0.7 Digital communication

Transmission consists of

1. pulse shaping: $\tilde{m}\left(t\right)={\sum }_{n}a\left[n\right]g(t-nT)$ ,
2. modulation: $s\left(t\right)=Re\left\{\tilde{m}\left(t\right)e^{j2\pi f_{c}t}\right\}$ .

Reception consists of

1. demodulation: $\tilde{v}\left(t\right)=\text{LPF}\left\{2r\left(t\right)e^{-j2\pi f_{c}t}\right\}$ ,
2. filtering: $y\left(t\right)=\tilde{v}\left(t\right)*q\left(t\right)$ ,
3. sampling: $y\left[m\right]=y\left(mT\right)$ .

Building on analog QAM mod/demod components, digital mod adds pulse shaping&demod adds filtering/sampling.

Simplifying via the complex-baseband equivalent channel:

Transmitter pulse shaping is used to convert the symbol sequence $\left\{a\right[n\left]\right\}$ into the continuous message $\tilde{m}\left(t\right)$ :

Thus, $\tilde{m}\left(t\right)$ can be seen to be a superposition of scaled and time-shifted copies of the pulse waveform $g\left(t\right)$ .

Example, if the symbol sequence $\left[a,\left[,0,\right],,,a,\left[,1,\right],,,a,\left[,2,\right],,,a,\left[,3,\right],,,a,\left[,4,\right]\right]$ equals $[1,3,-1,1,3]$ , then the square pulse $g\left(t\right)$ shown below left yields the message $\tilde{m}\left(t\right)$ shown below right.

Receiver filtering (via $q\left(t\right)$ ) has two goals:

1. noise suppression (i.e., SNR improvement),
2. inter-symbol interference (ISI) prevention.

Noise suppression was briefly discussed in Preliminaries and will soon be revisited in more detail. Next we describe ISI.

Realize that, in the ideal digital comm system, the $n^{th}$ output $y\left[n\right]$ would simply equal the $n^{th}$ input $a\left[n\right]$ . But in practice, $y\left[n\right]$ can be corrupted by interference from the other symbols ${\left\{a\left[m\right]\right\}}_{m\ne n}$ , known as “inter-symbol interference,” and noise.

Isi-prevention for the noiseless trivial channel

Consider the idealized system

Thus, the idealized system can be re-drawn as

where

To make $y\left[m\right]=a\left[m\right]$ (i.e., prevent ISI), we need

which is known as the “Nyquist Criterion.” This criterion can be simply stated as $p\left(mT\right)=\delta \left[m\right]$ using

Examples of Nyquist, and non-Nyquist, combined-pulses $p\left(t\right)$ for $\left[a,\left[,0,\right],,,a,\left[,1,\right],,,a,\left[,2,\right],,,a,\left[,3,\right],,,a,\left[,4,\right]\right]=[1,3,-1,1,3]$ :

There is an interesting frequency-domain interpretation. Since

we can see that

So, the time-domain Nyquist criterion $p\left(mT\right)=\delta \left[m\right]$ implies

which in turn implies

In other words, the superposition of ${\left\{\frac{1}{T},P,(f-\frac{k}{T})\right\}}_{k\in \mathbb{Z}}$ must sum to one. This frequency-domain version of the Nyquist Criterionwill soon come in handy...

A popular choice of combined pulse $p\left(t\right)=g\left(t\right)*q\left(t\right)$ is the “raised-cosine pulse” with rolloff parameter $\alpha \in [0,1]$ :

Tradeoff: larger $\alpha \Rightarrow$ less time-spread but more freq-spread:

So, we now know how to design the combined pulse $p\left(t\right)$ .

But what about the individual pulses $g\left(t\right)$ and $q\left(t\right)$ ?

Maximizing snr for isi-free pulses in white noise

Now let's bring the noise back into consideration. Given

we want a $\left\{g\right(t),q(t\left)\right\}$ pair that maximizes the SNR of $y\left[m\right]$ .

Separating the noise and signal contributions to $y\left[m\right]$ via

the SNR can be written

where E _s and E _n are average signal and noise energies.

Here, we treat both $\tilde{w}\left(t\right)$ and $a\left[n\right]$ as random, implying that $y_s\left[m\right]$ and $y_n\left[m\right]$ are both random.

Notice that, with an ISI-free combined pulse $p\left(t\right)$ , we get

so that

where σ _a ² denotes average symbol energy. Next, notice that

so that

Putting these together, we find

Cauchy-Schwarz says

which implies

Noting that SNR doesn't depend on K , we choose $K=1$ . Thus, given pulse $g\left(t\right)$ , the SNR-maximizing receiver filter is

We can write this in the frequency domain as

Summary: For SNR-maximizing ISI-free pulses, we need

1. $G\left(f\right)Q\left(f\right)=P\left(f\right)$ satisfying the Nyquist criterion,
2. $Q\left(f\right)=G^{*}\left(f\right)$ ,

which together imply ${\left|G\left(f\right)\right|}^2$ must satisfy the Nyquist criterion.

One option is $G\left(f\right)=\sqrt{P_{\text{RC}}\left(f\right)}$ , since $P_{\text{RC}}\left(f\right)$ was Nyquist. We call this the “square-root raised cosine” (SRRC) pulse.Working out the details of ${\mathcal{F}}^{-1}\left\{G_{\text{SRRC}}\left(f\right)\right\}$ , we find

At the receiver, we would use $q_{\text{SRRC}}\left(t\right)=g_{\text{SRRC}}^{*}(-t)=g_{\text{SRRC}}\left(t\right)$ ; the latter equality is due to $g_{\text{SRRC}}\left(t\right)$ being real and symmetric.