The quickest "explanation" of the FT I ever heard was in a casual aside from a professor once -- he referred to the Fourier domain as the "reciprocal domain". It took me a while to work out what he meant.
It was just that frequency = 1 / time. In this (barbarically reductive) conception, taking the FT is just a change of variable.
This relationship is one way to "derive" many of the standard Fourier facts.
For example, the scaling property, that if x(t) has transform X(f), then x(at) has transform (1/a) * X(f/a). It also "explains" why time signals concentrated around t=0 tend to have lots of high-frequency content (f = 1/t = 1/0 = infinity), and vice versa.
It also "explains" why the inverse FT formula looks just like the forward FT formula (since if f = 1/t, then t = 1/f). And, for the same reason, most of the duality relationships between the two domains.
All with just arithmetic! You can dispense with linear algebra, not to mention complex arithmetic, groups, or measure theory.
It was just that frequency = 1 / time. In this (barbarically reductive) conception, taking the FT is just a change of variable.
This relationship is one way to "derive" many of the standard Fourier facts.
For example, the scaling property, that if x(t) has transform X(f), then x(at) has transform (1/a) * X(f/a). It also "explains" why time signals concentrated around t=0 tend to have lots of high-frequency content (f = 1/t = 1/0 = infinity), and vice versa.
It also "explains" why the inverse FT formula looks just like the forward FT formula (since if f = 1/t, then t = 1/f). And, for the same reason, most of the duality relationships between the two domains.
All with just arithmetic! You can dispense with linear algebra, not to mention complex arithmetic, groups, or measure theory.