6. Spectral Estimation and Linear Prediction

Introduction and Context

The frequency-domain description of a random or deterministic signal is often more informative than its time-domain waveform. A vibration record may reveal the mechanical resonances of a machine, an electrocardiogram may expose periodic interference, and a communications signal may contain closely spaced spectral components that are difficult to distinguish directly in time. Spectral estimation provides methods for inferring this frequency-domain structure from a finite set of observations.

Unlike the Fourier transform of a known continuous-time signal, an estimated spectrum is subject to finite-data effects, measurement noise, leakage, variance, and model uncertainty. The central engineering problem is therefore not merely to compute a spectrum, but to select an estimator whose resolution, bias, variance, computational cost, and modelling assumptions are appropriate for the application.

This chapter develops the principal nonparametric, parametric, and subspace methods used in advanced digital signal processing. It also examines linear prediction, autoregressive modelling, model-order selection, resolution analysis, and uncertainty assessment. These topics are closely related: a spectrum can be estimated from correlations, correlations can be represented through prediction models, and signal-subspace methods can exploit the algebraic structure of sums of sinusoids.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain the statistical foundations of spectral estimation and distinguish bias from variance.

2. Compare periodogram, correlogram, Welch, multitaper, AR, MA, and ARMA estimators.

3. Derive and interpret linear-prediction equations and the Levinson–Durbin recursion.

4. Select an appropriate model order using criteria such as AIC, MDL, and prediction-error measures.

5. Explain the principles and assumptions of MUSIC and ESPRIT.

6. Assess frequency resolution, leakage, uncertainty, and estimator robustness.

7. Apply spectral estimators to engineering problems and identify common implementation errors.

6.1 Spectral Density as an Inference Problem

For a wide-sense stationary discrete-time random process x[n], the autocorrelation function is

r_x[k] = E{x[n]x*[n-k]},

and the power spectral density (PSD) is its discrete-time Fourier transform,

S_x(e^{jω}) = Σ_{k=-∞}^{∞} r_x[k]e^{-jωk}.

The PSD describes how average signal power is distributed over frequency. It is non-negative and periodic with period 2π. For a finite record x[0], x[1], ..., x[N−1], however, the true autocorrelation and PSD are unknown. Spectral estimation uses the observations to construct an estimate \hat S(ω).

Two statistical properties are especially important. Bias is the difference between the expected estimate and the true spectrum. Variance measures the random fluctuation of the estimate across repeated records. Increasing the record length generally improves frequency discrimination, but it does not automatically remove all estimation errors. A narrow spectral window may reduce smoothing and improve nominal resolution while increasing leakage and variance; stronger averaging may reduce variance but obscure closely spaced components.

The finite observation is equivalent to multiplying the underlying signal by a data window. In frequency, multiplication becomes convolution. Consequently, spectral lines are spread by the transform of the window. This phenomenon, called spectral leakage, is not a numerical defect; it is an unavoidable consequence of observing only a finite record.

6.2 Classical Nonparametric Estimators

The Periodogram

The basic periodogram is

\hat S_P(ω) = (1/N)|Σ_{n=0}^{N−1}x[n]e^{-jωn}|².

Scaling may be modified according to whether a two-sided PSD, one-sided PSD, or density per hertz is required. With a window w[n], the modified periodogram becomes

\hat S_w(ω) = \frac{1}{U N}|Σ_{n=0}^{N−1}w[n]x[n]e^{-jωn}|²,

where U is a window-power normalization factor.

The periodogram is attractive because it is simple, fast, and directly connected to the discrete Fourier transform. Its principal weakness is statistical inconsistency: even as N increases, the variance at a frequency does not generally converge to zero. A longer record improves the apparent resolution, but the raw periodogram remains noisy. Window selection introduces a trade-off between main-lobe width and side-lobe suppression. The rectangular window has the narrowest main lobe among common windows but relatively high side lobes; Hann, Hamming, Blackman, and Kaiser windows provide progressively different compromises.

