comnum.theory_differential_phase_tail#

comnum.theory_differential_phase_tail(psi, esn0_lin)#

The probability that the phase angle between two vectors of the same SNR perturbed by independent Gaussian noise exceeds psi in absolute value, 0 <= psi <= pi.

The integral form of Pawula, Rice & Roberts (IEEE Trans. Commun. 30(8), 1982, Sec. IV) for equal SNRs, by Simpson’s rule. It is the symbol error rate of M-DPSK for psi = pi / M, and ½ exp(-Es/N0) for psi = pi / 2.

Parameters:

  • psi (float) – the threshold angle (rad).

  • esn0_lin (float | array_like) – Es/N0 of each vector (linear).

Returns:

P(|dθ| > psi) at every Es/N0.

Return type:

float | numpy.ndarray

Raises:

ValueError – for values that are not real numbers.

Examples

>>> import comnum, math
>>> round(comnum.theory_differential_phase_tail(math.pi / 2, 5.0) / (0.5 * math.exp(-5.0)), 8)
1.0