Glossary term
Sampling Theorem
The principle that a band-limited signal can be reconstructed from samples taken above twice its highest frequency.
Definition
theoremThe sampling theorem states that an ideal band-limited signal can be reconstructed exactly from uniformly spaced samples when the sampling frequency is greater than twice the signal bandwidth.
The theorem defines the Nyquist condition for converting continuous-time signals into discrete-time data without aliasing. It underpins digital communications, instrumentation, audio, control, image acquisition, radar, and embedded data logging, but its exact reconstruction claim depends on ideal band limitation, uniform sampling, and suitable anti-alias filtering.
For a signal whose highest frequency component is B, the ideal sampling condition is:
where f_s is the sampling frequency. The frequency f_s/2 is called the Nyquist frequency. Components above the Nyquist frequency fold back into the sampled spectrum as lower-frequency components; this distortion is aliasing and cannot be removed after sampling because the sampled data no longer contain enough information to distinguish the original frequency.
Engineering interpretation
The theorem is exact only for ideal band-limited signals and ideal reconstruction filters. Real signals are not perfectly band-limited, and real filters have finite roll-off. Practical systems therefore sample above the theoretical minimum and use an analog anti-alias low-pass filter before the analog-to-digital converter. The sampling rate is chosen from signal bandwidth, filter transition band, timing jitter, latency, storage, processing cost, and required measurement uncertainty.
Sampling is not the same as quantization. Sampling discretizes time; quantization discretizes amplitude. A data-acquisition system can satisfy the sampling theorem and still have poor accuracy because of insufficient bit depth, noise, poor grounding, aperture jitter, sensor bandwidth, or front-end distortion.
Use in design
The sampling theorem guides oscilloscope setup, sensor logging, audio digitization, radio receivers, vibration monitoring, control systems, and digital signal processing. In rotating machinery, for example, the sample rate must cover not only shaft speed but also harmonics, bearing defect frequencies, transient events, and the bandwidth of the anti-alias filter.
Common mistakes
A common mistake is sampling at exactly twice the highest expected frequency and assuming the result is safe. In practice, filter roll-off, clock jitter, spectral leakage, uncertain bandwidth, and transient content require margin. Another mistake is using a high digital sample rate while leaving an analog front end that passes out-of-band noise into the converter. A strong sampling review states signal bandwidth, anti-alias filter cutoff and order, sampling frequency, clock accuracy, ADC resolution, expected noise, and reconstruction or analysis method.