The Correlogram

The correlogram first estimates the autocorrelation and then Fourier transforms it:

\hat r[k] = \frac{1}{N}\sum_{n=0}^{N-1-k}x[n+k]x^*[n], \quad 0\leq k<N,

followed by

\hat S_C(ω)=\sum_{k=-(N-1)}^{N-1}\hat r[k]e^{-jωk}.

The biased autocorrelation estimate, obtained by dividing by N for every lag, produces a non-negative spectral estimate because it corresponds to the Fourier transform of a positive semidefinite correlation matrix. The unbiased estimate divides by N−|k| and may have better lag-wise statistical interpretation, but its Fourier transform is not guaranteed to be non-negative.

Truncating the correlation sequence is equivalent to applying a lag window. Thus, the correlogram performs smoothing in the frequency domain through convolution with the transform of the lag window. It is useful when correlation structure is of primary interest, but its effective resolution depends strongly on the maximum retained lag.

Welch’s Method

Welch’s method divides the record into overlapping segments, applies a window to each segment, computes a modified periodogram for each, and averages the results. If K segment estimates are formed,

\hat S_W(ω)=\frac{1}{K}\sum_{i=1}^{K}\hat S_i(ω).

Averaging reduces variance approximately in proportion to the number of independent averages, although overlap and windowing reduce the exact improvement. The price is reduced frequency resolution because each segment is shorter than the complete record. Welch estimation is often a strong default for broadband random signals, vibration measurements, and noisy laboratory data because it provides a stable and interpretable PSD without imposing a rigid parametric model.

Segment length, overlap, and window are coupled design variables. Long segments offer finer frequency spacing, whereas many shorter segments offer more averaging. Fifty-percent overlap is common for Hann-windowed data, but the appropriate value depends on the window and computational constraints.

Multitaper Estimation

Multitaper methods use several mutually orthogonal tapers, commonly discrete prolate spheroidal sequences (DPSS), to form multiple independent or weakly dependent spectral estimates from the same record. The estimates are then averaged, often with adaptive weights based on local spectral concentration.

A taper is selected to concentrate as much energy as possible within a prescribed bandwidth. The time-bandwidth product controls the number of useful tapers and the smoothing bandwidth. Multitaper estimation reduces variance and leakage without relying on a single arbitrary window. It is particularly effective for short records, weak spectral components near strong components, and signals containing both broadband and line-like features.

Its interpretation must nevertheless be explicit: multitaper smoothing is not free resolution. A selected time-bandwidth product defines a frequency scale below which features are intentionally averaged. Adaptive weighting can improve performance but increases algorithmic complexity.

6.3 Parametric Spectral Models

Nonparametric estimators use relatively few assumptions about the signal. Parametric estimators instead assume that the data are generated by a model with a finite number of parameters. They can provide high resolution with short records when the model is appropriate, but model mismatch can produce misleading peaks.

Autoregressive Models

An AR model of order p represents the signal as

x[n] + a_1x[n−1]+...+a_px[n−p]=e[n],

where e[n] is a white-noise innovation with variance σ_e². Its spectrum is

S_{AR}(e^{jω})=\frac{σ_e²}{|1+\sum_{k=1}^{p}a_ke^{-jωk}|²}.

AR models are all-pole models. They are effective for resonant processes, speech spectra, and systems in which peaks are more important than spectral nulls. Poles close to the unit circle produce narrow resonances. Excessive order may create spurious peaks, while insufficient order may merge genuine resonances.

AR parameters can be estimated by the autocorrelation method, covariance method, modified covariance method, or Burg’s method. The autocorrelation method assumes unavailable samples outside the observation interval are effectively zero and leads naturally to a Toeplitz system. Burg’s method estimates forward and backward prediction errors and preserves a stable model under standard conditions, while often providing good resolution for short data records.

Moving-Average Models

An MA model is

x[n]=b_0e[n]+b_1e[n−1]+...+b_qe[n−q].

