When can we use real channel coefficient instead of complex one to understand communication scheme?

Question

For the traditional received signal model $y=hx+n$, the channel coeficient is usually considered as complex number.

But when analyzing the motivation for the Alamouti space time coding scheme here using real channel coeficient, the motivation become very easy to understand and extremely simple.

This just motivate me to use real channel coefficient for understanding communication system to get the gist more easily.

Therefore, my questions are

1/ When can we use real channel coefficient instead of complex one to understand communication scheme ?

2/ Are there any pitfalls to this way of thinking

Thank you for your enthusiasm !

Answer 1

Well, you can decompose any linear scheme into an imaginary and a real part.

But you'd not be making anything easier: Your channel can lead to any phase rotation, so any complex channel coefficient is possible, and so you need to always take into account the interaction between imaginary and real parts of transmit symbols and channel coefficients.

The fact that the explanation there uses real values to make the math for the didactic video easier has no effect on the math.

So:

1. Nowhere I can think of – even if you constrain yourself to only transmit symbols that are real, losing half your spectral efficiency (which is losing as much as you could maximally get through 2×2 MIMO! so you'd never do that), you can't control the channel phase, and thus, the equivalent channel is always complex

2. Pitfalls: see above.