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Geometric Representation of Modulation Signals

Module by: Tuan Do-Hong

Geometric representation of signals can provide a compact characterization of signals and can simplify analysis of their performance as modulation signals.
Orthonormal bases are essential in geometry. Let s 1 t s 2 t… s M t s 1 t s 2 t … s M t be a set of signals.
Define ψ 1 t= s 1 t E 1 ψ 1 t s 1 t E 1 where E 1 =∫0T s 1 2tdt E 1 t 0 T s 1 t 2 .
Define s 21 =< s 2 , ψ 1 >=∫0T s 2 t ψ 1 t¯dt s 21 s 2 ψ 1 t 0 T s 2 t ψ 1 t and ψ 2 t=1 E 2 ^ s 2 t- s 21 ψ 1 ψ 2 t 1 E 2 ^ s 2 t s 21 ψ 1 where E 2 ^ =∫0T s 2 t- s 21 ψ 1 t2dt E 2 ^ t 0 T s 2 t s 21 ψ 1 t 2
In general
ψ k t=1 E k ^ s k t-∑j=1k-1 s kj ψ j t ψ k t 1 E k ^ s k t j 1 k 1 s kj ψ j t (1)
where E k ^ =∫0T s k t-∑j=1k-1 s kj ψ j t2dt E k ^ t 0 T s k t j 1 k 1 s kj ψ j t 2 .
The process continues until all of the M M signals are exhausted. The results are N N orthogonal signals with unit energy, ψ 1 t ψ 2 t… ψ N t ψ 1 t ψ 2 t … ψ N t where N≤M N M . If the signals s 1 t… s M t s 1 t … s M t are linearly independent, then N=M N M .
The M M signals can be represented as
s m t=∑n=1N s mn ψ n t s m t n 1 N s mn ψ n t (2)
with m∈12…M m 1 2 … M where s mn =< s m , ψ n > s mn s m ψ n and E m =∑n=1N s mn 2 E m n 1 N s mn 2 . The signals can be represented by sm= s m1 s m2 ⋮ s mN s m s m1 s m2 ⋮ s mN
Example 1 
Figure 1
ψ 1 t= s 1 tA2T ψ 1 t s 1 t A 2 T (3)
s 11 =AT s 11 A T (4)
s 21 =-AT s 21 A T (5)
ψ 2 t= s 2 t- s 21 ψ 1 t1 E 2 ^ =-A+ATT1 E 2 ^ =0 ψ 2 t s 2 t s 21 ψ 1 t 1 E 2 ^ A A T T 1 E 2 ^ 0 (6)
Figure 2
Dimension of the signal set is 1 with E 1 = s 11 2 E 1 s 11 2 and E 2 = s 21 2 E 2 s 21 2 .
Example 2 
Figure 3
ψ m t= s m t E s ψ m t s m t E s where E s =∫0T s m 2tdt=A2T4 E s t 0 T s m t 2 A 2 T 4
s1= E s 000 s 1 E s 0 0 0 , s2=0 E s 00 s 2 0 E s 0 0 , s3=00 E s 0 s 3 0 0 E s 0 , and s4=000 E s s 4 0 0 0 E s
∀mn: d mn =|sm-sn|=∑j=1N s mj - s nj 2=2 E s m n d mn s m s n j 1 N s mj s nj 2 2 E s (7)
is the Euclidean distance between signals.
Example 3 
Set of 4 equal energy biorthogonal signals. s 1 t=st s 1 t s t , s 2 t= s ⊥ t s 2 t s ⊥ t , s 3 t=-st s 3 t s t , s 4 t=- s ⊥ t s 4 t s ⊥ t .
The orthonormal basis ψ 1 t=st E s ψ 1 t s t E s , ψ 2 t= s ⊥ t E s ψ 2 t s ⊥ t E s where E s =∫0T s m 2tdt E s t 0 T s m t 2
s1= E s 0 s 1 E s 0 , s2=0 E s s 2 0 E s , s3=- E s 0 s 3 E s 0 , s4=0- E s s 4 0 E s . The four signals can be geometrically represented using the 4-vector of projection coefficients s1 s 1 , s2 s 2 , s3 s 3 , and s4 s 4 as a set of constellation points.
Signal constellation
Figure 4
d 21 =|s2-s1|=2 E s d 21 s 2 s 1 2 E s (8)
d 12 = d 23 = d 34 = d 14 d 12 d 23 d 34 d 14 (9)
d 13 =|s1-s3|=2 E s d 13 s 1 s 3 2 E s (10)
d 13 = d 24 d 13 d 24 (11)
Minimum distance d min =2 E s d min 2 E s

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