Why Half-Power (−3 dB) Is Often Used
The short answer is: half-power is used because it is mathematically natural, physically meaningful, and robust in real systems. Below is the intuition—built step by step, without hand-waving.
1. Power vs Amplitude: Why “Half” Appears Naturally
Most physical signals behave as:
- Amplitude → field, voltage, pressure
- Power / intensity ∝ (amplitude)2
If power drops to one-half, amplitude becomes:
\[ \sqrt{\tfrac{1}{2}} \approx 0.707 \]
In decibels:
\[ 10\log_{10}(1/2) \approx -3.01\ \text{dB} \]
This is why the −3 dB point universally corresponds to half power.
2. Why Half-Power Defines a Natural Beamwidth (HPBW)
Near its maximum, most radiation or filter responses can be approximated by a second-order Taylor expansion:
\[ P(\theta) \approx P_{\max}(1 - a\theta^2) \]
Half-power occurs when:
\[ 1 - a\theta_{1/2}^2 = \tfrac{1}{2} \quad \Rightarrow \quad \theta_{1/2} = \sqrt{\frac{1}{2a}} \]
This width depends only on the curvature of the main lobe, making it stable and representative of the true beam spread.
Thresholds closer to the peak are unstable, while thresholds far from the peak are corrupted by sidelobes and noise.
3. Why Not Half-Amplitude?
Half-amplitude implies:
\[ P = (0.5)^2 = 0.25 \quad \text{(−6 dB)} \]
- Too narrow for realistic beams
- Less connected to energy transfer
- More sensitive to modeling assumptions
Since physical systems care about power flow, half-power is the meaningful reference.
4. Lambertian Model Intuition
A Lambertian radiator follows:
\[ I(\theta) = I_0 \cos^m(\theta) \]
Half-power angle is defined by:
\[ \cos^m(\theta_{1/2}) = \tfrac{1}{2} \]
Which gives:
\[ \theta_{1/2} = \cos^{-1}(2^{-1/m}) \]
This directly ties angular spread to emitted energy—again, half-power is not arbitrary.
5. Why Half-Power Is Robust to Noise and Sidelobes
Let the true power pattern be:
\[ P(\theta) = P_0 f(\theta), \quad f(0)=1 \]
Measured pattern includes noise:
\[ \tilde P(\theta) = P_0 f(\theta) + n(\theta) \]
Near the peak:
\[ f(\theta) = 1 - a\theta^2 + O(\theta^4) \]
Width is found by solving:
\[ P(\theta) = \alpha P_0 \]
Noise-induced error scales as:
\[ \delta\theta \approx \frac{n(\theta)}{P_0 |f'(\theta)|} \]
Since:
\[ |f'(\theta_\alpha)| = 2\sqrt{a(1-\alpha)} \]
Sensitivity behaves as:
\[ \delta\theta \propto \frac{1}{\sqrt{1-\alpha}} \]
6. Comparison of Different Thresholds
| Power Fraction | Behavior |
|---|---|
| 0.9 | Extremely noise-sensitive |
| 0.5 | Stable and representative |
| 0.1 | Sidelobe interference dominates |
| 0.01 | Noise-dominated, ambiguous |
7. Why Sidelobes Don’t Corrupt the −3 dB Point
Let sidelobe level be \(S \ll P_0\). At half-power:
\[ 0.5P_0 \gg S \]
So the equation \(P(\theta)=0.5P_0\) has only main-lobe solutions. At lower thresholds, sidelobes create multiple crossings.
8. Big Picture
- Physically meaningful → energy flow
- Mathematically stable → curvature-based
- Log-scale friendly → −3 dB
- Robust → minimal noise and sidelobe sensitivity
- Universal → same definition across disciplines
Half-power is not magic—it’s the point where math, physics, and engineering all agree.
Further Reading
- Physically meaningful → energy flow