# Optimal Fusion of Two Sensors
This demo makes inverse-variance fusion concrete. Two scalar sensors — a precise radar altimeter and a noisier barometric altimeter — both measure the same true altitude. The demo shows the three resulting probability densities (radar, baro, fused) on a common axis and a *variance-vs-weight* curve underneath. The variance curve is the headline pedagogy: drag the fusion weight $w_R$ around and the dot walks along a parabola whose minimum sits at exactly the inverse-variance optimum. Move away from that minimum in either direction and the fused $\sigma$ visibly inflates.
## The math
Two unbiased independent measurements with variances $\sigma_R^2$ and $\sigma_B^2$ combine as
$$
\hat{x} = w_R\, z_R + (1 - w_R)\, z_B, \qquad
\mathrm{Var}(\hat{x}) = w_R^2\, \sigma_R^2 + (1 - w_R)^2\, \sigma_B^2.
$$
Minimizing $\mathrm{Var}(\hat{x})$ with respect to $w_R$ gives the **inverse-variance** weights
$$
w_R^{\text{opt}} = \frac{1/\sigma_R^2}{1/\sigma_R^2 + 1/\sigma_B^2},
\qquad
\sigma_{\text{opt}}^2 = \left( \frac{1}{\sigma_R^2} + \frac{1}{\sigma_B^2} \right)^{-1}.
$$
Equivalently, in *information* form ($I = 1/\sigma^2$), $I_{\text{opt}} = I_R + I_B$. Independent sensors *add* information; you cannot lose information by fusing in a second sensor, only by refusing to use it.
## Interactive demo
Open in full screen
## Walkthrough
The demo opens at the canonical Block 3 scenario: $\sigma_R = 2$ m, $\sigma_B = 10$ m, no biases, fusion weight at the optimum $w_R = 25/26 \approx 0.962$. Try the following:
1. **Read the headline metrics.** $\sigma_{\text{fused}} \approx 1.961$ m at the optimum — slightly tighter than the radar alone ($\sigma_R = 2$ m). The information ratio $I_R / I_B = 25 : 1$ is the reason: the radar carries 25× the information of the baro. The optimum weight is essentially the radar's information share.
2. **Drag $w_R$ down to 0.5.** The fused PDF (green) widens dramatically; the dot on the variance-vs-weight curve walks up the parabola; $\sigma_{\text{fused}}$ jumps to about 5.1 m, **worse than either sensor alone** taken independently. The "Equal weights" chip is a fast way to land here. *Wrong weights actively destroy precision.*
3. **Click "Snap to optimal."** The fused PDF tightens back to its minimum width; the dot returns to the parabola's minimum.
4. **Slide $\sigma_B$ up to its slider max.** As the barometer gets noisier, the optimum weight pushes further toward 1.0 (radar) and the optimal $\sigma_{\text{fused}}$ approaches $\sigma_R$. In the limit $\sigma_B \to \infty$, the fused estimate is just the radar — a useless sensor contributes no information.
5. **Slide $\sigma_R$ and $\sigma_B$ to be equal**, say both at 5 m. The optimum is now $w_R = 0.5$ (the simple average is optimal in this case) and $\sigma_{\text{fused}} = 5/\sqrt{2} \approx 3.54$ m. The variance-vs-weight parabola is now symmetric about $w_R = 0.5$.
6. **Slide bias_R up to +5 m.** The radar PDF shifts right; the fused PDF inherits a weighted bias of $w_R \cdot 5 \approx 4.81$ m. The "Fused estimate vs truth" stat turns warning-color. **Optimal fusion only minimizes variance, not bias.** If your sensors disagree systematically, fusion will not save you — that is the entire motivation for fault detection in Block 8.
7. **Toggle "overlay Monte Carlo histograms."** 10,000 samples from each sensor and the fused estimate are drawn behind the smooth Gaussian PDFs. They should agree closely with the theoretical curves; "Reseed" cycles to a new realization.
## Key observations
- **The variance-vs-weight parabola is the fundamental picture.** A minimum exists; that minimum is the inverse-variance weighting; moving in either direction makes things strictly worse. This same picture is the mechanism behind the Kalman gain in Block 4, which trades off the prior variance $P_k^-$ against the measurement variance $R_k$ to land on an optimum every step.
- **You cannot lose information by adding an independent sensor.** $I_{\text{opt}} = I_R + I_B \geq I_R$ for any $I_B \geq 0$. The fused $\sigma$ is always less than or equal to the smaller of the two input $\sigma$s. Even a sensor with $\sigma_B = 50$ m still buys you a small but real precision improvement when fused optimally.
- **Information is what's additive, not variance.** $\sigma_{\text{opt}}^2 = (I_R + I_B)^{-1}$, not $\sigma_R^2 + \sigma_B^2$. This is why information form (Kalman filter cousin: the *information filter*) is the natural framework when you have many sensors to fuse.
- **Bias propagates through fusion, variance shrinks.** The fused mean is the weighted average of sensor biases; the fused variance is bounded above by the weighted-quadratic combination. The estimator only minimizes the second moment of the error, not the first.
## From fusion to the Kalman filter
The two-sensor fusion in this demo is the same operation a Kalman filter performs at every measurement update — but with one of the "sensors" replaced by the prior estimate $\hat{x}_k^-$ propagated forward from the last step. The Kalman gain $K_k = P_k^- / (P_k^- + R_k)$ is just the inverse-variance weight on the measurement. Block 4 derives this rigorously and adds the covariance update; the fusion math here is the building block.
## Source
MATLAB · code/OptimalFusionDemo.m↓