Analytic Signal - Examples

Examples

Example 1:, for some real parameter
(The 2nd equality is Euler's formula.)
This is a complex-valued signal with increasing phase (positive frequency).

It also follows from Euler's formula that So comprises both positive and negative frequency components. is just the positive portion. When dealing with simple sinusoids or sums of sinusoids, we can deduce directly, without determining first.

Example 2:
x_\mathrm{a}(t) = \begin{cases} \ \ e^{j |\omega| t}\cdot e^{j\theta}, & \mbox{if} \ \omega > 0, \\ \ \ e^{j |\omega| t}\cdot e^{-j\theta}, & \mbox{if} \ \omega < 0. \end{cases}

The removal of the negative frequency terms is also demonstrated in Example 3. We note that nothing prevents us from computing for a complex-valued But it might not be a reversible representation, because the original spectrum is not symmetrical in general. So except for this example, the general discussion assumes real-valued

Example 3:, for some real parameter

Read more about this topic:  Analytic Signal

Famous quotes containing the word examples:

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)