Engineering Reference
Filter Response, Q, Bandwidth and Damping Reference
Quick-reference filter response types, cutoff conventions, center frequency, bandwidth, Q factor, damping ratio, order, roll-off, and approximation terminology.
- Reading Time
- 14 min
- Format
- Response lookup
- Updated
- September 26, 2026
Filter Response Types
| Response | Passes | Rejects | Typical use |
|---|---|---|---|
| Low-pass | Below cutoff | Above cutoff | Smoothing, anti-aliasing, PWM ripple reduction |
| High-pass | Above cutoff | Below cutoff | AC coupling, drift or offset removal |
| Band-pass | Between lower and upper cutoffs | Below and above the passband | Tone, channel, and narrowband selection |
| Band-stop | Below and above the stopband | Between lower and upper cutoffs | Broad interference rejection |
| Notch | Most frequencies | A narrow band around the notch | Mains or known-tone rejection |
| All-pass | Magnitude ideally unchanged | None by magnitude | Phase and delay shaping |
Core Frequency Terms
| Term | Common expression | Boundary |
|---|---|---|
| Cutoff frequency | Defined response boundary, often −3.0103 dB | Definition depends on response and application |
| Lower / upper cutoff | fL and fH | Use the same amplitude or power criterion |
| Bandwidth | BW = fH − fL | Most useful when both cutoff points are defined |
| Geometric center | f0 = √(fL fH) | Common for resonant log-frequency responses |
| Angular frequency | ω = 2πf | rad/s is not interchangeable with Hz |
| Fractional bandwidth | FBW = BW/f0 | Dimensionless; multiply by 100 for percent |
Q and Damping
Bandwidth definition
Second-order damping
Butterworth section
Higher Q tradeoff
| f0 | BW | Q | ζ | Fractional BW |
|---|---|---|---|---|
| 10 kHz | 1 kHz | 10 | 0.05 | 10% |
| 10 kHz | 5 kHz | 2 | 0.25 | 50% |
| 1 kHz | 1.414 kHz | 0.7071 | 0.7071 | 141.4% (interpret with response context) |
Order and Asymptotic Slope
| Order | Poles | Asymptotic magnitude slope | Important qualification |
|---|---|---|---|
| 1 | 1 | 20 dB/decade ≈ 6 dB/octave | Transition is gradual around cutoff |
| 2 | 2 | 40 dB/decade ≈ 12 dB/octave | Q and approximation shape the corner |
| 3 | 3 | 60 dB/decade ≈ 18 dB/octave | Usually realized as first- and second-order sections |
| 4 | 4 | 80 dB/decade ≈ 24 dB/octave | Section Q values are generally not identical |
These are far-from-corner asymptotes. Do not use them as exact attenuation at cutoff or throughout the transition band.
Common Response Approximations
| Approximation | Passband | Stopband | Transition | Typical priority |
|---|---|---|---|---|
| Butterworth | Maximally flat magnitude | Monotonic | Moderate | General-purpose amplitude response |
| Bessel | Smooth, slower transition | Monotonic | Best waveform / delay behavior among common low orders | Pulse and transient preservation |
| Chebyshev I | Equal ripple | Monotonic | Faster than Butterworth for a given order | Sharper transition with passband ripple |
| Chebyshev II | Monotonic | Equal ripple | Faster than Butterworth for a given order | Flat passband with stopband ripple |
| Elliptic | Equal ripple | Equal ripple | Fastest of these for a given order | Minimum order when ripple is acceptable |
Topology Context
| Topology | Primary ideal terms | Real limits to check |
|---|---|---|
| RC / RL | Cutoff, time constant, source/load impedance | Loading, capacitor ESR/leakage, inductor DCR/SRF |
| RLC / LC | Resonance, Q, damping, characteristic impedance | Loss, saturation, ESR/DCR, parasitic resonance |
| Sallen-Key | Natural frequency, section Q, gain | Op-amp GBW, slew rate, output drive, component ratios |
| Multiple-feedback | Center frequency, Q, gain, bandwidth | Op-amp limits, resistor spread, input/output loading |
| Pi filter | Corner/resonance and attenuation | Source/load impedance, damping, inrush, component stress |
| Anti-alias | Sampling rate, Nyquist, pass/stop requirements | ADC drive, settling, alias bands, tolerance |
Worked Reference Examples
Q from bandwidth
Bandwidth from Q
Geometric center
Damping from Q
Half-power amplitude
Third-order asymptote
Common Interpretation Mistakes
- Treating cutoff as a brick wall.
