sinc1x

Sinc1x is not a standardized mathematical term with one universally accepted definition. In practice, the label may be used informally to denote the sinc function evaluated at an argument scaled by 1, that is, sinc(1x). Because different communities adopt different conventions for the sinc function, the exact form of sinc1x can vary between contexts.

Two common conventions for the sinc function are the unnormalized and normalized forms. The unnormalized sinc is defined as sinc_unnorm(x) = sin(x)/x, with the value at x = 0 defined by the limit sinc_unnorm(0) = 1. The normalized sinc is defined as sinc_norm(x) = sin(pi x)/(pi x), with sinc_norm(0) = 1. If sinc1x is interpreted under the normalized convention, it corresponds to sinc_norm(x); under the unnormalized convention, it corresponds to sinc_unnorm(x). The choice of convention affects locations of zeros and scaling.

Properties and applications common to sinc functions extend to sinc1x under each convention. The function is even, with zeros at nonzero multiples (for sinc_norm: at integer nonzero x; for sinc_unnorm: at x = nπ, n ∈ Z, n ≠ 0). It approaches zero as |x| grows and, in the time domain, serves as the ideal impulse response of a brick-wall low-pass filter. In digital signal processing, sinc-based interpolation (sinc interpolation) uses the sinc function to reconstruct or resample signals. In practice, implementations often use windowed or truncated versions to improve numerical behavior and finite support.