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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 j = 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 17th 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 Ren Descartes in the 17th 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.

Quadrature signals, represented by complex numbers, are used in just about every field of science and engineering. [ ] Quadrature signals are of interest to us because they describe the effects of Fourier analysis as well as the quadrature processing and implementations that take place in modern digital communications systems. In this chapter we'll review the fundamentals of complex numbers and get comfortable with how they're used to represent quadrature signals. Next we'll examine the notion of negative frequency as it relates to quadrature signal algebraic notation, and learn to speak the language of quadrature processing. In addition, we'll use three-dimensional time and frequency-domain plots to clarify and give physical meaning to quadrature signals.

[ ] That's because complex sinusoids are solutions to those second order linear differential equations used to describe so much of nature.

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Quadrature signal formats, also called complex signals , are used in many digital signal processing applications, such as:

• digital communications systems,

• time difference of arrival processing in radio direction finding schemes,

• coherent pulse measurement systems,

• antenna beamforming applications, and

• single sideband modulators.

These applications fall in the general category known as quadrature processing, and they provide additional processing power through the coherent measurement of the phase of sinusoidal signals.

A quadrature signal is a two-dimensional signal whose value at some instant in time can be specified by a single complex number having two parts : what we call the real part and the imaginary part . (The words real and imaginary, although traditional, are unfortunate because of their meanings in our everyday speech. Communications engineers use the terms in-phase and quadrature phase. More on that later.) Let's review the mathematical notation of these complex numbers .

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