- Calling −3 dB exactly half voltage.
- Using arithmetic instead of geometric center frequency.
- Applying Q = f0/BW without a compatible bandwidth definition.
- Applying ζ = 1/(2Q) outside the standard second-order convention.
- Mixing hertz and radians per second.
- Assuming higher Q is always better.
- Using asymptotic slope as exact corner attenuation.
- Assuming every section in a higher-order filter has the same Q.
- Ignoring source and load impedance.
- Ignoring capacitor ESR and inductor DCR/SRF.
- Ignoring op-amp GBW, slew rate, noise, and output limits.
- Ignoring tolerance sensitivity and response spread.
- Assuming an ideal calculated response replaces simulation or measurement.
Filter Review Workflow
- 1Define passband and stopband requirements.
- 2State every cutoff and attenuation criterion.
- 3Choose response type and approximation.
- 4Choose order from transition requirements.
- 5Choose topology for frequency, impedance, and implementation constraints.
- 6Assign section frequencies and Q values.
- 7Include source and load impedance.
- 8Select practical component values and tolerances.
- 9Check active-device and passive-component limits.
- 10Simulate magnitude, phase, and tolerance corners.
- 11Measure using appropriate source and probe loading.
- 12Compare measured results with the original requirements.
Support reference
FAQ
What is filter cutoff frequency?
Cutoff is a defined response point, often the −3.0103 dB half-power point for a standard first-order magnitude response. It is not a brick-wall boundary.
What does −3 dB mean?
Precisely half power is −3.0103 dB. At equal impedance this corresponds to a voltage or current magnitude ratio of 1/√2, about 0.7071.
How is filter bandwidth calculated?
For a band-pass or band-stop definition with lower and upper cutoff frequencies, bandwidth is commonly BW = fH − fL.
How is center frequency calculated?
For many resonant band-pass contexts, f0 = √(fL fH). Use the definition associated with the actual topology and response.
How is Q factor calculated?
For the applicable resonant bandwidth definition, Q = f0/BW. Q also has topology-specific energy and component relationships.
What is fractional bandwidth?
Fractional bandwidth is BW/f0 and is often expressed as a percentage. It is the reciprocal of Q only where Q = f0/BW applies.
How are Q and damping ratio related?
For the standard second-order denominator convention, ζ = 1/(2Q) and Q = 1/(2ζ). Do not apply that identity to unrelated definitions of Q.
What Q gives a second-order Butterworth response?
A normalized second-order Butterworth section has Q = 1/√2, about 0.7071, under the standard low-pass denominator convention.
Does higher Q always mean a better filter?
No. Higher Q narrows a resonant bandwidth but can increase peaking, ringing, settling time, tolerance sensitivity, and active-device demands.
How does filter order affect roll-off?
Each pole contributes an asymptotic 20 dB/decade, approximately 6 dB/octave. The exact response near cutoff depends on pole locations and approximation.
What is the difference between Butterworth and Bessel filters?
Butterworth prioritizes flat magnitude; Bessel prioritizes smooth delay and transient behavior, generally with a slower transition.
Do source and load impedance affect a passive filter?
Yes. They become part of the network and may shift cutoff, Q, insertion loss, and attenuation.
Can ideal filter equations predict the full hardware response?
No. Tolerances, ESR, DCR, parasitics, loading, op-amp limits, PCB layout, and measurement loading must be included.
When should I use angular frequency?
Use ω = 2πf when a formula is written in radians per second. Do not substitute hertz directly into a formula defined for ω.
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