# 4.1 Signalling

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(Blank Abstract)

Data symbols are "1" or "0" and data rate is $\frac{1}{T}$ Hertz.

## Example

Data symbols are "1" or "0" and the data rate is $\frac{2}{T}$ Hertz.

This strategy is an alternative to PAM with half the period, $\frac{T}{2}$ .

Relevant measures are energy of modulated signals

${E}_{m}=\forall , m\in \{1, 2, \dots , M\}\colon \int_{0}^{T} {s}_{m}(t)^{2}\,d t$
and how different they are in terms of inner products.
${s}_{m}\dot {s}_{n}=\int_{0}^{T} {s}_{m}(t)\overline{{s}_{n}(t)}\,d t$
for $m\in \{1, 2, \dots , M\}$ and $n\in \{1, 2, \dots , M\}$ .

antipodal
Signals ${s}_{1}(t)$ and ${s}_{2}(t)$ are antipodal if $\forall t, t\in \left[0 , T\right]\colon {s}_{2}(t)=-{s}_{1}(t)$
orthogonal
Signals ${s}_{1}(t)$ , ${s}_{2}(t)$ ,…, ${s}_{M}(t)$ are orthogonal if ${s}_{m}\dot {s}_{n}=0$ for $m\neq n$ .
biorthogonal
Signals ${s}_{1}(t)$ , ${s}_{2}(t)$ ,…, ${s}_{M}(t)$ are biorthogonal if ${s}_{1}(t)$ ,…, ${s}_{\frac{M}{2}}(t)$ are orthogonal and ${s}_{m}(t)=-{s}_{\frac{M}{2}+m}(t)$ for some $m\in \{1, 2, \dots , \frac{M}{2}\}$ .

It is quite intuitive to expect that the smaller (the more negative) the inner products, ${s}_{m}\dot {s}_{n}$ for all $m\neq n$ , the better the signal set.

Simplex signals
Let $\{{s}_{1}(t), {s}_{2}(t), \dots , {s}_{M}(t)\}$ be a set of orthogonal signals with equal energy. The signals $\stackrel{˜}{{s}_{1}}(t)$ ,…, $\stackrel{˜}{{s}_{M}}(t)$ are simplex signals if
$\stackrel{˜}{{s}_{m}}(t)={s}_{m}(t)-\frac{1}{M}\sum_{k=1}^{M} {s}_{k}(t)$

If the energy of orthogonal signals is denoted by

$\forall m, m\in \{1, 2, \mathrm{...}, M\}\colon {E}_{s}=\int_{0}^{T} {s}_{m}(t)^{2}\,d t$
then the energy of simplex signals
${E}_{\stackrel{˜}{s}}=(1-\frac{1}{M}){E}_{s}$
and
$\forall , m\neq n\colon \stackrel{˜}{{s}_{m}}\dot \stackrel{˜}{{s}_{n}}=\frac{-1}{M-1}{E}_{\stackrel{˜}{s}}$

It is conjectured that among all possible $M$ -ary signals with equal energy, the simplex signal set results in the smallest probability of error when used totransmit information through an additive white Gaussian noise channel.

The geometric representation of signals can provide a compact description of signals and can simplify performance analysis of communication systems usingthe signals.

Once signals have been modulated, the receiver must detect and demodulate the signals despite interference and noise and decide which of the set ofpossible transmitted signals was sent.

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