Nonparametric Spectrum Estimation: Periodogram, Bartlett, Welch, and Blackman-Tukey

The Estimation Problem

In theory the power spectrum is

\[P_{xx}(e^{j\omega})=\sum_{l=-\infty}^{\infty}r_{xx}[l]e^{-j\omega l}\]

In practice you have $N$ samples and no true autocorrelation. Nonparametric methods estimate the spectrum directly from the data without assuming a model — hence “nonparametric”.

Two unavoidable facts:

1. You only see a finite record, so the estimate is biased (spectral leakage).
2. The estimate is a random variable, so it has variance.

The whole game is trading bias against variance.

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The Periodogram

The simplest estimator replaces the expectation with a finite sum:

\[\hat P_{xx}(e^{j\omega})=\frac{1}{N}\left\vert\sum_{n=0}^{N-1}x[n]e^{-j\omega n}\right\vert^2=\frac{1}{N}\vert X(e^{j\omega})\vert^2\]

This is the periodogram. Computed at the DFT bins:

\[\hat P[k]=\frac{1}{N}\vert X[k]\vert^2,\qquad k=0,\dots,N-1\]

Properties

Bias. The expected value is the true spectrum convolved with the Fejér kernel (the squared magnitude of the rectangular window transform):

\[E\{\hat P(\omega)\}=\frac{1}{2\pi}\int_{-\pi}^{\pi}P_{xx}(\lambda)\,W_R(\omega-\lambda)\,d\lambda\]

The window smears sharp peaks. As $N\to\infty$ the kernel narrows and the bias vanishes — the periodogram is asymptotically unbiased.

Variance. For a Gaussian process,

\[\operatorname{var}\{\hat P(\omega)\}\approx P_{xx}^2(\omega)\]

The variance does not go to zero as $N\to\infty$. The standard deviation is as large as the mean, no matter how much data you collect.

This is the central fact of spectrum estimation: the periodogram is inconsistent. More data does not make it smoother. It only makes the estimate resolve finer detail while remaining equally noisy.

Why the Variance Stays

The periodogram at frequency $\omega$ is roughly the squared magnitude of a complex Gaussian with variance $P_{xx}$. A squared Gaussian has variance $2P^2$ regardless of $N$. Increasing $N$ adds more independent frequency bins, but each individual bin is still a single noisy measurement.

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Bartlett’s Method: Average Periodograms

Split the $N$-sample record into $K$ non-overlapping segments of length $L$ ($N=KL$):

\[\hat P^{(i)}(\omega)=\frac{1}{L}\left\vert\sum_{n=0}^{L-1}x[iL+n]e^{-j\omega n}\right\vert^2\]

Average:

\[\hat P_{\text{Bartlett}}(\omega)=\frac{1}{K}\sum_{i=0}^{K-1}\hat P^{(i)}(\omega)\]

The Trade-off

If the segments are independent,

\[\operatorname{var}\{\hat P_{\text{Bartlett}}\}=\frac{1}{K}\operatorname{var}\{\hat P\}\]

So the variance drops by a factor $K$ — the estimate becomes consistent as $K\to\infty$.

The price: each segment is only $L=N/K$ long, so the frequency resolution is

\[\Delta f_{\text{Bartlett}}=\frac{F_s}{L}=\frac{KF_s}{N}\]

Resolution degrades by the same factor $K$ that the variance improves. This is the bias–variance trade-off of spectrum estimation, and it cannot be beaten.

$K$	Variance	Resolution
1	$P^2$	$F_s/N$
4	$P^2/4$	$4F_s/N$
16	$P^2/16$	$16F_s/N$

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Welch’s Method: Overlap and Window

Welch improves on Bartlett in two ways:

1. Overlap the segments (typically 50%), so more segments fit in the same record.
2. Window each segment with a taper (Hamming, Hann) to reduce leakage.

\[\hat P_{\text{Welch}}(\omega)=\frac{1}{K}\sum_{i=0}^{K-1}\frac{1}{LU}\left\vert\sum_{n=0}^{L-1}w[n]x[iD+n]e^{-j\omega n}\right\vert^2\]

where $D$ is the hop size, $K$ is the number of segments, and

\[U=\frac{1}{L}\sum_{n=0}^{L-1}w^2[n]\]

normalizes the window power so the estimate stays unbiased.

Why Overlap Helps

With 50% overlap, an $N$-point record yields about $K=2N/L-1$ segments instead of $N/L$ — nearly twice the averaging for the same resolution. The segments are no longer independent, so the variance reduction is slightly less than $1/K$, but the gain is still substantial.

Why the Window Helps

The rectangular window has −13 dB sidelobes that leak strong tones across the whole spectrum. A Hann window drops the sidelobes to −31 dB. For a signal with a large dynamic range, this matters far more than the slightly wider main lobe.

