--- title: "Optimal equalizer weights using MSE algorithm" subject: "Wireless Communication" chapter: 5 key: "semester_7/wireless_communication:mse-optimal-weights" source: https://www.bctengineer.com/q/semester_7/wireless_communication/mse-optimal-weights site: BCT Engineering --- # Optimal equalizer weights using MSE algorithm Wireless Communication · Chapter 5: Equalization and Diversity - Asked in: 2079 Bhadra, 8 marks; 2080 Chaitra, 7 marks; 2079 Chaitra, 7 marks (asked 3 times) - Video: [Finding the MMSE Filter Optimum Weights](https://www.youtube.com/watch?v=NVB1j_1Jmf4) — Barry Van Veen - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/wireless_communication/mse-optimal-weights - Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-5 ## Question Determine the optimal solution of the weights using MSE algorithm with its appropriate diagram and derivation. ## Short answer The MSE criterion minimizes E[|e|^2] of an equalizer, giving the Wiener solution w_opt = R^-1 p, where R is the input correlation matrix and p the cross-correlation with the desired symbol. The error surface is a quadratic bowl with one global minimum; LMS or RLS iterate toward it. ## Answer The **minimum mean square error (MSE)** criterion chooses the equalizer tap weights that make the average squared error between the desired symbol and the equalizer output as small as possible. The optimum set is called the **Wiener solution**. #### Structure: adaptive linear combiner (transversal equalizer) ```text y_k ─┬─[z⁻¹]─┬─[z⁻¹]─┬─ ... ─[z⁻¹]─┐ │ │ │ │ (x)w₀ (x)w₁ (x)w₂ ... (x)w_N │ │ │ │ └───────┴──────(Σ)─────────────┘ │ d̂_k (output) v x_k ──►(+)─(−)──► e_k (error) │ [Adaptive algorithm] → updates w ``` #### Derivation Let the input vector and weight vector be: ```text y_k = [y_k y_(k−1) ... y_(k−N)]ᵀ w_k = [w₀ w₁ ... w_N]ᵀ ``` Equalizer output and error (x_k = desired/transmitted symbol): ```text d̂_k = y_kᵀ w_k = w_kᵀ y_k e_k = x_k − y_kᵀ w_k ``` Square the error: ```text |e_k|² = x_k² + w_kᵀ y_k y_kᵀ w_k − 2 x_k y_kᵀ w_k ``` Take expectation (w is treated as fixed): ```text ξ = E[|e_k|²] = E[x_k²] + wᵀ E[y_k y_kᵀ] w − 2 E[x_k y_kᵀ] w ``` Define: ```text R = E[y_k y_kᵀ] (input correlation matrix, (N+1)×(N+1)) p = E[x_k y_k] (cross-correlation vector) ``` Then the MSE is a quadratic ("bowl-shaped") function of w: ```text ξ = E[x_k²] + wᵀ R w − 2 pᵀ w ``` Set the gradient to zero to find the minimum: ```text ∇ξ = ∂ξ/∂w = 2 R w − 2 p = 0 R ŵ = p ŵ = R⁻¹ p ← optimum (Wiener) weights ``` Minimum MSE, by substituting ŵ: ```text ξ_min = E[x_k²] − pᵀ R⁻¹ p = E[x_k²] − pᵀ ŵ ``` #### Error surface ```text ξ |\ / | \ / | \ / quadratic bowl | \_____/ single global minimum | • ξ_min +------ŵ--------> w ``` Since R is positive definite, the surface has a single global minimum, so any gradient search (steepest descent, LMS) converges to ŵ. #### Practical note Computing R⁻¹ directly needs about (N+1)³ operations and knowledge of the statistics. Adaptive algorithms find ŵ iteratively: - **LMS:** w_(k+1) = w_k + α e_k y_k (simple, slow convergence). - **RLS:** faster convergence, more computation. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the MSE criterion choose? A. The weights that maximize output power B. The weights that make every error exactly zero C. The weights that maximize the number of taps D. The equalizer weights that minimize the average squared error E[|e_k|^2] between desired and output symbol Answer: D. The optimum set is the Wiener solution. (Key point: MSE criterion) 2. What is the MSE as a function of w? A. xi = E[x^2] + w^T R w - 2 p^T w, a quadratic bowl B. xi = E[x^2] - w^T R w C. A linear function of w D. A function with many local minima Answer: A. Since R is positive definite, the bowl has a single global minimum. (Key point: Quadratic form) 3. Setting the gradient to zero gives which result? A. w = R p B. R w = p, so w = R^-1 p C. w = p / E[x^2] D. w = 2 R p Answer: B. Gradient is 2Rw - 2p = 0, so w = R^-1 p. (Key point: Optimum weights) 4. Which statements about xi_min and practical algorithms is correct? A. xi_min = E[x^2] + p^T w, and LMS needs R^-1 B. xi_min is always zero C. xi_min = E[x^2] - p^T w_opt, and LMS finds w iteratively avoiding the matrix inverse D. RLS is simpler and slower than LMS Answer: C. LMS updates w(k+1) = w(k) + alpha e y, simple but slow; RLS converges faster with more computation. 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