4.1 Signalling

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 $s_m\cdot s_n=0$ 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, $s_m\cdot s_n$ 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 $\widetilde{s_1}(t)$ ,…, $\widetilde{s_M}(t)$ are simplex signals if

$\widetilde{s_m}(t)=s_m(t)-\frac{1}{M}\sum_{k=1}^M s_k(t)$

If the energy of orthogonal signals is denoted by

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.