# 0.4 Analog communication

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This module describes basic analog modulation techniques, including amplitude modulation (AM) with suppressed carrier, AM with a pilot tone or carrier tone, quadrature AM (QAM), vestigial sideband modulation (VSB), and frequency modulation (FM). Various demodulation techniques are also discussed, including envelope detection and the discriminator. Application examples include NTSC television and FM radio (both mono and stereo).
1. Amplitude modulation (AM)
3. Vestigial sideband modulation (VSB)
4. Frequency modulation (FM)

## Am with “suppressed carrier”

AM of real-valued message $m\left(t\right)$ (e.g., music) is

Euler's $cos\left(2\pi {f}_{c}t\right)=\frac{1}{2}\left[{e}^{j2\pi {f}_{c}t},+,{e}^{-j2\pi {f}_{c}t}\right]$ then implies

$\begin{array}{ccc}\hfill S\left(f\right)& =& {\int }_{-\infty }^{\infty }m\left(t\right)cos\left(2\pi {f}_{c}t\right){e}^{-j2\pi ft}dt\hfill \\ & =& \frac{1}{2}{\int }_{-\infty }^{\infty }m\left(t\right){e}^{-j2\pi \left(f-{f}_{c}\right)t}dt+\frac{1}{2}{\int }_{-\infty }^{\infty }m\left(t\right){e}^{-j2\pi \left(f+{f}_{c}\right)t}dt\hfill \\ & =& \frac{1}{2}M\left(f-{f}_{c}\right)+\frac{1}{2}M\left(f+{f}_{c}\right).\hfill \end{array}$

Because $m\left(t\right)\in \mathbb{R}$ , know $|M\left(f\right)|$ symmetric around $f=0$ , implying the AM transmitted spectrum below f c is redundant! This motivates the QAM and VSB modulation schemes...

With f c known, AM demodulation can be accomplished by:

For a trivial noiseless channel, we have $r\left(t\right)=s\left(t\right)$ , so that

$\begin{array}{ccc}\hfill v\left(t\right)& =& \text{LPF}\left\{s\left(t\right)·2cos\left(2\pi {f}_{c}t\right)\right\}\hfill \\ & =& \text{LPF}\left\{m\left(t\right)·\underset{1+cos\left(2\pi ·2{f}_{c}t\right)}{\underbrace{2{cos}^{2}\left(2\pi {f}_{c}t\right)}}\right\}\hfill \\ & =& \text{LPF}\left\{m\left(t\right)+m\left(t\right)cos\left(2\pi ·2{f}_{c}t\right)\right\}\hfill \\ & =& m\left(t\right),\hfill \end{array}$

assuming a LPF with passband cutoff ${B}_{p}\ge W$ Hz and stopband cutoff ${B}_{s}\le 2{f}_{c}-W$ Hz:

Note that we've assumed perfectly synchronized oscillators!

When the receiver oscillator has {freq,phase} offset $\left\{\gamma ,\phi \right\}$ :

$\begin{array}{ccc}\hfill v\left(t\right)& =& \text{LPF}\left\{m,\left(t\right),\underset{cos\left(2\pi \gamma t+\phi \right)\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}cos\left(2\pi \left(2{f}_{c}+\gamma \right)t+\phi \right)}{\underbrace{cos\left(2\pi {f}_{c}t\right)·2cos\left(2\pi \left({f}_{c}+\gamma \right)t+\phi \right)}}\right\}\hfill \\ & =& m\left(t\right)\phantom{\rule{-8.53581pt}{0ex}}\underset{\text{time-varying}\phantom{\rule{4.pt}{0ex}}\text{attenuation!}}{\underbrace{cos\left(2\pi \gamma t+\phi \right)}}.\hfill \end{array}$
a freq offset of $\lambda =\frac{\nu {f}_{c}}{c}$ Hz can occur when there is relative velocity of ν m/s between transmitter and receiver.

