Question E7C05:
Which filter type has ripple in the passband and a sharp cutoff?  

Choices:  
A: A Butterworth filter  
B: An active LC filter  
C: A passive op-amp filter  
D: A Chebyshev filter    

Correct Answer is  D: A Chebyshev filter  

To identify the filter with ripple in the passband and a sharp cutoff, remember the classic response families.

Why D (Chebyshev) is correct:
- Chebyshev Type I filters have an equiripple passband (ripples of equal height) and a steeper roll-off than Butterworth for the same order.
- Their magnitude response is:
  $$|H(j\omega)|^2 = \frac{1}{1 + \varepsilon^2\,T_n^2\!\left(\frac{\omega}{\omega_c}\right)}$$
  where:
  - $T_n$ is the Chebyshev polynomial of order $n$
  - $\varepsilon$ sets the passband ripple
  - $\omega_c$ is the cutoff frequency
- Passband ripple depth (in dB) is:
  $$A_r = 10\log_{10}(1+\varepsilon^2)$$

Why the others are incorrect:
- A: Butterworth filter
  - Butterworth has a maximally flat (no ripple) passband and a more gradual roll-off compared to Chebyshev.
  - Magnitude response:
    $$|H(j\omega)|^2 = \frac{1}{1+\left(\frac{\omega}{\omega_c}\right)^{2n}}$$
- B: An active LC filter
  - Describes implementation (active components with L and C), not the response shape. An LC-based design could be Butterworth, Chebyshev, elliptic, etc.
- C: A passive op-amp filter
  - “Op-amp” implies active, so this is not a well-defined category. Again, implementation does not define ripple behavior.

Useful rules of thumb:
- Butterworth: maximally flat passband, monotonic response, slower roll-off.
- Chebyshev Type I: equiripple passband, sharp cutoff.
- Chebyshev Type II (inverse): monotonic passband, equiripple stopband.
- Elliptic (Cauer): ripple in both passband and stopband, sharpest cutoff of all.