (very) pure mathematician, at least in this area. [1] Or whatever function you want to take the transform of, it doesn't have to be the time domain and is in fact often spatial. This is probably why OP used the phrase "signal domain". [2] https://en.m.wikipedia.org/wiki/Fourier_transform https://en.m.wikipedia.org/wiki/Fourier_transform