# Baseband Signals have zero imaginary component

I was revising a few basics on communication systems and I stumbled upon this line that I thought was incorrectly articulated.

I thought that baseband signals were complex and passband signals were real. If $$x_z (t)$$ is the baseband signal with in-phase component $$x_I (t)$$ and quadrature component $$x_Q (t)$$ such that $$x_z (t) = x_I (t) + j x_Q(t)$$ then passband signal is $$x_c (t) =\sqrt{2} Re(x_z (t)e^{2\pi f_c t})$$ which would imply that baseband signals may have non zero imaginary component and passband signals cannot have non-zero imaginary component.

But still the following statement from the book(Fundamentals of Communication System by Michael P.Fitz Chapter:Digital Communication Basics) seems to say it the other way around:

The only caveat that needs to be stated is that baseband data communication will always have a zero imaginary component, while for bandpass communication the imaginary component of the complex envelope might be nonzero.

Is this statement incorrect or am I misconstruing it somehow?

In signal processing, "baseband" has two related meanings.

The original notion of baseband (which is still used) is a real signal originating from some real process -- i.e., a voice recording, a video stream, a sensor's output over time. These processes are all characterized by having significant content down to zero Hz (or really close, in the case of audio).

Because this sort of "baseband" means, by implication, a real-valued signal, its Fourier transform must consist of a real part that's even-symmetric around zero (with, potentially, a real nonzero DC value), and an imaginary part that's odd-symmetric around zero (with, by definition, a zero DC value).

Then folks started using I/Q modulation & demodulation schemes. So you either generate a complex-valued signal* and up-convert directly to your desired RF frequency using a pair of carriers 90 degrees out of phase with each other, or you take an RF signal and demodulate it with those quadrature carriers and generate a baseband pair of signals that act like a real and an imaginary part.

With I/Q schemes, you can have an "imaginary" part that has non-zero DC content, because what you're representing is a real-valued signal that's been frequency shifted to some carrier frequency $$f_c$$, and you have spectra centered around $$\pm f_c$$ -- which resulting signal is single-valued, real, and obeys the rules for the Fourier transform of such a signal.

* Strictly speaking, an inphase/quadrature valued signal that acts like it's complex, but I'm picking nits here.

You're right, but you've misread the sentence:

Baseband Data Communication means that there's no $$f_c$$.

For example, your USB cable to your keyboard, that's baseband communication. It can only carrier real-valued signals, so you only get to choose one dimension, not two, of signal freedom.

This is opposed to passband systems, where you can modulate the baseband signal up to a carrier frequency, which means your negative (relative to $$f_c$$) frequency spectrum doesn't have to have the same content as the positive.

• So the original message signal is real and the modulation process converts the signal to a complex envelope which is then imposed on to the carrier....where the signal is real again? – Orpheus Jan 9 at 11:35
• there is no modulation process in baseband data communications! You're trying to apply mixing to something that just never gets mixed. – Marcus Müller Jan 9 at 11:37
• I get it now...they are two different modes of communication. Thanks a lot @Marcus Müller – Orpheus Jan 9 at 11:40