mathematics//signal processing//sampling//sampling theorem

The sampling theorem is the result of signal theory that a signal whose content lies entirely below half the sampling rate can be reconstructed exactly from its samples, and engineers use it to set the minimum rate of a converter and the cut-off of the filter in front of it. Half the sampling rate is the **Nyquist frequency**, \(f_N=f_s/2\), the boundary between what a stream of samples can represent and what it folds back as aliasing. A signal limited to bandwidth \(B\) therefore needs \(f_s>2B\): a sensor whose useful content ends at 100 Hz needs more than 200 samples per second.

The sampling theorem is the result of signal theory that a signal whose content lies entirely below half the sampling rate can be reconstructed exactly from its samples, and engineers use it to set the minimum rate of a converter and the cut-off of the filter in front of it. Half the sampling rate is the Nyquist frequency, fN=fs/2f_N=f_s/2fN​=fs​/2, the boundary between what a stream of samples can represent and what it folds back as aliasing. A signal limited to bandwidth BBB therefore needs fs>2Bf_s>2Bfs​>2B: a sensor whose useful content ends at 100 Hz needs more than 200 samples per second.

The result is exact, and that is surprising: between two samples the signal could in principle do anything, yet if it holds no frequency above fNf_NfN​ it cannot wiggle between them in more than one way, and a sum of interpolating pulses (the sinc reconstruction) recovers every intermediate value. The guarantee rests entirely on that condition; it says nothing about a signal that has even a little content above fNf_NfN​, which is every real signal, because noise and vibration have no natural upper limit.

Twice the bandwidth is the theoretical floor, and real designs sit well above it.

Physical filters cannot drop from full pass to full stop at one frequency, so practice samples at 5 to 10 times the band of interest and puts an anti-aliasing filter in the gap. A control loop samples faster still, for a reason the theorem does not mention: delay.

The theorem is about reconstruction, and control asks a different question. A loop acts on each sample as it arrives, without rebuilding anything between them, and the age of that sample (half a period on average, plus the converter and the filters) costs phase. That is why a drone's attitude loop, crossing over at a few hertz, runs its inner rate loop at 1 kHz or more, a hundred times above 2B2B2B (sampling rate selection).

Bandwidth means the content that matters, decided by physics. A spike lasting a millisecond carries energy to about 10 kHz, a bearing impact excites resonances of several kilohertz, a tank temperature changes in minutes; the rate follows from that judgment, then from the budget of data and bus (sampling).

The same counting appears in frequency resolution. A record of NNN samples at fsf_sfs​ resolves lines fs/Nf_s/Nfs​/N apart, so the rate fixes the highest frequency seen and the duration fixes how finely it is seen (discrete Fourier transform).