Question

Determine if any of the following transfer functions respresent a realizable filter:

a. H(s) = 1/ (s²+4)(s+1)
b. H(s) = s+1/s⁴+1
c. H(s) = s²+2s+2/(s+1) (s²+s+1)
Show if the corresponding realizable filters are low pass, high pass, band pass or band reject.

Answer 1

Out of the given transfer functions, a. and c. represent realizable filters that are low-pass due to the behavior of their magnitudes at zero and high frequencies. The transfer function b. is not realizable because it violates the physical realizability condition of systems.

Step-by-step explanation:

To determine if the provided transfer functions represent realizable filters and their types, let's analyze each one:

• a. H(s) = 1 / (s² + 4)(s + 1)
  
  This transfer function is realizable as it meets the condition of physically realizable systems, which is that the degree of the denominator should be greater than or equal to the numerator. This is a low-pass filter because the magnitude of H(s) approaches 0 as the frequency increases towards infinity, and it has a non-zero value at zero frequency (s=0).
• b. H(s) = (s + 1) / (s⁴ + 1)
  
  This transfer function is not realizable because the degree of the numerator is less than the degree of the denominator, which does not comply with the physical realizability requirement. Also, as the frequency tends towards infinity, the transfer function approaches infinity, which is non-physical.
• c. H(s) = (s² + 2s + 2) / ((s + 1)(s² + s + 1))
  
  This transfer function is realizable because the denominator degree is greater than the numerator’s degree. It is a low-pass filter, evident by the fact that as frequency increases, the magnitude of H(s) tends towards zero, and it has non-zero magnitude at zero frequency.