NyquistShannonstelling

NyquistShannonstelling, more commonly called the Nyquist–Shannon sampling theorem, is a foundational result in digital signal processing. It states that a continuous-time signal that is bandlimited to a maximum frequency B can be exactly reconstructed from its samples if the sampling rate f_s is greater than 2B. The quantity 2B is known as the Nyquist rate. This theorem underpins how analog signals are converted to digital form without loss of information, provided certain conditions are met.

Formally, if a signal x(t) has no spectral content above B Hz and is sampled at intervals T = 1/f_s with x[n] = x(nT), then x(t) can be recovered from the samples by ideal interpolation. In the time domain this reconstruction is achieved with the sinc interpolation formula x_hat(t) = sum_{n=-∞}^{∞} x[n] sinc((t - nT)/T). In the frequency domain, sampling creates copies of the spectrum; when f_s > 2B these copies do not overlap, allowing a perfect reconstruction by an ideal low-pass filter with cutoff B.

Practical applications require considerations beyond the ideal theorem. Real signals are not perfectly bandlimited, so anti-aliasing filters are used before sampling. Reconstruction with ideal sinc functions is not implementable, so engineers use practical filters and interpolation methods that approximate the ideal result. Quantization, noise, and nonuniform sampling introduce errors. Oversampling and advanced ADC architectures, such as delta-sigma converters, exploit these ideas to improve performance.

Historically, the theorem bears the names of Harry Nyquist, who studied sampling rates, and Claude Shannon, who formalized it in the mid-20th century. The Nyquist–Shannon sampling theorem applies broadly to one-dimensional time signals and extends to multi-dimensional signals under analogous conditions.