Final answer:
The frequency response of the discrete-time LTI system with impulse response h[n] = 0.5(0.85)ⁿ u[n] is H(ejω) = 0.5 / (1 - 0.85e-jω).
Step-by-step explanation:
To determine the frequency response of a discrete-time Linear Time-Invariant (LTI) system with the given impulse response, h[n] = 0.5(0.85)ⁿ u[n], where u[n] is the unit step function, we need to calculate its Z-transform which also represents its frequency response, H(z). The Z-transform is defined as H(z) = Σ (h[n] ∙ z⁻ⁿ), summed over all n from -∞ to ∞.
The impulse response given is only valid for n ≥ 0 due to the unit step function, u[n]. Therefore, the Z-transform simplifies to:
H(z) = Σ (0.5(0.85)ⁿ ∙ z⁻ⁿ), summed over n from 0 to ∞.
This is a geometric series with a common ratio r = 0.85 ∙ z⁻⁹. For the series to converge, we need |0.85 ∙ z⁻⁹| < 1. Assuming this condition holds, the sum of the geometric series is:
H(z) = Σ (0.5 ∙ (0.85 ∙ z⁻⁹)ⁿ) = 0.5 / (1 - 0.85 ∙ z⁻⁹)
Finally, to get the frequency response, we evaluate H(z) at z = ejω, where ω is the normalized angular frequency and j is the imaginary unit.
The frequency response of the system is then:
H(ejω) = 0.5 / (1 - 0.85e-jω)