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Course by: Tuan Do-Hong. E-mail the author

# Geometric Representation of Modulation Signals

Module by: Behnaam Aazhang. E-mail the author

Summary: Geometric representation of signals provides a compact, alternative characterization of signals.

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 2td t E 1 t 0 T s 1 t 2 .

Define s 21 = s 2 , ψ 1 =0T s 2 t ψ 1 t¯d t 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 t2d t 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 =1k1 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 =1k1 s kj ψ j t2d t 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 NM 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 m12M 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 s m = s m1 s m2 s mN s m s m1 s m2 s mN

## Example 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 t)1 E 2 ^ =(A+ATT)1 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)

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

ψ m t= s m t E s ψ m t s m t E s where E s =0T s m 2td t =A2T4 E s t 0 T s m t 2 A 2 T 4

s 1 = E s 000 s 1 E s 0 0 0 , s 2 =0 E s 00 s 2 0 E s 0 0 , s 3 =00 E s 0 s 3 0 0 E s 0 , and s 4 =000 E s s 4 0 0 0 E s

m n : d mn =| s m s n |= 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 2td t E s t 0 T s m t 2

s 1 = E s 0 s 1 E s 0 , s 2 =0 E s s 2 0 E s , s 3 = E s 0 s 3 E s 0 , s 4 =0 E s s 4 0 E s . The four signals can be geometrically represented using the 4-vector of projection coefficients s 1 s 1 , s 2 s 2 , s 3 s 3 , and s 4 s 4 as a set of constellation points.

d 21 =| s 2 s 1 |=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 =| s 1 s 3 |=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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