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

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