Question

Which of the following is a recursive form of a non-recursive system described by the equation y(n)=\(\frac{1}{M+1} \sum_{k=0}^Mx(n-k)\)?

(a) y(n)=y(n-1)+\(\frac{1}{M+1}\)[x(n)+x(n-1-M)]

(b) y(n)=y(n-1)+\(\frac{1}{M+1}\)[x(n)+x(n-1+M)]

(c) y(n)=y(n-1)+\(\frac{1}{M+1}\)[x(n)-x(n-1+M)]

(d) y(n)=y(n-1)+\(\frac{1}{M+1}\)[x(n)-x(n-1-M)]

This question was posed to me in an international level competition.

This interesting question is from Implementation of Discrete Time Systems topic in division Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

The correct option is (d) y(n)=y(n-1)+\(\frac{1}{M+1}\)[x(n)-x(n-1-M)]

The explanation: The given system equation is y(n)=\(\frac{1}{M+1} \sum_{k=0}^M x(n-k)\)

It can be expressed as follows

\(y(n)=\frac{1}{M+1} \sum_{k=0}^M x(n-1-k)+\frac{1}{M+1}[x(n)-x(n-1-M)]\)

=\(y(n-1)+\frac{1}{M+1}[x(n)-x(n-1-M)]\)