Näytteenottoteorema

Näytteenottoteorema, also known as the sampling theorem, is a fundamental principle in digital signal processing that describes the conditions under which a continuous-time signal can be perfectly reconstructed from its discrete samples. It states that if a signal has a maximum frequency component (bandwidth) of f_max, then it can be uniquely represented by samples taken at a rate (sampling frequency) f_s, provided that f_s > 2 * f_max. This minimum sampling rate, 2 * f_max, is called the Nyquist rate.

The theorem implies that if the sampling rate is too low, information about the original signal will be lost, leading to aliasing. Aliasing occurs when high-frequency components of the signal are misinterpreted as lower frequencies after sampling, distorting the reconstructed signal. To avoid aliasing, the sampling rate must be at least twice the highest frequency present in the signal.

Conversely, if the sampling rate is sufficiently high, the original continuous-time signal can be perfectly recovered from its discrete samples using appropriate reconstruction filters. These filters, often low-pass filters, effectively remove any frequencies above half the sampling rate, ensuring that only the original signal's components are retained.

The sampling theorem is crucial for the conversion of analog signals, such as audio or video, into digital formats. It underpins technologies like digital audio recording, digital television, and telecommunications, enabling the efficient storage, transmission, and processing of real-world information. Understanding the sampling theorem is essential for anyone working with digital representations of continuous signals.