Digital modulation

When \(m(t) \in \mathbb{Z}_2\) as in (2), we have to make the \(0\)s and \(1\)s logical values to some real-world representation of them. These representations are often called symbols.

Any scheme modulating digital data requires at least that

\[ 0 \mapsto s_0(t) \]

and

\[ 1 \mapsto s_1(t) \]

This scheme assumes one bit per symbol.

It’s possible to map multiple bits to a single symbol so, for example:

\[ 00 \mapsto s_{00}(t) \]

\[ 01 \mapsto s_{01}(t) \]

\[ 10 \mapsto s_{10}(t) \]

\[ 11 \mapsto s_{11}(t) \]

would map two bits to each symbol. In this case

\[ m(t) \in \mathbb{Z}_2^2 \]

Some modulation schemes map many more bits to each symbol, \(B\), so that

\[ m(t) \in \mathbb{Z}_2^B. \]

Baseband

Baseband is the case where the signal is not modulated [Rutenbeck, 2012]:

Refers to a basic set of frequencies of an RF signal prior to modulation.

For baseband signals, the usual selection of \(s_0(t)\) and \(s_1(t)\) is

\[\begin{split} \begin{align} s_0(t) &= 0 V \\ s_1(t) &= 5 V \end{align} \end{split}\]

However, we might just as well choose

\[\begin{split} \begin{align} s_0(t) &= 5 V \\ s_1(t) &= 0 V \end{align} \end{split}\]

This choice is sometimes called “inverted logic”.

We might also choose

\[\begin{split} \begin{align} s_0(t) &= -2.5 V \\ s_1(t) &= 2.5 V \end{align} \end{split}\]

because this choice allows \(m(t)\) to be zero mean, provided the number of \(1\)s and \(0\)s are equal.