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Why Path Loss Modeling Is Critical

Before worrying about fading or noise, you need to know how much power arrives at the receiver. Path loss determines the coverage area, the required transmit power, and ultimately the system capacity. Every link budget starts with a path loss model.

Definition:
Free-Space Path Loss (Friis)

The free-space path loss (FSPL) at distance $d$ and frequency $f_c$ is:

$PL_{\text{FS}}(d) = 20\log_{10}\!\left(\frac{4\pi d f_c}{c}\right) = 32.4 + 20\log_{10}(f_c/\text{MHz}) + 20\log_{10}(d/\text{km}) \;\text{dB}$

where $c$ is the speed of light. Power decays as $1/d^2$ (20 dB/decade).

def fspl(d_m, fc_hz):
    """Free-space path loss in dB."""
    return 20 * np.log10(4 * np.pi * d_m * fc_hz / 3e8)

FSPL applies only in free space (no reflections). Real environments always have higher path loss due to obstructions and multipath.

Definition:
Log-Distance Path Loss Model

The log-distance model generalizes FSPL with an adjustable path loss exponent $n$ :

$PL(d) = PL(d_0) + 10n\log_{10}(d/d_0) + X_\sigma \;\text{dB}$

where $d_0$ is a reference distance, $n$ is the path loss exponent (typically 2-6), and $X_\sigma \sim \mathcal{N}(0, \sigma^2)$ is log-normal shadow fading.

def log_distance_pl(d, d0=1.0, PL0=None, n=3.5, sigma_sf=8.0, fc=2.4e9):
    if PL0 is None:
        PL0 = 20 * np.log10(4 * np.pi * d0 * fc / 3e8)
    sf = np.random.randn(*np.shape(d)) * sigma_sf
    return PL0 + 10 * n * np.log10(d / d0) + sf

Definition:
3GPP TR 38.901 Path Loss Models

The 3GPP standard defines path loss models for different deployment scenarios. For Urban Macro (UMa) LOS:

$PL = 28.0 + 22\log_{10}(d) + 20\log_{10}(f_c) \;\text{dB}$

where $d$ is in meters and $f_c$ in GHz. For UMa NLOS:

$PL = 13.54 + 39.08\log_{10}(d) + 20\log_{10}(f_c) - 0.6(h_{UT} - 1.5)$

def uma_los(d, fc_ghz):
    return 28.0 + 22*np.log10(d) + 20*np.log10(fc_ghz)

def uma_nlos(d, fc_ghz, h_ut=1.5):
    return 13.54 + 39.08*np.log10(d) + 20*np.log10(fc_ghz) - 0.6*(h_ut - 1.5)

Definition:
Log-Normal Shadow Fading

Shadow fading models the slow, location-dependent fluctuations in received power due to obstructions. In dB, it is modeled as:

$X_\sigma \sim \mathcal{N}(0, \sigma_{\text{SF}}^2)$

Typical values: $\sigma_{\text{SF}} = 4$ - $8$ dB for outdoor, $3$ - $6$ dB for indoor. Shadow fading is spatially correlated with decorrelation distance $d_{\text{corr}} \approx 50$ - $100$ m.

Definition:
Link Budget Equation

The received power in dBm is:

$P_r = P_t + G_t + G_r - PL - L_{\text{misc}} + X_\sigma$

where $P_t$ is transmit power, $G_t, G_r$ are antenna gains, $PL$ is path loss, and $L_{\text{misc}}$ includes cable losses, body loss, etc. The received SNR is:

$\text{SNR} = P_r - N_{\text{floor}} = P_r - (-174 + 10\log_{10}(BW) + NF) \;\text{dB}$

Theorem: Two-Ray Ground Reflection Model

For transmitter/receiver heights $h_t, h_r$ and distance $d \gg h_t h_r$ :

$P_r \approx P_t G_t G_r \frac{h_t^2 h_r^2}{d^4}$

The path loss exponent transitions from $n = 2$ (free space) to $n = 4$ beyond the breakpoint distance $d_{\text{bp}} = 4h_t h_r / \lambda$ .

Close to the transmitter, the direct and reflected paths add constructively and destructively (fast fading). Beyond the breakpoint, the reflected path always partially cancels the direct path, doubling the exponent.

Theorem: Coverage Probability with Shadow Fading

The probability that the received SNR exceeds a threshold $\gamma_{\text{th}}$ at distance $d$ is:

$P_{\text{cov}}(d) = Q\!\left(\frac{PL(d) + \gamma_{\text{th}} + N_{\text{floor}} - P_t - G}{\sigma_{\text{SF}}}\right)$

where $G$ lumps antenna gains and $Q$ is the Gaussian tail function.

Shadow fading creates a "soft" coverage boundary instead of a hard circle. At the cell edge, coverage probability is typically 50%.

Theorem: COST-231 Hata Model

For urban environments at 1500-2000 MHz:

$PL = 46.3 + 33.9\log_{10}(f_c) - 13.82\log_{10}(h_b) - a(h_m) + (44.9 - 6.55\log_{10}(h_b))\log_{10}(d) + C_m$

where $a(h_m)$ is a correction factor for mobile antenna height, $h_b$ is base station height, and $C_m = 0$ dB (suburban) or $3$ dB (urban).

The Hata model is an empirical fit to Okumura's measurements. It captures the key dependencies on frequency, distance, and antenna heights.

