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In Fourier Transform, what does the alternate axis mean





lightworker88
In Fourier Transform, what does the alternate dimension represent?

It is my understanding that the square root of minus one is essentially ignored and its term instead represents the value of another axis.
kelseymh
lightworker88 wrote:
In Fourier Transform, what does the alternate dimension represent?


Do you mean the conjugate variable? In the classical FT, the conjugate variables are wavenumber and position, or frequency and time. In quantum mechanics, the conjugate variables are position and momentum, or time and energy.

Quote:
It is my understanding that the square root of minus one is essentially ignored and its term instead represents the value of another axis.


I'm not sure what this means. If you're referring to the use of Eulerian trig terms [exp(iwt)], that's a well-defined notational convention: exp(iwt) = cos(wt) + i sin(wt) in the complex plane is equivalent to the Cartesian pair [cos(wt),sin(wt)].
lightworker88
kelseymh wrote:

Do you mean the conjugate variable? In the classical FT, the conjugate variables are wavenumber and position, or frequency and time. In quantum mechanics, the conjugate variables are position and momentum, or time and energy.


I was thinking classical. Why not use time*cos(frequency), or iterations*cos(frequency)? (sine could substitute for cosine. Either would produce a wave function with consistent amplitude and consistent wavelength.)
kelseymh
lightworker88 wrote:
kelseymh wrote:

Do you mean the conjugate variable? In the classical FT, the conjugate variables are wavenumber and position, or frequency and time. In quantum mechanics, the conjugate variables are position and momentum, or time and energy.


I was thinking classical. Why not use time*cos(frequency), or iterations*cos(frequency)? (sine could substitute for cosine. Either would produce a wave function with consistent amplitude and consistent wavelength.)


Time * cos(w) isn't dimensionally correct. The argument to sine or cosine must be dimensionless, which is why we write cos(t*w) [or equivalently, cos(2pi*f*t)]. The conjugate form is cos(x*k) or equivalently cos(x/lambda) (where k is wavenumber, lambda is wavelength), which again has the proper dimension.
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