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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

$\displaystyle \Psi_T(t) \isdef T\sum_{n=-\infty}^\infty \delta(t-nT) \protect$ (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 $ x(nT)$ . Thus, we replace $ x(t)$ 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 $ x_s$ isB.3

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.

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

&=& \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

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,

$\textstyle \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 $x$.}}$

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University