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[cosϕ+jsin(ϕ)] |
| Polar | c=Mejϕ |
| Mag-angle | c=M∠ϕ |
| Euler's | $ej φ |
Last is from euler's identity,
These complex exponentials should be viewed as phase transformations, or rotations on the complex plane.
Multiplying by ejπ2=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)=ej2πft 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)=ej2πft
q(f)+q(−f)=2cos2πft
The dual of euler's other identity is q(f)−q(−f)=2jsin2πft
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.