Geometric Representation of Modulation Signals2.132001/06/012005/09/19 13:36:15.537 GMT-5BehnaamAazhangaaz@ece.rice.eduDineshRajandinesh@ece.rice.eduMohammadJaberBorranmohammad@ece.rice.eduRoyHarha@rice.eduShawnStewartmrshawn@alumni.rice.eduBehnaamAazhangaaz@ece.rice.edubasesbasisgeometric representationorthogonalsignalGeometric 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
s1ts2t…sMt
be a set of signals.
Define
ψ1ts1tE1
where
E1t0Ts1t2.
Define
s21s2ψ1t0Ts2tψ1t
and
ψ2t1E2^s2ts21ψ1
where
E2^t0Ts2ts21ψ1t2
In general
ψkt1Ek^sktj1k1skjψjt
where
Ek^t0Tsktj1k1skjψjt2.
The process continues until all of the M signals are exhausted. The results are
N orthogonal signals with unit
energy,
ψ1tψ2t…ψNt
where
NM.
If the signals
s1t…sMt
are linearly independent, then
NM.
The M signals can be represented
as
smtn1Nsmnψnt
with
m12…M
where
smnsmψn
and
Emn1Nsmn2.
The signals can be represented by
smsm1sm2⋮smNψ1ts1tA2Ts11ATs21ATψ2ts2ts21ψ1t1E2^AATT1E2^0
Dimension of the signal set is 1 with
E1s112
and
E2s212.
ψmtsmtEs
where
Est0Tsmt2A2T4s1Es000,
s20Es00,
s300Es0, and
s4000Esmndmnsmsnj1Nsmjsnj22Es
is the Euclidean distance between signals.
Set of 4 equal energy biorthogonal signals.
s1tst,
s2ts⊥t,
s3tst,
s4ts⊥t.
The orthonormal basis
ψ1tstEs,
ψ2ts⊥tEs
where
Est0Tsmt2s1Es0,
s20Es,
s3Es0,
s40Es. The four signals can be geometrically represented using the
4-vector of projection coefficients
s1,
s2,
s3, and
s4 as a set of constellation points.
Signal constellationd21s2s12Esd12d23d34d14d13s1s32Esd13d24
Minimum distance
dmin2Es