I have been trying to understand the introduction of fourier series and fourier transformation for quite sometime. But I am still not able to understand.
Can anyone please help me to understand what does fourier series really tell us? Why we needed to introduce that?
I have knowledge of basic calculus , Analysis and Topology.
Thank You.
There are many answers I'm sure that could be given, but I can offer a very practical example of the application of Fourier Transforms as an illustration of the use.
Fourier Transforms can be thought of as a way to easily observe the frequency spectrum of a general time series. In other words, imagine I have a sinusoidal signal generator of a particular frequency that is putting out a signal which I am measuring. I can use the Fourier transform to quickly determine the signal frequency from the detected time series.
So, one practical application is in the case of Magnetic Resonance Imaging (MRI) in clinical settings. The signal generators in this case are actually the atoms in the body responding to magnetic fields and as a result they generate a radio frequency time series signal which is detected by the receiver of the MRI machine. Now, this time series is Fourier Transformed to directly create the images of the human anatomy that you have probably seen pictures of. The various atoms are generating signals at different frequencies in the body, but the Fourier Transform properly maps these frequencies to their spatial locations.
This is one practical example, but I hope illustrates the use of Fourier methods for you.
I hope this helps.