I am learning DSP (with Digital Images) and I have some elementary confusion about the convolution between two discrete periodic signals. Specifically, I have learnt that when filtering an image, we apply convolution between the image (as discrete 2D signal) and a kernel filter. To visualize what happens in the frequency domain, we calculate the DFT of the image and the kernel and since we are convolving in the space domain, in the frequency domain this corresponds to a point per point multiplication of the two spectrums.
Now, the DFT assumes that the signal to transform is discrete AND periodic, resulting in a spectrum which is discrete and periodic. So here we are assuming that both the signal and the filter are discrete and periodic signals which repeat (usually) after the same M samples. My confusion is, how can convolution work for two discrete periodic signals? Since they repeat infinitely, applying the convolution usual definition as a flipped-shifting multiplication and sum, we would end up with an infinite sum in some single points if the signal to be filtered and the filter overlap. (Because they would overlap in the current repetition, in the next, and the next ad infinitum)
As an example in 1D, let $x[n]$ be our discrete function, which is periodic with a period of $N$ samples in the spatial domain. And let $h[n]$ be the filter impulse response in the spatial domain, also periodic of $N$ samples. We want to filter $x[n]$ with $h[n]$ and get the filtered signal $y[n]$. To do this we convolve these two periodic discrete signals:
$$y[n] = \sum_{k = -\infty}^\infty x[k]h[n - k]$$
However this convolution for a certain $n$, e.g. $n=0$, can lead to an $y[0]$ which is infinite, as both $x[n]$ and $h[n]$ are periodic.
When we convolve between the image and the kernel, we are only considering one repetition of $h[n]$ and one repetition of $x[n]$. Like this:
$$y[n] = \sum_{k = 0}^{N-1} x[k]h[n - k]$$
Why can we do this? Will this still give us a periodic discrete $y[n]$ whose representation in the frequency domain is the point-per-point multiplication of the original signal and filter?