The Welch method is a technique for estimating the power spectral density (PSD) of a signal by computing averaged modified periodograms of overlapping, windowed segments of the input data. It trades frequency resolution for reduced variance in the spectral estimate compared to a single periodogram computed from the full data record.
In practice
In embedded signal processing, the Welch method appears most often when you need a stable noise floor or spectral shape estimate rather than a snapshot of instantaneous frequency content. Typical applications include characterizing ADC noise, evaluating sensor output quality, measuring vibration spectra from accelerometers, and profiling audio or audio-adjacent signals. Because the method averages multiple periodograms, random spectral fluctuations are suppressed and the estimate converges toward the true PSD as the number of segments grows.
The core trade-off is resolution versus variance. Shorter segments mean more segments to average (lower variance, smoother estimate) but coarser frequency resolution. Longer segments improve resolution but yield fewer averages and a noisier estimate. A common starting point is 50% overlap with a Hann window, which is also the default behavior of MATLAB's pwelch function -- covered in detail in the EmbeddedRelated post "Use Matlab Function pwelch to Find Power Spectral Density -- or Do It Yourself" and a companion post "A Simplified Matlab Function for Power Spectral Density". Window choice matters: rectangular windows leak badly, while Hann, Hamming, or Chebyshev windows suppress spectral leakage at the cost of slightly widened main lobes.
On resource-constrained targets, the method is often applied offline to data captured from the embedded system, though real-time use is possible depending on hardware and sample rate. When a real-time estimate is needed on an MCU or DSP, the overlapping-segment accumulation loop can be implemented with a circular buffer and a fixed-size FFT, but memory and cycle budgets need careful planning. On Cortex-M4/M7 parts with a hardware FPU and CMSIS-DSP FFT routines, real-time Welch estimation at moderate sample rates is feasible; on 8-bit or 16-bit MCUs without hardware multiply it is generally impractical without significant decimation first.
A subtle pitfall when using Welch PSD estimates to judge system performance is that a smooth, well-converged spectrum can still mislead. The EmbeddedRelated post "The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance" illustrates how spectral plots can obscure important time-domain or phase behavior, a caution that applies whenever PSD estimates are used as the sole design metric.
Frequently asked
How does the Welch method differ from a plain periodogram?
A plain
periodogram computes a single
FFT magnitude-squared over the entire data record. The result has high variance -- spectral estimates at each frequency bin fluctuate widely, and this variance does not reliably shrink simply by lengthening the record at a fixed frequency. The Welch method splits the record into overlapping windowed segments, computes a periodogram for each, and averages them. This averaging reduces variance as more segments are accumulated, at the cost of frequency resolution because each segment is shorter than the full record.
What window function should I use with the Welch method?
The
Hann window is the most common default and a reasonable first choice: it provides good sidelobe suppression (-31 dB first sidelobe) with moderate main-lobe widening, and it works well with 50% overlap because adjacent Hann-windowed segments sum to a constant. Hamming is similar. Flat-top windows give more accurate amplitude readings at the expense of wider main lobes. Chebyshev windows offer
equiripple sidelobe control and are useful when you need a specific out-of-band rejection level; see the EmbeddedRelated post 'Computing Chebyshev Window Sequences' for implementation details. Avoid the rectangular window unless spectral leakage is known to be irrelevant, which is uncommon in practice.
What overlap percentage should I use between segments?
50% overlap is the conventional default and is what
MATLAB's pwelch uses. It works well with the
Hann window because the window weights sum to a near-constant across adjacent segments, making efficient use of every sample. Overlaps up to 75% can be used to squeeze more segments from a short record, further reducing variance, but the additional segments become increasingly correlated so the variance reduction per extra segment diminishes. Zero overlap (non-overlapping segments) is also valid and simpler to implement, but wastes some data near segment boundaries when tapered windows are used.
How do I choose segment length?
Segment length sets both
FFT size and frequency resolution. The FFT bin spacing is approximately
fs / N, where fs is the sample rate and N is the segment length in samples, but true resolvable
bandwidth also depends on the window shape and any zero-padding used. A 1024-point segment at 10 kHz gives a bin spacing of roughly 9.8 Hz. Start with a segment length that gives the finest resolution you need, then check how many segments fit in your data record. If you have fewer than roughly 8 to 16 segments to average, the variance reduction may be insufficient and the estimate will still be noisy. Shorten segments if you need a smoother estimate and can tolerate coarser resolution.
Can the Welch method run in real time on an embedded target?
It depends heavily on the hardware and
sample rate. On Cortex-M4 or Cortex-M7 parts with a hardware FPU and CMSIS-DSP
FFT routines (e.g., STM32F4, STM32H7, i.MX RT), real-time Welch estimation with moderate segment sizes (256 to 1024 points) at audio or low RF sample rates is feasible. On 8-bit or 16-bit MCUs without hardware
floating-point, it is generally impractical at anything but very low sample rates unless you use
fixed-point arithmetic throughout and keep segment sizes small. A common embedded pattern is to capture a burst of samples into RAM, then compute the Welch estimate offline or during a low-activity period.
Differentiators vs similar concepts
The Welch method is often confused with the Bartlett method. Both average multiple periodograms, but Bartlett uses non-overlapping segments with no
window function (equivalent to a rectangular window), while Welch adds both overlap and windowing. The overlap recovers data efficiency lost to tapering at segment edges, and the window suppresses spectral leakage. In practice, the Welch method almost always produces better estimates than Bartlett for real-world signals that are not perfectly stationary or integer-periodic in the segment length. The Welch method is also sometimes loosely equated with pwelch, which is simply
MATLAB's implementation of the Welch algorithm with specific defaults (Hann window, 50% overlap, and a segment length chosen automatically by MATLAB that may vary by release and calling pattern).
