# What Does "Reduced Modulo N" mean in this context?

I am trying to understand a piece of notation used in several papers, the simplest/shortest of which is this paper by Crochiere. The equation in question is Equation 7 on the second page:

$$x_m(sR) = \tilde{x}_{((m-sR))_M}(sR)$$

followed by the remark:

where ((n)) denotes n reduced modulo M

In the context of Fourier transforms, DFTs and FFTs, where $$x$$, $$\tilde{x}$$, etc are sampled signals, what does "n reduced modulo M" mean? I've seen that notation elsewhere with essentially the same non-explanation, so it must have been something very obvious in context (and maybe still is) but I am completely failing to grasp it.

• \begin{align} ((n))_N \ &\triangleq \quad n \mod N \\ \\ &= n - N\left\lfloor \frac{n}{N} \right\rfloor \\ \end{align} where $\lfloor x \rfloor$ is the floor() operator, the largest integer that is no greater than the argument $x$. May 5, 2022 at 3:39

It just means that the number in the double parentheses is changed to be between $$0$$ and $$M-1$$ by adding to or subtracting from it $$M$$ an integer number of times.

See equation (5) of the paper:

$$\stackrel{\large x}{\sim}_m(sR) = \sum_{l=-\infty}^{+\infty} x(sR+lm+m)h(-lm-m) \tag{5}$$

so the modulo version is just the time-aliased copies of $$x$$ summed over the interval of interest, $$M$$.

The diagram below attempts to illustrate this with $$M=4$$ (though the image only shows four subsequences, all subsequences should be summed). • Bear with me: So if I had some signal $x(n)$ with thousands of samples, but M was some small number like 16, then I'm just cycling through a set of 16 points even as the index increases well outside that range? May 3, 2022 at 2:46
• @Novak No worries! Please see my addition to attempt to explain the detail of what's happening.
– Peter K.
May 3, 2022 at 14:33
• Ah, I think I see-- that's just a shorthand for the combined operation of "Window, and slide window back to origin". Which, in context, makes sense. Thank you! May 4, 2022 at 0:34