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Launch BFSK Simulator Launch FM Simulator BFSK Orthogonality SimulatorDerivation of Frequency Separation for Orthogonality
Step 1: Define BFSK Signals
s₁(t) = √(2Eb/T) cos(2πf₁t)
s₂(t) = √(2Eb/T) cos(2πf₂t)
Defined over: 0 ≤ t ≤ T
For orthogonality:
∫₀ᵀ s₁(t)s₂(t) dt = 0
Step 2: Remove Constants
∫₀ᵀ cos(2πf₁t) cos(2πf₂t) dt = 0
Step 3: Use Trigonometric Identity
cos A cos B = ½ [ cos(A − B) + cos(A + B) ]
Applying identity:
½ ∫₀ᵀ [ cos(2π(f₁ − f₂)t) + cos(2π(f₁ + f₂)t) ] dt
Step 4: Focus on Frequency Difference
The second term integrates to zero for high carrier frequencies.
∫₀ᵀ cos(2πΔf t) dt
Where:
Δf = f₁ − f₂
Step 5: Integrate
∫₀ᵀ cos(2πΔf t) dt = sin(2πΔfT) / (2πΔf)
For orthogonality:
sin(2πΔfT) = 0
Step 6: Solve Condition
2πΔfT = nπ → ΔfT = n/2
For minimum separation (n = 1):
Δf = 1 / (2T) (Coherent BFSK, symbol-duration orthogonality)
Case 2: Full-Period Orthogonality (Δf = 1/T)
If the signals are considered as full-period sinusoidal basis functions, i.e., over one or more complete periods, orthogonality occurs when:
Δf = 1 / T
This is typical for general orthogonal sinusoidal signals used in signal-space representation, not limited to BFSK symbol duration.
Summary
- Δf = 1 / (2T) → Minimum frequency separation for coherent BFSK over symbol duration T
- Δf = 1 / T → Frequency separation for full-period orthogonality of sinusoidal basis signals