Spectral Efficiency in M-ary PSK
Definition
Spectral efficiency tells how efficiently bandwidth is used.
η = Rb / B
Where:
- η = Spectral efficiency (bits/s/Hz)
- Rb = Bit rate (bps)
- B = Bandwidth (Hz)
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Launch M-ary PSK ToolM-PSK Relationships
Rb = Rs × log₂(M)
Where:
- Rs = Symbol rate
- M = Number of PSK phases
Using Nyquist Bandwidth
Nyquist minimum bandwidth:
B = Rs / 2
Spectral efficiency:
η = Rb / B
Substitute the values:
η = (Rs × log₂(M)) / (Rs / 2)
η = 2 × log₂(M)
This result is obtained for ideal double-sided baseband interpretation.
Bandpass M-PSK System
In many communication textbooks, transmission bandwidth is approximated as:
B ≈ 2Rs
Therefore:
η = Rb / 2Rs
Substitute:
η = (Rs × log₂(M)) / 2Rs
η = log₂(M) / 2
Final Common Formula
η = log₂(M) / 2 bits/s/Hz
Example: 8-PSK
η = log₂(8) / 2
η = 3 / 2
η = 1.5 bits/s/Hz
Why Different Answers Exist
| Bandwidth Assumption | Spectral Efficiency |
|---|---|
| B ≈ Rs | log₂(M) |
| B ≈ 2Rs | log₂(M) / 2 |
Different bandwidth definitions lead to different spectral efficiency formulas.