--- title: "Elliptic response vs Chebyshev and inverse Chebyshev" subject: "Filter Design" chapter: 2 key: "semester_5/filter_design:elliptic-response-comparison" source: https://www.bctengineer.com/q/semester_5/filter_design/elliptic-response-comparison site: BCT Engineering --- # Elliptic response vs Chebyshev and inverse Chebyshev Filter Design · Chapter 2: Approximation Methods - Asked in: 2073 Shrawan, 3+2+2 marks; 2071 Shrawan, 6 marks (asked 2 times) - Video: [Filter Approximations: Overview of Butterworth, Chebyshev, Elliptic and Bessel Filter Approximation](https://www.youtube.com/watch?v=JXcbbBMrLYE) — ALL ABOUT ELECTRONICS - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/elliptic-response-comparison - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2 ## Question What are the characteristics of Elliptical (elliptic) Response? Compare it with Chebyshev and Inverse Chebyshev response. ## Short answer Elliptic response has equiripple in both passband and stopband with finite zeros, providing the sharpest transition and lowest order. Compared to Chebyshev (passband ripple only) and inverse Chebyshev (stopband ripple only), it has the poorest phase linearity and greatest complexity. ## Answer #### Characteristics of the elliptic (Cauer) response The elliptic low-pass response is $$ |T(j\omega)|^2 = \frac{1}{1 + \varepsilon^2 R_n^2(\omega)} $$ where $R_n(\omega)$ is a Chebyshev rational function (Jacobi elliptic function). - **Equiripple in both the passband and the stopband.** - Has **finite zeros of transmission** on the $j\omega$ axis in the stopband, where the attenuation becomes infinite. - **Sharpest transition** (narrowest transition band) of all classical approximations for a given order; so for given $\alpha_{max}$, $\alpha_{min}$, $\omega_s/\omega_p$ it needs the **lowest order**. - Stopband attenuation does not keep increasing; it ripples down to $\alpha_{min}$ between the zeros. - **Poorest phase linearity**: group delay varies strongly near the band edge, causing pulse distortion. - Design is complex (elliptic integrals); normally done with tables or software. - Realization needs both poles and finite zeros (e.g. LC ladders with parallel-resonant series arms, or notch biquads). ```text |T| 1 |~v~v~~. | \ elliptic | \ | \ /\ /\ | \/ \__/ \___ +--------------------------> w wp ws (zeros at dips) ``` #### Comparison | Feature | Chebyshev | Inverse Chebyshev | Elliptic | |---|---|---|---| | Passband | Equiripple | Flat, monotonic | Equiripple | | Stopband | Monotonic | Equiripple | Equiripple | | Transmission zeros | All at $\infty$ (all-pole) | Finite, on $j\omega$ axis | Finite, on $j\omega$ axis | | Transition sharpness | Good | Good (same as Chebyshev) | Best | | Order for given specs | Medium | Medium (same formula) | Lowest | | Phase / delay | Poor | Better than Chebyshev | Worst | | Complexity | Simple | Moderate | Most complex | The elliptic filter is chosen when a very narrow transition band is needed with minimum components and phase is not important; Chebyshev/inverse Chebyshev are simpler alternatives. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Where is the elliptic response equiripple? A. Only in the passband B. In both passband and stopband C. Only in the stopband D. Nowhere Answer: B. It has ripples in both bands and finite transmission zeros. (Key point: Elliptic ripple) 2. How does the order of an elliptic filter compare with Chebyshev for the same specs? A. Lowest, because it has the sharpest transition B. Highest, because of finite zeros C. Same as Butterworth D. Twice Chebyshev Answer: A. Finite zeros on the jω axis give the narrowest transition. (Key point: Order) 3. Which has the worst phase/delay behavior? A. Elliptic B. Inverse Chebyshev C. Chebyshev, never elliptic D. All three equal Answer: A. Group delay varies strongly near the band edge for elliptic. (Key point: Phase) 4. Which statement about transmission zeros is correct? A. All three have finite zeros B. Only Chebyshev has finite zeros C. Chebyshev has all zeros at infinity; inverse Chebyshev and elliptic have finite zeros on the jω axis D. None has any zeros Answer: C. Chebyshev is all-pole. 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