Question

An LTI discrete system is characterized by the difference equation

y(n)=0.5y(n−2)+x(n)+x(n−2).
Find the eigenvalue that corresponds to the eigenfunction eʲθⁿ = eʲπⁿ/⁴

Answer 1

An LTI discrete system is characterized by the difference equation y(n) = 0.5y(n-2) + x(n) + x(n-2). However, we cannot find the eigenvalue that corresponds to the given eigenfunction using this difference equation.

Step-by-step explanation:

An LTI (Linear Time-Invariant) discrete system is characterized by the difference equation y(n) = 0.5y(n-2) + x(n) + x(n-2). To find the eigenvalue that corresponds to the eigenfunction eʲθⁿ = eʲπⁿ/⁴, we substitute the eigenfunction into the difference equation:

y(n) = 0.5y(n-2) + x(n) + x(n-2)

eʲπⁿ/⁴ = 0.5eʲπ(n-2)/⁴ + xeʲπⁿ/⁴ + xeʲπ(n-2)/⁴

Simplifying the equation, we get:

eʲπⁿ/⁴ - 0.5eʲπ(n-2)/⁴ - xeʲπⁿ/⁴ - xeʲπ(n-2)/⁴ = 0

This equation represents a discrete difference equation, not an eigenvalue equation. Therefore, we cannot find the eigenvalue that corresponds to the given eigenfunction using this difference equation.