Chapter Eight. Quadrature Signals

Quadrature signals are based on the notion of complex numbers. Perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j-operator, complex, analytic, imaginary, real, and orthogonal. If you’re a little unsure of the physical meaning of complex numbers and the operator, don’t feel bad because you’re in good company. Sixteenth-century Italian mathematician Girolamo Cardano described an imaginary number “as subtle as it is useless.” In the seventeenth century Gottfried Leibniz described imaginary numbers as being “amphibian, halfway between existence and nonexistence.” (The term imaginary was first used by the brilliant mathematician/philosopher René Descartes in the seventeenth century and was meant to be derogatory. That’s because not only was the notion of the square root of a negative number dubious at best, surprisingly there was no consensus at that time as to the true meaning of negative real numbers.) Even Karl Gauss, one the world’s greatest mathematicians, called the j-operator the “shadow of shadows.” Here we’ll shine some light on that shadow so you’ll never have to call the Quadrature Psychic Hotline for help.