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Fourier transform need help?

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I need help to understand use of fourier transform. Fourier transform means the conversation of time domain into frequency domain. I can solve example by the formula but I don't understand actually what is happening. Here is example
Can someone help how to compare the answer with given questions how does time domain convert into frequency domain?


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Hello there,

It's harder to see with the FT than with the Fourier Series so you should really start with the Fourier Series and work your way along.

It's basically saying that when we take a bunch of sine waves of the right frequency and phase and add them together, the result is a time domain representation of the same information.

In the analysis what we are doing is looking for the amplitudes and phase shifts of those sine waves so we get the right ones to match the time domain wave, because the time domain wave can be reconstructed from the sine waves we find. So rather than use the time domain wave we can use the frequency domain waves and if we do it right we get the same thing.

With the series we are talking about integer multiples of the fundamental frequency like 2,3,4,5, etc.
With the transform we are talking about the continuous frequency range.

With the series though it is easier to see what is happening.

In a manner of speaking we can view this as a curve fitting algorithm. We have a curve in the time domain, and we want to find a fit to that curve using sine waves with various amplitudes and phase shifts. Once we find them we have a way to reproduce the curve without having to work in the time domain. Frequency domain problems are often simpler than time domain problems so it's often an advantage.

History tells us that Fourier discovered this while looking for solutions to the heat equation ut=k*uxx or similar.

You might be more familiar with finding a power series such as:

and here we try to find the 'amplitudes' of the powers of x instead of for sine waves.
We might call that the "algebraic power series domain".
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