# A question about aliasing and sampling

Imagine an ADC unit with a sampling rate fs is not used with an anti-aliasing filter and the sampled data is logged. If in FFT of this logged file one sees a suspicious frequency component call it fx, what can we say about the value of the down sampled high frequency component fh which appears as a false fx? Mathematically speaking.

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One formula is

fx = |fh - N fs|

or if you prefer

fx = abs(fh - N fs)

where N is whatever integer you need to get a result between 0 and fs.

If fh < fs, then N is 0 and there's no aliasing. If fs/2 < fh < fs, then N is 1 and you get "folding" of the frequency.

If in FFT of this logged file one sees a suspicious frequency component call it fx, what can we say about the value of the down sampled high frequency component fh which appears as a false fx?

• Firstly, if you don't have an adequate anti-alias filter then all you can have are "suspicions".
• Secondly, you cannot say it is down-sampled because you have no other reference point. In other words it could be a proper signal in the baseband or it could be an aliased artefact
• Thirdly, you cannot say it is a false signal.

The above image shows how two signals at frequencies H1 and H2 are converted into the base band by aliasing (mirror imaging about the nyquist frequency FN). You cannot tell if they are real base band signals or aliased images that were mirrored from above the nyquist frequency.