7. Optimal Signal EstimationIntroduction and ContextDigital signal processing rarely operates on perfectly known signals. Measurements are contaminated by sensor noise, quantization, interference, model uncertainty, missing samples, and time-varying disturbances. The central task of signal estimation is therefore to infer an unknown signal, parameter, or system state from incomplete and noisy observations. An estimator is called optimal only with respect to a stated criterion, a specified signal model, and assumptions about uncertainty. There is no universally optimal estimator independent of the problem formulation. This chapter develops the principal estimation methods used in advanced digital signal processing: Wiener filtering, least-squares estimation, recursive least squares, Kalman filtering, and smoothing. Wiener methods are primarily statistical and often assume stationarity; least-squares methods minimize an observation error without requiring a probabilistic model; recursive least squares updates estimates as data arrive; and Kalman filtering combines a dynamic state model with sequential measurements. Smoothers extend filtering by using observations after the time being estimated. The methods are closely related. Wiener filtering and linear minimum mean-square estimation optimize mean-square error. Least squares can be interpreted geometrically as projection onto a data subspace. The Kalman filter is a recursive minimum mean-square estimator for linear Gaussian state-space models, while its covariance recursion is closely connected to recursive least squares. Understanding these relationships is more valuable than memorizing isolated algorithms. Learning ObjectivesAfter completing this chapter, the reader should be able to: 1. Formulate signal estimation problems using signal, observation, and noise models. 2. Derive the Wiener filter from the minimum mean-square error criterion. 3. Solve batch and constrained least-squares estimation problems. 4. Explain the operation and numerical issues of recursive least squares. 5. Derive and apply the prediction and correction equations of the Kalman filter. 6. Distinguish filtering, prediction, and smoothing. 7. Interpret estimation error covariance and assess estimator reliability. 8. Select an appropriate estimator under computational, statistical, and uncertainty constraints. 9. Identify practical, ethical, and implementation risks in automated estimation systems. Estimation as Inference Under UncertaintyLet an unknown quantity be represented by a scalar or vector \(x\), and let the available measurement be \(y\). A general observation model is \[ y = h(x) + v, \] where \(h(\cdot)\) describes the measurement process and \(v\) is measurement noise. In linear estimation, \[ y = Hx + v. \] When the quantity evolves with time, a state-space model is more appropriate: \[ x_k = F_kx_{k-1}+B_ku_k+w_k, \] \[ y_k = H_kx_k+v_k. \] Here, \(x_k\) is the state, \(u_k\) is a known input, \(F_k\) is the state-transition matrix, \(B_k\) is the input matrix, and \(H_k\) maps the state into the measurement space. The process noise \(w_k\) represents unmodeled dynamics, while \(v_k\) represents measurement uncertainty. Two error criteria occur repeatedly. The mean-square error is \[ J=E\{|x-\hat{x}|^2\}, \] for a scalar estimate, or \(E[(x-\hat{x})(x-\hat{x})^T]\) for a vector. Least squares instead minimizes a data-dependent sum such as \[ J(\theta)=\|y-\Phi\theta\|_2^2. \] The first criterion requires statistical expectations or their estimates; the second requires measured data and a model relating data to unknown parameters. Important terminology includes prior estimate, formed before the current observation; posterior estimate, formed after incorporating the observation; innovation, the difference between an actual measurement and its predicted value; error covariance, a measure of estimation uncertainty; observability, the ability to infer internal states from outputs; and identifiability, the ability to determine unknown parameters uniquely from available data. Wiener Filtering: The Stationary Minimum-Mean-Square SolutionA Wiener filter estimates a desired signal from an observed process by minimizing steady-state mean-square error. Consider a finite impulse response filter \[ \hat{d}(n)=\sum_{k=0}^{M-1}w_kx(n-k)=\mathbf{w}^T\mathbf{x}(n), \] where \(x(n)\) is the input and \(d(n)\) is the desired signal. The error is \(e(n)=d(n)-\mathbf{w}^T\mathbf{x}(n)\). The objective is \[ J(\mathbf{w})=E[e^2(n)]. \] Expanding the expression gives \[ J(\mathbf{w})=\sigma_d^2-2\mathbf{w}^T\mathbf{p}+\mathbf{w}^T\mathbf{R}\mathbf{w}, \] where \(\mathbf{R}=E[\mathbf{x}(n)\mathbf{x}^T(n)]\) is the input autocorrelation matrix and \(\mathbf{p}=E[\mathbf{x}(n)d(n)]\) is the cross-correlation vector. Setting the gradient to zero produces the Wiener–Hopf equation: \[ \mathbf{R}\mathbf{w}_{opt}=\mathbf{p}. \] If \(\mathbf{R}\) is nonsingular, \[ \mathbf{w}_{opt}=\mathbf{R}^{-1}\mathbf{p}. \] The minimum mean-square error is \[ J_{min}=\sigma_d^2-\mathbf{p}^T\mathbf{R}^{-1}\mathbf{p}. \] The orthogonality principle explains the result: at the optimum, the estimation error is orthogonal to every component of the input vector, so \(E[\mathbf{x}(n)e(n)]=0\). This principle is also fundamental to linear minimum mean-square estimation and the Kalman filter. For a desired signal corrupted by additive noise, spectral descriptions are often more convenient. If \(y(n)=s(n)+v(n)\), and the desired signal is \(s(n)\), the noncausal frequency-domain Wiener filter is commonly written as \[ H_{opt}(e^{j\omega})=\frac{S_{sy}(e^{j\omega})}{S_{yy}(e^{j\omega})}. \] For additive noise independent of the signal, this becomes \(S_s/(S_s+S_v)\). Frequencies with high signal-to-noise ratio are passed more strongly, while unreliable frequency components are attenuated. A causal realizable filter requires spectral factorization and introduces a delay or prediction structure. Wiener filtering is effective when correlation functions or power spectra are reasonably stable. In practice, \(\mathbf{R}\) and \(\mathbf{p}\) are estimated from finite data. Poor conditioning can make direct matrix inversion unstable; regularization, diagonal loading, or numerically stable factorizations are then preferable. Least-Squares Estimation and Geometric InterpretationSuppose measurements satisfy \[ \mathbf{y}=\mathbf{\Phi}\boldsymbol{\theta}+\mathbf{e}, \] where \(\boldsymbol{\theta}\) is an unknown parameter vector. Ordinary least squares chooses the parameter vector minimizing the squared residual: \[ \hat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}\|\mathbf{y}-\mathbf{\Phi}\boldsymbol{\theta}\|_2^2. \] Differentiation gives the normal equations \[ \mathbf{\Phi}^T\mathbf{\Phi}\hat{\boldsymbol{\theta}}=\mathbf{\Phi}^T\mathbf{y}. \] When the columns of \(\mathbf{\Phi}\) are linearly independent, \[ \hat{\boldsymbol{\theta}}=(\mathbf{\Phi}^T\mathbf{\Phi})^{-1}\mathbf{\Phi}^T\mathbf{y}. \] The geometric meaning is important. The vector \(\mathbf{\Phi}\hat{\boldsymbol{\theta}}\) is the orthogonal projection of \(\mathbf{y}\) onto the column space of \(\mathbf{\Phi}\). Consequently, the residual is orthogonal to every column of the regression matrix. Direct use of the normal equations may square the condition number of \(\mathbf{\Phi}\). QR factorization is generally more numerically stable, and singular-value decomposition is especially useful when the model is rank deficient or nearly unidentifiable. If prior information or a preference for small parameter values is required, ridge-regularized least squares uses \[ \hat{\boldsymbol{\theta}}=(\mathbf{\Phi}^T\mathbf{\Phi}+\lambda\mathbf{I})^{-1}\mathbf{\Phi}^T\mathbf{y}, \] with \(\lambda>0\). Regularization reduces variance but introduces bias, so its value should reflect noise level, conditioning, and application risk. Weighted least squares accounts for unequal measurement reliability: \[ \hat{\boldsymbol{\theta}}=(\mathbf{\Phi}^T\mathbf{W}\mathbf{\Phi})^{-1}\mathbf{\Phi}^T\mathbf{W}\mathbf{y}. \] If \(\mathbf{W}\) is proportional to the inverse noise covariance, the estimator gives less influence to uncertain measurements. Constraints such as known parameter bounds, conservation laws, or physical positivity can be incorporated through constrained optimization rather than applied as an informal post-processing step. Recursive Least Squares for Streaming DataBatch least squares assumes that all data are simultaneously available. In adaptive filtering and online system identification, observations arrive sequentially. Recursive least squares (RLS) updates the estimate without recomputing a complete matrix inverse. For the model \[ y_k=\boldsymbol{\phi}_k^T\boldsymbol{\theta}+e_k, \] RLS minimizes an exponentially weighted objective: \[ J_k(\boldsymbol{\theta})=\sum_{i=1}^{k}\lambda^{k-i}(y_i-\boldsymbol{\phi}_i^T\boldsymbol{\theta})^2, \] where \(0<\lambda\leq1\) is the forgetting factor. A value of one treats all observations equally; a smaller value tracks changing systems more rapidly but increases sensitivity to noise. Define the estimate covariance matrix \(\mathbf{P}_{k-1}\). The standard update is \[ \mathbf{K}_k=\frac{\mathbf{P}_{k-1}\boldsymbol{\phi}_k}{\lambda+\boldsymbol{\phi}_k^T\mathbf{P}_{k-1}\boldsymbol{\phi}_k}, \] \[ e_k=y_k-\boldsymbol{\phi}_k^T\hat{\boldsymbol{\theta}}_{k-1}, \] \[ \hat{\boldsymbol{\theta}}_k=\hat{\boldsymbol{\theta}}_{k-1}+\mathbf{K}_ke_k, \] \[ \mathbf{P}_k=\lambda^{-1}(\mathbf{I}-\mathbf{K}_k\boldsymbol{\phi}_k^T)\mathbf{P}_{k-1}. \] The gain determines how strongly the new residual changes the estimate. Large uncertainty or an informative regressor produces a larger update. RLS converges faster than basic LMS adaptation in many correlated-input problems, but it requires more computation and can become numerically fragile. Square-root RLS, QR-based implementations, covariance symmetrization, and sensible initialization improve reliability. RLS is appropriate for slowly changing channel characteristics, acoustic paths, sensor calibration parameters, and online plant identification. It should not be interpreted as a general remedy for an incorrect model. Persistent excitation of the regressors is needed to identify all parameters adequately. Kalman Filtering: Dynamic State EstimationThe Kalman filter addresses a different but related problem: estimating a hidden state that evolves over time and is observed indirectly. For the linear model \[ \mathbf{x}_k=\mathbf{F}_k\mathbf{x}_{k-1}+\mathbf{B}_k\mathbf{u}_k+\mathbf{w}_k, \] \[ \mathbf{y}_k=\mathbf{H}_k\mathbf{x}_k+\mathbf{v}_k, \] assume zero-mean process and measurement noise with covariances \[ E[\mathbf{w}_k\mathbf{w}_k^T]=\mathbf{Q}_k,\qquad E[\mathbf{v}_k\mathbf{v}_k^T]=\mathbf{R}_k. \] The usual formulation assumes that the noises are white, mutually independent, and independent of the initial state. Under Gaussian assumptions, the Kalman estimate is the conditional mean and is minimum mean-square optimal. The algorithm has two stages. First, prediction: \[ \hat{\mathbf{x}}_{k|k-1}=\mathbf{F}_k\hat{\mathbf{x}}_{k-1|k-1}+\mathbf{B}_k\mathbf{u}_k, \] \[ \mathbf{P}_{k|k-1}=\mathbf{F}_k\mathbf{P}_{k-1|k-1}\mathbf{F}_k^T+\mathbf{Q}_k. \] Second, correction: \[ \mathbf{r}_k=\mathbf{y}_k-\mathbf{H}_k\hat{\mathbf{x}}_{k|k-1}, \] \[ \mathbf{S}_k=\mathbf{H}_k\mathbf{P}_{k|k-1}\mathbf{H}_k^T+\mathbf{R}_k, \] \[ \mathbf{K}_k=\mathbf{P}_{k|k-1}\mathbf{H}_k^T\mathbf{S}_k^{-1}, \] \[ \hat{\mathbf{x}}_{k|k}=\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_k\mathbf{r}_k. \] The covariance can be updated using \[ \mathbf{P}_{k|k}=(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}_{k|k-1}, \] although the Joseph form \[ \mathbf{P}_{k|k}=(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}_{k|k-1}(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)^T+\mathbf{K}_k\mathbf{R}_k\mathbf{K}_k^T \] is often safer in finite-precision arithmetic. The innovation covariance \(\mathbf{S}_k\) measures expected uncertainty in the new residual. Innovation monitoring is useful for detecting model mismatch, sensor faults, and abnormal data. The relative sizes of \(\mathbf{Q}\) and \(\mathbf{R}\) determine behavior. Large \(\mathbf{R}\) makes the filter distrust measurements and rely on its model. Large \(\mathbf{Q}\) indicates uncertain or rapidly changing dynamics and causes the filter to respond more readily to observations. Incorrect covariance tuning can cause excessive lag, noisy estimates, divergence, or false confidence. The extended Kalman filter linearizes nonlinear state and measurement functions locally. The unscented Kalman filter propagates selected sigma points to capture nonlinear transformations without explicitly forming Jacobians. Both are approximations and can fail when the posterior distribution is strongly non-Gaussian or multimodal. Particle filters address broader nonlinear, non-Gaussian settings at substantially greater computational cost. Filtering, Prediction, and SmoothingA filtered estimate \(\hat{x}_{k|k}\) uses data through time \(k\). A predicted estimate \(\hat{x}_{k+j|k}\) forecasts a future state using data only through \(k\). A smoothed estimate \(\hat{x}_{k|N}\), where \(N>k\), uses the complete data record through a later time. Smoothing is therefore appropriate when latency is acceptable and retrospective accuracy is more important than immediate response. The Rauch–Tung–Striebel smoother begins with a forward Kalman-filter pass and proceeds backward. For a time-invariant notation, the backward gain is \[ \mathbf{C}_k=\mathbf{P}_{k|k}\mathbf{F}_{k+1}^T\mathbf{P}_{k+1|k}^{-1}. \] The smoothed state and covariance are updated by \[ \hat{\mathbf{x}}_{k|N}=\hat{\mathbf{x}}_{k|k}+\mathbf{C}_k(\hat{\mathbf{x}}_{k+1|N}-\hat{\mathbf{x}}_{k+1|k}), \] \[ \mathbf{P}_{k|N}=\mathbf{P}_{k|k}+\mathbf{C}_k(\mathbf{P}_{k+1|N}-\mathbf{P}_{k+1|k})\mathbf{C}_k^T. \] Smoothing is used in offline denoising, trajectory reconstruction, speech analysis, biomedical signal interpretation, and post-event fault diagnosis. It cannot replace filtering in safety-critical real-time control because future observations are unavailable at the decision instant. Choosing an EstimatorThe