Now, if we want to describe the transformation of
winding up a wave around a cricle, we just need to
time the amplitude of the wave introns of this new
expression, which can be denoted as g(t)( as shown on
the to of the graph on the left.). This is awesome. Now
we can find the geometric centre (or the centre of
mass) of a winded up wave, by finding an average
value of of the distance between the origin and each
point on the plotted curve. We can do this by
summing their distance together and then deciding it by the total number of points. As the there is more and
more points collected, the value of the centre of mass will be more accurate. And if we take every single
point, this can be expressed as a integral.
So now we have built, a kind of complicated, but still surprisingly short expression for the transformation of
winding a wave around a centre. Now, there is only one difference between this expression and the Fourier
Transform, that is the fact that we don't need to divide the expression by the time period that was recorded.
Instead of looking at the center of mass, you would scale it up by some multiple amount. If you are looking
at an input of a wave in a time interval of 3 second, the outcome of the Fourier Transform would have a
center of mass which is 3 times away from the axis. If the time interval is 10 seconds, you would multiply
the centre of mass by 10.
Physically, by doing this step, we achieve the effect that when a certain frequency persists for a long time,
then the magnitude of the Fourier transform at that frequency is scaled up more and more. Then center of
mass would stay in the same spot, but the long the
signal persists, the larger the value of the Fourier
transform at that frequency.
That was a lot of information, now let us summarize.
The Fourier Transform of an intensity vs time function,
like g(t), is a new function, which doesn't have time as
an input, but instead it has winding frequency on the x-
axis. In terms of notation, let us call this new function
'g hat’ with a lite circumflex on the top of g. The out
put of this function is a complex number, some point in
the 2d plane that correspond to the strength of a given frequency in the original signal. The final graph
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