Kalman Filter: Mathematical Description and Example

Kalman Filter: Mathematical Description

1. System Model (State-Space Representation)

State Equation

\[
\mathbf{x}_{k} = \mathbf{F}_{k-1}\mathbf{x}_{k-1} + \mathbf{B}_{k-1}\mathbf{u}_{k-1} + \mathbf{w}_{k-1}
\]

Measurement Equation

\[
\mathbf{z}_{k} = \mathbf{H}_{k}\mathbf{x}_{k} + \mathbf{v}_{k}
\]

where:

\(\mathbf{x}_k\): system state vector
\(\mathbf{u}_k\): control input
\(\mathbf{z}_k\): measurement vector
\(\mathbf{F}_k\): state transition matrix
\(\mathbf{B}_k\): control input matrix
\(\mathbf{H}_k\): observation matrix
\(\mathbf{w}_k \sim \mathcal{N}(0,\mathbf{Q}_k)\): process noise
\(\mathbf{v}_k \sim \mathcal{N}(0,\mathbf{R}_k)\): measurement noise

2. Kalman Filter Assumptions

Linear system dynamics
Gaussian, white, zero-mean noise
Gaussian initial state

\[
\mathbf{x}_0 \sim \mathcal{N}(\hat{\mathbf{x}}_0, \mathbf{P}_0)
\]

3. Kalman Filter Algorithm

The Kalman filter consists of two recursive steps: Prediction and Update.

4. Prediction (Time Update)

State Prediction

\[
\hat{\mathbf{x}}_{k|k-1} =
\mathbf{F}_{k-1}\hat{\mathbf{x}}_{k-1|k-1} +
\mathbf{B}_{k-1}\mathbf{u}_{k-1}
\]

Error Covariance Prediction

\[
\mathbf{P}_{k|k-1} =
\mathbf{F}_{k-1}\mathbf{P}_{k-1|k-1}\mathbf{F}_{k-1}^T +
\mathbf{Q}_{k-1}
\]

5. Update (Measurement Correction)

Innovation (Residual)

\[
\mathbf{y}_k = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k|k-1}
\]

Innovation Covariance

\[
\mathbf{S}_k =
\mathbf{H}_k\mathbf{P}_{k|k-1}\mathbf{H}_k^T +
\mathbf{R}_k
\]

Kalman Gain

\[
\mathbf{K}_k =
\mathbf{P}_{k|k-1}\mathbf{H}_k^T
\mathbf{S}_k^{-1}
\]

6. State Update

\[
\hat{\mathbf{x}}_{k|k} =
\hat{\mathbf{x}}_{k|k-1} +
\mathbf{K}_k \mathbf{y}_k
\]

7. Covariance Update

\[
\mathbf{P}_{k|k} =
(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k)
\mathbf{P}_{k|k-1}
\]

Joseph stabilized form:

\[
\mathbf{P}_{k|k} =
(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}_{k|k-1}
(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)^T
+ \mathbf{K}_k\mathbf{R}_k\mathbf{K}_k^T
\]

8. Optimality Property

\[
\hat{\mathbf{x}}_k =
\mathbb{E}[\mathbf{x}_k \mid \mathbf{z}_{1:k}]
\]

The Kalman filter minimizes the mean square error:

\[
\mathbb{E}\left[
(\mathbf{x}_k - \hat{\mathbf{x}}_k)
(\mathbf{x}_k - \hat{\mathbf{x}}_k)^T
\right]
\]

9. Extensions

Extended Kalman Filter (EKF)
Unscented Kalman Filter (UKF)
Ensemble Kalman Filter (EnKF)

Kalman Filter: Simple Numerical Example for Wireless Communication

1. Problem Setup (Wireless Channel Estimation)

We consider a slowly varying wireless channel gain \(h_k\) tracked at the receiver using noisy pilot measurements.

State (Channel) Model

\[
h_k = h_{k-1} + w_k
\]

Measurement Model

\[
y_k = h_k + v_k
\]

where \(w_k \sim \mathcal{N}(0,Q)\) represents channel variation and \(v_k \sim \mathcal{N}(0,R)\) represents receiver noise.

2. Numerical Data (Arrays)

True Channel (Unknown)

h_true = [1.0, 1.1, 0.9, 1.05, 1.0]

Noisy Pilot Measurements

y = [0.95, 1.20, 0.85, 1.10, 0.98]

Noise Statistics

Q = 0.01   (process noise variance)
R = 0.04   (measurement noise variance)

Initial Conditions

ĥ0 = 0.8
P0 = 0.1

3. Kalman Filter Equations (1D Case)

Prediction

\[
\hat{h}_{k|k-1} = \hat{h}_{k-1}
\]

\[
P_{k|k-1} = P_{k-1} + Q
\]

Update

\[
K_k = \frac{P_{k|k-1}}{P_{k|k-1} + R}
\]

\[
\hat{h}_k = \hat{h}_{k|k-1} + K_k (y_k - \hat{h}_{k|k-1})
\]

\[
P_k = (1 - K_k) P_{k|k-1}
\]

4. Step-by-Step Numerical Computation

Iteration 1

Prediction:
ĥ⁻ = 0.8
P⁻ = 0.11

Kalman Gain:
K = 0.11 / (0.11 + 0.04) = 0.733

Update:
ĥ1 = 0.8 + 0.733 × (0.95 − 0.8) = 0.91
P1 = 0.029

Iteration 2

ĥ⁻ = 0.91
P⁻ = 0.039
K  = 0.494

ĥ2 = 1.05
P2 = 0.020

Iteration 3

ĥ⁻ = 1.05
P⁻ = 0.030
K  = 0.43

ĥ3 = 0.96
P3 = 0.017

Iteration 4

ĥ⁻ = 0.96
P⁻ = 0.027
K  = 0.40

ĥ4 = 1.02
P4 = 0.016

Iteration 5

ĥ⁻ = 1.02
P⁻ = 0.026
K  = 0.39

ĥ5 = 1.00
P5 = 0.016

5. Final Results

True channel:      [1.00, 1.10, 0.90, 1.05, 1.00]
Measured (noisy): [0.95, 1.20, 0.85, 1.10, 0.98]
Kalman estimate:  [0.91, 1.05, 0.96, 1.02, 1.00]

The Kalman filter smooths measurement noise and accurately tracks the time-varying wireless channel.

6. Relevance to Wireless Communication

Channel estimation using pilot symbols
Doppler and mobility tracking
SNR and RSSI estimation
Adaptive modulation and coding