Its spectrum is

S_{MA}(e^{jω})=σ_e²|\sum_{k=0}^{q}b_ke^{-jωk}|².

MA models are all-zero models and can represent spectral nulls naturally. Their parameters are generally more difficult to estimate than AR parameters because the equations are nonlinear in the coefficients. Numerical optimization or iterative methods are commonly required.

ARMA Models

An ARMA(p,q) model combines both structures:

x[n]+\sum_{k=1}^{p}a_kx[n−k]=\sum_{k=0}^{q}b_ke[n−k].

The corresponding spectrum is

S_{ARMA}(e^{jω})=σ_e²\frac{|\sum_{k=0}^{q}b_ke^{-jωk}|²}{|1+\sum_{k=1}^{p}a_ke^{-jωk}|²}.

ARMA models are more economical than pure AR models when both resonances and antiresonances are present. Their additional flexibility creates a more difficult identification problem, including the need to ensure causality, stability, and invertibility. In practice, ARMA modelling may be estimated through prediction-error minimization, maximum likelihood, or state-space identification.

6.4 Linear Prediction and Its Computational Structure

Linear prediction estimates a sample from a weighted combination of previous samples. A pth-order predictor is

\hat x[n]=−\sum_{k=1}^{p}a_kx[n−k],

with prediction error

ε[n]=x[n]−\hat x[n]=x[n]+\sum_{k=1}^{p}a_kx[n−k].

The coefficients are selected to minimize the mean-square error E{|ε[n]|²}. Differentiating this error with respect to the coefficients gives the normal, or Yule–Walker, equations:

\sum_{k=1}^{p}a_kr[m−k]=−r[m], \quad m=1,...,p.

The matrix of these equations is Hermitian Toeplitz. Direct solution requires cubic computational complexity in p, but the Levinson–Durbin recursion solves successive orders efficiently, typically with O(p²) complexity.

At each recursion stage, a reflection coefficient is computed. For a valid positive-definite autocorrelation sequence, the magnitudes of reflection coefficients are less than one. This condition is closely related to stability of the resulting AR model. The recursion also updates the minimum prediction-error variance. A rapidly decreasing error variance suggests that additional model order is explaining meaningful correlation; a slowly changing variance suggests that further order may mainly fit noise.

Linear prediction has interpretations beyond spectrum estimation. In speech coding, the AR coefficients describe the vocal-tract envelope. In system identification, they approximate the dynamics of a stable process. In detection, the prediction error can expose transients or departures from a learned background model.

6.5 Choosing Model Order

Model order determines the balance between underfitting and overfitting. An order that is too small cannot represent the relevant spectral structure. An order that is too large may fit random fluctuations, produce unstable or nearly cancelling poles and zeros, and degrade extrapolation.

Information criteria penalize model complexity. A general form is

IC(p)=N\ln(\hat σ_p²)+c_N d_p,

where \hat σ_p² is the residual variance, d_p is the number of estimated parameters, and c_N determines the penalty. AIC uses a relatively light penalty and is often prediction-oriented. MDL and BIC use penalties that grow approximately with ln N and are more conservative for identifying a parsimonious structure. For an AR model, the parameter count is commonly related to p, with adjustments depending on whether the innovation variance and mean are estimated.

Order selection should not be based on an information criterion alone. Residuals should be examined for whiteness, model stability, and sensitivity to changes in the data record. Cross-validation or blocked validation is valuable when forecasting performance matters. Domain knowledge can also constrain the order: a physical system with a known number of resonant modes should not be assigned an arbitrarily high order merely because an information criterion decreases slightly.

6.6 High-Resolution Subspace Methods

When the signal consists of a small number of complex sinusoids in additive white noise, covariance-based subspace methods can resolve frequencies more sharply than conventional periodograms.

