Library / Signal processing and digital filters / DSP-37
Signal processing and digital filters
Decay time constant from a pole radius
Time for the envelope rⁿ of a simple pole of magnitude r (0 < r < 1) to fall to 1/e of its initial value, at sampling rate fs.
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- DSP-37
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- Signal processing and digital filters
- Result
- τ [s]
A single pole of magnitude r (0 < r < 1) gives an impulse response that decays as rⁿ. The envelope falls to 1/e after n = −1/ln r samples, which is a time τ = −1/(fs ln r). For r close to 1 this is about 1/(fs(1 − r)), so a pole at r = 0.99 rings for about 100 samples, and a pole nearer the unit circle rings much longer.
Time for the envelope rⁿ of a simple pole of magnitude r (0 < r < 1) to fall to 1/e of its initial value, at sampling rate fs. A pole near the unit circle gives a long ringing time.
Calculator
- Time constant in samples, −1/ln r100 samples
- Time to decay to 1 % of the start, 4.6τ0.4605 s
- Decay per sample, 1 − r0.995 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A pole of magnitude 0.99005 rings with a time constant of 0.1 s, 100 samples at 1,000 Hz.
- It takes about 0.4605 s to fall to 1 % of its start, which is the settling time to budget for in the filter’s response.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fs Sampling rate | 10 to 1,000,000 Hz | 1e-4 | 10 | 10 | +10% | −9.09 % |
| r Pole magnitude | 0.5 to 0.9999 | 0.001443 | 9.999 | 9.998 | +10% | — |
Sweep of Pole magnitude
| Pole magnitude | Decay time constant (s) |
|---|---|
| 0.5 | 0.001443 |
| 0.5625 | 0.001738 |
| 0.625 | 0.002127 |
| 0.6875 | 0.002668 |
| 0.75 | 0.003475 |
| 0.8124 | 0.004814 |
| 0.8749 | 0.007484 |
| 0.9374 | 0.01547 |
| 0.9999 | 9.999 |
Sweep one variable
Variables and how each one contributes
Decay time constant
What the equation gives you. Everything else feeds it.
Pole magnitude
The magnitude of the pole, between 0 and 1. A pole closer to 1 rings much longer; the time constant grows roughly as 1/(1 − r). Raising it by 10 % (to 1.089) gives no valid result at the other default values.
Sampling rate
The sampling rate, in Hz. A higher rate makes each sample shorter, so the same r gives a shorter time constant. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 0.1 to 0.09091 s.
Worked example with the values above
- Logarithm: ln r = ln(0.99) = −0.01.
- Samples: −1/ln r = 100 samples.
- τ = 100 / 1,000 Hz = 0.1 s.
Assumptions and limits
- A real, linear, time-invariant second-order section with the leading denominator coefficient normalised to one.
- Exact arithmetic; fixed-point overflow and round-off noise are not included.
Common mistakes
- Using the unnormalised a₀ ≠ 1 coefficients without dividing through.
- Taking the pole angle as the frequency of peak gain.
- Treating a stability margin close to zero as safe in finite precision.
Statistics and uncertainty
Coefficient quantisation moves poles: in fixed point or single precision, poles close to the unit circle (r above about 0.999) can drift to instability, so check the margin with the rounded coefficients.
Where it is used
- Checking and designing recursive filters such as notches and resonators.
- Estimating ringing time and oscillation frequency from pole locations.
Maths you need
- Poles: roots of the denominator polynomial in z
- Unit circle: |z| = 1, the stability boundary
- Difference equation: the time-domain form of a recursive filter