3.7  Baseband Representations of Signals and Passband Channels

The complex-envelope and analytic representation can be used not only for QAM signals, as in Eq. (3.8) for example, but any passband signal. Similarly, it can be used for impulse responses and, therefore, represent a passband channel by its baseband-equivalent channel.

In summary, given the impulse response $h(t)$ of a passband channel (or any passband signal $x(t)$), its analytic-equivalent channel is

which in frequency-domain can be written as

where $H(f)=F\{h(t)\}$.

The baseband-equivalent channel is then given by

which depends on the carrier frequency value $f_c$ and assumes that $h_{\mathrm{\text{bb}}}(t)$ does not have significant energy at frequencies larger than $f_c$. In other words, given $H_{\mathrm{\text{bb}}}(f)=F\{h_{\mathrm{\text{bb}}}(t)\}$, the baseband representation is valid only if $H_{\mathrm{\text{bb}}}(f)=0$, for $|f|>f_c$.

The frequency-domain version of Eq. (3.13) can be written as

or

which is equivalent to Eq. (3.14) when one considers only the frequencies $f>-f_c$.

With these representations of signals and systems, it is possible to define the baseband equivalent system for a passband channel as

or

for $f>-f_c$.

It is also possible to use baseband representations for passband random processes. Hence, the AWGN can have a baseband representation to simplify its analysis and simulation when the involved signals are passband. Note that in these cases the noise is not strictly white but can be considered flat over the frequency range of interest.