Relation of the DFT to Fourier Series

We now show that the DFT of a sampled signal $x(n)$ (of length $N$ ), is proportional to the Fourier series coefficients of the continuous periodic signal obtained by repeating and interpolating $x$ . More precisely, the DFT of the $N$ samples comprising one period equals $N$ times the Fourier series coefficients. To avoid aliasing upon sampling, the continuous-time signal must be bandlimited to less than half the sampling rate (see Appendix D); this implies that at most $N$ complex harmonic components can be nonzero in the original continuous-time signal.

If $x(t)$ is bandlimited to $\omega T\in(-\pi,\pi)$ , it can be sampled at intervals of $T$ seconds without aliasing (see §D.2). One way to sample a signal inside an integral expression such as Eq.(B.5) is to multiply it by a continuous-time impulse train

$$\Psi_T(t) \isdef T\sum_{n=-\infty}^\infty \delta(t-nT) \protect$$ (B.6)

where $\delta(t)$ is the continuous-time impulse signal defined in Eq.(B.3).

We wish to find the continuous-time Fourier series of the sampled periodic signal $x(nT)$ . Thus, we replace $x(t)$ in Eq.(B.5) by

$$x_s(t) \isdef x(t)\cdot \Psi_T(t).$$

By the sifting property of delta functions (Eq.(B.4)), the Fourier series of $x_s$ is^B.3

$$\begin{eqnarray*} X_s(\omega_k) = \frac{1}{P} \int_0^P x_s(t) e^{-j\omega_k t} dt &=& \frac{1}{P} \sum_{n=0}^{\lceil P/T\rceil-1} x(nT) e^{-j\omega_k nT} T. \end{eqnarray*}$$

If the sampling interval $T$ is chosen so that it divides the signal period $P$ , then the number of samples under the integral is an integer $N\isdef P/T$ , and we obtain

$$\begin{eqnarray*} X_s(\omega_k) &=& \frac{T}{P} \sum_{n=0}^{N-1} x(nT) e^{-j\omega_k nT}\\ &\isdef & \frac{1}{N}\hbox{\sc DFT}_{N,k}(x_p), \quad k=0,\pm 1, \pm 2, \dots \end{eqnarray*}$$

where $x_p\isdef [x(0),x(T),\dots,x((N-1)T)]$ . Thus, $X_s(\omega_k)=X(\omega_k)$ for all $k$ at which the bandlimited periodic signal $x(t)$ has a nonzero harmonic. When $N$ is odd, $X(\omega_k)$ can be nonzero for $k\in[-(N-1)/2,(N-1)/2]$ , while for $N$ even, the maximum nonzero harmonic-number range is $k\in[-N/2+1,N/2-1]$ .

In summary,

$\parbox{0.8\textwidth}{% WHY IS THIS NEEDED??? \emph{the Fourier series coefficients of a periodic, bandlimited signal $x$\ are given by the DFT of one period of the samples of $x$, divided by $N$, where $N$\ is the DFT length, and $N$\ is also the number of samples in each period of $x$.}}$