Andy Resnick said:Usually, one transforms between a time-varying signal and a temporal frequency, or a spatially-varying signal and 'wavenumber' (equivalently, spatial frequency or angle).
Does that help?
A Fourier transform is a way of writing a given function as a sum of sinusoids, so I can Fourier transform just about any function that meets some minimal standards for well-behavedness. The interesting question is whether that's useful: do the sinusoids correspond to any physically interesting function? For example, Fourier transforming a sound signal tells me what frequencies have been superimposed to produce that signal... but Fourier transforming the elevation above sea level along a path is unlikely to tell me anything interesting (unless the topography happens to include some unusually evenly spaced and symmetrical hills, which would show up as a spike in the transform).Higgsono said:But why? What pair of variables are allowed and why?
Higgsono said:But why? What pair of variables are allowed and why?
Andy Resnick said:I'm not sure what you are getting at. For one thing, the product of whatever conjugate variables you choose (call them 'K' and 'L') must be dimensionless.
Higgsono said:What about the dimensions? RHS should have the same dimension as the LHS?
A Fourier transform is a mathematical operation that decomposes a function into its individual frequency components. It allows us to analyze a complex signal in terms of simpler sinusoidal signals. This is useful in many applications such as signal processing, image analysis, and quantum mechanics.
A Fourier transform can be used to transform between any two variables that are related by a linear transformation. This includes variables such as time and frequency, position and momentum, and voltage and current.
Yes, a Fourier transform can be applied to both continuous and discrete signals. For continuous signals, we use the continuous Fourier transform, while for discrete signals, we use the discrete Fourier transform.
Yes, a Fourier transform is reversible. This means that we can use the inverse Fourier transform to reconstruct the original signal from its frequency components. This property is useful in applications such as data compression.
There are some limitations to using a Fourier transform. One limitation is that it assumes the signal is periodic, which may not always be the case in real-world applications. Additionally, the Fourier transform can only be applied to signals that have finite energy or power.