BFSK Orthogonality Online Simulator

 

BFSK Orthogonality Simulator

T: f₁: f₂:

Signals

📘 Mathematical Model Behind the Simulator

The simulator generates two sinusoidal signals used in Binary Frequency Shift Keying (BFSK):

s₁(t) = cos(2πf₁t)

s₂(t) = cos(2πf₂t)

To check orthogonality, it computes the inner product:

∫₀ᵀ s₁(t)s₂(t) dt

If the result is approximately zero, the signals are orthogonal.

⚙️ What Each Input Means

T (Symbol Duration)

This defines the time interval over which signals are observed. Orthogonality depends directly on this value.

0 ≤ t ≤ T

f₁ (Frequency 1)

Frequency of the first BFSK signal (represents binary "0" or "1").

cos(2πf₁t)

f₂ (Frequency 2)

Frequency of the second BFSK signal.

cos(2πf₂t)

Δf (Frequency Separation)

Difference between the two frequencies:

Δf = |f₁ − f₂|

Orthogonality Condition

For the signals to be orthogonal over time T:

Δf = 1 / (2T)

This is the minimum frequency separation required in coherent BFSK systems.

Another important case:

Δf = 1 / T

This corresponds to full-period orthogonality used in general signal basis functions.

🔄 Workflow of the Simulator

1. User enters values for T, f₁, f₂
2. Simulator generates both cosine signals
3. Signals are multiplied point-by-point
4. Numerical integration is performed:

Integral ≈ Σ s₁(t)s₂(t) · Δt

5. Result is compared with zero
6. System declares:
   • Orthogonal ✅ (if ≈ 0)
   • Not Orthogonal ❌

1. Signal Orthogonality (Sine Waves)

T: f₁: f₂:

2. Vector Orthogonality (Dot Product)

v₁: v₂:

3. Signal Projection (Energy Overlap)

Phase Shift:

📘 Mathematical Model

s₁(t) = cos(2πf₁t) s₂(t) = cos(2πf₂t)

Orthogonality condition: ∫₀ᵀ s₁(t)s₂(t) dt = 0

⚙️ Input Fields

• T → Time duration (integration interval)
• f₁ → Frequency of signal 1
• f₂ → Frequency of signal 2
• v₁, v₂ → Vector components for dot product check

Δf = |f₁ − f₂|

🔄 Simulator Workflow

1. Generate signals using cosine functions
2. Multiply signals point-by-point
3. Numerically integrate using summation:

Integral ≈ Σ s₁(t)s₂(t) Δt

4. If result ≈ 0 → Orthogonal
5. Otherwise → Not orthogonal

Orthogonality means zero overlap in energy, not just different shapes. Even similar signals can be orthogonal if spacing is correct.