Signals & SystemsDesign workbenchPreliminary engineering
Frequency Response of an LTI System
Analyze a GIVEN continuous-time LTI system H(s): its magnitude (dB) and phase response H(jω) on a log-frequency axis, evaluated exactly, with the value at any chosen frequency.
About this calculator
Every continuous-time LTI system has a frequency response H(jω) — how it scales and phase-shifts a sinusoid of each frequency. This tool takes a transfer function H(s) you provide and evaluates H(jω) exactly (by complex polynomial division, no straight-line asymptote approximation), giving the magnitude in decibels and the unwrapped phase in degrees across frequency, plus the exact value at any single frequency you choose.
The framing here is deliberately narrow: this is ANALYSIS of a system you already have — a measured plant, a filter you designed elsewhere, a textbook H(s) — not the design of a controller or a physical system. It shares the same tested transfer-function engine as the Control Systems toolkit, so H(s) means the same thing here as there; if you are closing a feedback loop and need gain/phase margins, use the Bode Plot Generator in Control Systems, which adds the loop-stability machinery this tool intentionally omits.
Enter the numerator and denominator coefficients in descending powers of s. The magnitude curve reveals passbands, corner frequencies, and roll-off slopes (−20 dB/decade per pole, +20 per zero); the phase curve shows the lag each pole contributes. Read the low-frequency asymptote for the DC gain and the high-frequency slope for the relative degree.
Design notes & common mistakes
- This analyzes a GIVEN H(s) — it is not controller design and reports no loop margins. For gain/phase margin of a feedback loop, use the Control Systems Bode Plot Generator.
- Magnitude uses 20·log₁₀ (an amplitude ratio); each pole adds −20 dB/decade of roll-off and up to −90° of phase, each zero the reverse.
- The phase is unwrapped, so a third-order roll-off honestly reaches −270° instead of wrapping at ±180°.
- Read the low-frequency flat region for DC gain and the high-frequency slope for the relative degree (poles − zeros).
Assumptions
- The system is continuous-time, linear, time-invariant, and proper (deg N ≤ deg D).
- H(jω) is evaluated exactly by complex polynomial division — no asymptotic straight-line approximation.
- Phase is unwrapped by continuity along the frequency sweep.
- The transfer function is taken as given; parameter accuracy is the user's responsibility.
When to use this calculator
Appropriate for
- Plotting the exact magnitude and phase of a known transfer function
- Reading passband, corner frequencies, roll-off slope, and DC gain of a given system
- Teaching frequency response and reusing the same H(s) as the Control toolkit
Not suitable for
- Feedback-loop stability analysis needing gain and phase margins (use the Control Bode Plot Generator)
- Discrete-time or delay systems
- Final filter or system design sign-off without a full specification and verification
What this calculator does not cover
- Analysis of a given H(s) only — no loop-stability margins (that is controller design; see the Bode Plot Generator in Control Systems).
- Continuous-time rational transfer functions only; no transport delay e^(−sT) and no sampled-data (z-domain) systems.
- Proper systems only (deg N ≤ deg D); improper input is rejected rather than approximated.
- Does not design a filter or a system — it reports the response of the one you enter.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
How is this different from the Bode Plot Generator in Control Systems?
This tool analyzes the frequency response of a system you already have — magnitude and phase of H(jω). The Control Systems Bode Plot Generator adds feedback-loop machinery: it treats H(s) as an open-loop L(s) and computes gain and phase margins for closed-loop stability. Use this for analysis of a given system, that one for loop design.
What does the high-frequency slope tell me?
The asymptotic roll-off is −20 dB/decade for each excess pole over zeros (the relative degree). A slope of −40 dB/decade at high frequency means two more poles than zeros; the phase approaches −90° times the relative degree.
Why is the phase unwrapped?
So the accumulated lag is honest. A wrapped plot renders −181° as +179° and hides how far the phase has actually fallen. Unwrapping keeps the phase continuous, so a third-order system reads −270° rather than appearing to jump back up.
References
- Oppenheim, A. V. & Willsky, A. S., Signals and Systems, 2nd ed., Ch. 6 & 9 (frequency response, Laplace)
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 10 (frequency response of H(jω))
- Lathi, B. P., Linear Systems and Signals, 2nd ed. (frequency response of LTI systems)
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