Abtastindex

Abtastindex is the index used to enumerate samples in a discretely sampled signal. In German-language signal processing texts, the discrete signal is typically denoted x[n], where n is an integer referred to as the Abtastindex. The continuous-time signal x(t) is converted to the discrete sequence x[n] by sampling at times t = nT, with T being the sampling period and f_s = 1/T the sampling frequency. The Abtastindex thus provides a direct link between the discrete sequence and physical time.

The Abtastindex also underpins common DSP conventions and formulas. Algorithms such as digital filtering, convolution, and resampling are expressed in terms of sums and operations on samples indexed by n. In the frequency domain, the discrete-time representation involves operations that depend on the Abtastindex, for example through exponentials e^{-jωn} in the discrete-time Fourier transform. The index itself is dimensionless; any physical time information comes from the sampling period T associated with the index.

Convention about the starting value of n varies by source; it is common to use n = 0, but some texts begin at n = 1. This choice affects initial conditions and alignment when implementing filters or reconstructing signals, so clear documentation of the starting Abtastindex is important.

Example: a sinusoid x[n] = sin(2π f0 n T) uses the Abtastindex n to specify which sample corresponds to which point in time, with f0 constrained by the Nyquist criterion (f0 ≤ f_s/2).

See also: sampling theorem, discrete-time signal, sampling period, digital filter, discrete Fourier transform.