If your signal is not band-limited prior to sampling, then without any further information (such as a copy of the signal sampled at a time offset, which could synthesize a higher sampling rate), irreversible “damage” is done to the waveform via aliasing: each sample contains energy from all Nyquist zones and the FFT will properly reflect this result. It is simply a many to one mapping so nothing further can be done without other information. An analogy is x+ y = 5; we know the sun but without a second formula we cannot distinguish x from y —- so in this case without other information about the signal we cannot know what Nyquist zone in the continuous time domain the signal originated. Other information can be what filtering was used or other copies of the waveform with different sampling that was done concurrently (or if we know other details of the waveform such as ergodicity and stationarity). In most cases we are sampling the waveform to determine its characteristics but there are situations where other information can be used. This last point can be more confusing and should not distract from truly understanding the aliasing mechanism and need for band selection filtering prior to sampling.

The bottom line is generalized typical cases for waveform capture is that the signal must be band-limited to be within one Nyquist zone prior to sampling  (including the possibility of higher Nyquist zones when the analog input bandwidth of the ADC allows for bandpass sampling).

The same condition exists when decimating or resampling the signal- prior to down-sampling, the higher sampled waveform must first be band-limited to be within one Nyquist zone for the lower sampling rate.

If your signal is not band-limited prior to sampling, then without any further information (such as a copy of the signal sampled at a time offset, which could synthesize a higher sampling rate), irreversible “damage” is done to the waveform via aliasing: each sample contains energy from all Nyquist zones and the FFT will properly reflect this result. It is simply a many to one mapping so nothing further can be done without other information. An analogy is x+ y = 5; we know the sum but without a second formula we cannot distinguish x from y —- so in this case without other information about the signal we cannot know what Nyquist zone in the continuous time domain the signal originated. "Other information" can be what filtering was used or other copies of the waveform with different sampling that was done concurrently (or repeatably sampled if we know other details of the waveform such as ergodicity and stationarity). In most cases we are sampling the waveform to determine its characteristics but noting that there are situations where other information can be used. This last point can be more confusing so should not distract from understanding the aliasing mechanism when sampling signals and understanding the need for band selection filtering prior to sampling.

The bottom line to avoid aliasing is that the signal must be band-limited to be within one Nyquist zone prior to sampling (including the possibility of bandpass filtering higher Nyquist zones when the analog input bandwidth of the ADC allows for bandpass sampling).

The same condition exists when decimating or resampling the signal- prior to down-sampling, the higher sampled waveform must first be band-limited to be within one Nyquist zone for the lower sampling rate.

If your signal is not band-limited prior to sampling, then without any further information (such as a copy of the signal sampled at a time offset, which could synthesize a higher sampling rate), irreversible “damage” is done to the waveform via aliasing: each sample contains energy from all Nyquist zones and the FFT will properly reflect this result.

The bottom line is that the signal must be band-limited to be within one Nyquist zone prior to sampling (including the possibility of higher Nyquist zones when the analog input bandwidth of the ADC allows for bandpass sampling).

The same condition exists when decimating or resampling the signal- prior to down-sampling, the higher sampled waveform must first be band-limited to be within one Nyquist zone for the lower sampling rate.

If your signal is not band-limited prior to sampling, then without any further information (such as a copy of the signal sampled at a time offset, which could synthesize a higher sampling rate), irreversible “damage” is done to the waveform via aliasing: each sample contains energy from all Nyquist zones and the FFT will properly reflect this result. It is simply a many to one mapping so nothing further can be done without other information. An analogy is x+ y = 5; we know the sun but without a second formula we cannot distinguish x from y —- so in this case without other information about the signal we cannot know what Nyquist zone in the continuous time domain the signal originated. Other information can be what filtering was used or other copies of the waveform with different sampling that was done concurrently (or if we know other details of the waveform such as ergodicity and stationarity). In most cases we are sampling the waveform to determine its characteristics but there are situations where other information can be used. This last point can be more confusing and should not distract from truly understanding the aliasing mechanism and need for band selection filtering prior to sampling.

The bottom line is generalized typical cases for waveform capture is that the signal must be band-limited to be within one Nyquist zone prior to sampling (including the possibility of higher Nyquist zones when the analog input bandwidth of the ADC allows for bandpass sampling).

The same condition exists when decimating or resampling the signal- prior to down-sampling, the higher sampled waveform must first be band-limited to be within one Nyquist zone for the lower sampling rate.