--- title: "Define passband, stopband, αmax, αmin, ωp, ωs" subject: "Filter Design" chapter: 1 key: "semester_5/filter_design:define-passband-stopband-terms" source: https://www.bctengineer.com/q/semester_5/filter_design/define-passband-stopband-terms site: BCT Engineering --- # Define passband, stopband, αmax, αmin, ωp, ωs Filter Design · Chapter 1: Introduction - Asked in: 2070 Asar, 7 marks - Video: [Low-pass High-pass Band-pass Band-stop Filter Basics](https://www.youtube.com/watch?v=-P3oZluajrY) — Academia - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/define-passband-stopband-terms - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1 ## Question Define the terms: Passband, Stopband, αmax, αmin, ωp, ωs and Bandwidth with necessary figures. ## Short answer Passband: frequencies with attenuation ≤ αmax; stopband: frequencies with attenuation ≥ αmin. ωp is passband edge, ωs stopband edge. Bandwidth: passband width; for LPF it is ωp; for BPF it is ω2−ω1. αmax is max allowed passband loss, αmin is minimum required stopband attenuation. ## Answer Attenuation is $\alpha(\omega) = -20\log_{10}|T(j\omega)|$ dB. The figure shows the specification of a low-pass filter: ```text alpha(dB) | : //////////////// amin |- - - - - - - - -:-+ stopband | passband : | | :/ <- transition amax |=================+ 0 +-----------------+--+---------------> w wp ws ``` - **Passband:** the band of frequencies the filter passes with little attenuation, $\alpha \le \alpha_{max}$. For a LPF it is $0 \le \omega \le \omega_p$. - **Stopband:** the band the filter rejects, with $\alpha \ge \alpha_{min}$. For a LPF it is $\omega \ge \omega_s$. - **$\alpha_{max}$:** the maximum attenuation allowed in the passband (passband ripple), e.g. 0.5 dB or 1 dB. - **$\alpha_{min}$:** the minimum attenuation that must be achieved in the stopband, e.g. 40 dB. - **$\omega_p$:** passband edge frequency, the highest frequency (for LPF) where attenuation is still within $\alpha_{max}$. - **$\omega_s$:** stopband edge frequency, from where attenuation is at least $\alpha_{min}$. - **Bandwidth:** the width of the passband. For a LPF it is $\omega_p$ (or $\omega_{3dB}$ for half-power bandwidth). For a BPF it is $BW = \omega_2 - \omega_1$, with centre frequency $\omega_o = \sqrt{\omega_1\omega_2}$. ```text |T| band-pass 1 + .-----. | / \ | / \ +---+---+---+---+---> w w1 wo w2 <--- BW ---> ``` The region between $\omega_p$ and $\omega_s$ is the transition band; the ratio $\omega_s/\omega_p$ (selectivity) together with $\alpha_{max}$ and $\alpha_{min}$ fixes the filter order. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a LPF, which definitions are correct? A. Passband 0 ≤ ω ≤ ωp with α ≤ αmax; stopband ω ≥ ωs with α ≥ αmin B. Passband ω ≥ ωs; stopband 0 ≤ ω ≤ ωp C. Passband α ≥ αmin; stopband α ≤ αmax D. Passband ω ≤ ωs; stopband ω ≥ ωp Answer: A. The passband has small attenuation and the stopband large attenuation. (Key point: Passband and stopband) 2. What do αmax and αmin specify? A. Minimum passband attenuation and maximum stopband attenuation B. The attenuation at the cut-off and at the centre frequency C. Maximum allowed attenuation in the passband, and minimum required attenuation in the stopband D. The passband and stopband edge frequencies Answer: C. αmax limits ripple/droop in the passband; αmin is the attenuation that must be reached beyond ws. (Key point: αmax and αmin) 3. For a band-pass filter, how are bandwidth and centre frequency given? A. BW = √(ω1ω2) and ωo = ω2 - ω1 B. BW = ω2 - ω1 and ωo = √(ω1ω2) C. BW = ω2 - ω1 and ωo = (ω1 + ω2)^2 D. BW = ω1 + ω2 and ωo = ω2/ω1 Answer: B. Centre frequency is the geometric mean of the band edges. (Key point: BPF bandwidth and centre) 4. What fixes the filter order? A. Only the cut-off frequency B. Only the load resistance C. The centre frequency alone D. αmax, αmin and the selectivity ωs/ωp Answer: D. These three specifications determine the needed order. 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