selection should follow the information structure and the engineering objective. Wiener filtering is suitable when stationary second-order statistics are available and a fixed linear filter is desired. Batch least squares is natural for static parameters and offline data analysis. RLS is useful when parameters must be updated online and system variation is gradual. Kalman filtering is preferable when a dynamic state model, uncertainty covariances, and sequential measurements are available. Smoothing is chosen when delayed processing is acceptable. The choice also depends on dimensionality, latency, numerical precision, model credibility, and the cost of wrong decisions. A computationally sophisticated estimator with poorly identified noise statistics may perform worse than a simpler, carefully validated method. Estimation quality must be assessed not only by mean-square error but also by bias, transient response, robustness to outliers, calibration of uncertainty, and consequences of failure. Applied Case: Sensor Fusion for Vehicle Motion EstimationBackgroundConsider an illustrative autonomous-vehicle subsystem that estimates longitudinal position and velocity. A wheel encoder provides frequent incremental motion measurements, while a satellite-based position sensor provides an absolute but noisier position measurement. The vehicle may also experience wheel slip and changing acceleration. Situation and ProblemUsing only the encoder causes drift because small velocity errors accumulate. Using only the satellite position produces a delayed and visibly noisy velocity estimate. The design requires a real-time state estimate and a quantified uncertainty for control and collision-risk decisions. AnalysisDefine the state as \(\mathbf{x}_k=[p_k\;v_k]^T\), where \(p_k\) is position and \(v_k\) is velocity. With sampling interval \(T\), a constant-velocity model is \[ \mathbf{F}=\begin{bmatrix}1&T\\0&1\end{bmatrix}. \] The process covariance \(\mathbf{Q}\) represents unmodeled acceleration and wheel slip. The satellite measurement can be represented as \(y_k=[1\;0]\mathbf{x}_k+v_k\), with \(\mathbf{R}\) determined from validated sensor data. Approach and SolutionA Kalman filter predicts position and velocity from the previous state, then corrects position using the satellite observation. During satellite obstruction, the filter continues prediction but its covariance grows, explicitly indicating reduced confidence. When the signal returns, the innovation is checked before accepting the correction. An innovation that is statistically implausible may indicate multipath, spoofing, or a sensor fault rather than genuine vehicle motion. OutcomeThe fused estimate is smoother than the raw satellite signal and less prone to long-term drift than encoder-only integration. The covariance provides downstream control software with a measure of confidence. Performance depends on realistic tuning: setting \(\mathbf{Q}\) too small causes the filter to resist real acceleration and slip; setting \(\mathbf{R}\) too small makes it follow noisy or corrupted satellite measurements. Learning PointsThe case illustrates that a filter is not merely a smoothing device. It is a model-based inference system. Sensor diversity improves observability, covariance values encode trust, and residual monitoring is essential when measurements may be faulty or adversarial. Limitations, Risks, and Professional ResponsibilitiesAll estimators depend on assumptions. Stationarity may fail in Wiener filtering; least squares may be distorted by outliers and collinear regressors; RLS may become unstable under poor excitation or inappropriate forgetting factors; and Kalman filters may be misleading when the model is wrong, the noise is non-Gaussian, or sensors are biased. A low reported covariance does not guarantee accuracy: it may simply reflect overconfident noise specifications. Common mistakes include confusing measurement noise with process noise, using matrix inversion where a factorization is safer, ignoring units and scaling, treating correlated noise as independent, failing to initialize covariance sensibly, and evaluating only average error while ignoring rare catastrophic failures. Data leakage is a serious error in offline smoothing or model validation: future samples must not be used when assessing a real-time filter. Professional systems should log innovations, covariance behavior, parameter settings, and estimator resets. Safety-related applications require fault detection, fallback modes, traceable validation, and clear responsibility for decisions influenced by estimates. Privacy is relevant when estimation uses location, physiological, or behavioral data. Security considerations include spoofed measurements, denial of sensor access, and deliberate manipulation of statistical assumptions. Current Developments and Best PracticesModern estimation systems increasingly combine classical state estimators with machine-learning models. A learned model may represent difficult dynamics, while a Kalman-like uncertainty framework preserves temporal structure and supports residual checking. Neural filters and differentiable estimators are active research areas, but their uncertainty calibration and failure behavior require careful validation. Recommended practice is to: - define the state, measurements, objective, and timing constraints explicitly;
