SME
Communication Systems
Digital modulation
Use this for “A 16-QAM link sends 1 Msymbol/s with roll-off 0.25. Find the bit rate and bandwidth.”
Diagram
16-QAM: 16 points, log₂16 = 4 bits each, Gray-coded so the neighbours across a dashed boundary differ in one bit.
Rb = Rₛ × log₂M = 1 × 4 = 4 Mb/s.
B
=
Rₛ(1
+
α)
=
1
×
(1
+
0.25)
=
1.25
MHz.
η
=
Rb
B
=
3.2
b/s/Hz.
This is an example. Type your own values to start a new problem.
k
=
log₂
M
Rb (Mb/s)
=
Rₛ (MBd)
×
k
B (MHz)
=
Rₛ (MBd)
(1
+
α
)
η
Spectral efficiency
calculated · b/s/Hz
Formulas
k
=
log₂(
M
)
4 = log₂(16)
Rb
=
Rₛ
×
k
4 = 1 × 4
B
=
Rₛ
×
(1
+
α
)
1.25 = 1 × (1 + 0.25)
η
=
Rb
B
3.2
=
4
1.25
Assumptions
•
Each symbol is one of M points, so it carries log₂M bits; M is 2, 4, 8, 16, 64 or 256.
•
Rates are in millions a second: Rₛ in Msymbol/s, Rb in Mb/s and B in MHz; η is in bits per second per hertz.
•
Raised-cosine pulses with roll-off α: the signal fills Rₛ(1 + α) of bandwidth.
•
The picture draws PSK up to 8 points and square QAM from 16, each Gray-coded.
Step-by-step
WE KNOW
M = 16, Rₛ = 1 MBd, α = 0.25
FIND
bits per symbol (k), bit rate (Rb), bandwidth (B), spectral efficiency (η)
Step 1 · Find bits per symbol (k)
Use k = log₂(M)
Each
bit
doubles
the
choices,
so
M
points
name
log₂
M
bits.
k
=
log₂(16)
k
=
4
Step 2 · Find bit rate (Rb)
Use Rb = Rₛ × k
Every
symbol
carries
k
bits,
so
the
bits
flow
k
times
as
fast
as
the
symbols.
Rb = 1 × 4
Rb = 4 Mb/s
Step 3 · Find bandwidth (B)
Use B = Rₛ × (1 + α)
The pulses need Rₛ of bandwidth, plus the roll-off’s share of it.
B
=
1
×
(1
+
0.25)
B
=
1.25
MHz
Step 4 · Find spectral efficiency (η)
Use η = Rb ÷ B
The bits each second carried by each hertz of bandwidth.
η
=
4
1.25
η
=
3.2
b/s/Hz
Check
Put the numbers back in. Both sides should match.
4 = log₂(16) ✓
4 = 1 × 4 ✓
1.25 = 1 × (1 + 0.25) ✓
3.2
=
4
1.25
✓
Topic check
See what you know: 5 questions on this topic
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