Analytic Signal - Definition

Definition

If x(t) is a real-valued signal with Fourier transform X(f), and u(f) is the Heaviside step function, then the function:

\begin{align} X_\mathrm{a}(f) & \ \stackrel{\mathrm{def}}{=}\ \begin{cases} \ \ 2 X(f), & \mbox{for } f > 0, \\ \ \ X(f), & \mbox{for } f = 0, \\ \ \ 0, & \mbox{for } f < 0, \end{cases} \\ &= X(f)\cdot \underbrace{2 \mathrm{u}(f)}_{1 + \sgn(f)} = X(f) + X(f)\cdot\sgn(f) \end{align}

contains only the non-negative frequency components of X(f). And the operation is reversible, due to the Hermitian property of X(f):

\begin{align} X(f) &= \begin{cases} \ \ \frac{1}{2} X_\mathrm{a}(f), & \mbox{for } f > 0,\\ \ \ X_\mathrm{a}(f) & \mbox{for} f = 0,\\ \ \ \frac{1}{2} X_\mathrm{a}(-f)^*, & \mbox{for } f < 0 \end{cases} \\ &= \frac{1}{2} \left( X_\mathrm{a}(f) + X_\mathrm{a}(-f)^* \right) \ . \end{align}

X(f)* denotes the complex conjugate of X(f) .

The inverse Fourier transform of Xa(f) is the analytic signal:

\begin{align} x_\mathrm{a}(t) &= \mathcal{F}^{-1}\{X(f) + X(f)\cdot\sgn(f)\}\\ &= \mathcal{F}^{-1}\{X(f)\} + \underbrace{\mathcal{F}^{-1}\{X(f)\} * \mathcal{F}^{-1}\{\sgn(f)\}}_{convolution}\\ &= x(t) + j\underbrace{\left}_{\hat{x}(t)}, \end{align}

where is the Hilbert transform of and is the imaginary unit.

Read more about this topic:  Analytic Signal

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