# Equivalent baseband communication channel

I was reading on my lecture slides (I do not put them here since they are not in English) this statement about equivalent baseband channel models:

In typical wireless applications, communication occurs in a passband [fc W/2; fc +W/2] of bandwidth W around a center frequency fc. However, most of the processing, such as coding/decoding, modulation/demodulation, synchronization, etc., is actually done at the baseband. Therefore from a communication system design point of view, it is most useful to have a baseband equivalent representation of the system.

Since

it results:

and

and

I do not understand what it wants to get and why. Precisely:

• what does it mean with "equivalent channel model"? I'd say that it means a channel with same input, same output, and same transfer function

• which is the equivalent baseband channel model between the last two pictures?

• So are baseband signal really used inside a channel?

For many cases where you have a passband signal (meaning, a signal with some carrier frequency, $$\f_c\$$, and a signal with bandwidth $$\W\$$ around it, typical in communications systems), analysis is more convenient working with the so-called baseband equivalent version of that signal. It is called baseband equivalent, because it takes out the dependence on $$\f_c\$$, and can be viewed as a sort of baseband signal that is equivalent to the passband signal in a way. In your example, in the frequency domain, the passband is $$\S(f)\$$ and the baseband equivalent is $$\S_b(f)\$$.
What is the baseband equivalent of the passband signal? Your example shows a baseband equivalent signal, $$\s(t)\$$ being derived. Notice that it is no longer real-valued, but complex valued in general. However, it is not a function of $$\f_c\$$ anymore, which helps with analysis. At the output of the system, you can always convert back to passband if desired.