Fast Fourier Transform- Why does the help section of Matlab double the values in FFT?

Hi, I'm new to using fft in Matlab. I do not understand why they do the following in their help section- why do they cut off half of P2, and then double the P1 values? Could someone give me the mathematical justification for this?

Thank you!

Answer 1

[Matt J]

The example is in fact a demonstration of using the FFT to compute a standard Fourier series (i.e., positive frequency harmonics only). The standard Fourier series coefficients are related to the complex Fourier series coefficients by a factor of 2 because the standard Fourier series only contains positive frequency harmonics whereas the complex series splits the same contribution into positive and negative frequencies.

As a very simple example, the standard Fourier series of the function f(x)=A*cos(x) is itself. The equivalent complex series representation is

f(x)=(A/2)*exp(j*x)+(A/2)*exp(-j*x)

As you can see, this is now a 2-term series with coefficients that are half that of the standard series.

[Avishek Mondal]

Hey thank you for your answer!