# quadrature signal

## Table of Contents

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 | $e^{j φ} |

Last is from euler's identity,

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.