0.6 Complex-baseband equivalent channel

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  $h_{\text{bp}}\left(t\right)=Re\{\tilde{h}\left(t\right)·2e^{j2\pi f_{c}t}\}$ .

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!