Question

Consider the DT-LTI system with impulse response: h[n]=(0.8)ⁿu[n].

Derive on paper the response of the system (y[n]ₚₐₚₑᵣ ) to the input signal x[n]=u[n]. Recall the following series: ∑k=0M aᵏ =1-a ᴹ⁺¹ /1-a, a ≠ 0

Answer 1

The student's question concerns deriving the response of a DT-LTI system with a specific impulse response to a unit step function input by applying the convolution sum and using the geometric series formula.

Step-by-step explanation:

The student is asking about the response of a Discrete-Time Linear Time-Invariant (DT-LTI) system with a given impulse response h[n]=(0.8)ⁿ u[n] to the input signal x[n]=u[n]. To find the response y[n]ᵐᵃₚ†ₙ, we utilize the convolution sum of the input signal with the system's impulse response.

Since the input is a unit step function u[n], we apply the convolution as follows:

• Express the convolution sum (∑k=0∞ x[k]h[n-k]).
• Implement the summation considering x[k] is non-zero for k ≥ 0.
• Simplify using the geometric series sum formula given in the question.

Finally, the output y[n]ᵐᵃₚ†ₙ will be expressed in terms of n.

It is important to note that this is a basic application of the convolution sum in signal processing, which is a fundamental concept in systems and signals analysis.