Question

Which of the following statements about digital filters is/are CORRECT?

• A. Non-recursive filters do not have poles
• B. Recursive filters can have only poles OR both poles and zeros
• C. The order of the digital filter defines the presence of minimum number of previous inputs only
• D. The ratio of output to input defines the transfer function of the digital filter

Hint:

Write the general recursive-filter transfer function \(H(z)=Y(z)/X(z)\) and check each claim about poles, zeros, and filter order against it.

Answer 1

The Correct Option is A, B, D

Go through each statement using the standard definitions of digital filter structures.

(A) Non-recursive filters do not have poles - CORRECT. A non-recursive (FIR) filter computes its output purely from the current and past INPUT samples:
\[y_n=\sum_{k=0}^{M}b_k\,x_{n-k}\]
Its transfer function \(H(z)=\sum b_k z^{-k}\) is a polynomial in \(z^{-1}\) with only zeros (roots of the numerator); there is no feedback denominator, so there are no non-trivial poles. This statement is true.

(B) Recursive filters can have only poles OR both poles and zeros - CORRECT. A recursive (IIR) filter feeds back past OUTPUT values as well:
\[y_n=\sum_{k}b_k x_{n-k}+\sum_{k}a_k y_{n-k}\]
giving a transfer function \(H(z)=\dfrac{\sum b_kz^{-k}}{1-\sum a_kz^{-k}}\). The denominator always contributes poles. If the numerator is just a constant (an all-pole filter), the filter has poles only; if the numerator also has multiple terms, the filter has both poles and zeros. Either situation is possible for a recursive filter, so this statement is true.

(C) The order of the digital filter defines the presence of minimum number of previous inputs only - INCORRECT. The order of a filter is set by the number of delay elements needed, which for a recursive filter includes both previous INPUT samples and previous OUTPUT samples (the feedback terms), not inputs alone. Restricting the definition to "previous inputs only" is therefore wrong.

(D) The ratio of output to input defines the transfer function of the digital filter - CORRECT. By definition, the (z-domain) transfer function is exactly \(H(z)=Y(z)/X(z)\), the ratio of the z-transform of the output to that of the input. This is textbook-correct.

So the correct statements are (A), (B) and (D), i.e. options \(\boxed{1,2,4}\).