# 0.6 Complex-baseband equivalent channel

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In this module, the "complex-baseband equivalent channel" is derived from the three-part cascade of modulation, wideband channel, and demodulation.

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

$\begin{array}{ccc}\hfill \left[s,\left(t\right),*,{h}_{\text{bp}},\left(t\right)\right]2{e}^{-j2\pi {f}_{c}t}& =& \int s\left(\tau \right){h}_{\text{bp}}\left(t-\tau \right)d\tau ·2{e}^{-j2\pi {f}_{c}t}\hfill \\ & =& \int s\left(\tau \right)2{e}^{-j2\pi {f}_{c}\tau }{h}_{\text{bp}}\left(t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\tau \right){e}^{-j2\pi {f}_{c}\left(t-\tau \right)}d\tau \hfill \\ & =& \left[s,\left(t\right),2,{e}^{-j2\pi {f}_{c}t}\right]*\left[{h}_{\text{bp}},\left(t\right),{e}^{-j2\pi {f}_{c}t}\right],\hfill \end{array}$

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 $\stackrel{˜}{m}\left(t\right)$ is bandlimited to W Hz, there is no need to model the left component of ${H}_{\text{bp}}\left(f+{f}_{c}\right)=\mathcal{F}\left\{{h}_{\text{bp}}\left(t\right){e}^{-j2\pi {f}_{c}t}\right\}$ :

Replacing ${h}_{\text{bp}}\left(t\right){e}^{-j2\pi {f}_{c}t}$ with the complex-baseband response $\stackrel{˜}{h}\left(t\right)$ gives the “complex-baseband equivalent” signal path:

The spectrums above show that  ${h}_{\text{bp}}\left(t\right)=Re\left\{\stackrel{˜}{h}\left(t\right)·2{e}^{j2\pi {f}_{c}t}\right\}$ .

Next consider the noise path on it's own:

From the diagram, ${\stackrel{˜}{v}}_{n}\left(t\right)$ is a baseband version of the bandpass noise spectrum that occupies the frequency range $f\in \left[{f}_{c}-{B}_{s},{f}_{c}+{B}_{s}\right]$ .
Since ${\stackrel{˜}{v}}_{n}\left(t\right)$ is complex-valued, ${\stackrel{˜}{V}}_{n}\left(f\right)$ 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 ${\stackrel{˜}{v}}_{n}\left(t\right)$ will be constant over the LPF passband, i.e., $f\in \left[-{B}_{p},{B}_{p}\right]$ :

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 $f\in \left[-W,W\right]$ will get totally suppressed, it doesn't matter how we model it! Thus, we choose to replace the lowpass complex noise ${\stackrel{˜}{v}}_{n}\left(t\right)$ with something simpler to describe: white complex noise $\stackrel{˜}{w}\left(t\right)$ with PSD ${S}_{\stackrel{˜}{w}}\left(f\right)={N}_{0}$ :

We'll refer to $\stackrel{˜}{w}\left(t\right)$ 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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