Signal average power (complex signal)

[O.J.]

Homework Statement

Evaluate the average signal power of the complex signal x(t) = e ^ jwt

Homework Equations

The Attempt at a Solution

I know how to solve this but I have a couple of questions on why we have the take the magnitude of the complex signal in the integral? Besides, does a complex signal mean anything in reality? because the way I figured it is complex analysis is just a mathematical tool to help us analyse sinusoidal signals and a signal is always a real or an imaginery part of a complex function (e ^ jwt) and can't be both.

And as such I tried evaluating the integral of e ^ jwt from 0 to To using two ways:

1. write the exponential e ^ jwt in terms of cos and j sin then square that to give cos^2 wt +j2cos (wt) sin (wt) - sin^2 wt where it evaluates to some finite value
2. square the funciton so it becomes e ^ j 2wt and write that in terms of cos and j sin where it evaluates to 0

this is too mathematical I know, but how come the two mathematically valid manipulations yield integrals with different values?

 

[O.J.]

y am i being ignored?

 

[Mene]

I understand your questions and concerns about the calculation of average power for a complex signal. First, let's define what we mean by a complex signal. A complex signal is a signal that has both a real and an imaginary component. In other words, it can be represented by a complex number, where the real part represents the signal's magnitude and the imaginary part represents its phase.

Now, when we are calculating the average power of a complex signal, we are essentially calculating the average value of the squared magnitude of that signal. This is why we take the magnitude of the complex signal in the integral. The magnitude represents the signal's overall power, and squaring it gives us a positive real number that can be averaged.

To answer your question about whether a complex signal means anything in reality, the answer is yes. Complex signals are commonly used in many fields, including signal processing, electrical engineering, and physics. They are used to represent and analyze signals that have both amplitude and phase components, such as electromagnetic waves or audio signals.

Regarding your two different approaches to evaluating the integral, both are mathematically valid, but they yield different results because they are evaluating different things. In the first approach, you are taking the average of the squared magnitude of the complex signal, while in the second approach, you are taking the average of the squared signal itself. These are two different quantities, and therefore, they give different results.

In conclusion, complex signals are an important tool in signal analysis, and understanding their properties and how to calculate their average power is essential for any scientist or engineer working with signals.