I am reading the paper, Design of an energy-efficient accelerator for training of convolutional neural networks using frequency-domain computation, and I came across the following definition of sinc interpolation.
Warning. I don't have a strong background in signal processing. Also, I have no clue what that bar on $\bar{F}$ means. I don't know why that would be the conjugate. Would it actually be the conjugate?
Context. In this part, for a given kernel $f_k$ of size $k$, we want to expand $F_k = DFT\{f_k\}$ to $F_N = DFT\{f_N\}$ have size $N$, without going to the spatial domain, padding and back to the spectral domain.
Let's discuss on a 1D scenario for notation simplification.
$$ G(u) = F(u) \ast \left [ e^{\frac{-j 2 \pi u}{N}(\frac{k-1}{2})} \frac{sin(\frac{\pi u k}{N})}{sin(\frac{\pi u}{N})} \right ] $$
Q1. I am assuming that $G(u)$ and $F(u)$ are discrete, having the respective sizes of $N$ and $k$. Is this assumption right?
Q2. Would that second term actually be the $sinc$ kernel, i.e. would the expression below be true? If not, what is their between that term and the sinc function? $$ sinc(u) = \frac{sin(u)}{u} = \left [ e^{\frac{-j 2 \pi u}{N}(\frac{k-1}{2})} \frac{sin(\frac{\pi u k}{N})}{sin(\frac{\pi u}{N})} \right ] $$
Another way I know to avoid using the DFT to compute $F_N$, is by representing $f_N$ in terms of $f_k$ as
$$f_n = \sum_{i=0}^{k-1} \delta(n-i) f_i$$
Thus, having that $$DFT\{\delta[n]\} = 1$$
and $$ DFT\{x[n-n_o]\} = e^{-j \frac{2 \pi}{N} k n_o} DFT\{x[n]\}$$
Hence $$F_N(k) = \sum_{n_o=0}^{k-1} e^{-j \frac{2 \pi}{N} k n_o } f_{n_o}$$
I am not sure if that the convolution on the frequency domain $G(u)$, and the impulse based composition $F_N(k)$ are similar approaches of doing the same thing. I did notice the similarity between
$$ e^{\frac{-j 2 \pi u}{N}(\frac{k-1}{2})} \sim e^{\frac{-j 2 \pi k}{N}n_o} $$
But, if this is the case, then,
$$ \frac{sin(\frac{\pi u k}{N})}{sin(\frac{\pi u}{N})} \sim F\{\delta(k-n_o)\} $$
Q3. Are these the same approaches? Is there any relation between these expressions? $$\frac{k-1}{2} \sim n_o$$