Power Spectral Density (PSD)

Power Spectral Density describes the distribution of signal power as a function of frequency, normalized per unit bandwidth. Unlike the Auto Spectrum (which depends on the FFT block size), the PSD provides a bandwidth-independent representation that enables direct comparison between measurements made with different frequency resolutions.

Definition

The PSD is obtained by dividing the Auto Spectrum (power spectrum) by the effective noise bandwidth (ENBW) of the analysis:

S_xx(f) = |X(f)|² / (ENBW · Δf)

The ENBW accounts for the broadening effect of the window function. For a Hanning window, ENBW = 1.5 · Δf. The resulting PSD has units of power per hertz, for example Pa²/Hz for sound pressure or (m/s²)²/Hz for acceleration.

The ENBW correction factor is 1.5 for the Hanning window. Omitting or double-applying this factor introduces a systematic error of approximately 1.76 dB (= 10 · log₁₀(1.5)). If you observe this offset, check whether the ENBW normalization is applied correctly.

Relationship to the FFT / Auto Spectrum

The Auto Spectrum and the PSD contain the same information but differ in normalization:

Property	Auto Spectrum	PSD
Normalization	Per frequency bin	Per Hz
Units (acoustic)	Pa²	Pa²/Hz
Units (vibration)	(m/s²)²	(m/s²)²/Hz
Depends on Δf	Yes	No
Integration over band	Sum of bins	Integral yields total power

To convert from Auto Spectrum to PSD, divide each bin value by the frequency resolution Δf (after ENBW correction). Conversely, to recover the Auto Spectrum from the PSD, multiply by Δf.

Parameters

The PSD shares all parameters with the FFT analysis. The only additional consideration is the normalization mode:

Parameter	Options	Description
Spectrum Size	256 – 65536	Determines the frequency resolution Δf = f_s/N.
Overlap	0% – 95%	Block overlap percentage.
Window Type	Hanning, Hamming, Flat Top, etc.	Each window has a different ENBW correction factor.
Averaging	Linear / Exponential	Averaging mode for variance reduction.
Output Scaling	PSD / ASD	PSD in power/Hz; ASD (Amplitude Spectral Density) is the square root of PSD.

Use Cases

Random Vibration Testing

PSD is the standard representation for random vibration profiles in environmental testing (MIL-STD-810, ISO 16750). Test specifications define the required acceleration PSD as a piecewise function of frequency, and control systems regulate the shaker output to match this target profile.

Noise Floor Characterization

Because the PSD is independent of frequency resolution, it provides a consistent measure of the noise floor of a measurement system. Comparing PSD values at different block sizes verifies that the system noise is correctly characterized.

Broadband Energy Comparison

When comparing the energy content of two signals across a specific frequency band, integrate the PSD over that band:

P_band = ∫_f1^f2 S_xx(f) df

This integral is independent of the FFT resolution used, making PSD ideal for standardized comparisons.

Practical Tips

Use PSD when comparing spectra measured with different block sizes or sampling rates.
For tonal signals (narrowband peaks), the Auto Spectrum is usually more intuitive than PSD.
When reporting overall levels in dB, always specify whether the value comes from the Auto Spectrum or the PSD to avoid ambiguity.
The Amplitude Spectral Density (ASD = √PSD) is useful when working in linear amplitude units rather than power units.

Do not confuse PSD with the Energy Spectral Density (ESD), which is used for transient signals and has units of energy per Hz (e.g., Pa²·s/Hz). The PSD applies to stationary signals with defined power.