A quadrature signal is a complex signal.

It is represented by two values, a real part and complex part.

The following are equivalent notations

 Rectangular $$c = a + jb$$ Trig $$c = M[\cos{\phi} + j\sin(\phi)]$$ Polar $$c = Me^{j\phi}$$ Mag-angle $$c = M \angle \phi$$ Euler's \$ej φ

Last is from euler's identity,

$$e^{j\phi} = \cos{\phi} + j\sin{\phi} e^{-j\phi} = \cos{\phi} - j\sin{\phi}$$

These complex exponentials should be viewed as phase transformations, or rotations on the complex plane.

Multiplying by $$e^{j\frac{\pi}{2}} = j$$ rotates a number 90 degrees (lit, pi/2 rads) on the complex plane.

Quadrature signals have some rotation aspect, given a frequency and are a function of time. Signal $$s(f, t) = e^{j2\pi ft}$$ where $$f$$ is freq.

As time increases, the exponent increases so the rotation factor increases.

## 1. Euler's identity

From now, let $$q(f)$$ be the function $$s(t) = e^{j2\pi ft}$$

$q(f) + q(-f) = 2\cos{2\pi f t}$. Think of this as two phasors rotating in opposite directions. The y-values cancel leaving only real values.

The dual of euler's other identity is $q(f) - q(-f) = 2j\sin{2\pi ft}$.

The result that cos and sin can be independently recovered from a single complex phasor (they are functional bases) is central to quadrature shift keying.