At present, I am learning the theory of operation of resampling for bandlimited periodic discrete signals using $\mathrm{sinc}$ interpolation. I am developing a design flow and having difficulty in understanding a particular expression/equation, hence I am stuck! Here I go:
Suppose we have a samples $x(nT_s)$ of a continuous time signal $x(t)$, where $t$ is time in seconds and $n$ is sample points that ranges over integers and ofcourse $T_s$ is the sampling period. $x(t)$ is bandlimited to $\pm F_s/2$, where $F_s = 1/T_s$.
Now, we know that the original signal $x(t)$ can be reconstructed or synthesized by the samples of $x(nT_s)$ using the equation:
$$ x'(t) = x(nT_s)+H_s(t-nT_s) = x(t), \quad \text{for}\quad n =\ldots,-3,-2,-1,0,1,2,3,\ldots\tag1 $$
and
$$ H_s(t) = \mathrm{sinc}(\pi F_s t) $$
To resample $x(t)$ for the new rate i.e. $F'_s = 1/T'_s$, we have to evaluate equation $(1)$ at integer multiples of $T'_s$ i.e. $nT'_s$.
We know that, when the new sampling rate $F'_s$ is less than the original sampling rate or frequency $F_s$, the lowpass cutoff must be placed below half the new lower sampling rate.
Doubt - Based on the highlighted statement made above, the ideal low pass is given by:
$$ H_s(t) = \min\left\{1,F'_s/F_s\right\}\mathrm{sinc}\left(\pi\min\left\{F'_s,F_s\right\}t\right) \tag2 $$
where the scale factor maintains unity gain in the passband.
What does the equation $(2)$ represent? I understand that, if $F'_s$ is lower than $F_s$, then the filters cut off frequency should be half the $F'_s$, however, I am having trouble understanding this equation $(2)$ and how to benefit from it in the design process. Any explanation regarding this would be appreciated.
Source/Reference:
$\scriptstyle{\textrm{Julius O.Smith & Phil Gosett "A Flexible Sampling Rate Conversion Method" IEEE, 1984}}$.