## Am with “pilot tone” or “carrier tone”

It's common to include a pilot/carrier tone with frequency| f c :

$\begin{array}{ccc}\hfill s\left(t\right)& =& m\left(t\right)cos\left(2\pi {f}_{c}t\right)+\underset{\text{pilot/carrier}\phantom{\rule{4.pt}{0ex}}\text{tone}}{\underbrace{Acos\left(2\pi {f}_{c}t\right)}}\hfill \\ & =& \left[m\left(t\right)+A\right]cos\left(2\pi {f}_{c}t\right)\hfill \\ \hfill S\left(f\right)& =& \frac{1}{2}\left[M,\left(f\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{f}_{c}\right),\phantom{\rule{-0.166667em}{0ex}},+,\phantom{\rule{-0.166667em}{0ex}},M,\left(f\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}{f}_{c}\right),\phantom{\rule{-0.166667em}{0ex}},+,\phantom{\rule{-0.166667em}{0ex}},A,\delta ,\left(f\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{f}_{c}\right),\phantom{\rule{-0.166667em}{0ex}},+,\phantom{\rule{-0.166667em}{0ex}},A,\delta ,\left(f\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}{f}_{c}\right)\right]\hfill \end{array}$

While modern systems choose $A\ll max|m\left(t\right)|$ , many older systems use $A>max|m\left(t\right)|$ , known as “large carrier AM,” allowing reception based on envelope detection :

$\begin{array}{ccc}\hfill v\left(t\right)& =& \frac{\pi }{2}\text{LPF}\left\{\phantom{\rule{0.166667em}{0ex}}|r\left(t\right)|\phantom{\rule{0.166667em}{0ex}}\right\}-A\hfill \\ & \approx & m\left(t\right)\phantom{\rule{1.em}{0ex}}\text{(with}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{trivial}\phantom{\rule{4.pt}{0ex}}\text{channel)}\hfill \end{array}$

where $|·|$ can be easily implemented using a diode.

The gain $\frac{\pi }{2}$ above makes up for the loss incurred when LPFing the rectified signal:

QAM is motivated by unwanted redundancy in the AM spectrum, which was symmetric around f c .

QAM sends two real-valued signals $\left\{{m}_{I}\left(t\right),{m}_{\text{Q}}\left(t\right)\right\}$ simultaneously, resulting in a non-symmetric spectrum.

QAM demodulation is accomplished by:

where the LPF specs are the same as in AM, i.e., passband edge ${B}_{p}\ge W$ Hz and stopband edge ${B}_{s}\le 2{f}_{c}-W$ Hz.

For a trivial channel, we have $r\left(t\right)=s\left(t\right)$ , so that

$\begin{array}{ccc}\hfill {v}_{I}\left(t\right)& =& \text{LPF}\left\{r\left(t\right)·2cos\left(2\pi {f}_{c}t\right)\right\}\hfill \\ & =& \text{LPF}\left\{{m}_{I}\left(t\right)\underset{1+cos\left(4\pi {f}_{c}t\right)}{\underbrace{2{cos}^{2}\left(2\pi {f}_{c}t\right)}}\hfill \\ & & -{m}_{\text{Q}}\left(t\right)\underset{sin\left(4\pi {f}_{c}t\right)}{\underbrace{2sin\left(2\pi {f}_{c}t\right)cos\left(2\pi {f}_{c}t\right)}}\right\}\hfill \\ & =& {m}_{I}\left(t\right)\hfill \\ \hfill {v}_{\text{Q}}\left(t\right)& =& \text{LPF}\left\{-r\left(t\right)·2sin\left(2\pi {f}_{c}t\right)\right\}\hfill \\ & =& \text{LPF}\left\{-{m}_{I}\left(t\right)\underset{sin\left(4\pi {f}_{c}t\right)}{\underbrace{2cos\left(2\pi {f}_{c}t\right)sin\left(2\pi {f}_{c}t\right)}}\hfill \\ & & +{m}_{\text{Q}}\left(t\right)\underset{1-cos\left(4\pi {f}_{c}t\right)}{\underbrace{2{sin}^{2}\left(2\pi {f}_{c}t\right)}}\right\}\hfill \\ & =& {m}_{Q}\left(t\right),\hfill \end{array}$

assuming synchronized oscillators.

When the oscillators are not synchronized, one gets coupling between the I&Q components as well as attenuation of each. Writing the I&Q signals in the “complex-baseband” form

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