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The Q -function can be expressed in terms of the error function , or the complementary error function, as . An alternative form of the Q -function known as Craig's formula, after its discoverer, is expressed as: . This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad  for the Q -function of the sum of two non-negative variables, as follows:. The inverse Q -function can be related to the inverse error functions :. It is usually expressed in dB and generally called Q-factor :.
For instance, for QPSK in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y. Some values of the Q -function are given below for reference. The Q -function can be generalized to higher dimensions: . As in the one dimensional case, there is no simple analytical formula for the Q -function.
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