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

A quick way to sort these out is to write down the general difference equation and transfer function once, then check each claim against it.

General recursive filter: \(y_n=\sum_{k=0}^{M}b_kx_{n-k}+\sum_{k=1}^{N}a_ky_{n-k}\), with transfer function
\[H(z)=\frac{Y(z)}{X(z)}=\frac{b_0+b_1z^{-1}+\cdots+b_Mz^{-M}}{1-a_1z^{-1}-\cdots-a_Nz^{-N}}\]

Setting all \(a_k=0\) collapses this to the non-recursive (FIR) case, \(H(z)=\sum b_kz^{-k}\) - a pure numerator, hence only zeros and no poles: this confirms (A).

Keeping the denominator active but setting all \(b_k=0\) except \(b_0\) gives an all-pole recursive filter (poles only); keeping several \(b_k\) nonzero along with the denominator gives a pole-zero filter. Both are legitimate recursive filters, so "only poles OR both poles and zeros" correctly covers all recursive cases: this confirms (B).

The filter order \(N\) (or \(\max(M,N)\)) counts the number of unit-delay memory elements required to implement the difference equation, and for a recursive filter that memory has to hold BOTH the past \(x\) values and the past \(y\) values - the order is not solely about how many previous inputs are stored. This shows (C) is an incomplete, and hence incorrect, statement.

Finally, \(H(z)=Y(z)/X(z)\) is the very definition of a discrete-time transfer function, so (D) is unquestionably correct.

Collecting the correct statements gives (A), (B), (D) - options \(\boxed{1,2,4}\).