Technology / Technical guide
Notch Filters: Selective Attenuation & Phase Consequences
A notch filter reduces a selected frequency region while retaining more outside it. It changes a represented signal, rather than removing physical excitation or adding structural damping. Its usefulness depends on which information is suppressed and what remains necessary to interpret the signal.
A narrow rejection band has a particular purpose.
Magnitude, width and phase describe different aspects of the same filter; none independently establishes a physical fault or remedy.
01 / Selective response
A notch differs from a general low-pass transformation.
A low-pass response broadly attenuates content above a transition region.
A low-pass response broadly attenuates content above a transition region. A notch concentrates attenuation around a chosen centre frequency and can retain both lower and higher frequencies. This selectivity is useful mathematically when a narrow component is distinct from other represented information.
MathWorks describes notching filters by their rejection band and quality factor. That description concerns frequency response. It does not establish that a peak in a measurement is unwanted, nor that removing the peak leaves every meaningful component unchanged.
02 / Ideal model
Zeros and poles explain the frequency shape.
Consider the illustrative continuous model H(s) = (s² + ω0²)/(s² + (ω0/Q)s + ω0²), with ω0 positive and Q positive.
Consider the illustrative continuous model H(s) = (s² + ω0²)/(s² + (ω0/Q)s + ω0²), with ω0 positive and Q positive. The input and output share units, so H is dimensionless. This chosen model is not a MARSTON filter specification.
At s = jω0, its numerator is zero and its denominator remains nonzero, producing ideal rejection at the centre. At zero frequency the gain is one; its high-frequency limit is also one. The poles have negative real parts for the stated positive parameters.
03 / Bandwidth
Quality factor compares the centre with the rejection width.
For this ideal model, Q = ω0/Δω, where Δω is the width between the half-power magnitude points around the notch.
For this ideal model, Q = ω0/Δω, where Δω is the width between the half-power magnitude points around the notch. The same ratio is f0/Δf when both frequencies use hertz. It is a ratio, not a frequency or a damping coefficient.
In a hypothetical example, f0 = 8 Hz and Q = 4 give a width of 2 Hz. Those numbers describe the selected mathematical response. They are neither recommended settings nor evidence that an actual vibration at 8 Hz should be suppressed.
04 / Phase
Magnitude rejection does not describe the entire response.
Away from the exact zero, the filter introduces a frequency-dependent phase relationship.
Away from the exact zero, the filter introduces a frequency-dependent phase relationship. A narrow magnitude feature can coincide with a rapid phase change nearby. At the exact ideal zero the output amplitude vanishes, so a meaningful output phase is not defined there.
If the transformed signal belongs to a feedback model, its phase contribution participates in that model’s loop relationship. Attenuation alone cannot establish the stability consequence. The location of the filter within the model matters as much as its standalone magnitude curve.
05 / Mismatch
A moving component can leave a fixed rejection centre.
The model rejects exactly at its stated centre.
The model rejects exactly at its stated centre. A tone at a different frequency generally has nonzero output, even if it lies near that centre. When a represented frequency varies, a fixed notch therefore produces changing attenuation rather than universal removal.
Multiple close components also complicate interpretation. Suppressing one region may remove information shared by several causes, while a nearby component remains visible. A quieter plot can be the result of the transformation rather than a reduction in physical motion or force.
06 / Scope
Filtering a measurement does not dissipate structural energy.
Mechanical damping concerns energy loss in a physical response.
Mechanical damping concerns energy loss in a physical response. A notch acting on a signal concerns the information passed through a mathematical mapping. These mechanisms should remain distinct even when both reduce a plotted amplitude at one frequency.
No MARSTON notch algorithm, centre, quality factor or adaptive function is claimed. This explanation gives no filter settings, tuning sequence or live-signal modifications; it identifies what a declared rejection response can and cannot mean.
Related technical guides: Phase-Locked Loops: Phase, Frequency & Synchronisation · Vibration Envelopes: Modulation, Sidebands & Interpretation · Modal Superposition: Coordinates, Coupling & Truncation.
General background: MathWorks: peaking and notching filter concepts. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Centre f0 | Frequency of ideal rejection | A defined filter model and frequency unit |
| Quality factor Q | Centre divided by bandwidth | Use the same frequency convention in numerator and denominator |
| Magnitude |H| | Amplitude ratio | Input and output quantity must be comparable |
| Phase arg H | Angle relationship away from zeros | Undefined where the ideal output amplitude is zero |
| Physical damping | Energy dissipation in a structure | Not established by signal attenuation |
Engineering questions.
Is a notch the same as a low-pass filter?
No. A notch targets a bounded frequency region rather than all higher content.
Does a notch remove the vibration source?
Signal rejection does not establish any change in the physical excitation.
Is Q measured in hertz?
No. It is a dimensionless frequency ratio.
Does an ideal notch reject every nearby tone equally?
No. Attenuation varies with frequency relative to the centre.
What happens to phase at the exact zero?
A zero-amplitude output does not have a defined phase angle.
Is this an actual converter filter design?
No. The transfer model and example are illustrative only.
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