Question

A digital signal processing system is described by the expression \[ y(n) = 2x(n) + x(n-1) + 2y(n-1) \] The system is

• A. A stable IIR filter
• B. A stable FIR filter
• C. An unstable FIR filter
• D. An unstable IIR filter

Hint:

Any system with a recursive (feedback) term involving \( y(n-k) \) is an IIR filter. If the characteristic roots of its homogeneous equation lie outside the unit circle, the system is unstable.

Answer 1

The Correct Option is D

The given difference equation is: \[ y(n) = 2x(n) + x(n-1) + 2y(n-1) \] This system has a feedback term \( 2y(n-1) \), so it is an Infinite Impulse Response (IIR) filter.
To assess stability, we examine the homogeneous part: \[ y(n) - 2y(n-1) = 0 \] Its characteristic equation is: \[ r - 2 = 0 \Rightarrow r = 2 \] Since the root of the characteristic equation lies outside the unit circle (\( |r| = 2>1 \)), the system is unstable.
Thus, it is an unstable IIR filter.