Periodogram & Welch's Method | DSP-101

The Periodogram — Revisited

In the previous lesson we introduced the periodogram as the simplest estimator of the Power Spectral Density (PSD). Given N samples x[0], x[1], …, x[N−1], the periodogram computes the squared magnitude of the DFT, normalized by N:

The Periodogram

\hat{S}(\omega) = \frac{1}{N}\left|\sum_{n=0}^{N-1} x[n]\, e^{-j\omega n}\right|^2

X_N(ω) is the DFT of the N-sample record. Normalizing by N makes the estimate asymptotically unbiased — its expected value approaches the true PSD as N → ∞.

The periodogram is appealingly simple, but it suffers from a fundamental statistical problem: its variance does not decrease as the data length N increases. In fact, the variance of each spectral estimate is approximately equal to the square of the true PSD value — roughly 100% relative error at every frequency. No matter how much data you collect, the periodogram remains a noisy estimate.

To understand why, consider that collecting more data with a single DFT gives you finer frequency resolution (smaller Δf = f_s/N), but each frequency bin still corresponds to only one complex-valued DFT sample. The "sample size" per bin never increases — you're always computing one noisy measurement per frequency.

Bias and Variance of the Periodogram

Statistical estimators are characterized by two properties: bias (systematic error) and variance (random fluctuation). For the periodogram:

Property	Behavior	Effect
Bias	Decreases as N → ∞	Asymptotically unbiased — approaches true PSD
Variance	≈ [S(ω)]² — constant!	Does NOT decrease with more data
Consistency	Not consistent	Never converges to the true PSD

The periodogram is therefore an inconsistent estimator — in statistics, an estimator that does not converge to the true value as sample size grows. This is a critical limitation. For reliable PSD estimates, we need methods that reduce variance by averaging multiple independent spectral estimates.

Why Variance Stays High

Each DFT bin X[k] involves contributions from all N samples — so it's not "one sample" in an intuitive sense. But mathematically, each |X[k]|² follows a scaled chi-squared distribution with 2 degrees of freedom, regardless of N. Averaging chi-squared variables reduces variance — and that's exactly what Bartlett's and Welch's methods do.

Var[periodogram] ≈ S²(ω) — independent of N

Bartlett's Method — Averaging Non-Overlapping Segments

The key insight to reducing periodogram variance is averaging. If you could compute the periodogram independently on K different data records and average the results, the variance would decrease by a factor of K. Bartlett's method (1948) achieves this from a single long data record by splitting it into non-overlapping segments.

Given N total samples and a segment length of M, Bartlett's method creates K = ⌊N/M⌋ non-overlapping segments. The periodogram is computed for each segment, then the K periodograms are averaged:

Bartlett's Estimator

\hat{S}_B(\omega) = \frac{1}{K}\sum_{i=0}^{K-1} \frac{1}{M}\left|\sum_{n=0}^{M-1} x_i[n]\, e^{-j\omega n}\right|^2

x_i[n] is the i-th segment of length M. Averaging K periodograms reduces variance by a factor of K — but frequency resolution also decreases (Δf = f_s/M, not f_s/N).

Bartlett's method reveals the fundamental variance-resolution trade-off in PSD estimation. Shorter segments → more segments → lower variance, but poorer frequency resolution. Longer segments → fewer segments → better resolution, but higher variance. You cannot have both simultaneously with a fixed amount of data.

Welch's Method — Overlapping Windowed Segments

Peter Welch (1967) improved Bartlett's method in two key ways: overlapping segments and window functions. By allowing segments to overlap, more periodograms can be computed from the same data, increasing K and further reducing variance without throwing away data at segment edges.

Welch's Estimator

\hat{S}_W(\omega) = \frac{1}{KU}\sum_{i=0}^{K-1} \left|\sum_{n=0}^{M-1} x_i[n]\, w[n]\, e^{-j\omega n}\right|^2

w[n] is the window function applied to each segment. U = (1/M)Σw²[n] normalizes for the window's energy. Windowing reduces spectral leakage at the cost of slightly wider peaks.

The window function applied to each segment reduces spectral leakage — the bleeding of energy from a strong frequency component into neighboring bins. Without windowing (i.e., the rectangular window), the sharp edges at segment boundaries create large sidelobes in the frequency domain. Applying a smooth window (Hann, Hamming, Blackman) tapers the edges and dramatically reduces leakage.

Bartlett vs. Welch — Key Differences

Bartlett's Method

Non-overlapping segments only
Rectangular window (no tapering)
K = ⌊N/M⌋ segments
Variance reduced by factor K
Susceptible to spectral leakage
Simpler to implement

Welch's Method

Overlapping segments (typically 50%)
Smooth window function applied
More segments for same data length
Greater variance reduction
Reduced spectral leakage
Industry standard (pwelch, scipy)

Choosing Welch Parameters in Practice

Three parameters control the Welch estimator's behavior. Choosing them correctly is an engineering judgment that depends on your application:

Parameter	Effect on Resolution	Effect on Variance	Typical Value
Segment length M	↑M → finer Δf = f_s/M	↑M → fewer segments → higher variance	256–2048 samples
Overlap %	No effect on resolution	↑overlap → more segments → lower variance	50% (Hann) or 75%
Window type	Wider main lobe = coarser resolution	Lower sidelobes = less leakage bias	Hann (general purpose)

The 50% Overlap Rule

With a Hann window and 50% overlap, consecutive segments share half their data. This overlap percentage is optimal because the Hann window has zero amplitude at the edges — samples near segment boundaries are downweighted and overlap lets them contribute fully in the adjacent segment. Increasing overlap beyond 50–75% provides diminishing returns since successive segments become highly correlated.

Hann window + 50% overlap = the most common Welch configuration

Implementation — pwelch and scipy

Welch's method is the default PSD estimation tool in MATLAB (pwelch) and Python (scipy.signal.welch). Both functions implement the same algorithm but use slightly different default conventions:

Parameter	MATLAB pwelch	scipy.signal.welch
Default window	Hamming	Hann
Default segment length	⌊N/8⌋	256
Default overlap	50%	50%
Output convention	One-sided by default	One-sided by default
Normalization	Power per Hz (density)	Power per Hz (density)

A key subtlety: both tools output a one-sided PSD by default with units of power/Hz (also called spectral density). If you want total power in a frequency band, integrate the PSD over that band: P_band = ∫ S(f) df ≈ Σ S[k] · Δf.

Interpreting the Welch Output

When comparing PSD plots from different sources, always check: (1) one-sided vs. two-sided, (2) linear vs. dB scale, (3) power/Hz vs. power/sample (normalized). A 3 dB offset often indicates one-sided/two-sided confusion; a frequency axis mismatch indicates normalized vs. physical frequency.

Always verify: one-sided or two-sided? Power/Hz or power/sample?

Key Takeaways

The periodogram (|DFT|²/N) is asymptotically unbiased but has constant variance ≈ S²(ω) — it is an inconsistent estimator.
Bartlett's method splits data into K non-overlapping segments and averages their periodograms, reducing variance by K at the cost of frequency resolution.
Welch's method adds two improvements: overlapping segments (more segments → lower variance) and window functions (reduce spectral leakage).
The three Welch parameters — segment length M, overlap %, and window — trade off frequency resolution against variance reduction.
Hann window with 50% overlap is the most common practical configuration; used by default in pwelch and scipy.signal.welch.
Welch output is one-sided by default; integrate over a band to compute band power.
There is no free lunch: given fixed total data, you cannot simultaneously improve resolution and reduce variance.