Signalling 2.11 2001/06/20 2005/09/19 13:00:41.412 GMT-5 Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Roy Ha rha@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Behnaam Aazhang aaz@ece.rice.edu antipodal biorthogonal orthogonal PAM PPM pulse amplitude modulation pulse position modulation signal simplex (Blank Abstract) Data symbols are "1" or "0" and data rate is 1 T Hertz.

Pulse amplitude modulation (PAM)

Pulse position modulation

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

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

Relevant measures are energy of modulated signals E m m 1 2 … M t 0 T s m t 2 and how different they are in terms of inner products. s m s n t 0 T s m t s n t for m 1 2 … M and n 1 2 … M . antipodal Signals s 1 t and s 2 t are antipodal if t t 0 T s 2 t s 1 t orthogonal Signals s 1 t , s 2 t ,…, s M t are orthogonal if s m s n 0 for m n . biorthogonal Signals s 1 t , s 2 t ,…, s M t are biorthogonal if s 1 t ,…, s M 2 t are orthogonal and s m t s M 2 + m t for some m 1 2 … M 2 . It is quite intuitive to expect that the smaller (the more negative) the inner products, s m s n for all m n , the better the signal set. Simplex signals Let s 1 t s 2 t … s M t be a set of orthogonal signals with equal energy. The signals s 1 ˜ t ,…, s M ˜ t are simplex signals if s m ˜ t s m t 1 M k 1 M s k t If the energy of orthogonal signals is denoted by m m 1 2 ... M E s t 0 T s m t 2 then the energy of simplex signals E s ˜ 1 1 M E s and m n s m ˜ s n ˜ -1 M 1 E 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 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.