- characterize sensor noise experimentally rather than selecting covariance values arbitrarily;
- use QR, Cholesky, or square-root methods when numerical conditioning matters;
- test observability, sensitivity, bias, and outlier response;
- validate uncertainty calibration, not only point-estimate accuracy;
- monitor innovations and establish fault-handling thresholds;
- compare against simple baselines and evaluate performance under distribution shift;
- document assumptions, tuning decisions, and operating limits.
Chapter SummaryOptimal signal estimation infers unknown signals, parameters, or states from noisy and incomplete observations. Wiener filtering minimizes steady-state mean-square error using correlation statistics and leads to the Wiener–Hopf equations. Least squares minimizes a residual norm and has a useful projection interpretation; weighted, regularized, and constrained forms address unequal reliability, ill-conditioning, and prior restrictions. Recursive least squares updates parameter estimates for streaming data through a gain, residual, and covariance recursion, with the forgetting factor controlling tracking behavior. The Kalman filter applies recursive minimum mean-square estimation to linear dynamic systems. Its prediction stage propagates the state and covariance, while its correction stage uses the innovation and innovation covariance to incorporate measurements. Extended and unscented variants address nonlinear models approximately. Smoothers use future data to improve past estimates and are valuable in offline analysis. Across all methods, model validity, covariance tuning, numerical stability, observability, fault detection, and uncertainty interpretation are as important as the equations themselves. Key TermsWiener filter: A linear filter that minimizes mean-square estimation error using signal correlation statistics. Wiener–Hopf equation: The normal equation \(\mathbf{R}\mathbf{w}=\mathbf{p}\) defining the optimum Wiener filter. Mean-square error: The expected squared difference between an unknown quantity and its estimate. Orthogonality principle: The condition that the optimum estimation error is orthogonal to the data used for estimation. Least squares: Estimation based on minimizing the squared residual between observations and model predictions. Normal equations: The matrix equations obtained by setting the gradient of a least-squares objective to zero. Regularization: A penalty or prior constraint introduced to improve conditioning or limit excessive parameter values. Weighted least squares: Least squares in which observations receive different weights according to reliability. Recursive least squares: An online least-squares method that updates parameters and covariance as new data arrive. Forgetting factor: The RLS parameter that controls how strongly older observations are discounted. State-space model: A dynamic representation relating hidden states, inputs, process noise, and observations. Process noise: Random uncertainty representing unmodeled state evolution or disturbances. Measurement noise: Random error corrupting an observation. Innovation: The difference between an actual measurement and its predicted value. Innovation covariance: The predicted covariance of the innovation. Kalman gain: The matrix determining how an innovation changes the state estimate. Error covariance: A matrix quantifying uncertainty and cross-correlation in estimation error. Filtering: Estimating the current state using observations available up to the current time. Smoothing: Estimating a past state using observations collected both before and after that state. Observability: The ability to infer internal states from available input-output observations. Review QuestionsShort Answer Questions1. State the Wiener–Hopf equation and define its two principal matrices or vectors. 2. What is the geometric interpretation of an ordinary least-squares estimate? 3. Explain the role of the forgetting factor in recursive least squares. 4. Distinguish process noise from measurement noise in a Kalman model. 