Assume

x[n]=\sum_{r=1}^{K}A_re^{j(ω_r n+φ_r)}+v[n],

where v[n] is spatially or temporally white noise. The covariance matrix can be decomposed as

R_x=U_sΛ_sU_s^H+U_nΛ_nU_n^H,

where U_s spans the signal subspace and U_n spans the noise subspace. For a steering vector a(ω) corresponding to a true sinusoidal frequency, a(ω) lies in the signal subspace and is orthogonal to the noise subspace.

MUSIC

The MUSIC pseudospectrum is

P_{MUSIC}(ω)=\frac{1}{a^H(ω)U_nU_n^Ha(ω)}.

Large peaks occur when the steering vector is nearly orthogonal to the estimated noise subspace. MUSIC can resolve closely spaced tones when the number of sensors or effective correlation lags is sufficient, the model order K is correctly specified, and the noise assumptions are reasonably satisfied.

MUSIC is not an ordinary PSD. Its peak heights do not directly represent signal power, and its frequency grid affects numerical peak localization. Correlated sources, coloured noise, finite samples, and incorrect model order can substantially reduce performance. Spatial smoothing or prewhitening may be required in specialized settings.

ESPRIT

ESPRIT exploits shift invariance in a sensor or data array. If the signal subspace is represented by U_s and two overlapping subarrays are formed, then

U_{s1}Φ \approx U_{s2}.

The eigenvalues of Φ contain the phase shifts associated with the source frequencies. Frequencies are obtained from the arguments of these eigenvalues. Unlike MUSIC, ESPRIT does not require a frequency search grid and is often computationally efficient. However, it requires a suitable shift-invariant array or data structure and shares the sensitivity of subspace methods to model-order errors and noise covariance assumptions.

6.7 Resolution, Leakage, and Uncertainty

Frequency resolution has several meanings. The DFT bin spacing is f_s/N, but bin spacing is not identical to the ability to distinguish two tones. Practical resolution depends on record duration, window main-lobe width, signal-to-noise ratio, relative amplitude, estimator bias, and the separation between frequencies. Parametric and subspace methods may show sharper peaks than nonparametric methods, but their apparent resolution is conditional on model validity.

Uncertainty should be reported rather than hidden. For averaged periodograms, the estimate at a frequency is approximately related to a scaled chi-square random variable, with degrees of freedom determined by the effective number of averages. Confidence intervals can therefore be formed on a logarithmic or linear scale. For Welch and multitaper methods, the effective degrees of freedom depend on overlap, taper correlation, weighting, and leakage.

Parametric uncertainty can be assessed from residual variance, asymptotic covariance approximations, bootstrap resampling, or repeated-record analysis. Bootstrap methods must respect temporal dependence; independently resampling individual samples from a correlated time series is generally inappropriate. Block bootstrap, parametric simulation, and subsampling are more defensible alternatives.

A useful engineering report should state the sampling rate, record duration, window or taper, segment length, overlap, detrending procedure, normalization, model order, and uncertainty method. Without these details, two apparently different spectra may simply reflect different analysis settings.

6.8 Practical Workflow for Estimator Selection

A reliable analysis begins with the measurement rather than the algorithm. The analyst should inspect missing values, saturation, quantization, transients, nonstationarity, and sampling-clock errors. Detrending and removal of the mean are important when low-frequency leakage is not part of the phenomenon under study.

For exploratory analysis of an unknown broadband process, Welch or multitaper estimation is usually appropriate. For a short record dominated by resonances, AR estimation may provide useful compact structure, provided order and stability are checked. MA or ARMA models become attractive when spectral zeros or mixed pole-zero behaviour are physically meaningful. For a small number of tones in white noise, MUSIC or ESPRIT can offer high resolution, but only after verifying the sinusoidal model and estimating the number of sources.

Results should be compared across reasonable settings. A peak that survives changes in window, segment length, model order, and record subinterval is more credible than one that appears only under a single highly tuned configuration. In industrial monitoring, this robustness check can be automated as part of a condition-monitoring pipeline.

