# Can a cosine modulated complex valued baseband signal be transmitted over physical channels?

I am reading Digital Communication by Lee and Messerschmitt. In Chapter 6 on modulation, the authors say:

"Modulating a complex-valued baseband signal with a cosine wave yields complex-valued passband signals. We can say that this signal is complex-valued in the time domain because it does not have conjugate symmetry about zero frequency. This complex-valued passband signal is not particularly useful because our physical media accepts only real-valued signals."

I can't get this point, can we not transmit complex-valued passband signal over the real communication channels?

In theory, also, I have always seen the process to convert an analytic signal into a real signal before transmission, why it is so? Why I cannot transmit a complex signal as such?

## 1 Answer

There are two interpretations of this problem.

One is that only real voltages and currents can exist in the real world. Complex signals exist to simplify calculations, and in the DSP realm, where computers can easily handle complex numbers.

The other interpretation is that a complex signal is just two real signals, one of which is labeled "imaginary", and where all systems involved agree to use complex arithmetic. Two wires (or two wireless links) are required to transmit a complex signal: one for the real part, and the other for the imaginary part.

These two interpretations are, aside from philosophical matters, functionally the same.

In communications, it is simpler to assume that you have one single wireless link, not two, and that single link is used to transmit a real signal. Then, complex baseband signal is converted to a single real signal with simultaneous amplitude and phase modulation. This model is simple, easy to understand and implement, and thus universally adopted in communications engineering.

• But then how will I define a situation when I can send a sine and cosine carrier wave on a single link and then can demodulate them successfully. Both of them are orthogonal to each other and can be considered as real and imaginary parts of a complex signal. There I do not need two links and the waves are sent simultaneously. Commented Oct 2, 2021 at 16:54
• Yeah, but a quadrature signal (what you describe) requires double the bandwidth of the individual real and imaginary parts of the complex baseband signal. That's why I said you can interpret it in two ways: As a single link with a signal with amplitude and phase modulation (quadrature), or as two links, each with modulation in amplitude only. The transmitted signal is always real, obviously; it's just the interpretation that is slightly different.
– MBaz
Commented Oct 2, 2021 at 17:11