Time / Frequency Separation for Orthogonality

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Derivation of Frequency Separation for Orthogonality

Step 1: Define BFSK Signals

s₁(t) = √(2E_b/T) cos(2πf₁t)

s₂(t) = √(2E_b/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

Further Reading

1. Sine and Cosine orthogonality
2. Orthogonality in OFDM
3. Orthogonal Time Frequency Space (OTFS)