Illustrative Case Study: Closely Spaced Vibrational Modes

Background

Consider an illustrative laboratory case in which an accelerometer samples a rotating machine at 2 kHz. A 4-second record contains broadband structural vibration, a dominant component near 240 Hz, and a weaker mode near 247 Hz. The objective is to determine whether the second component is a genuine mechanical mode or a leakage artefact.

Situation and Analysis

A single rectangular-window periodogram shows a broad asymmetric peak. Because the record length is 4 seconds, the nominal DFT spacing is 0.25 Hz, but the leakage pattern of the strong component obscures the weaker one. A Hann-windowed periodogram suppresses side lobes but broadens the main lobe, so the two modes remain difficult to distinguish.

Welch estimation with eight short, overlapped segments produces a stable broadband floor, but the reduced segment length sacrifices the resolution needed for a 7-Hz separation. A multitaper estimate using a moderate time-bandwidth product reduces leakage and reveals a shoulder near 247 Hz. Several AR models are then fitted. Low orders merge the modes; a moderate order produces two stable resonant peaks; higher orders introduce additional narrow peaks that move substantially when the record is divided into halves.

Approach and Solution

The analyst compares AR orders using MDL, residual whiteness, and stability rather than selecting the order solely because it gives the tallest peak. A moderate-order AR model is retained because its poles are stable, its residuals show little remaining correlation, and both peaks persist across record subdivisions. A multitaper spectrum is used as the nonparametric reference. The agreement between the multitaper shoulder and the stable AR peak supports the interpretation of a real secondary mode.

Outcome

The analysis does not claim that the AR peak alone proves the existence of a second mode. Instead, the mode is accepted provisionally because it is supported by an estimator with controlled leakage, a parametric model with stable poles, and repeatability across data segments. A follow-up measurement at a different machine speed is recommended to test whether the component follows the expected physical behaviour of a mechanical resonance.

Learning Points

The case illustrates that nominal frequency-bin spacing is not practical resolution, model order must be validated, and agreement between methods is more informative than an isolated high-resolution plot. It also shows why uncertainty and sensitivity analysis are essential when a weak component lies near a strong one.

6.9 Limitations, Risks, and Professional Practice

The most common error in nonparametric analysis is confusing a visually smooth spectrum with an accurate spectrum. Smoothing reduces variance but can conceal narrow structure. Conversely, a highly peaked parametric spectrum may reflect overfitting rather than genuine resolution.

Another error is applying AR models to data with strong moving-average structure or severe nonstationarity. A high-order AR model can approximate many processes, but the resulting poles may have no physical interpretation. MUSIC and ESPRIT are frequently misused on broadband signals, coloured noise, or insufficiently sampled data. Their assumptions should be stated explicitly.

Aliasing, poor anti-alias filtering, spectral mirroring, incorrect one-sided scaling, and failure to remove a large mean can all lead to incorrect conclusions. For complex-valued data, conjugation and covariance conventions must be handled consistently. For real-valued data, the relationship between one-sided and two-sided spectra must preserve total power.

Professional practice also requires traceability. Analysis software should record parameter settings and version information, and automated systems should flag unstable models, inadequate sample counts, and order-selection ambiguity. In safety-critical applications, a spectral alarm should not be based on a single unverified peak; decision thresholds should account for false alarms, missed detections, and changing operating conditions.

6.10 Contemporary Developments

Modern spectral analysis increasingly combines classical estimators with statistical learning. Bayesian spectral methods represent uncertainty directly through posterior distributions, while sparse spectral estimation promotes a small number of active frequencies under suitable assumptions. Time-frequency and state-space models address signals whose spectral characteristics evolve with time. Neural networks are being used for denoising, feature extraction, and anomaly detection, but they do not eliminate the need for physically interpretable validation.

In embedded and edge systems, efficient FFT libraries, recursive covariance updates, and reduced-order subspace tracking permit near-real-time spectral monitoring. Streaming implementations must address concept drift, memory limits, and changing noise conditions. The most reliable systems retain classical diagnostic quantities—residuals, confidence intervals, stability indicators, and model-order checks—even when machine-learning components are added.

Best-Practice Checklist

A defensible spectral-estimation study should:

• verify stationarity or explicitly analyse time variation;
• remove mean and trend only when justified;
• document sampling, window, normalization, and averaging parameters;
• distinguish frequency-bin spacing from true resolution;
• use Welch or multitaper methods as robust references;
• validate AR, MA, and ARMA models through residuals and stability;
• select model order using both statistical criteria and engineering knowledge;
• use MUSIC or ESPRIT only when their signal and noise assumptions are credible;
• quantify uncertainty and test sensitivity to analysis choices;
• preserve reproducibility through parameter and software records.

Chapter Summary

Spectral estimation infers the frequency distribution of signal power from finite, noisy observations. The periodogram is simple but has high variance; the correlogram estimates the spectrum through finite-lag correlation; Welch’s method reduces variance by averaging segmented periodograms; and multitaper methods control leakage using orthogonal, spectrally concentrated tapers. Parametric AR, MA, and ARMA models impose a model structure and can provide compact or high-resolution descriptions, but they are sensitive to model mismatch and order selection.

Linear prediction leads directly to AR modelling through the Yule–Walker equations and Levinson–Durbin recursion. A suitable model order balances explanatory power and complexity and should be assessed with information criteria, residual analysis, stability checks, and domain knowledge. MUSIC and ESPRIT exploit signal–noise subspace separation to estimate closely spaced sinusoidal frequencies, although their assumptions must be respected. Resolution is determined by more than DFT spacing, and uncertainty arises from finite data, windowing, averaging, model selection, and noise. Reliable engineering practice therefore combines estimator comparison, documented settings, validation, and uncertainty assessment.

Key Terms

Spectral density: Frequency-domain distribution of the average power of a stationary process.

Periodogram: Squared magnitude of a finite-record Fourier transform, suitably normalized.

Correlogram: Spectral estimate obtained by Fourier transforming an estimated autocorrelation sequence.

Spectral leakage: Spreading of energy caused by finite observation and windowing.

Window: A finite sequence applied to data to control leakage and frequency-domain smoothing.

Welch method: Averaged periodogram method using windowed, usually overlapped, data segments.

Multitaper estimation: Spectral estimation using several orthogonal, spectrally concentrated tapers.

Bias: Difference between the expected estimated spectrum and the true spectrum.

Variance: Random fluctuation of an estimator across repeated data records.

Autoregressive model: Model in which the current sample depends linearly on previous samples and an innovation.

Moving-average model: Model expressing the signal as a finite weighted sum of present and past innovations.

ARMA model: Combined autoregressive and moving-average model.

Linear prediction: Estimation of a sample from a weighted combination of other samples.

Reflection coefficient: Recursion parameter in lattice or Levinson–Durbin prediction algorithms.

Model order: Number of dynamic parameters used in a statistical signal model.

MUSIC: Subspace method that locates frequencies through orthogonality to the estimated noise subspace.

ESPRIT: Shift-invariance subspace method that estimates frequencies from eigenvalues of a transformation.

Resolution: Ability of an estimator to distinguish nearby spectral components.

Whiteness: Absence of significant correlation in a residual or innovation sequence.

Uncertainty assessment: Quantification of variability or confidence associated with an estimate.

Review Questions

Short Answer Questions

1. Why does the raw periodogram remain statistically variable even for long records?

2. What is the principal trade-off introduced by Welch segmentation?

3. State the difference between an AR model and an MA model.

4. Why are reflection coefficients important in linear prediction?

5. What orthogonality principle is used by MUSIC?

Long Answer Questions

1. Compare the periodogram, correlogram, Welch, and multitaper methods in terms of leakage, resolution, variance, and computational requirements.

2. Derive the relationship between an AR model and its power spectral density, and explain the effect of pole location on spectral peaks.

3. Explain the Yule–Walker equations and the role of the Levinson–Durbin recursion in estimating AR coefficients.

4. Discuss AIC, MDL, residual analysis, and stability checks as tools for model-order selection.

5. Explain the assumptions, operation, strengths, and limitations of MUSIC and ESPRIT.

Analytical / Application Questions

1. A short vibration record contains two closely spaced tones and substantial broadband noise. Design an estimation strategy and justify your choices of estimator, window or taper, model order, and validation procedure.

2. An AR spectrum shows three narrow peaks, but only one peak remains when the record is divided into two halves. Analyse possible causes and propose tests before accepting the peaks as physical modes.

3. A MUSIC implementation produces unstable frequency estimates when the noise becomes coloured and two sources are correlated. Explain why this occurs and recommend suitable modifications or alternative methods.

22. Multiple Choice Questions with Answers

Question 1

Which statement best describes the principal statistical weakness of the raw periodogram?

A. It cannot be computed using the DFT

B. Its variance does not generally vanish as record length increases

C. It always produces negative power values

D. It requires an AR model before estimation

Correct Answer: B. Its variance does not generally vanish as record length increases

Question 2

Welch’s method reduces spectral variance primarily by

A. Increasing the pole order of the signal model

B. Replacing the Fourier transform with an autocorrelation recursion

C. Averaging periodograms from windowed data segments

D. Removing all frequency-domain smoothing

Correct Answer: C. Averaging periodograms from windowed data segments

Question 3

What is the main purpose of using multiple DPSS tapers in multitaper estimation?

A. To maximize aliasing between frequency bands

B. To control leakage while obtaining several spectral estimates

C. To force the signal to be white noise

D. To estimate only the signal mean

Correct Answer: B. To control leakage while obtaining several spectral estimates

Question 4

An AR spectrum with a pole close to the unit circle generally exhibits

A. A narrow resonant peak near the pole angle

B. A guaranteed spectral null at DC

C. Zero prediction error for every finite record

D. No dependence on model order

Correct Answer: A. A narrow resonant peak near the pole angle

Question 5

Which criterion generally imposes a stronger complexity penalty than AIC for large sample sizes?

A. The raw periodogram

B. MDL or BIC

C. The rectangular window

D. The covariance lag

Correct Answer: B. MDL or BIC

Question 6

In linear prediction, the Levinson–Durbin recursion is valuable because it

A. Solves successive Toeplitz prediction problems efficiently

B. Eliminates the need for autocorrelation information

C. Guarantees that every estimated model is minimum phase

D. Converts every MA process into white noise without estimation

Correct Answer: A. Solves successive Toeplitz prediction problems efficiently

Question 7

MUSIC identifies a true sinusoidal frequency because its steering vector is approximately

A. Parallel to the noise subspace

B. Orthogonal to the noise subspace

C. Uncorrelated with the signal subspace only

D. Equal to the innovation sequence

Correct Answer: B. Orthogonal to the noise subspace

Question 8

A key computational distinction between ESPRIT and MUSIC is that ESPRIT

A. Does not require a frequency search grid

B. Cannot use covariance matrices

C. Assumes the signal contains no noise

D. Always estimates broadband PSD directly

Correct Answer: A. Does not require a frequency search grid

Question 9

Why can a high-order AR model produce misleading spectral peaks?

A. Higher order always decreases frequency resolution

B. The model may fit random fluctuations and create spurious resonances

C. AR models cannot represent resonances

D. The PSD of an AR model is independent of its coefficients

Correct Answer: B. The model may fit random fluctuations and create spurious resonances

Question 10

Which conclusion is most defensible when a weak spectral peak appears in one tuned analysis but disappears under reasonable changes of record segment and model order?

A. It must be a physical resonance

B. It should be accepted because its amplitude is high

C. It requires further validation before physical interpretation

D. It proves that the sampling frequency is incorrect

Correct Answer: C. It requires further validation before physical interpretation