Library / Signal processing and digital filters / DSP-45
Signal processing and digital filters
Phase delay at a specified frequency
Delay in samples of a single sinusoid passing through a filter, from the unwrapped phase φ in radians at frequency ω (rad/sample, nonzero), where the response magnitude is nonzero.
- Ref
- DSP-45
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- D_φ [samples]
Phase delay is how much a single sinusoid is shifted in time by the filter: D_φ = −φ(ω)/ω samples, from the unwrapped phase φ at ω rad/sample. It differs from the group delay, which measures the delay of the envelope, and it depends on the chosen phase branch, since phase is only known to a multiple of 2π.
Delay in samples of a single sinusoid passing through a filter, from the unwrapped phase φ in radians at frequency ω (rad/sample, nonzero), where the response magnitude is nonzero. The chosen phase branch changes the reported delay.
Calculator
- Phase in degrees−60 °
- Period of the sinusoid, 2π/ω12 samples
- Delay as a fraction of a period, −φ/2π0.1667 cycles
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 tone at 0.5236 rad/sample is delayed by 2 samples, which is 0.167 of its 12-sample period.
- If the phase were wrapped by another 2π the reported delay would jump by 12 samples, so make sure the phase branch is the one that matches the physical delay.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (samples) | Local step | Change in result |
|---|---|---|---|---|---|---|
| φ Unwrapped phase | −50 to 50 rad | −95.49 | 95.49 | 191 | +2 rad | −191 % |
| ω Angular frequency | 0.01 to 3.142 rad/sample | 0.3333 | 104.7 | 104.4 | +10% | −9.09 % |
Sweep of Unwrapped phase
| Unwrapped phase (rad) | Phase delay (samples) |
|---|---|
| −50 | 95.49 |
| −37.5 | 71.62 |
| −25 | 47.75 |
| −12.5 | 23.87 |
| 0 | 0 |
| 12.5 | −23.87 |
| 25 | −47.75 |
| 37.5 | −71.62 |
| 50 | −95.49 |
Sweep one variable
Variables and how each one contributes
Phase delay
What the equation gives you. Everything else feeds it.
Unwrapped phase
The unwrapped phase, in rad. A more negative phase raises the delay in direct proportion. Raising it by 10 % (to −1.152) moves the result by +10 %, from 2 to 2.2 samples.
Angular frequency
The angular frequency, in rad/sample (nonzero). A higher frequency lowers the delay for a fixed phase. Raising it by 10 % (to 0.576) moves the result by −9.09 %, from 2 to 1.818 samples.
Worked example with the values above
- Phase: φ = −1.0472 rad (−60°); frequency: ω = 0.5236 rad/sample.
- Ratio: φ/ω = −2.
- D_φ = −(−2) = 2 samples.
Assumptions and limits
- A stable, linear, time-invariant filter at a steady sinusoidal frequency in rad/sample.
- Phase that is consistently unwrapped, with magnitude not equal to zero at the frequency used.
Common mistakes
- Using wrapped phase, which jumps by 2π and ruins the delay.
- Mixing hertz and rad/sample in the delay formulas.
- Applying the forward–backward result to a single pass or to a record with large end transients.
Statistics and uncertainty
Phase and group delay come from phase differences, so they amplify noise where the magnitude is small and fail at zeros of the response. A secant estimate also carries an error of order Δω² times the third derivative of phase.
Where it is used
- Evaluating gain, phase and delay of a digital filter at a chosen frequency.
- Combining sections and predicting zero-phase filtering.
Maths you need
- Frequency response: H(ω) evaluated on the unit circle
- Group delay: the negative slope of phase against frequency
- Phase delay: the negative phase divided by frequency