Example: Comparing Path Loss Models

Plot free-space, log-distance ( $n = 3.5$ ), and 3GPP UMa path loss from 10 m to 1 km at 3.5 GHz.

Solution

Implementation

import numpy as np
d = np.logspace(1, 3, 200)  # 10 to 1000 m
fc = 3.5e9
pl_fs = 20*np.log10(4*np.pi*d*fc/3e8)
pl_ld = pl_fs[0] + 35*np.log10(d/d[0])  # n=3.5
pl_uma = 28 + 22*np.log10(d) + 20*np.log10(3.5)  # UMa LOS

Key observation

Free space is always the most optimistic. The UMa model sits between free space and log-distance with $n = 3.5$ .

Example: Generating a Correlated Shadow Fading Map

Generate a 2D spatially-correlated shadow fading map on a 500 m x 500 m grid with decorrelation distance 50 m and standard deviation 8 dB.

Solution

Exponential correlation

grid_size = 100
d_corr = 50  # m
sigma_sf = 8  # dB
x = np.linspace(0, 500, grid_size)
X, Y = np.meshgrid(x, x)
# Distance matrix
dist = np.sqrt((X[:,:,None,None]-X[None,None,:,:])**2
               + (Y[:,:,None,None]-Y[None,None,:,:])**2)
# This is expensive; use FFT-based approach instead

FFT-based generation

# Efficient: filter white noise with exponential kernel
kernel_1d = np.exp(-x / d_corr)
kernel_2d = np.outer(kernel_1d, kernel_1d)
kernel_2d /= np.sqrt(np.sum(kernel_2d**2))
from scipy.signal import fftconvolve
white = np.random.randn(grid_size, grid_size)
sf_map = sigma_sf * fftconvolve(white, kernel_2d, mode='same')

Example: 5G NR Link Budget Calculation

Compute the maximum cell radius for a 5G NR base station at 3.5 GHz with 43 dBm transmit power, 18 dBi antenna gain, targeting -100 dBm receiver sensitivity.

Solution

Calculation

Pt = 43        # dBm
Gt = 18        # dBi
Gr = 0         # dBi (mobile)
sens = -100    # dBm
margin = 10    # dB (shadow + interference)
max_PL = Pt + Gt + Gr - sens - margin  # = 151 dB
# Using UMa LOS: PL = 28 + 22*log10(d) + 20*log10(3.5)
# 151 = 28 + 22*log10(d) + 10.88
d_max = 10**((151 - 28 - 10.88) / 22)
print(f"Max cell radius: {d_max:.0f} m")

Path Loss Model Comparison

Compare FSPL, log-distance, and 3GPP UMa models across frequency and distance.

Parameters

Path Loss Exponent in Different Environments — Typical path loss exponents for different wireless environments.

Quick Check

What path loss exponent does free-space propagation correspond to?

1

2

3

4

Correction:

2

Free-space: power decays as $1/d^2$ , so $n = 2$ (20 dB/decade).

Common Mistake: Mixing dB and Linear Scales

Mistake:

Adding path loss in dB to power in watts, or multiplying path loss in dB with distance.

Correction:

All link budget arithmetic is done in dB/dBm: additions and subtractions only. Convert to linear scale only for SNR-dependent computations like capacity: snr_linear = 10**(snr_dB / 10).

Key Takeaway

Path loss determines the fundamental coverage and capacity limits of a wireless system. Use the appropriate model for your scenario: FSPL for satellite, log-distance for general analysis, and 3GPP models for standards-compliant simulations.

Historical Note: Okumura's Measurements (1968)

1968

Yoshihisa Okumura conducted extensive field measurements in Tokyo at frequencies from 150 MHz to 1920 MHz. His empirical curves became the basis for the Hata model (1980) and its COST-231 extension, which remain widely used for coverage planning even today.

Path Loss

The reduction in signal power as it propagates through space, measured in dB. Includes free-space spreading and environmental attenuation.

Related: Shadow Fading

Shadow Fading

Slow, log-normal fluctuations in received power due to large obstacles (buildings, terrain). Also called large-scale fading.

Related: Path Loss

Link Budget

An accounting of all gains and losses in a communication link, from transmitter to receiver, used to determine system feasibility.

Path Loss and Shadowing

Why Path Loss Modeling Is Critical

Definition: Free-Space Path Loss (Friis)

Definition: Log-Distance Path Loss Model

Definition: 3GPP TR 38.901 Path Loss Models

Definition: Log-Normal Shadow Fading

Definition: Link Budget Equation

Theorem: Two-Ray Ground Reflection Model

Theorem: Coverage Probability with Shadow Fading

Theorem: COST-231 Hata Model

Example: Comparing Path Loss Models

Implementation

Key observation

Example: Generating a Correlated Shadow Fading Map

Exponential correlation

FFT-based generation

Example: 5G NR Link Budget Calculation

Calculation

Path Loss Model Comparison

Parameters

Path Loss Exponent in Different Environments

Quick Check

Common Mistake: Mixing dB and Linear Scales

Key Takeaway

Historical Note: Okumura's Measurements (1968)

Path Loss

Shadow Fading

Link Budget

Definition:
Free-Space Path Loss (Friis)

Definition:
Log-Distance Path Loss Model

Definition:
3GPP TR 38.901 Path Loss Models

Definition:
Log-Normal Shadow Fading

Definition:
Link Budget Equation