--- title: "Passband, transition band, roll-off and design steps" subject: "Filter Design" chapter: 1 key: "semester_5/filter_design:passband-rolloff-design-steps" source: https://www.bctengineer.com/q/semester_5/filter_design/passband-rolloff-design-steps site: BCT Engineering --- # Passband, transition band, roll-off and design steps Filter Design · Chapter 1: Introduction - Asked in: 2079 Bhadra, 4+3 marks - Video: [Practical Low-Pass Filters](https://www.youtube.com/watch?v=cSWnRKwLV8A) — Precision Filters - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/passband-rolloff-design-steps - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1 ## Question Define the following terms with the help of illustrations: Passband, Stopband, Transition band, Roll-off and band width. Write down the basic steps to be followed in the design of a filter. ## Short answer Passband allows signals with attenuation ≤ αmax; stopband attenuates by ≥ αmin; transition band lies between them; roll-off is the attenuation slope (20n dB/decade); bandwidth is the passband width. Design steps: 1. Specifications, 2. Normalization, 3. Approximation, 4. Realization, 5. Denormalization, 6. Non-idealities study, 7. Build and test. ## Answer #### Terms ```text |T|(dB) 0 |~~~~~~~~~~~~~~. | passband \ transition band | \ (roll-off slope) -As |- - - - - - - - -\_______________ | : stopband +-------------+----+--------------> w wp ws ``` - **Passband:** the range of frequencies passed with attenuation not more than $\alpha_{max}$ (LPF: $0$ to $\omega_p$). - **Stopband:** the range attenuated by at least $\alpha_{min}$ (LPF: $\omega \ge \omega_s$). - **Transition band:** the region between $\omega_p$ and $\omega_s$, where attenuation rises from $\alpha_{max}$ to $\alpha_{min}$. An ideal filter has zero transition width. - **Roll-off:** the rate at which the gain falls beyond the cut-off, in dB/decade or dB/octave. An $n$th-order all-pole LPF rolls off at $20n$ dB/decade ($6n$ dB/octave); a steeper roll-off needs a higher order. - **Bandwidth:** the width of the passband: $\omega_p$ (or $\omega_{3dB}$) for LPF, $\omega_2 - \omega_1$ between half-power points for BPF. #### Basic steps in filter design 1. **Specifications:** state the filter type (LP, HP, BP, BS) and $\alpha_{max}, \alpha_{min}, \omega_p, \omega_s$, gain, impedance levels. 2. **Normalization:** convert to a normalized low-pass prototype ($\omega_p$ = 1 rad/s, 1 Ω), using frequency transformation for HP/BP/BS. 3. **Approximation:** choose a response (Butterworth, Chebyshev, inverse Chebyshev, elliptic, Bessel), find the order $n$ and the transfer function $T(s)$ that meets the specifications. 4. **Realization (synthesis):** build a circuit for $T(s)$: a passive LC ladder, or cascaded active biquads (Sallen–Key, MFB, Tow–Thomas), or switched-capacitor sections. 5. **Denormalization (scaling):** apply frequency and magnitude scaling (and the inverse frequency transformation) to get practical element values. 6. **Study of non-idealities:** check sensitivity to element tolerances, op-amp limits and temperature; simulate. 7. **Construction and testing:** build, measure and tune the filter. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the transition band? A. The region below ωp B. The region above ωs C. The region between ωp and ωs where attenuation rises from αmax to αmin D. The region around the centre frequency of a BPF Answer: C. An ideal filter has a zero-width transition band. (Key point: Transition band) 2. What is the roll-off of an nth-order all-pole LPF? A. 20 dB/decade regardless of n B. 20n dB/decade (6n dB/octave) C. n dB/decade D. 6 dB/decade for each decade of order Answer: B. Each order adds 20 dB/decade, so steeper roll-off needs higher order. (Key point: Roll-off) 3. In the filter design procedure, which order is correct? A. Specification, normalization, approximation, realization, denormalization, non-idealities, construction B. Specification, realization, approximation, normalization, denormalization, testing C. Approximation, specification, denormalization, realization, normalization, testing D. Specification, approximation, scaling, normalization, realization, testing Answer: A. You normalize to a prototype, find T(s), realize a circuit, and only then denormalize to actual values. (Key point: Design step order) 4. What is done in the approximation step of filter design? A. Choose op-amps and resistor tolerances B. Scale the elements to real values C. Measure and tune the built filter D. Choose a response (e.g. Butterworth) and find the order n and T(s) meeting the specifications Answer: D. Approximation picks Butterworth, Chebyshev, elliptic and so on, and finds the order and transfer function. (Key point: Approximation step) ## Related in Filter Design - [Define passband, stopband, αmax, αmin, ωp, ωs](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adefine-passband-stopband-terms) — Filter Design, chapter 1 - [Frequency scaling relations and filter design steps](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afrequency-scaling-design-steps) — Filter Design, chapter 1 - [How transformation reduces design steps; band stop design](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abandstop-design-steps-reduction) — Filter Design, chapter 3 - [Band stop filter (2000 rad/s, 400 rad/s) from 3rd order](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abandstop-2000-400-third-order) — Filter Design, chapter 3, asked 4× - [Steps in designing a constant delay filter](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-design-steps) — Filter Design, chapter 2, asked 2× - [Tow-Thomas band pass biquad design](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atow-thomas-bandpass-design) — Filter Design, chapter 6 ## Related in other subjects - [Filter design steps and triple-pad HPF](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atriple-pad-hpf-microstrip) — RF and Microwave Engineering, chapter 6 - [Filter design steps and double pad T-type filter](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-pad-t-filter-microstrip) — RF and Microwave Engineering, chapter 6 - [Fuse terms with cut-off characteristic sketch](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Afuse-terms-cutoff-sketch) — Switchgear and Protection, chapter 3 - [Bilinear Butterworth, 0.25π passband, −15 dB at 0.45π](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Abilinear-butter-0-25-1db-0-45) — Digital Signal Analysis and Processing, chapter 6, asked 3×