Question

Which of the following is the rational system function of an LTI system characterized by the difference equation y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)?

(a) \(\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=0}^N a_k y(n-k)}\)

(b) \(\frac{1+\sum_{k=1}^N a_k y(n-k)}{\sum_{k=0}^N b_k x(n-k)}\)

(c) \(\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}\)

(d) \(\frac{1+\sum_{k=0}^N a_k y(n-k)}{\sum_{k=0}^N b_k x(n-k)}\)

I had been asked this question in an interview.

My question is based upon Structures for Realization of Discrete Time Systems in division Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

Correct choice is (c) \(\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}\)

Best explanation: The difference equation of the LTI system is given as

y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)

By applying the z-transform on both sides of the above equation and by rearranging the obtained equation, we get the rational system function as H(z)=\(\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}\)