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When we are dealing with the problem of changing the sampling rate by a noninteger factor like L/M, we first need to the do the upsampling, and then the downsampling.

My question is that under which condition can we reverse this order?

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    $\begingroup$ Are you familiar with the reason behind the ordering of these operations? $\endgroup$ – A_A Feb 18 at 16:35
  • $\begingroup$ I know about the reason why we use a low-pass filter, roughly speaking before the compressor and after the expander. $\endgroup$ – Niousha Feb 18 at 16:42
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The problem with downsampling is that it can be lossy -- since you're reducing the sampling rate, you can introduce aliasing. So, you can reverse the order whenever downsampling does not result in aliasing.

For example, say your discrete-time signal $x[n]$ contains energy in frequencies up to $f_N/3$, where $f_N$ is the Nyquist frequency. Then, downsampling by a factor of 2 will not affect the signal: the "new" Nyquist frequency is $f_N/2$, so there is no aliasing.

See also Which order to perform downsampling and filtering?.

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