Analytic Signals

The real sinusoid $x(t) = A\cos(\omega t + \phi)$ can be converted to an analytic signal, by generating a phase quadrature component,

$\displaystyle y(t) = A\sin(\omega t + \phi),$

to serve as the imaginary part.

Consider the positive and negative frequency components of a real sinusoid at frequency $\omega_0$ :

$\displaystyle x_{+}$ $\displaystyle \triangleq$ $\displaystyle e^{j\omega_0 t}$

$\displaystyle x_{-}$ $\displaystyle \triangleq$ $\displaystyle e^{-j\omega_0 t}.$
Apply a phase shift of $-\pi/2$ radians to the positive-frequency component,

$\displaystyle y_{+} = e^{-j\pi/2}e^{j\omega_0 t} = -je^{j\omega_0 t}$
and a phase shift of $\pi/2$ to the negative-frequency component,

$\displaystyle y_{-} = e^{j\pi/2}e^{-j\omega_0 t} = je^{-j\omega_0 t}.$
Form a new complex signal by adding them together:

$\displaystyle z_{+}(t)$ $\displaystyle \triangleq$ $\displaystyle x_{+}(t) + jy_{+}(t) = e^{j\omega_0t} - j^2e^{j\omega_0t}=2e^{j\omega_0t}$

$\displaystyle z_{-}(t)$ $\displaystyle \triangleq$ $\displaystyle x_{-}(t) + jy_{-}(t) = e^{-j\omega_0t} + j^2e^{-j\omega_0t} = 0.$

``Music 270a: Complex Exponentials and Spectrum Representation'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).

$\displaystyle x_{+}$	$\displaystyle \triangleq$	$\displaystyle e^{j\omega_0 t}$
$\displaystyle x_{-}$	$\displaystyle \triangleq$	$\displaystyle e^{-j\omega_0 t}.$

$\displaystyle z_{+}(t)$	$\displaystyle \triangleq$	$\displaystyle x_{+}(t) + jy_{+}(t) = e^{j\omega_0t} - j^2e^{j\omega_0t}=2e^{j\omega_0t}$
$\displaystyle z_{-}(t)$	$\displaystyle \triangleq$	$\displaystyle x_{-}(t) + jy_{-}(t) = e^{-j\omega_0t} + j^2e^{-j\omega_0t} = 0.$