Relation of the DFT to Fourier Series

We now show that the DFT of a sampled signal (of length ), is proportional to the Fourier series coefficients of the continuous periodic signal obtained by repeating and interpolating . More precisely, the DFT of the samples comprising one period equals 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 complex harmonic components can be nonzero in the original continuous-time signal.

If is bandlimited to $\omega T\in(-\pi,\pi)$ , it can be sampled at intervals of 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

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

(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 . Thus, we replace in Eq. $\,$ (B.5) by

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

By the sifting property of delta functions (Eq. $\,$ (B.4)), the Fourier series of

is^B.3

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

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

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

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

In summary,

$\textstyle \parbox{0.8\textwidth}{% WHY IS THIS NEEDED??? \emph{the Fourier ser... ...he DFT length, and $N$\ is also the number of samples in each period of $x$.}}$

Previous: Fourier Series (FS) and Relation to DFT
Next: Selected Continuous-Time Fourier Theorems

Order a Hardcopy of Mathematics of the DFT

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Sponsor

Chapters

See Also

Mathematics of the DFT
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Series (FS) and Relation to DFT
          Relation of the DFT to Fourier Series