sgncos

sgncos is a mathematical function defined as the sign of the cosine function, written as sgncos(x) = sgn(cos x). Here the sign function sgn(y) returns -1 if y < 0, 0 if y = 0, and 1 if y > 0. Consequently, sgncos evaluates to 1 when cos(x) > 0, to -1 when cos(x) < 0, and to 0 when cos(x) = 0.

As a result, sgncos is a 2π-periodic, even function with discontinuities at points where cos(x) = 0, namely x = π/2 + kπ for integers k. It forms a square-wave-like pattern that alternates between 1 and -1, with instantaneous zeros at the odd multiples of π/2. An equivalent closed form for cos x ≠ 0 is sgncos(x) = cos(x)/|cos(x)|.

In terms of representations, sgncos can be expressed by a Fourier series consisting of odd harmonics: sgncos(x) = (4/π) [cos x − (1/3) cos 3x + (1/5) cos 5x − …], summing over all odd integers. This series converges to the function at all points of continuity and to the average of the left- and right-hand limits at discontinuities.

Applications of sgncos arise in signal processing and digital waveform generation as a simple, zero-mean, binary-valued transformation of a cosine input. It is related to other sign-based functions such as sgn(sin x) and is often used to study non-linear effects and waveform shaping.

See also: sign function, cosine, square wave, Fourier series, sgn(sin x).