Question E7C06:
What are the characteristics of an elliptical filter?  

Choices:  
A: Gradual passband rolloff with minimal stop-band ripple  
B: Extremely flat response over its pass band with gradually rounded stop-band corners  
C: Extremely sharp cutoff with one or more notches in the stop band  
D: Gradual passband rolloff with extreme stop-band ripple    

Correct Answer is  C: Extremely sharp cutoff with one or more notches in the stop band  

To understand an elliptical (Cauer) filter:

1) Why the correct answer is correct (C)
- Elliptical filters have equal ripple in both the passband and the stopband and include finite-frequency transmission zeros.
- Those zeros create very deep “notches” (theoretically infinite attenuation) in the stopband.
- They also provide the sharpest transition (smallest transition bandwidth) for a given filter order.
- Hence: “Extremely sharp cutoff with one or more notches in the stop band.”

2) Why the other choices are incorrect
- A: “Gradual passband rolloff with minimal stop-band ripple”  
  - Describes Butterworth-like behavior (monotonic passband/stopband, gentle rolloff), not elliptical. Elliptical is not gradual and does have ripple in both bands.
- B: “Extremely flat response over its pass band with gradually rounded stop-band corners”  
  - This is the classic Butterworth description (maximally flat magnitude, no ripple, monotonic stopband).
- D: “Gradual passband rolloff with extreme stop-band ripple”  
  - Elliptical filters do not have gradual rolloff; they are very sharp and also have passband ripple and distinct notches (zeros), not just generic “extreme ripple.”

3) Useful formulas and rules of thumb
- Elliptical (Cauer) filters show equal-ripple behavior in both bands and have transmission zeros where:
  $$|H(j\omega_z)| = 0$$
  producing stopband notches.
- Passband ripple in dB, $A_p$, relates to the ripple factor:
  $$\epsilon = \sqrt{10^{A_p/10} - 1}$$
- Sharpness/transition-width ranking for a given order:
  $$\text{Bessel} < \text{Butterworth} < \text{Chebyshev} < \text{Elliptic}$$
- Quick ID:
  - Chebyshev I: passband ripple only.
  - Chebyshev II (inverse): stopband ripple only (with zeros).
  - Elliptic: ripple in both and finite-frequency zeros → notches and sharpest cutoff.