Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.7 SIGNUM FUNCTION

Definition: Signum Function The signum function is

(B.36)

although usually in practice sgn(0) = 1.

The signum function can also be expressed as

(B.37)

where in the last expression, and its derivative can be written in terms of the Dirac delta function as follows:

(B.38)

For complex-valued x:

(B.39)

where {xR, xI} are the real and imaginary parts of x. Note that (even though the right-hand side is the maximum likelihood (ML) detector for quadrature phase shift keying (QPSK) signals in an additive white Gaussian noise (AWGN) channel: see Chapter 10). Instead, the complex signum function is the point on the unit circle of the complex plane that is closest to x, as depicted in Figure B.7. Complex sgn(x) could be any point on the unit circle as x is varied, whereas sgn(xR)+jsgn(xI) is one of only four points: .

Figure B.7 Signum function for a complex argument: sgn( ...