Geometric Representation of Modulation Signals. Digital Modulation involves Choosing a particular signal waveform for transmission for a particular symbol For M possible symbols, the set of all signal waveforms are:.
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Geometric Representation of Modulation Signals
Let {j(t) j = 1,2,…,N} represent a basis ofSsuch that
(1) Any symbol, si(t)
si(t)=
(2) Basis signals are orthogonal to each other in time
(3) Each basis signal is normalized to have unit energy
E =
Basis signals Coordinate system for S
GramSchmidt process systematic way to obtain basis for S
I
ExampleTwo signal
waveforms to
be used for
transmission
The basis signal
One dimensional
Constellation Diagram
Q
/2
M1 =
I
I
0
3/4
/4
3/2
7/4
54
M2 =
QPSK Constellation Diagram
Rotation by /4 obtain new QPSK signal set
Es = 2Eb
grey coded
QPSK signal
si1
si2
10
7π/4
11
5π/4
01
3π/4
00
π/4
binary symbol
grey coded
QPSK signal
si1
si2
10
3π/2
0
11
π
0
01
π/2
0
00
0
0
si(t) = si1,1(t) + si22(t)
Signal Space Characterization of
QPSK Signal Constellations
ithQPSK signal, based on message points (si1, si2) defined in tables
for i = 1,2 and 0 ≤ t ≤ Ts
= possible states forkfork1= n/4
= possible states for kfork1= n/2
I
possible signal transitions
/4 QPSKmodulation
phase jumps
cos w1t
switch
cos w2t
=
sBFSK(t)= vH(t)
binary 1
binary 0
sBFSK(t)= vL(t)
=
Discontinuous Phase FSK
Switching between 2 independent oscillators for binary 1 & 0
single carrier that is frequency modulated using m(t)
sBFSK(t) =
=
where (t) =
Spectrum & Bandwidth of BFSK Signals
Spectrum & Bandwidth of BFSK Signals
General FSK signal and orthogonality
?
?
vH(t) =
vL(t) =
and
then
and
vH(t)vL(t) =
=
=
=
An FSK signal for 0 ≤ t ≤ Tb
vH(t)vL(t) are orthogonal if Δf sin(4πfcTb) = fc(sin(4πΔf Tb)
s(t) =
θ(t) = θ(0) ±
‘+’ corresponds to ‘1’ symbol
‘’ corresponds to ‘0’ symbol
h = deviation ratio of CPFSK
CPFSK Modulation
elimination of phase discontinuity improves
spectral efficiency & noise performance
0 ≤ t ≤ T
2πfct +θ(0) +
= 2πf2 t+θ(0)
f1 =
2πfct +θ(0) 
= 2πf1t+θ(0)
fc=
f2 =
yields
and
thus
h = T(f2 – f1)
To determine fc and h by substitution
At t = T
θ(T) = θ(0) ± πh
kFSK=
symbol‘1’ θ(T)  θ(0) = πh
symbol‘0’ θ(T)  θ(0) = πh
‘1’ sent increases phase of s(t) by πh
‘0’ sent decreases phase of s(t) by πh
FSK modulation index =kFSK (similar to FM modulation index)
θ(t)  (0) rads
3πh
2πh
πh
0
πh
2πh
3πh
0 T 2T 3T 4T 5T 6T t
Phase Tree
θ(t)  (0)
3π
2π
π
0
π
2π
3π
0 T 2T 3T 4T 5T 6T t
θ(t) = θ(0) ±
0 ≤ t ≤ T
Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK
1 0 0 0 0 1 1
fi=
nc = fixed integer
assume fi given by as
si(t) =
0 ≤ t ≤ T
for i = 1, 2
si(t) =
0 ≤ t ≤ T
for i = 1, 2
= 0 otherwise
= 0 otherwise
1(t) =
2(t) =
0 ≤ t ≤ T
0 ≤ t ≤ T
= 0 otherwise
= 0 otherwise
for i = 1, 2
i(t) =
0 ≤ t ≤ T
= 0 otherwise
BFSK constellation: define two coordinates as
let nc = 2 and T = 1s (1Mbps) then f1= 3MHz,f2 = 4MHz
0 ≤ t ≤ T
s1(t) =
= 0 otherwise
0 ≤ t ≤ T
=
=
s2(t) =
= 0 otherwise
2(t)
0
1
1(t)
BFSK Constellation
cos wct
output
+

Decision
Circuit
r(t)
sin wct
Probability of error in coherent FSK receiver given as:
Pe,BFSK =
Coherent BFSK Detector
fH
Envelope
Detector
+

r(t)
output
Tb
Decision
Circuit
Envelope
Detector
Matched Filter
fL
Pe,BFSK, NC =
Noncoherent Detection of BFSK
Average probability of error in noncoherent FSK receiver: