DSP 101
M9 · L4
Module 9 — Spectral Analysis
Parametric Spectral Estimation
Beyond the DFT limit — fitting signal models to achieve super-resolution spectral estimates.
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DSP 101
M9 · L4
The Problem
Why Non-Parametric Fails
Periodogram and Welch's method are limited by 1/(NT) — no matter how much post-processing, you cannot resolve two sinusoids closer than this.
- Short records: N is small → resolution is coarse
- Closely spaced tones: merge into one broad peak in the DFT
- Parametric idea: fit a model (AR, subspace) → derive PSD analytically
- Model imposes structure → can extract super-resolution estimates
- Tradeoff: super-resolution comes at the cost of model assumptions
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DSP 101
M9 · L4
The Core Model
Autoregressive AR(p) Model
Each sample = linear combination of p past samples + white noise. Equivalent to an all-pole filter driven by white noise.
AR(p)
x[n] = -\sum_{k=1}^{p} a_k\, x[n-k] + w[n]
AR PSD
P_AR(ω) = σ²/|A(ω)|² — peaks near poles of the all-pole filter. Resolution limited only by model order and SNR, not data length.
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DSP 101
M9 · L4
Fitting the Model
Yule-Walker & Burg
Yule-Walker: relate AR coefficients to the autocorrelation sequence. Solve 𝐑𝐚 = −𝐫 using the O(p²) Levinson-Durbin recursion.
- Yule-Walker: uses sample autocorrelation — simple, standard
- Burg method: minimizes forward + backward prediction errors — preferred for short records, always stable
- Model order p: chosen by AIC, MDL, or FPE criteria
- AIC tends to overestimate; MDL is asymptotically consistent
- Rule of thumb: p ≈ 2 × (number of expected spectral peaks)
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DSP 101
M9 · L4
High-Resolution Methods
MUSIC & ESPRIT
Decompose the covariance matrix into signal subspace (M large eigenvalues) + noise subspace (N−M equal eigenvalues = σ²).
- Key insight: true frequency steering vectors are ⊥ to noise subspace
- MUSIC pseudospectrum: 1/‖𝐄ₙᴴ𝐞(ω)‖² → infinity at true frequencies
- ESPRIT: exploits shift-invariance — extracts frequencies directly by eigendecomposition, no spectral search needed
- Both need M (number of sinusoids) as prior input
- Coherent signals (multipath) degrade performance — fix with spatial smoothing
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DSP 101
M9 · L4
MUSIC Pseudospectrum
The MUSIC Formula
P_MUSIC has sharp peaks at the true sinusoidal frequencies — not a power estimate, but a frequency locator.
MUSIC
P_{MUSIC}(\omega) = \dfrac{1}{\mathbf{e}^H(\omega)\,\mathbf{E}_N\mathbf{E}_N^H\,\mathbf{e}(\omega)}
Interpretation
𝐄_N = noise eigenvectors. When ω = ωₖ, 𝐞ᴴ(ω)𝐄_N𝐄_Nᴴ𝐞(ω) → 0, so P_MUSIC → ∞. Peaks locate frequencies, not power levels.
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DSP 101
M9 · L4
Choosing the Right Tool
Parametric vs. Non-Parametric
AR/MUSIC
Super-resolution, short records, line spectra
Welch
General, broadband, no model needed
- Parametric: speech LPC, radar DOA, biomedical HRV, NMR
- Non-parametric: audio analysis, exploratory spectral inspection
- Model mismatch → parametric can fail catastrophically
- When unsure: start non-parametric, go parametric if needed
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DSP 101
M9 · L4
Key Takeaways
What You Learned
- Parametric methods fit a model first, then compute PSD — enabling super-resolution beyond 1/(NT)
- AR(p) model: x[n] = −∑ aₖ x[n−k] + w[n]; PSD = σ²/|A(ω)|²
- Yule-Walker: solve 𝐑𝐚 = −𝐫 via O(p²) Levinson-Durbin; Burg is preferred for short records
- Model order via AIC (overestimates) or MDL (consistent)
- MUSIC: pseudospectrum peaks at sinusoidal frequencies via noise subspace orthogonality
- ESPRIT: direct frequency estimation from signal subspace — no spectral search
- Use parametric for line spectra + short records; non-parametric for broadband / unknown signals
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