A channel-select filter in a radio receiver must cut off sharply just outside the desired channel. The Chebyshev filter achieves a much steeper roll-off than Butterworth by allowing a controlled amount of ripple in the pass-band.

Figure 1: Fourth-order Chebyshev Type I versus Butterworth, demonstrating equiripple pass-band and sharper transition

Core Concept

The Chebyshev filter uses Chebyshev polynomials to distribute the approximation error equally across the pass-band, rather than concentrating all the flatness near DC as Butterworth does. This equiripple characteristic allows a given order Chebyshev to roll off faster than the same order Butterworth.

There are two types. Type I has equiripple in the pass-band and a monotonically decreasing response in the stop-band. Type II (inverse Chebyshev) has a flat pass-band and equiripple in the stop-band. For GATE and most practical filter design, Type I is the default when the term Chebyshev is used without qualification.

The ripple in the pass-band is specified in dB, typically 0.5 dB, 1 dB, or 3 dB. A 1 dB ripple Chebyshev filter achieves the same 40 dB stopband attenuation at fs = 2fc using only 3rd order, whereas a Butterworth would require 7th order for the same specification. This is the fundamental advantage of Chebyshev design.

Key Equations

Magnitude response: |H(jΩ)|^2 = 1 / (1 + ε^2 × Cn^2(Ω)) where Cn(Ω) is the nth-order Chebyshev polynomial and ε is related to ripple.

Ripple factor ε from ripple dB: ε = sqrt(10^(Rp/10) - 1) where Rp is pass-band ripple in dB. For 1 dB ripple: ε = sqrt(10^0.1 - 1) = 0.5088.

Chebyshev polynomials: C0 = 1, C1 = Ω, C2 = 2Ω^2-1, C3 = 4Ω^3-3Ω, Cn = 2Ω×C(n-1) - C(n-2)

Minimum order for given specs: n = cosh^-1(sqrt((10^(As/10)-1)/(10^(Rp/10)-1))) / cosh^-1(Ωs) where Ωs = fs/fc.

Roll-off rate beyond fc: steeper than Butterworth. At frequencies well into the stopband, -20n dB/decade asymptotically, same as Butterworth, but the transition from pass-band to stopband is faster.

Example

Given:
  Pass-band edge: fc = 1 kHz
  Pass-band ripple: Rp = 1 dB
  Stopband attenuation: As = 40 dB at fs = 3 kHz
  Ωs = fs/fc = 3 kHz / 1 kHz = 3

Why this formula:
  Minimum Chebyshev order:
  n = cosh^-1(sqrt((10^(As/10)-1)/(10^(Rp/10)-1))) / cosh^-1(Ωs)

Substitution:
  10^(40/10) - 1 = 9999
  10^(1/10) - 1 = 0.2589
  Ratio = 9999 / 0.2589 = 38,620
  sqrt(38620) = 196.5
  cosh^-1(196.5) = ln(196.5 + sqrt(196.5^2-1)) ≈ ln(393) ≈ 5.974

  cosh^-1(3) = ln(3 + sqrt(9-1)) = ln(3 + 2.828) = ln(5.828) = 1.763

Calculation:
  n = 5.974 / 1.763 = 3.388
  Round up: n = 4

Final Answer:
  Minimum order n = 4 for 1 dB ripple Chebyshev
  providing 40 dB attenuation at 3× fc.
  Compare: Butterworth needs n = 8 for same spec.

Exam Tip: GATE compares Butterworth and Chebyshev directly. The key fact: for the same order, Chebyshev achieves greater stopband attenuation than Butterworth. For the same stopband spec, Chebyshev requires lower order. The trade-off is pass-band ripple. Also know that all Chebyshev Type I poles lie on an ellipse in the s-plane (not a circle as in Butterworth). The ellipse has its real-axis radius as a × sinh(β/n) and imaginary-axis radius as a × cosh(β/n), where β = sinh^-1(1/ε).

Key Properties

Equiripple magnitude response in the pass-band. The ripple amplitude is constant throughout the pass-band, unlike Butterworth.
For the same filter order n, Chebyshev provides sharper roll-off than Butterworth in the transition band.
Pass-band ripple Rp specified in dB (commonly 0.5 dB, 1 dB, or 3 dB). Higher ripple = steeper roll-off.
Ripple factor ε = sqrt(10^(Rp/10) - 1). For 1 dB: ε = 0.5088.
Type I poles lie on an ellipse in the s-plane, unlike Butterworth poles which lie on a circle.
Type II (inverse Chebyshev) has flat pass-band and equiripple stop-band, useful when pass-band flatness matters more than sharpness.
Chebyshev filters have more group delay variation than Butterworth, which causes greater phase distortion of wideband signals like pulses.

Quick Revision

Chebyshev Type I: equiripple pass-band, monotone stopband. Sharper roll-off than Butterworth for same order.
|H|^2 = 1/(1 + ε^2 Cn^2(Ω)). ε sets ripple level.
ε = sqrt(10^(Rp/10) - 1). For 1 dB ripple, ε = 0.5088.
Order formula uses cosh^-1 functions. Always round n UP.
Poles on ellipse in s-plane, not circle.
Type II: flat pass-band, equiripple stopband. Poles on different ellipse configuration.
More phase distortion and group delay variation compared to Butterworth or Bessel.
Exam trap: Students use the Butterworth order formula n = log(sqrt(10^(As/10)-1))/log(Ωs) for Chebyshev. This is wrong. Chebyshev order uses cosh^-1 in both numerator and denominator, not log. Using the Butterworth formula gives a higher n than necessary.