Library / Signal processing and digital filters / DSP-42
Signal processing and digital filters
Response of cascaded second-order sections
Combined response of up to three filter sections at the same frequency, as the complex product of their responses.
- Ref
- DSP-42
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- |Htotal|
Filters in cascade multiply their frequency responses: H_total(ω) = H₁(ω)H₂(ω)H₃(ω). The magnitudes multiply and the phases add. Use 1 + 0i for an unused section and include any separate scalar gain once. Cascading second-order sections is the standard way to build high-order IIR filters, because it keeps each section’s poles well conditioned.
Combined response of up to three filter sections at the same frequency, as the complex product of their responses. Set unused sections to 1 + 0i, and include any separate scalar gain once. The product keeps each section’s phase. The page returns the gain; the real part, imaginary part and phase are in the derived rows.
Calculator
- Real part of H_total0
- Imaginary part of H_total2
- Phase of H_total90 °
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.
- The cascade has a gain of 2 and a phase of 90°, which is the sum of the three section phases.
- The magnitudes 1.41, 0.707 and 2 multiply to 2.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Re H2 Real part, section 2 | −10 to 10 | 1.414 | 28.32 | 26.91 | +0.4 | +45.6 % |
| Im H2 Imaginary part, section 2 | −10 to 10 | 1.414 | 28.32 | 26.91 | +0.4 | −27.9 % |
| Re H1 Real part, section 1 | −10 to 10 | 1.414 | 14.21 | 12.8 | +0.4 | +21.7 % |
| Im H1 Imaginary part, section 1 | −10 to 10 | 1.414 | 14.21 | 12.8 | +0.4 | +21.7 % |
| Im H3 Imaginary part, section 3 | −10 to 10 | 0 | 10 | 10 | +0.4 | +20 % |
| Re H3 Real part, section 3 | −10 to 10 | 2 | 10.2 | 8.198 | +0.4 | +1.98 % |
Sweep of Imaginary part, section 3
| Imaginary part, section 3 | Total gain |
|---|---|
| −10 | 10 |
| −7.5 | 7.5 |
| −5 | 5 |
| −2.5 | 2.5 |
| 0 | 0 |
| 2.5 | 2.5 |
| 5 | 5 |
| 7.5 | 7.5 |
| 10 | 10 |
Sweep one variable
Variables and how each one contributes
Total gain
What the equation gives you. Everything else feeds it.
Real part, section 1
The real part of the response of section 1. Rotates and scales the total response together with the other sections. Raising it by 10 % (to 1.1) moves the result by +5.12 %, from 2 to 2.102.
Imaginary part, section 1
The imaginary part of the response of section 1. Rotates and scales the total response together with the other sections. Raising it by 10 % (to 1.1) moves the result by +5.12 %, from 2 to 2.102.
Real part, section 2
The real part of the response of section 2. Rotates and scales the total response together with the other sections. Raising it by 10 % (to 0.55) moves the result by +5.12 %, from 2 to 2.102.
Imaginary part, section 2
The imaginary part of the response of section 2. Rotates and scales the total response together with the other sections. Raising it by 10 % (to −0.55) moves the result by +5.12 %, from 2 to 2.102.
Real part, section 3
The real part of the response of section 3. Rotates and scales the total response together with the other sections. Raising it by 10 % (to 0) moves the result by +0 %, from 2 to 2.
Imaginary part, section 3
The imaginary part of the response of section 3. Rotates and scales the total response together with the other sections. Raising it by 10 % (to 2.2) moves the result by +10 %, from 2 to 2.2.
Worked example with the values above
- First two sections: (1 + 1i) × (0.5 − 0.5i) = 1 + 0i.
- Times section 3, 0 + 2i: 0 + 2i.
- |H_total| = 2.
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