FSK Constant Envelope – Mathematical Explanation

FSK Constant Envelope – Mathematical Explanation

The key idea behind FSK (Frequency Shift Keying) having a constant envelope comes from how the signal is mathematically constructed.

Mathematical Form of FSK

FSK signal:

s(t) = A cos(2π f_i t + φ)

• A = constant amplitude
• f_i = frequency (f₀ or f₁ depending on bit)
• φ = phase

Binary FSK uses:
Bit 0 → f₀
Bit 1 → f₁

Why the Envelope is Constant

The envelope is the instantaneous magnitude:

|s(t)| = √(s²(t) + ŝ²(t))

Since:

s(t) = A cos(θ(t))

The envelope becomes:

|s(t)| = A

• Amplitude A is constant
• Only frequency changes

Comparison with Other Modulations

• ASK → amplitude changes (non-constant envelope)
• QAM → amplitude + phase change (non-constant)
• FSK → frequency changes only (constant envelope)

Why This Matters for RF Hardware

Non-linear amplifiers distort amplitude variations but handle frequency variations well. Since FSK keeps amplitude constant, it works efficiently with such amplifiers.

Instantaneous Power

P(t) ∝ A² cos²(...)

Average Power = A² / (2R)

Power remains constant across symbols → no fluctuation.

Summary

Same loudness (amplitude), different pitch (frequency). The “volume” stays constant, only the tone changes.