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Linear communication schemes (e.g., AM, QAM, VSB) can all be represented (using complex-baseband mod/demod) as:
It turns out that this diagram can be greatly simplified...
First, consider the signal path on its own:
Since $s\left(t\right)$ is a bandpass signal, we can replace the wideband channel response $h\left(t\right)$ with its bandpass equivalent ${h}_{\text{bp}}\left(t\right)$ :
Then, notice that
which means we can rewrite the block diagram as
We can now reverse the order of the LPF and ${h}_{\text{bp}}\left(t\right){e}^{-j2\pi {f}_{c}t}$ (since both are LTI systems), giving
Since mod/demod is transparent (with synched oscillators), it can be removed, simplifying the block diagram to
Now, since $\tilde{m}\left(t\right)$ is bandlimited to W Hz, there is no need to model the left component of ${H}_{\text{bp}}(f+{f}_{c})=\mathcal{F}\left\{{h}_{\text{bp}}\left(t\right){e}^{-j2\pi {f}_{c}t}\right\}$ :
Replacing ${h}_{\text{bp}}\left(t\right){e}^{-j2\pi {f}_{c}t}$ with the complex-baseband response $\tilde{h}\left(t\right)$ gives the “complex-baseband equivalent” signal path:
The spectrums above show that ${h}_{\text{bp}}\left(t\right)=Re\{\tilde{h}\left(t\right)\xb72{e}^{j2\pi {f}_{c}t}\}$ .
Next consider the noise path on it's own:
From the diagram,
${\tilde{v}}_{n}\left(t\right)$ is a baseband version of the bandpass noise
spectrum that occupies the frequency range
$f\in [{f}_{c}-{B}_{s},{f}_{c}+{B}_{s}]$ .
Since
${\tilde{v}}_{n}\left(t\right)$ is complex-valued,
${\tilde{V}}_{n}\left(f\right)$ is
non-symmetric.
Say that $w\left(t\right)$ is real-valued white noise with power spectral density (PSD) ${S}_{w}\left(f\right)={N}_{0}$ . Since ${S}_{w}\left(f\right)$ is constant over all f , the PSD of the complex noise ${\tilde{v}}_{n}\left(t\right)$ will be constant over the LPF passband, i.e., $f\in [-{B}_{p},{B}_{p}]$ :
A well-designed communications receiver will suppress all energy outside the signal bandwidth W , since it is purely noise. Given that the noise spectrum outside $f\in [-W,W]$ will get totally suppressed, it doesn't matter how we model it! Thus, we choose to replace the lowpass complex noise ${\tilde{v}}_{n}\left(t\right)$ with something simpler to describe: white complex noise $\tilde{w}\left(t\right)$ with PSD ${S}_{\tilde{w}}\left(f\right)={N}_{0}$ :
We'll refer to $\tilde{w}\left(t\right)$ as " complex baseband equivalent " noise.
Putting the signal and noise paths together, we arrive at the complex baseband equivalent channel model :
The diagrams above should convince you of the utility of the complex-baseband representation in simplifying the system model!
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