As far as I understand, you can have two different continuous-time signals with the same discrete-time frequency spectrum after they are sampled and it may be possible these shifted replicas in the discrete frequency domain do not overlap with each other. In textbooks aliasing is associated with overlapping of shifted replicas. is the case I described a case of aliasing?


2 Answers 2


What you describe is indeed aliasing. If there are frequency components above half the sampling frequency then they will be folded back to the interval $[0,f_s/2]$. This folding back is called aliasing, no matter whether these aliased components overlap with other parts of the signal spectrum or not.

Note that you can also use aliasing to your advantage if you're careful that no overlap occurs. An example is bandpass sampling, where you deliberately choose a sampling frequency that is much smaller than twice the highest signal frequency.


I'm not going to be formal about this, but the easy way to think about aliasing is in terms of Laplace vs. z-transform. The sampling process essentially substitutes $z\leftarrow e^{sT}$ (where $T$ is the sampling interval) while the reconstruction does the inverse $s\leftarrow \frac{\ln{z}}{T}$. The way to think about it is that the Fourier transform found on the s-plane imaginary axis is wrapped to get the DTFT on the z-plane unit circle.

Now, for perfect reconstruction to be feasible, we need the logarithm to be unique and the basic version of Nyquist-Shannon sampling theorem then essentially says that if we pick some interval of angles no wider than $2\pi$ this makes the logarithm unique and we're good.

What you are asking then is whether we can take multiple smaller intervals instead. The answer is yes. As long as the logarithm remains unique (ie. the intervals don't overlap modulo $2\pi$) we're good. In theory this is the only requirement, although in practice some transition bands are obviously required (as usual).

Finally to directly answer your question: You can sample multiple intervals at the same time as long as any frequencies from those intervals remain unique after sampling and "aliasing" in the informal sense is really just anything that doesn't belong to any of the chosen intervals.


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