I'm currently trying to understand Fourier transform and I've got curious about why Fourier transform exists.
Let's suppose that we have a 10 seconds of non-periodic wave. For example:
As far as I understand, the concept of Fourier transform is to think the wave's period is infinite. If I describe what is in my mind with a picture, it is as same as the following:
And if we apply Fourier transform to that nonperiodic wave, we will get a continuous spectrum which shows components of the nonperiodic wave.
Meanwhile, let's just treat the nonperiodic wave's period as 10 seconds. Then we become to be able to calculate Fourier coefficients. And if we make a discrete spectrum with the coefficients we just got, we can still describe the nonperiodic wave.
And, I guess the shape of both spectrum will be same. (Please correct me if this is false)
So... I wonder why we need Fourier transform...
Of course, unlike Fourier coefficients, which give us nth multiples of fundamental frequency, Fourier transform can give us any frequency... but is it important? I think Fourier coefficients are enough information!