In a discrete-time LTI system, which expression corresponds to the eigenvalue for input e^{jωn}?

• A. H(e^{jω})
• B. H(jω)
• C. H(s)
• D. H(-ω)

Answer: (A)

In a discrete-time LTI system, complex exponentials e^{jωn} are eigenfunctions. If you feed the system x[n] = e^{jωn}, the output becomes y[n] = (h * x)[n] = ∑ h[k] e^{jω(n−k)} = e^{jωn} ∑ h[k] e^{−jωk}. The sum ∑ h[k] e^{−jωk} is the discrete-time frequency response evaluated on the unit circle, written as H(e^{jω}). Therefore, the output is simply a scaled copy of the input: y[n] = H(e^{jω}) e^{jωn}. The factor H(e^{jω}) is the eigenvalue corresponding to the input e^{jωn}.

This H(e^{jω}) is the system’s frequency response for discrete time. The other forms reference continuous-time representations or alternate frequency axes (like H(jω) for continuous-time, H(s) for Laplace-domain, or evaluating at negative frequency H(-ω)), which don’t give the correct eigenvalue for a discrete-time input e^{jωn}.