A prior \((\hat{x}_k^-,\,P_k^-)\) propagated forward from the previous step, and a new measurement \((z_k,\,R_k)\). The filter fuses them optimally to produce a posterior \((\hat{x}_k^+,\,P_k^+)\).
Innovation: \(\nu_k = z_k - \hat{x}_k^-\). It is the new information the measurement carries: the difference between what the prediction expected to see and what the sensor actually reported. Small innovation means prediction was already close; large innovation means the measurement strongly corrects the estimate.
\(S_k = P_k^- + R_k\): the prediction uncertainty plus the measurement uncertainty. \(S_k\) appears in the Kalman gain denominator and is also the test statistic in the Mahalanobis-distance fault detection rule we will see in Block 8.
\(K_k = \dfrac{P_k^-}{P_k^- + R_k}\). This is exactly the inverse-variance weight from Block 3 with the prior playing the role of one sensor and the measurement playing the role of the other.
\(P_k^- \gg R_k \Rightarrow K_k \to 1\). The filter pulls the posterior estimate strongly toward the measurement, because the prior is unreliable and the new information dominates.
\(R_k \gg P_k^- \Rightarrow K_k \to 0\). The filter mostly ignores the measurement and trusts the prediction, because the measurement carries little information relative to what the filter already knows.
\(\hat{x}_k^+ = \hat{x}_k^- + K_k\,(z_k - \hat{x}_k^-) = \hat{x}_k^- + K_k\,\nu_k\). The posterior is the prior plus a fraction \(K_k\) of the innovation.
\(P_k^+ = (1 - K_k)\,P_k^-\). Because \(0 \le K_k \le 1\), the posterior covariance is always smaller than the prior. A measurement update never increases uncertainty.
\(\hat{x}_{k+1}^- = F\,\hat{x}_k^+\) and \(P_{k+1}^- = F\,P_k^+\,F^\top + Q_k\). The state transitions deterministically; the covariance grows by the process-noise variance \(Q_k\).
Everything the deterministic motion model fails to capture: unmodeled dynamics, model mismatch, neglected sensor noise. \(Q\) too small makes the filter over-confident and slow to react to real measurements; \(Q\) too large makes the filter chase measurement noise. Tuning \(Q\) against truth data is part of every real deployment.
Predict: \(\hat{x}^- = F \hat{x}^+_\text{prev}\), \(P^- = F P^+_\text{prev} F^\top + Q\). Update: \(K = P^- / (P^- + R)\), \(\hat{x}^+ = \hat{x}^- + K(z - \hat{x}^-)\), \(P^+ = (1 - K) P^-\). Run this loop once per time step; if no measurement is available, skip the update.
\(K = 25/(25 + 9) = 25/34 \approx 0.735\). The measurement is twice as trustworthy as the prediction by variance ratio, so the gain is well above 0.5.
\(\hat{x}^+ = 1000 + 0.735\cdot 10 = 1007.35\). \(P^+ = (1 - 0.735)\cdot 25 \approx 6.62\), so \(\sigma^+ \approx 2.57\). The posterior std is smaller than either the prior std (5) or the measurement std (3) — the standard inverse-variance fusion result.
Because \(Q\) grows the covariance between updates and the measurement update shrinks it. Those two forces balance at a steady-state \(P^-\), which gives a steady-state \(K = P^-/(P^- + R)\). After a few cycles the filter "warms up" to that steady state and the gain stops changing.
You skip the update equations. The state propagates forward by the prediction equation, and the covariance grows by \(Q\): \(P^+_k = P^-_k\) for that step. The next time a measurement arrives, the gain is a little larger because the prior covariance built up while the filter was waiting.
Mathematically it is identical: it is inverse-variance weighting of two sources. The difference is operational. Static fusion combines two sensors that fired simultaneously. Recursive fusion (Kalman) combines a single new measurement with the prior summary of all past measurements, propagated forward in time. Recursion lets the filter run with constant memory regardless of how many measurements have come in.