quadrature signal

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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,

\begin{equation} e^{j\phi} = \cos{\phi} + j\sin{\phi} e^{-j\phi} = \cos{\phi} - j\sin{\phi} \end{equation}

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.

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