näytteenottoteoreemaan

The sampling theorem, known in Finnish as näytteenottoteoreemaan, is a fundamental principle in digital signal processing. It addresses the question of how to accurately represent a continuous-time signal using a discrete sequence of samples. The theorem states that if a signal contains no frequencies higher than f_max Hertz, then it can be perfectly reconstructed from its samples, provided that the sampling rate is greater than twice the highest frequency, i.e., at a rate of f_s > 2 * f_max. This minimum sampling rate, 2 * f_max, is called the Nyquist rate.

If the sampling rate is below the Nyquist rate, a phenomenon called aliasing occurs. Aliasing causes higher frequencies in the original signal to be misrepresented as lower frequencies in the sampled signal, leading to distortion that cannot be removed during reconstruction. Therefore, to avoid aliasing and ensure faithful reconstruction, the sampling rate must be chosen appropriately.

The theoretical basis for the sampling theorem lies in Fourier analysis. A continuous-time signal can be decomposed into its constituent frequencies. The theorem essentially states that if a signal is band-limited, meaning its frequency content is confined to a finite range, then all the information required to reconstruct the original signal is contained within the samples taken at a sufficiently high rate. The reconstruction process typically involves interpolating between the sampled points using a sinc function, which effectively filters out any unwanted higher frequencies that might have been introduced during sampling.