What is the difference in significance of using complex representation in signal processing and communication(transmission)?

Question

I got to know that in signal processing we use \$j\$ to represent a phase of \$\pi/2\$ in between two real signals which are being transmitted. This also conforms to the fact that to find signal energy we find the magnitude first and then square it to time integrate.

But, in transmission lines when we represent a wave as a complex function in time and space, for power delivered to the load we take only the real part of \$VI^∗\$ and half it . So my question is what is the significance of the complex part in the wave and how is it different from signals?

Above is a attached screenshot of a question from a book(Electromagnetic Waves - By R K Shevgaonkar, published by Tata McGraw-Hill, fourth reprint 2010, pg 18-19) which further clarifies my question as we see that only real part of the general wave equation is taken even for calculating voltage and current at a certain instant of time and position in space. So why do we not consider the complex part?

Answer 1

Phasors (complex numbers with an associated unit) are used to represent sinusoidally time-variant signals for purposes of efficient computation. They are used in a great many contexts- for voltages, currents, mechanical displacements, pressure, etc. etc. In all cases a real physical quantity is being represented. At the end of a computation the complex quantity is converted to a real sinusoidally varying quantity by multiplying by e^jwt and taking the real part.

Computing the (real, dissipated) power as 0.5xVxI* is a shortcut to computing i(t) and v(t), multiplying the two, and taking the average over one cycle. Most textbooks do this derivation once and then use the shortcut afterwards.

So phasors are used in exactly the same way and represent real quantities in exactly the same way in different contexts.