5. Define innovation and explain why it is useful for fault detection. Long Answer Questions1. Derive the finite-length Wiener filter by minimizing the mean-square error and explain the orthogonality principle. 2. Derive the ordinary least-squares estimator and discuss why QR or singular-value decomposition may be preferable to direct normal-equation inversion. 3. Present the recursive least-squares algorithm and explain the roles of the gain, residual, covariance, and forgetting factor. 4. Derive the prediction and correction stages of the Kalman filter and interpret each covariance term. 5. Compare filtering, prediction, and smoothing, including an explanation of the Rauch–Tung–Striebel smoother. Analytical / Application Questions1. A sensor-fusion system has a highly trusted dynamic model but a noisy measurement sensor. Predict how increasing the measurement-noise covariance will affect the Kalman gain, estimate, and covariance. 2. An RLS estimator tracks a slowly varying communication channel but responds too slowly after an abrupt change. What parameter and implementation choices could improve tracking, and what trade-offs would result? 3. A least-squares design matrix has nearly dependent columns. Explain the likely consequences for parameter estimates and recommend a numerically and statistically appropriate remedy. 22. Multiple Choice Questions with AnswersQuestion 1For a finite-length Wiener filter, which equation determines the optimum coefficient vector? A. R w = p B. R p = w C. H x = y D. P F = Q Correct Answer: A. R w = p Question 2The least-squares residual is geometrically characterized by being A. parallel to every column of the regression matrix B. orthogonal to the column space of the regression matrix C. equal to the measurement noise covariance D. independent of the model matrix Correct Answer: B. orthogonal to the column space of the regression matrix Question 3In recursive least squares, decreasing the forgetting factor generally makes the estimator A. ignore all new measurements B. track changes more rapidly while becoming more sensitive to noise C. equivalent to batch least squares with all data equally weighted D. independent of the covariance initialization Correct Answer: B. track changes more rapidly while becoming more sensitive to noise Question 4The innovation in a Kalman filter is A. the difference between the actual and predicted measurement B. the difference between process and measurement covariance C. the predicted state before time propagation D. the inverse of the Kalman gain Correct Answer: A. the difference between the actual and predicted measurement Question 5If measurement-noise covariance R is increased while all other quantities remain fixed, the Kalman filter will generally A. trust the measurement more strongly B. produce a larger Kalman gain in every case C. trust the model prediction relatively more strongly D. set process noise to zero Correct Answer: C. trust the model prediction relatively more strongly Question 6Which estimator is most directly suited to estimating a static parameter from a complete offline data set? A. Batch least squares B. Rauch–Tung–Striebel smoothing only C. A frequency-domain Wiener filter only D. A predictor without measurements Correct Answer: A. Batch least squares Question 7A smoothed estimate differs from a filtered estimate because the smoother A. uses future observations relative to the state being estimated B. does not require a state-space model C. can operate only when measurement noise is zero D. always has greater real-time latency-free performance Correct Answer: A. uses future observations relative to the state being estimated Question 8Why can direct inversion of the normal-equation matrix be problematic for an ill-conditioned regression matrix? A. It removes all measurement noise B. It squares the condition number and can amplify numerical errors C. It guarantees a biased estimate D. It prevents use of regularization Correct Answer: B. It squares the condition number and can amplify numerical errors Question 9In a Kalman filter, a large process-noise covariance usually causes the filter to A. become more responsive to new measurements B. discard all measurements C. assume the state is perfectly constant D. reduce state uncertainty before prediction Correct Answer: A. become more responsive to new measurements Question 10Which situation most strongly motivates a square-root or QR-based recursive implementation? A. The estimator has no covariance matrix B. Numerical conditioning and preservation of covariance properties are important C. The signal is known exactly in advance D. Only one measurement is available Correct Answer: B. Numerical conditioning and preservation of covariance properties are important
|