---
title: "Approximation and inverse Chebyshev order (100/140 krad/s)"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:approximation-inverse-chebyshev-order"
source: https://www.bctengineer.com/q/semester_5/filter_design/approximation-inverse-chebyshev-order
site: BCT Engineering
---

# Approximation and inverse Chebyshev order (100/140 krad/s)

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2081 Baisakh, 1+5+3 marks
- 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/approximation-inverse-chebyshev-order
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

What is approximation in filter design? Derive an expression to calculate the order of Inverse Chebyshev low pass filter. Use this formula to estimate the order of Chebyshev low pass filter with the following specifications; ωp = 100 Krad/s, ωs = 140 Krad/s, αmax = 0.25 dB, αmin = 18 dB

## Short answer

Approximation finds a realizable transfer function that meets passband and stopband attenuation specs. The inverse Chebyshev order is n ≥ cosh⁻¹√((10^(0.1αmin)−1)/(10^(0.1αmax)−1)) / cosh⁻¹(ωs/ωp). For ωp=100 krad/s, ωs=140 krad/s, αmax=0.25 dB, αmin=18 dB, ωs/ωp=1.4 gives n ≥ 4.81, so n = 5.

## Answer

#### Approximation in filter design

**Approximation** is the step of finding a realizable transfer function $T(s)$ (a ratio of polynomials with real coefficients and poles in the left half plane) whose magnitude (or phase) stays within the given specifications ($\alpha_{max}$ up to $\omega_p$, $\alpha_{min}$ beyond $\omega_s$). An ideal brick-wall response cannot be built, so it is approximated by standard functions such as Butterworth, Chebyshev, inverse Chebyshev, elliptic and Bessel-Thomson.

#### Order of the inverse Chebyshev low-pass filter

The inverse Chebyshev (Chebyshev Type II) response, normalized to the stopband edge ($\Omega = \omega/\omega_s$, stopband begins at $\Omega=1$), is

$$
|T(j\Omega)|^2 = \frac{\varepsilon^2 C_n^2(1/\Omega)}{1+\varepsilon^2 C_n^2(1/\Omega)}
$$

It is obtained by taking a Chebyshev response, subtracting it from 1 (turning it into a high-pass with equiripple stopband) and replacing $\Omega$ by $1/\Omega$.

**Step 1: stopband edge.** At $\Omega = 1$, $C_n(1) = 1$:

$$
\alpha_{min} = 10\log_{10}\left(1+\frac{1}{\varepsilon^2}\right) \;\Rightarrow\; \varepsilon^2 = \frac{1}{10^{0.1\alpha_{min}}-1}
$$

**Step 2: passband edge.** At $\Omega_p = \omega_p/\omega_s$, $1/\Omega_p = \omega_s/\omega_p > 1$, and the attenuation must not exceed $\alpha_{max}$:

$$
\alpha_{max} \ge 10\log_{10}\left(1+\frac{1}{\varepsilon^2 C_n^2(\omega_s/\omega_p)}\right)
\;\Rightarrow\; C_n^2\!\left(\frac{\omega_s}{\omega_p}\right) \ge \frac{1}{\varepsilon^2(10^{0.1\alpha_{max}}-1)} = \frac{10^{0.1\alpha_{min}}-1}{10^{0.1\alpha_{max}}-1}
$$

**Step 3: solve.** With $C_n(x) = \cosh(n\cosh^{-1}x)$ for $x>1$:

$$
n \ge \frac{\cosh^{-1}\sqrt{\dfrac{10^{0.1\alpha_{min}}-1}{10^{0.1\alpha_{max}}-1}}}{\cosh^{-1}\left(\dfrac{\omega_s}{\omega_p}\right)}
$$

This is the same expression as for the Chebyshev filter: for the same specification both need the same order.

#### Order for the given specification

$\omega_p = 100$ krad/s, $\omega_s = 140$ krad/s, $\alpha_{max} = 0.25$ dB, $\alpha_{min} = 18$ dB, so $\omega_s/\omega_p = 1.4$.

$$
\begin{aligned}
10^{0.025}-1 &= 0.05925 \\
10^{1.8}-1 &= 62.0957 \\
\sqrt{62.0957/0.05925} &= 32.372, \quad \cosh^{-1}(32.372) = 4.1702 \\
\cosh^{-1}(1.4) &= 0.8670 \\
n &\ge \frac{4.1702}{0.8670} = 4.810
\end{aligned}
$$

**Answer: n = 5.** Because the order formula is identical, a Chebyshev (Type I) filter for this specification also needs $n = 5$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the approximation step in filter design?
   A. Rounding component values to standard sizes
   B. Finding a realizable T(s) with left-half-plane poles whose response meets the specification
   C. Measuring the real circuit response
   D. Converting analog to digital
   Answer: B. An ideal brick-wall response cannot be built, so a realizable rational function that stays within the specs is found. (Key point: Meaning of approximation)

2. For the inverse Chebyshev, epsilon^2 is fixed by:
   A. alpha_max: eps^2 = 10^(0.1 alpha_max) - 1
   B. alpha_min: eps^2 = 1/(10^(0.1 alpha_min) - 1)
   C. The order n
   D. wp only
   Answer: B. At the stopband edge alpha_min = 10log(1+1/eps^2), so eps^2 = 1/(10^(0.1 alpha_min)-1). (Key point: Epsilon from the stopband edge)

3. For ws/wp = 1.4, alpha_max = 0.25 dB, alpha_min = 18 dB, the order is 5 (4.810). Compared with a Chebyshev Type I filter for the same spec:
   A. Type I needs a lower order
   B. Type I needs the same order since the formula is identical
   C. Type I needs a higher order
   D. Type I cannot meet the spec
   Answer: B. The order formula is identical, so both need n = 5. (Key point: Order result)

## Close variants set in other exams

- [Chebyshev order (0.25 dB, 20 dB, 1500 rad/s)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Achebyshev-order-0-25db-1500) — Filter Design, chapter 2, asked 6×

## Related in Filter Design

- [Inverse Chebyshev order (0.25 dB, 18 dB, 1400 rad/s)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-order-1400) — Filter Design, chapter 2, asked 5×
- [Inverse Chebyshev order (0.5 dB, 25 dB, 1800 rad/s)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-order-1800-25db) — Filter Design, chapter 2, asked 5×
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- [Chebyshev order (0.25 dB, 18 dB, 1650 rad/s)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Achebyshev-order-0-25db-1650) — Filter Design, chapter 2, asked 2×
- [Chebyshev order, poles and TF (1 dB, 18 dB)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Achebyshev-order-poles-tf-1db-18db) — Filter Design, chapter 2
- [Inverse Chebyshev characteristics and order (0.4 dB, 16 dB)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-characteristics-order-16db) — Filter Design, chapter 2, asked 2×

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