# Is digital up and down sampling linear/causal/time-invariant?

So I am trying to determine whether the process of up sampling (zero padding) and downsampling are linear, causal and/or time-invariant.

Based on some resources I found online, I am under the impression that it is linear, but not time-invariant. However I am unsure about causality (since it depends neither on future nor past values?).

## 1 Answer

Upsampling and downsampling are both causal, since the output at time $m$ depends only on samples you've seen up time $m$.

It is also linear (apply upsampling/downsampling to a x[n]+y[n] and see that you get a upsample/downsample x[n] + upsample/downsample y[n]).

Downsampling not time invariant though - what happens if you downsample by 2 and consider inputting x[n] and x[n] delayed by 1 time unit? Now, what about upsampling?