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

Famous quotes containing the word definition:

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)