# the DFT of a periodic signal represented by a fourier series

If I have a signal represented by a Fourier series(like in the photo), which is sampled with $T_s$:

$$x[n]=x(t=nT_s) = \sum_{m=-\infty}^{\infty}a[m]e^{j2\pi m(nT_s)/T_0}$$

How do I find its DFT? I can't understand that. Sorry for the English mistakes.

• What is the relationship (if any) between the period of the signal (or, if you prefer, the fundamental frequency in the Fourier series representation) and the sampling interval $T_s$? – Dilip Sarwate Sep 27 '15 at 13:53
• i ask about the general solution but you can assume T0/Ts is an integer – Ori Barak Sep 27 '15 at 14:03
• If you are using Matlab/Octave, it's as easy as dct(x) where x is the data. – Mohit Sep 27 '15 at 14:23
• Then perhaps you could clarify whether you want the DFT of $(x[0], x[1],\ldots, x[N-1])$ or the DTFT of the infinitely long (but periodic) sequence of samples $\ldots, x_{-2T_s}, x_{-T_s}, x_0, x_{T_s}, x_{2T_s},\ldots$ – Dilip Sarwate Sep 27 '15 at 14:28
• If Ts is the reciprocal of the data sampling frequency then what is that strange T0 value? Do those a[m] samples represent some sort of frequency-domain samples? Are you really saying each x[n] sample is equal to the sum of an infinite number of complex numbers? Are you sure your equation is correct? By the way, in order to compute the DFT of an x[n] sequence, that sequence must be finite in length. – Richard Lyons Sep 27 '15 at 21:11