DFT as Sampled Version of DTFT

Consider an example where $x[n] = e^{j2\pi f_0 n}$ for $n=0, 1, 2, \cdots, L-1$. This is a length $L$ sequence. The DTFT is

Also the $N$-point DFT is $X[k]=\left.X(f)\right|_{f=\frac{k}{N}}$ for $k=0, 1, 2, \cdots, N-1$. Note that when $N>L$ there is zero padding, whereas when $N=L$ there is no zero padding.

Now consider the four cases.

1. Take $f_0=\frac{k_0}{L}$ for $k_0\in \{ 0, 1, 2, \cdots, L-1\}$ and let $N=L$ (no zero padding). In this case, the DFT is

   $$\begin{align} X[k] &= \frac{\sin(\pi [k-k_0])}{\sin(\frac{\pi}{L} [k-k_0])} e^{-j \pi [k-k_0]\frac{L-1}{L}} \\ &= \begin{cases} L, & k=k_0 \\ 0, & \text{otherwise} \end{cases} \\ &= L \delta[k-k_0] \end{align}$$

   The special set of frequencies $f_0=\frac{k_0}{L}$ for $k_0\in \{ 0, 1, 2, \cdots, L-1\}$ are called "bin center" frequencies. The DFT of a complex exponential with a bin center frequency is a Kronecker delta. The plot below shows the $k_0=25$ and $L=N=100$ case.

   

2. Take $f_0=\frac{k_0+\frac{1}{2}}{L}$ for $k_0 \in \{0, 1, 2, \cdots, L-1\}$ (a non-bin center frequency) and let $N=L$ (no zero padding). The plot below is for $k_0=25.5$ and $L=N=100$.

   

3. Take $f_0 = \frac{k_0}{L} \neq \frac{l_0}{N}$ (not bin center for $N$ point transform) and $N>L$ (zero padding). The plot below shows the $k_0=25$ and $L=100$ and $N=256$ case.

   

4. Take $f_0 = \frac{l_0}{N}$ (bin center for $N$ point transform) and $N>L$ (zero padding). The plot below shows the $l_0=25$ and $L=100$ and $N=256$ case.

   

   In all of these cases, the underlying shape of the transform (the DTFT) was a peridoic sinc function. The only difference in these examples is where theh samples fall on the sinc function.