Chebyshev Filters

The Chebyshev filter achieves a sharper transition from passband to stopband compared to Butterworth for the same filter order, but it does so by allowing a controlled amount of ripple in either the passband or stopband. This deliberate ripple is the mechanism that enables steeper rolloff.

There are two distinct types. Type I Chebyshev filters place ripple in the passband and are monotonic in the stopband. Type II Chebyshev filters are monotonic in the passband and place ripple in the stopband. For GATE, Type I is more frequently tested, but understanding both types conceptually is important.

Core Concept Explanation

The sharpness advantage of Chebyshev filters comes from the mathematical properties of Chebyshev polynomials, denoted T_N(x). These polynomials oscillate uniformly between -1 and +1 for |x| <= 1 and grow rapidly outside this range. By constructing the magnitude squared response using these polynomials, the filter equiripple behavior in one band and rapid rolloff simultaneously.

For Type I, the magnitude squared response is |H(jΩ)|^2 = 1 / (1 + epsilon^2 * T_N^2(Ω/Ω_p)), where epsilon controls the ripple magnitude. The ripple is equiripple, meaning it oscillates with the same amplitude throughout the passband. This equal ripple allocation is optimal in the sense that no other same-order filter can have a steeper transition while keeping the ripple bounded.

For Type II, the construction uses an inverse mapping. The stopband has equiripple behavior and the passband is monotone. Type II is less commonly implemented in practice but is relevant conceptually and for exam distinctions.

Mathematical Expression

The Chebyshev polynomial of order N is defined recursively:

T_0(x) = 1, T_1(x) = x, T_N(x) = 2x*T_(N-1)(x) - T_(N-2)(x)

The Type I magnitude squared response is:

|H(jΩ)|^2 = 1 / (1 + epsilon^2 * T_N^2(Ω / Ω_p))

The ripple parameter epsilon is related to the passband ripple R_p in dB: epsilon^2 = 10^(R_p/10) - 1. The minimum order N for Chebyshev Type I given passband and stopband specifications is:

N >= cosh^(-1)(sqrt((10^(A_s/10)-1)/(10^(A_p/10)-1))) / cosh^(-1)(Omega_s / Omega_p)

Practical Understanding

For a given order N and passband ripple, Chebyshev Type I provides a steeper transition than Butterworth. This means you can meet a given stopband attenuation with a lower order filter, reducing implementation cost and computational load. The trade-off is the ripple in the passband, which can cause distortion in applications where flat passband response is critical.

In audio equalization and radio frequency bandpass filters, Chebyshev filters are common. When the application can tolerate a small passband ripple but demands a very narrow transition band, Chebyshev outperforms Butterworth significantly.

Given:
Passband: A_p = 1 dB at Omega_p = 1000 rad/s
Stopband: A_s = 30 dB at Omega_s = 3000 rad/s

Why this formula applies:
Chebyshev Type I order formula uses cosh inverse unlike Butterworth.

Formula:
N >= cosh^(-1)(sqrt((10^(As/10)-1)/(10^(Ap/10)-1))) / cosh^(-1)(Omega_s/Omega_p)

Substitution:
10^(30/10) - 1 = 999
10^(1/10) - 1 = 0.2589
sqrt(999/0.2589) = sqrt(3858) = 62.1
cosh^(-1)(62.1) = ln(62.1 + sqrt(62.1^2 - 1)) ≈ 4.82
cosh^(-1)(3000/1000) = cosh^(-1)(3) ≈ 1.763

Calculation:
N >= 4.82 / 1.763 = 2.73

Final Answer with units:
N = 3 (round up)
A 3rd order Chebyshev Type I meets both specifications with 1 dB passband ripple.

Exam Tip: Chebyshev order formula uses cosh inverse; Butterworth uses log. For GATE numerical problems, remember the ripple parameter epsilon^2 = 10^(Rp/10) - 1 where Rp is in dB. A 3 dB ripple gives epsilon = 1, same as Butterworth at cutoff.

Mechanism Summary

• Type I: equiripple in passband, monotone in stopband. Type II: monotone in passband, equiripple in stopband.
• Magnitude squared uses Chebyshev polynomial: |H|^2 = 1/(1 + epsilon^2 * T_N^2(Omega/Omega_p)).
• Epsilon^2 = 10^(Rp/10) - 1 where Rp is the passband ripple in dB.
• Order formula uses cosh inverse, unlike Butterworth which uses log.
• Chebyshev poles lie on an ellipse in the s-plane, not a circle like Butterworth.
• For same order N, Chebyshev gives steeper rolloff than Butterworth. Equiripple structure is the reason.

Quick Revision

• Type I: passband ripple, monotone stopband. Type II: monotone passband, stopband ripple.
• |H|^2 = 1/(1 + epsilon^2 * T_N^2(Omega/Omega_p)) for Type I.
• epsilon^2 = 10^(Rp/10) - 1. For 3 dB ripple, epsilon = 1.
• Order: N >= cosh^(-1)(sqrt((10^(As/10)-1)/(10^(Ap/10)-1))) / cosh^(-1)(Omega_s/Omega_p). Round up.
• Poles on an ellipse in s-plane. Ellipse axes depend on epsilon and Omega_p.
• Trap: do not use Butterworth log formula for Chebyshev. The cosh inverse formula is different.
• Trap: Type I and Type II have different stopband/passband behavior. Know which type is specified.