---
title: "Chebyshev characteristics and order (1500/4500 rad/s)"
subject: "Filter Design"
chapter: 2
key: "semester_5/filter_design:chebyshev-characteristics-order-4500"
source: https://www.bctengineer.com/q/semester_5/filter_design/chebyshev-characteristics-order-4500
site: BCT Engineering
---

# Chebyshev characteristics and order (1500/4500 rad/s)

Filter Design · Chapter 2: Approximation Methods

- Asked in: 2081 Baisakh, 3+4+3 marks
- Video: [Problem #1 - An Analog Type-I Chebyshev LowPass Filter Design - DTSP](https://www.youtube.com/watch?v=8M05FuKXIaw) — Padmasri Naban
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/chebyshev-characteristics-order-4500
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2

## Question

What are the characteristics of the Chebyshev magnitude response? Derive an expression to calculate the order of a Chebyshev filter for given low-pass specifications. Using your expression, calculate the order of a Chebyshev filter for following lowpass specifications: αmax = 0.5 dB, αmin = 20 dB, ωp = 1500 rad/sec, ωs = 4500 rad/sec

## Short answer

Chebyshev filter has equiripple passband, monotonic stopband, -20n dB/dec roll-off, and lower order than Butterworth. The order is n ≥ arcosh(√[(10^(0.1αmin)-1)/(10^(0.1αmax)-1)]) / arcosh(ωs/ωp). For αmax=0.5 dB, αmin=20 dB, ωs/ωp=3, n=3.

## Answer

#### Characteristics of the Chebyshev magnitude response

- **Equiripple passband:** $|T|$ oscillates between $1$ and $1/\sqrt{1+\varepsilon^2}$ in $0 \le \Omega \le 1$; the ripple height is $\alpha_{max}$ dB and is set by $\varepsilon$.
- **Number of ripples:** the passband contains $n$ half-cycles of ripple between $\Omega = 0$ and $\Omega = 1$, so a higher order gives more ripples (of the same height).
- **DC value:** $|T(0)| = 1$ for odd $n$ and $1/\sqrt{1+\varepsilon^2}$ for even $n$.
- **Monotonic stopband:** beyond $\Omega = 1$ the response falls monotonically; the asymptotic roll-off is $-20n$ dB/decade, but the transition is much sharper than Butterworth of the same order (about $6(n-1)$ dB more stopband attenuation for the same $\varepsilon$).
- **All-pole function:** no finite transmission zeros; poles lie on an **ellipse** in the s-plane.
- **Phase/delay:** phase is more non-linear and group delay has a larger peak near the band edge than Butterworth (poorer transient response, more ringing).
- For a given specification it needs a **lower order** than Butterworth, so fewer components.

#### Derivation of the order

The Chebyshev low-pass magnitude response (normalized so that the passband edge is $\Omega = \omega/\omega_p = 1$) is

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

where $C_n(\Omega)$ is the Chebyshev polynomial of order $n$:

$$
C_n(\Omega) = \begin{cases} \cos(n\cos^{-1}\Omega), & |\Omega| \le 1 \\ \cosh(n\cosh^{-1}\Omega), & |\Omega| > 1 \end{cases}
$$

The attenuation in dB is $\alpha(\Omega) = 10\log_{10}\left[1+\varepsilon^2 C_n^2(\Omega)\right]$.

**Step 1: ripple factor from the passband edge.** At $\Omega = 1$, $C_n(1) = 1$ for every $n$, so the attenuation is exactly $\alpha_{max}$:

$$
\alpha_{max} = 10\log_{10}(1+\varepsilon^2) \;\Rightarrow\; \varepsilon^2 = 10^{0.1\alpha_{max}} - 1
$$

**Step 2: stopband condition.** At $\Omega_s = \omega_s/\omega_p\ (>1)$ the attenuation must be at least $\alpha_{min}$:

$$
10\log_{10}\left[1+\varepsilon^2 C_n^2(\Omega_s)\right] \ge \alpha_{min}
\;\Rightarrow\; C_n^2(\Omega_s) \ge \frac{10^{0.1\alpha_{min}}-1}{\varepsilon^2}
$$

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

$$
\cosh(n\cosh^{-1}\Omega_s) \ge \sqrt{\frac{10^{0.1\alpha_{min}}-1}{10^{0.1\alpha_{max}}-1}}
$$

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

The order is the next whole number above this value. (Useful identity: $\cosh^{-1}x = \ln\left(x+\sqrt{x^2-1}\right)$.)

#### Order for the given specification

Given $\alpha_{max} = 0.5$ dB, $\alpha_{min} = 20$ dB, $\frac{\omega_s}{\omega_p} = \frac{4500}{1500} = 3$.

$$
\begin{aligned}
10^{0.1\alpha_{max}} - 1 &= 10^{0.05} - 1 = 0.12202 \\
10^{0.1\alpha_{min}} - 1 &= 10^{2} - 1 = 99.0000 \\
\sqrt{\frac{99.0000}{0.12202}} &= 28.4843 \\
\cosh^{-1}(28.4843) &= 4.0422 \\
\cosh^{-1}(3) &= 1.7627 \\
n &\ge \frac{4.0422}{1.7627} = 2.293
\end{aligned}
$$

**Answer: n = 3** (next integer above 2.293).

## Check yourself

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

1. What does the Chebyshev magnitude do in the passband?
   A. Falls monotonically from 1
   B. Oscillates between 1 and 1/sqrt(1+eps^2) with equal ripple height
   C. Oscillates with ripple growing toward the edge
   D. Stays exactly flat at 1
   Answer: B. C_n swings between -1 and 1 in the passband, so |T| ripples equally between 1 and 1/sqrt(1+eps^2). (Key point: Equiripple passband)

2. What is |T(0)| for a Chebyshev filter?
   A. 1 for even n and 1/sqrt(1+eps^2) for odd n
   B. Always 1
   C. 1 for odd n and 1/sqrt(1+eps^2) for even n
   D. Always 1/sqrt(1+eps^2)
   Answer: C. Odd n starts a ripple maximum at DC (gain 1); even n starts at a ripple minimum (1/sqrt(1+eps^2)). (Key point: DC value and order)

3. Which statement about Chebyshev filters is correct?
   A. Poles lie on a circle and the filter needs a higher order than Butterworth
   B. Poles lie on an ellipse and the filter has finite transmission zeros
   C. Poles lie on an ellipse and a lower order than Butterworth meets the same spec
   D. Poles lie on a straight line and the response is monotonic everywhere
   Answer: C. Chebyshev is all-pole with poles on an ellipse, and its sharper transition needs a lower order than Butterworth. (Key point: Poles on an ellipse and comparison with Butterworth)

4. For alpha_max = 0.5 dB, alpha_min = 20 dB, ws/wp = 3, the formula gives 4.0422/1.7627 = 2.293. The order is:
   A. 3
   B. 2
   C. 4
   D. 5
   Answer: A. The next whole number above 2.293 is 3. (Key point: Order for given spec)

## Close variants set in other exams

- [Inverse Chebyshev characteristics, order, poles and zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-poles-zeros) — Filter Design, chapter 2, asked 2×

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