## A Hitchhiker's Guide to the FFT

## IV. Relation between CFT, DTFT and \(P\)-DFT

Let us consider a continuous function \(f_c\) and create the vector \(f_v\) by sampling \(f_c\) as follows: $$\begin{aligned} f_v [n] & = f_c (nT), \end{aligned}$$ where \(T\) is a given positive real number called the sampling interval. For example, if we want to sample a function defined on the interval \([- L, L]\) with \(N + 1\) equidistant points within this interval, the sampling interval would be given by \(T = \frac{2 L}{N}\). The relation between the CFT \(F_c\) and the DTFT \(F^{\text{DTFT}}\) is the following: $$\begin{aligned} F^{\text{DTFT}} (\omega) & = \frac{1}{T} \sum_{m = - \infty}^{\infty} F_c \left( \frac{\omega}{T} + \frac{2 m \pi}{T} \right) \end{aligned}$$ This explains the fact that on Figure 4, \(F^{\text{DTFT}}\) looks a bit like \(F_c\) repeated for each period. Note that by the Definition III.1, we also have the following relation between the \(P\)-DFT associated to \(f_v\) and \(F_c\): $$\begin{aligned} F^{P \text{-} \text{DFT}} [k] & = \frac{1}{T} \sum_{m = - \infty}^{\infty} F_c \left( \frac{2 \pi k}{TP} + \frac{2 m \pi}{T} \right), \end{aligned}$$ for \(k = 0, \ldots, P - 1\).