Wireless Notes
Complete reference of wireless communication formulas including path loss, Shannon capacity, antenna gain, modulation BER, cellular reuse, fading channel, and link budget equations for engineering students.
How to Use This Reference
This chapter consolidates all important wireless communication formulas in one place for quick reference during exam revision, lab calculations, and system design. Each formula includes a brief explanation of when and why you would use it. Understanding the physical meaning behind each equation is more important than memorizing them — if you understand the concept, the formula follows naturally.
📊 Decibel Conversions
Decibels are the language of wireless engineering. Every gain, loss, power level, and ratio is expressed in dB for easier calculation (multiplication becomes addition):
| Formula | Use | Example |
|---|---|---|
| P(dBm) = 10×log₁₀(P_mW) | Power in dBm (reference: 1 mW) | 100 mW = 20 dBm |
| P(dBW) = 10×log₁₀(P_W) | Power in dBW (reference: 1 W) | 1 W = 0 dBW = 30 dBm |
| dBm = dBW + 30 | Convert between dBW and dBm | -10 dBW = 20 dBm |
| Gain(dB) = 10×log₁₀(P_out/P_in) | Power gain/loss in dB | 2× power = +3 dB |
| Gain(dB) = 20×log₁₀(V_out/V_in) | Voltage gain in dB | 2× voltage = +6 dB |
Common dB values to memorize:
- +3 dB = double the power
- +10 dB = 10× the power
- -3 dB = half the power
- +20 dB = 100× the power
📡 Propagation and Path Loss
These formulas determine how much signal strength is lost as a wave travels from transmitter to receiver:
| Formula | Description | Application |
|---|---|---|
| FSPL(dB) = 20log(d_km) + 20log(f_MHz) + 32.44 | Free Space Path Loss | Satellite, LOS microwave links |
| FSPL = (4πd/λ)² | FSPL in linear form | Theoretical derivations |
| Pᵣ = Pₜ × Gₜ × Gᵣ × (λ/4πd)² | Friis transmission equation | Point-to-point link calculations |
| PL(d) = PL(d₀) + 10n×log(d/d₀) + X_σ | Log-distance path loss model | Real-world urban/indoor propagation |
| d_LOS = 4.12(√h₁ + √h₂) km | Line-of-sight distance (h in meters) | Microwave link planning |
Path loss exponents (n) for different environments:
| Environment | Path Loss Exponent (n) | Explanation |
|---|---|---|
| Free space | 2.0 | Ideal — power spreads as 1/d² |
| Open area/rural | 2.0-2.5 | Near free-space, minimal obstacles |
| Suburban | 2.5-3.5 | Some buildings, trees |
| Urban line-of-sight | 2.7-3.5 | Buildings create reflections |
| Urban non-line-of-sight | 3.0-5.0 | Heavy obstruction, diffraction |
| Indoor (same floor) | 1.6-3.5 | Waveguide effect can reduce loss |
| Indoor (through floors) | 4.0-6.0 | Floors cause severe attenuation |
📊 Capacity and Information Theory
| Formula | Description | Significance |
|---|---|---|
| C = B × log₂(1 + SNR) | Shannon channel capacity (bps) | Maximum error-free data rate for given B and SNR |
| C = B × log₂(1 + 10^(SNR_dB/10)) | Shannon with SNR in dB | Practical calculation form |
| η = C/B = log₂(1 + SNR) bps/Hz | Spectral efficiency | How efficiently bandwidth is used |
| C_MIMO ≈ min(Nₜ,Nᵣ) × B × log₂(1 + SNR/min(Nₜ,Nᵣ)) | MIMO capacity (approximate) | Capacity scales linearly with min(antennas) |
| R_Nyquist = 2B × log₂(M) | Maximum symbol rate (noiseless channel) | Theoretical maximum without ISI |
Shannon capacity insight: Doubling bandwidth doubles capacity. But increasing SNR by 3 dB (doubling power) only adds 1 bps/Hz. This is why bandwidth is more valuable than power for increasing data rates — and why mmWave (with hundreds of MHz bandwidth) is so attractive for 5G.
