Example 1

Data symbols are "1" or "0" and data rate is 1T 1 T Hertz.

Figure 1

Pulse amplitude modulation (PAM)

Figure 2

Pulse position modulation

Example 2: Example

Data symbols are "1" or "0" and the data rate is 2T 2 T Hertz.

Figure 3

This strategy is an alternative to PAM with half the period, T2 T 2 .

Figure 4

Relevant measures are energy of modulated signals

E m =∫0T s m 2td t   ,   m∈12…M    E m m 1 2 … M t 0 T s m t 2

(1)

and how different they are in terms of inner products.

〈 s m , s n 〉=∫0T s m ⁢t⁢ s n ⁢t*d t s m s n t 0 T s m t s n t

(2)

for m∈12…M m 1 2 … M and n∈12…M n 1 2 … M .

Definition 1: antipodal

Signals s 1 ⁢t s 1 t and s 2 ⁢t s 2 t are antipodal if s 2 ⁢t=− s 1 ⁢t  ,   t∈ 0 T    t t 0 T s 2 t s 1 t

Definition 2: orthogonal

Signals s 1 ⁢t s 1 t , s 2 ⁢t s 2 t ,…, s M ⁢t s M t are orthogonal if 〈 s m , s n 〉=0 s m s n 0 for m≠n m n .

Definition 3: biorthogonal

Signals s 1 ⁢t s 1 t , s 2 ⁢t s 2 t ,…, s M ⁢t s M t are biorthogonal if s 1 ⁢t s 1 t ,…, s M 2 ⁢t s M 2 t are orthogonal and s m ⁢t=− s M 2 + m ⁢t s m t s M 2 + m t for some m∈12…M2 m 1 2 … M 2 .

It is quite intuitive to expect that the smaller (the more negative) the inner products, 〈 s m , s n 〉 s m s n for all m≠n m n , the better the signal set.

Definition 4: Simplex signals

Let s 1 ⁢t s 2 ⁢t… s M ⁢t s 1 t s 2 t … s M t be a set of orthogonal signals with equal energy. The signals s 1 ˜ ⁢t s 1 ˜ t ,…, s M ˜ ⁢t s M ˜ t are simplex signals if

s m ˜ ⁢t= s m ⁢t−1M⁢∑ k =1M s k ⁢t s m ˜ t s m t 1 M k 1 M s k t

(3)

If the energy of orthogonal signals is denoted by

It is conjectured that among all possible M M-ary signals with equal energy, the simplex signal set results in the smallest probability of error when used to transmit 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 using the signals.

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

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Last edited by Elizabeth Gregory on Sep 19, 2005 1:01 pm -0500.