Frequency Bandwidth & PSD Resolution

What Is It?

Frequency resolution is the spacing between adjacent frequency points in a PSD. It determines how accurately the PSD represents the true spectral content of the signal. Coarse resolution can smear narrow peaks, reducing their amplitude and overestimating the surrounding values. Fine resolution captures narrow peaks accurately but requires more computation. Understanding frequency resolution is essential for both PSD measurement and PSD-based analysis.

Why It Matters

In random vibration analysis, the frequency resolution of the response PSD determines how accurately resonant peaks are captured. A coarse resolution can underestimate the peak response at a resonance, leading to an underestimated RMS and non-conservative conclusions. In PSD measurement, the resolution is determined by the sample length and windowing. In FEA, the resolution is controlled by the solver output frequency spacing.

Coarse frequency resolution can smear narrow resonant peaks, underestimating the response. The resolution must be fine enough to resolve the narrowest mode of interest.

Frequency Resolution

Frequency resolution is the spacing between adjacent frequency points in the PSD. In measurement, it is determined by the length of the time record and the FFT. In FEA, it is the output frequency spacing specified by the analyst. The resolution determines the smallest spectral feature that can be distinguished.

Frequency resolution:

Δf  =  f_s / N  =  1 / T

where:
Δf  =  frequency resolution  [Hz]
f_s  =  sample rate  [Hz]
N   =  number of samples in the FFT block
T   =  duration of the time record  [s]

Finer resolution requires longer time records (measurement)
or more frequency output points (FEA).

Bandwidth

The bandwidth of a spectral feature — such as a resonant peak — is the frequency width over which it is significant. The half-power bandwidth of a resonance is the width at the level 3 dB below the peak. For a mode with damping ratio ζ and natural frequency fn, the half-power bandwidth is approximately 2·ζ·fn. The frequency resolution must be significantly finer than this bandwidth to capture the peak accurately.

Half-power bandwidth of a resonance:

Δf_HP  ≈  2 · ζ · f_n

where:
ζ    =  damping ratio
f_n  =  natural frequency  [Hz]

Rule of thumb:  Δf  ≤  Δf_HP / 5  to  Δf_HP / 10
(fine enough to resolve the peak)

FFT Bins

In PSD measurement from time-domain data, the PSD is computed using the Fast Fourier Transform (FFT). The FFT divides the frequency range into bins, each with width Δf = fs/N. Each bin represents the spectral content within that frequency band. A narrow peak that falls within a single bin is represented by a single value — if the bin is wider than the peak, the peak is smeared and its amplitude is underestimated.

• FFT bins — discrete frequency intervals in the computed PSD
• Bin width = sample rate / number of samples
• Narrow peaks within a single bin are smeared — amplitude underestimated
• Windowing (Hanning, etc.) affects bin shape and leakage

Narrow Peaks and Modal Resolution

Structural resonances produce narrow peaks in the response PSD. The width of these peaks is determined by the damping — lightly damped modes produce very narrow peaks. If the frequency resolution is too coarse, the peak may fall between output points and be missed entirely, or be smeared across adjacent points with reduced amplitude. This produces an underestimated RMS response.

Numerical Implications

In FEA, the frequency resolution of the random vibration analysis output affects both accuracy and computational cost. Finer resolution requires more frequency response evaluations. For structures with many closely spaced modes, fine resolution is needed to resolve each peak. The cost can be managed by using finer resolution only near resonances and coarser resolution elsewhere.

• Finer resolution = more frequency points = more computation time
• Coarse resolution risks missing narrow peaks — non-conservative
• Adaptive frequency spacing — fine near resonances, coarse elsewhere
• Check RMS convergence by re-running with finer resolution

Resolution in Measurement

In PSD measurement, the resolution is determined by the sample length: longer records produce finer resolution. A record of T seconds produces Δf = 1/T Hz. To achieve 1 Hz resolution, a 1-second record is needed; for 0.1 Hz resolution, 10 seconds. Averaging multiple PSDs improves the statistical reliability but does not change the resolution.

Measurement resolution:

Δf  =  1 / T

For 1 Hz resolution:  T = 1 s
For 0.1 Hz resolution:  T = 10 s
For 0.01 Hz resolution:  T = 100 s

Averaging N PSDs reduces variance but does not change Δf.

Resolution in FEA

In random vibration FEA, the solver computes the response PSD at discrete frequency points. The analyst specifies the frequency range and the number of output points (or the frequency spacing). The spacing should be fine enough to resolve the narrowest mode. Many solvers offer logarithmic or adaptive spacing that concentrates points near resonances.

In FEA, specify output frequency points fine enough to resolve the narrowest resonance. Check that response PSD peaks align with natural frequencies and that the RMS does not change with finer resolution.

Key Takeaways

• Frequency resolution is the spacing between PSD frequency points — it determines the smallest feature that can be distinguished
• Coarse resolution smears narrow peaks, underestimating the response — non-conservative
• Resolution should be at least 5-10 times finer than the half-power bandwidth of the narrowest mode
• In measurement, resolution = 1/T (longer records = finer resolution)
• In FEA, use fine resolution near resonances and check RMS convergence with finer spacing