📶 Modulation Formulas
| Formula | Description |
|---|---|
| m = Aₘ/Aᶜ | AM modulation index |
| BW_AM = 2fₘ | AM bandwidth (DSB-FC) |
| P_total = Pᶜ(1 + m²/2) | AM total transmitted power |
| η_AM = m²/(2 + m²) | AM power efficiency |
| β = Δf/fₘ | FM modulation index |
| BW_FM ≈ 2(Δf + fₘ) = 2fₘ(β + 1) | Carson's rule (FM bandwidth) |
| BER_BPSK = ½ × erfc(√(Eᵦ/N₀)) | BPSK bit error rate in AWGN |
| BER_QPSK = ½ × erfc(√(Eᵦ/N₀)) | QPSK BER (same as BPSK per bit!) |
| BER_M-QAM ≈ (4/log₂M)(1-1/√M) × Q(√(3log₂M × Eᵦ/(N₀(M-1)))) | M-QAM approximate BER |
| BER_Rayleigh ≈ 1/(4 × Eᵦ/N₀) | BPSK over Rayleigh fading (high SNR) |
Key insight: In AWGN, BER decreases exponentially with SNR (steep waterfall curve). In Rayleigh fading, BER decreases only inversely with SNR (much slower) — this is why fading channels are so challenging and why diversity/MIMO techniques are essential.
📡 Antenna Formulas
| Formula | Description |
|---|---|
| G = η × D | Gain = radiation efficiency × directivity |
| D ≈ 41,253 / (θ₁° × θ₂°) | Directivity from 3-dB beamwidths (degrees) |
| G_array ≈ G_element + 10×log₁₀(N) | Array gain with N elements |
| Aₑ = G × λ² / (4π) | Effective aperture area |
| r_ff > 2D²/λ | Far-field (Fraunhofer) distance |
| EIRP = Pₜ(dBm) + Gₜ(dBi) | Effective Isotropic Radiated Power |
| G/T = Gᵣₓ(dBi) - 10×log₁₀(T_sys) | Receiver figure of merit (dB/K) |
| Half-wave dipole: G = 2.15 dBi, Z = 73 Ω | Reference antenna parameters |
📱 Cellular Network Formulas
| Formula | Description |
|---|---|
| N = i² + ij + j² | Valid hexagonal cluster sizes |
| D = R × √(3N) | Co-channel reuse distance |
| Q = D/R = √(3N) | Co-channel reuse ratio |
| SIR ≈ (3N)^(n/2) / 6 | Signal-to-Interference Ratio (first tier) |
| Cell capacity = Total_channels / N | Channels per cell |
| Cell area = (3√3/2) × R² | Hexagonal cell area |
| Erlang B: P_block = B(A,C) | Blocking probability (A=traffic, C=channels) |
| Traffic intensity: A = λ × T_hold (Erlangs) | Offered traffic per cell |
🌊 Fading and Channel Formulas
| Formula | Description |
|---|---|
| Bᶜ ≈ 1/(5×σ_τ) | Coherence bandwidth (50% correlation) |
| Tᶜ ≈ 9/(16π×f_m) | Coherence time (50% correlation) |
| f_m = v×f/c = v/λ | Maximum Doppler frequency |
| f_d = (v/c)×f×cos(θ) | Doppler shift at angle θ |
| N = kTB | Thermal noise power (W) |
| N₀ = kT | Noise spectral density (W/Hz) |
| Noise floor = -174 + 10×log₁₀(B_Hz) dBm | Receiver noise floor at 290K |
| NF = SNR_in - SNR_out (dB) | Noise figure of receiver |
Flat fading condition: Signal bandwidth < Coherence bandwidth (B_signal < B_c) Frequency selective fading: Signal bandwidth > Coherence bandwidth (needs OFDM equalization)
💰 Complete Link Budget Template
Received signal power
Pᵣ(dBm) = Pₜ(dBm) + Gₜ(dBi) - L_cable(dB) - FSPL(dB) - L_atm(dB) + Gᵣ(dBi) - L_rx_cable(dB)
Link Margin = Pᵣ - Receiver_Sensitivity - Fade_Margin
System works if: Link Margin > 0 dB (target ≥ 10 dB for reliability)
Example (WiFi link)
Pₜ = 20 dBm, Gₜ = 3 dBi, FSPL(50m, 2.4GHz) = 60 dB, Gᵣ = 0 dBi
Pᵣ = 20 + 3 - 60 + 0 = -37 dBm
Sensitivity = -70 dBm → Margin = -37 - (-70) = 33 dB ✅ (excellent)
📝 Summary
These formulas form the mathematical foundation of wireless communication engineering. The most frequently used in practice are: FSPL for link calculations, Shannon capacity for system dimensioning, SIR for cellular planning, and BER expressions for modulation performance evaluation. Master the physical meaning behind each formula rather than memorizing blindly — understanding why path loss increases with frequency (shorter wavelength means less effective antenna aperture) or why Shannon capacity is logarithmic in SNR (diminishing returns of power) makes the formulas intuitive rather than arbitrary.
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