Why fft results of discrete Fourier series coefficients are different from the actual definition

I tried to find the fourier coefficients of this discrete time singal x[n] = [1 1 0 0 0] using fft to get the coeffiecits.

However, they differ from the definition of fourier series by the factor 1/N.

for example the output of fft for c_0 is 2, but by the defintion of fourier series, it has to be 2/N = 2/5

Could someone explain which one is true?

Answer 1

[David Goodmanson]

Hi Romio,

I believe you are referriing to the convention where (informally speaking)

f(x) =      Sum c_n  exp( 2*pi*i*f0*n*x)     % ifft direction
c_n = (1/n) Sum f(x) exp(-2*pi*i*f0*n*x)     % fft direction

(where f0 is chosen to give one oscillation over the length of the x record).

Then the c_n are straight amplitudes of the complex oscillatory functions. However, Matlab fft does not do things that way. Instead, it's

f(x) = (1/n) Sum c_n  exp( 2*pi*i*f0*n*x)     % ifft direction
c_n =        Sum f(x) exp(-2*pi*i*f0*n*x)     % fft direction

so if you want the first convention, you have to multiply the fft by 1/n. And multiply the ifft by n if you want to get back to where you were.

[Romio]

I see! Thanks