--- title: "Ideal vs practical filters and scaling" subject: "Filter Design" chapter: 1 key: "semester_5/filter_design:ideal-practical-filters-scaling" source: https://www.bctengineer.com/q/semester_5/filter_design/ideal-practical-filters-scaling site: BCT Engineering --- # Ideal vs practical filters and scaling Filter Design · Chapter 1: Introduction - Asked in: 2082 Bhadra, 2+5 marks; 2080 Baisakh, 3+4 marks (asked 2 times) - Video: [4. FIR Session 2_Part 2-Ideal Vs Practical filter](https://www.youtube.com/watch?v=u95Zf2KbpC8) — Dr Ashwini Deshpande - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/ideal-practical-filters-scaling - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1 ## Question Compare and contrast between ideal and practical filters. Derive the necessary formulae for magnitude scaling and frequency scaling used in the design of a filter. ## Short answer An ideal filter has flat passband gain, infinite stopband attenuation, vertical transition, and is non-causal. A practical filter exhibits passband ripple (αmax), finite stopband attenuation (αmin), a transition band, and is realizable with finite order. Magnitude scaling: R' = km·R, L' = km·L, C' = C/km. Frequency scaling: L' = L/kf, C' = C/kf, R' = R. Combined: R' = km·R, L' = (km/kf)·L, C' = C/(km·kf). ## Answer #### Ideal vs practical filters An **ideal filter** passes all frequencies in the passband with constant gain and zero phase distortion and completely blocks all frequencies in the stopband, with a vertical edge between them. It cannot be built, because its impulse response is non-causal and infinitely long. A **practical filter** only approximates this. | Ideal filter | Practical filter | |---|---| | Flat gain in passband | Gain varies within $\alpha_{max}$ (ripple or droop) | | Zero gain (infinite attenuation) in stopband | Finite attenuation, at least $\alpha_{min}$ | | No transition band (brick wall) | Transition band between $\omega_p$ and $\omega_s$ | | Linear phase / constant delay | Non-linear phase, delay varies | | Non-causal, not realizable | Causal, realizable with R, L, C, op-amps | | Infinite order | Finite order $n$ | ```text |T| ideal |T| practical 1 +------+ 1 +~~~~~. <- ripple (amax) | | | \ | | | \ transition | | | `--..__ (amin) 0 +------+------> w 0 +------+---+------> w wc wp ws ``` #### Magnitude (impedance) scaling Every impedance in the network is multiplied by $k_m$, at the same frequency. A voltage transfer function is a ratio of impedances, so it does not change. $$ \begin{aligned} R' &= k_m R \\ sL' = k_m (sL) &\Rightarrow L' = k_m L \\ \frac{1}{sC'} = k_m \frac{1}{sC} &\Rightarrow C' = \frac{C}{k_m} \end{aligned} $$ #### Frequency scaling The response at the old frequency $\omega$ must appear at the new frequency $k_f\omega$, where $k_f = \omega_{new}/\omega_{old}$. So each element must have at $k_f\omega$ the same impedance that the old element had at $\omega$: $$ \begin{aligned} j(k_f\omega)L' = j\omega L &\Rightarrow L' = \frac{L}{k_f} \\ \frac{1}{j(k_f\omega)C'} = \frac{1}{j\omega C} &\Rightarrow C' = \frac{C}{k_f} \\ R' &= R \quad (\text{frequency independent}) \end{aligned} $$ The new transfer function is $T'(s) = T(s/k_f)$: the response keeps its shape but is shifted along the frequency axis. #### Combined scaling Applying both: $$ R' = k_m R, \qquad L' = \frac{k_m}{k_f}L, \qquad C' = \frac{C}{k_m k_f} $$ **Example:** the normalized third-order Butterworth ladder (1 Ω, 1 H, 2 F, 1 H, 1 Ω, $\omega_o$ = 1 rad/s) scaled to $\omega_o = 10^4$ rad/s ($k_f = 10^4$) and 1 kΩ terminations ($k_m = 1000$) gives $R$ = 1 kΩ, $L = 1\times1000/10^4 = 0.1$ H and $C = 2/(1000\times10^4) = 0.2\ \mu$F. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can an ideal brick-wall filter not be built? A. Its passband gain is greater than 1 B. Its impulse response is non-causal and infinitely long (infinite order) C. It needs a negative resistance D. It requires an infinite power supply Answer: B. A vertical edge needs a non-causal, infinite-duration impulse response, so only approximations are realizable. (Key point: Ideal filter) 2. Which feature belongs to a practical filter but not to an ideal one? A. A transition band between wp and ws B. Flat gain in the passband C. Zero gain in the stopband D. Linear phase in the passband Answer: A. A practical filter has a finite transition band between the passband edge wp and the stopband edge ws; an ideal one has none. (Key point: Practical vs ideal) 3. Under magnitude scaling by km (same frequency), how do R, L and C change? A. R' = km R, L' = L/km, C' = km C B. R' = R, L' = km L, C' = km C C. R' = km R, L' = km L, C' = C/km D. R' = R/km, L' = km L, C' = C/km Answer: C. Every impedance is multiplied by km: R and sL scale by km, while 1/(sC) scaling by km means C is divided by km. (Key point: Magnitude scaling) 4. For frequency scaling by kf = w_new/w_old, how do R, L and C change? A. R' = kf R, L' = L/kf, C' = C/kf B. R' = R, L' = kf L, C' = kf C C. R' = R, L' = L/kf, C' = C kf D. R' = R, L' = L/kf, C' = C/kf Answer: D. Equal impedance at kf*w requires kf*L' = L and kf*C' = C; R does not depend on frequency. (Key point: Frequency scaling) ## Related in Filter Design - [Ideal filter and magnitude/frequency scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aideal-filter-magnitude-frequency-scaling) — Filter Design, chapter 1 - [Filter importance in communication; ideal vs practical](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afilter-communication-ideal-practical) — Filter Design, chapter 1, asked 2× - [Relevance of analog filters; frequency scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aanalog-filter-relevance-frequency-scaling) — 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 - [Frequency and magnitude scaling to 10⁵ rad/s](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ascaling-ladder-1e5-numerical) — Filter Design, chapter 1 - [Analog filter applications and element scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aanalog-filter-applications-scaling) — Filter Design, chapter 1 ## Related in other subjects - [Practical vs ideal transformer, phasor diagram](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Apractical-vs-ideal-transformer-phasor) — Electrical Machines, chapter 2 - [Compare and contrast pollution in a city](https://www.bctengineer.com/question.md?key=semester_5%2Fcommunication_english%3Acompare-contrast-pollution-city-nepal) — Communication English, chapter 1 - [Ideal low pass and high pass filters](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afrequency-response-lp-hp-filters) — Signals and Systems, chapter 5 - [Ideal band pass and low pass filters (DT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-bandpass-lowpass-dt) — Signals and Systems, chapter 6