Constellation Distance and Power in M-ary PSK

Constellation Point Distance and Power Requirement in M-ary PSK

In digital communication systems, especially M-ary PSK (Phase Shift Keying), the arrangement of constellation points plays a crucial role in determining error performance and power efficiency.

1. Constellation Point Distance

In an M-ary PSK system, the constellation points are equally spaced on a circle of radius √E_s, where:

• E_s = Energy per symbol
• M = Number of constellation points

The minimum distance between adjacent constellation points is:

d_min = 2 √E_s sin(π / M)

Key Observations:

• As M increases, sin(π/M) decreases.
• The minimum distance between points decreases.
• Smaller distance → constellation points are closer → higher probability of error.

2. Probability of Error and Symbol Energy

The probability of symbol error (P_e) in M-PSK depends on the minimum distance between constellation points.

d_min ∝ √E_s

This means:

• Increasing symbol energy E_s increases the distance between constellation points.
• Larger distance reduces the probability of error.
• To maintain the same error performance at higher M, more power is required.

3. Extra Power Required for Higher Order Constellations

Consider two modulation schemes:

• Scheme 1 → distance d₁, energy E₁
• Scheme 2 → distance d₂, energy E₂

Since:

d ∝ √E_s

We get:

E₁ / E₂ = (d₁ / d₂)²

Therefore:

• Higher order constellations (larger M) reduce distance.
• To maintain the same minimum distance (same error rate), E_s must increase.
• Higher-order modulation requires extra transmit power.

4. Important Insight

• Increasing M increases spectral efficiency (more bits per symbol).
• But it reduces minimum distance.
• To compensate, symbol energy must increase.
• This creates a trade-off between bandwidth efficiency and power efficiency.

Conclusion

d_min = 2 √E_s sin(π / M)

Higher-order constellations improve bandwidth efficiency but require more power to maintain the same error performance.