Typical Parameter Choices

Parameter	Typical value	Effect
Segment length $L$	256–4096	Sets resolution
Overlap	50% (Hann) or 75%	More averages
Window	Hann or Hamming	Low leakage
FFT length	$\ge L$, usually power of 2	Zero-pad for smooth curve

Rule of Thumb

• Narrowband tones, low noise: long segments, little overlap, rectangular or Hann.
• Broadband noise, smooth spectrum: short segments, 50% overlap, many averages.
• Transients: short segments with high overlap (spectrogram).

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Blackman–Tukey: Window the Autocorrelation

The classical method estimates the autocorrelation first, then transforms:

\[\hat r_{xx}[l]=\frac{1}{N}\sum_{n=0}^{N-1-\vert l\vert}x[n]x[n+\vert l\vert]\] \[\hat P_{BT}(\omega)=\sum_{l=-(M-1)}^{M-1}w[l]\,\hat r_{xx}[l]\,e^{-j\omega l}\]

where $w[l]$ is a lag window of length $2M-1$ and $M\ll N$.

The Lag Window

Truncating the autocorrelation at lag $M-1$ is equivalent to multiplying by a rectangular lag window. That produces negative sidelobes in the spectral estimate (the Gibbs phenomenon of the triangular window). A smoother lag window (Bartlett, Parzen, Hamming) tapers the autocorrelation and removes the negatives.

Lag window	Spectral window	Sidelobe
Rectangular	Fejér (sinc²)	−13 dB, can go negative
Bartlett	Triangular	−27 dB
Hann	Hann	−31 dB
Parzen	Parzen	−53 dB

Relation to the Periodogram

The Blackman–Tukey estimate is the periodogram smoothed by the spectral window:

\[\hat P_{BT}(\omega)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\hat P_{\text{per}}(\lambda)\,W(\omega-\lambda)\,d\lambda\]

So it is a frequency-domain smoothing operation. Smoothing reduces variance exactly the way averaging does, and it costs resolution the same way.

Choosing $M$

• Large $M$: good resolution, high variance.
• Small $M$: smooth estimate, poor resolution.
• Typical: $M\approx N/5$ to $N/10$.

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Comparing the Methods

Method	How it reduces variance	Resolution	Leakage	Cost
Periodogram	Does not	$F_s/N$	High (rectangular)	1 FFT
Bartlett	$K$ non-overlapping averages	$K F_s/N$	High	$K$ FFTs
Welch	$K$ overlapping, windowed	$\approx K F_s/N$	Low	$K$ FFTs
Blackman–Tukey	Lag-window smoothing	$\approx F_s/M$	Controlled by window	1 FFT + autocorr

Welch is the default in practice: it gives a consistent, low-leakage estimate with a single parameter (segment length) to tune.

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Worked Example: Resolving Two Tones

Two tones at 100 Hz and 110 Hz, $F_s=1000$ Hz, $N=1000$ samples.

Resolution needed: $\Delta f=10$ Hz.

Periodogram: resolution $F_s/N=1$ Hz, so the tones are resolvable — but the estimate is noisy, with $\operatorname{std}\approx P$.

Bartlett with $K=10$: each segment $L=100$, resolution $F_s/L=10$ Hz. The tones are just barely resolvable, and the variance is 10× lower. Any more averaging and they merge.

Welch with $L=256$, 50% overlap: resolution $\approx3.9$ Hz, about 7 segments. Both tones resolved with much lower variance. This is why Welch is the practical choice for this kind of measurement.

Lesson: if you need both resolution and low variance, you need more data, not a better algorithm. The uncertainty principle of spectrum estimation is as firm as the one in time–frequency analysis.

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Practical Code

import numpy as np
from scipy.signal import welch, periodogram

fs = 1000
t = np.arange(0, 2, 1/fs)
x = np.cos(2*np.pi*100*t) + np.cos(2*np.pi*110*t) + 0.1*np.random.randn(len(t))

# Periodogram: noisy
f1, P1 = periodogram(x, fs=fs)

# Welch: smooth, overlapping, windowed
f2, P2 = welch(x, fs=fs, nperseg=256, noverlap=128, window='hann')

Plot both on a dB scale and the difference is immediate: the periodogram looks like grass; the Welch estimate shows two clean peaks.

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Summary

1. The periodogram is $\vert X\vert^2/N$; it is asymptotically unbiased but its variance never shrinks with $N$.
2. Bartlett averages non-overlapping periodograms: variance drops by $K$, resolution degrades by $K$.
3. Welch overlaps and windows the segments — the practical default.
4. Blackman–Tukey smooths the periodogram with a lag window; it is frequency-domain smoothing by another name.
5. Every nonparametric method trades resolution for variance; only more data improves both.