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I am wondering about physical interpretation of energy / power for complex signals. E.g. for signal energy definition: \begin{equation} E=\int\limits_{-\infty}^{\infty}|x(t)|^2 \, \mathrm{d}t \end{equation} For $x(t)$ real, when representing voltage (or current) waveform, above definition would give energy dissipated across a 1 ohm resistor. Does such intepretation exist for complex $x(t)$?

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3 Answers 3

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If you have a complex signal

$$x(t)=x_R(t)+jx_I(t)$$

with real-valued $x_R(t)$ and $x_I(t)$, we have

$$|x(t)|^2=x_R^2(t)+x_I^2(t)$$

and, consequently, the energy of the complex signal $x(t)$ is just the sum of the energies of its real and imaginary parts (assuming they are energy signals, i.e., the corresponding integrals converge):

$$E_x=\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{-\infty}^{\infty}x^2_R(t)dt+\int_{-\infty}^{\infty}x^2_I(t)dt=E_{x_R}+E_{x_I}$$

If that makes physical sense or not depends on how you defined the complex signal.

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Yes and no.

"No": there are no physical signal that are "complex". Complex is a mathematical construct, not a physical one. For convenience, we can use complex numbers to represent physical signal. For example we often use complex phasors to represent voltages and currents but that doesn't make the signals themselves complex.

"Yes": you can always create a complex signal by combining two real signals and call one the real part and one the imaginary part. Whether that has a meaningful physical interpretation depends on the signal and why you combine them.

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  • $\begingroup$ Hil, I am in agreement with you, but like other issues (such as the meaning of the DFT), I have gotten into fights about whether or not there exist physical quantities that are complex. I agree with you that there do not. But some folks just as adamantly assert that there are. $\endgroup$ Commented Jun 20, 2023 at 17:51
  • $\begingroup$ For me, complex signals are just as physical as real signals (or both equally not physical, take your pick). Both can be described as are mathematical constructs. Point is there is no difference to me in distinguishing the two as being different with either "physical" or "mathematical" descriptions. If I represent a real signal as voltage levels or digital counts on a single trace, or I represent a complex signal as voltage levels or digital counts on two traces-- more generally math or physical a real wfm is a single stream of real numbers, and a complex wfm is 2 streams of real numbers. $\endgroup$ Commented Jun 21, 2023 at 3:50
  • $\begingroup$ Yep, that's probably more of a philosophical than a physical question. You can call any pair of real signals a complex signal if you want to . $\endgroup$
    – Hilmar
    Commented Jun 21, 2023 at 12:29
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Adding an additional answer, the solution may be simpler and more intuitive in polar form (magnitude and angle) as the energy is determined by the complex conjugate product, where with the real signal it is simply the product. Consider the simpler case of the power of a DC signal for the real and complex case:

Real DC Signal (product)

$$P= V^2/R$$

Complex DC Signal (complex conjugate product)

$$P = \frac{VV^*}{R}$$

The same would apply in the OP's case where for a complex signal we would have:

$$|x(t)|^2 = x(t)x^*(t)$$

And therefore:

\begin{equation} E=\int\limits_{-\infty}^{\infty}x(t)x^*(t) \, \mathrm{d}t \end{equation}

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