Question

Consider a non-periodic discrete-time signal xₐ:Vₙ = {O,5ⁿ; 0 ≤ n ≤ N₀⁻¹, 0; otherwise}

Determine the discreate time fourierconsider transform X(e⁻ʲΩ) = ∑4[infinity], n = -[infinity], xₙe⁻ʲΩⁿ

Answer 1

The question requires calculation of the Discrete-Time Fourier Transform for a non-periodic discrete-time signal defined over a certain non-zero range. The Discrete-Time Fourier Transform is found by summing up the value of 5^n multiplied by the complex exponential e^(-jΩn) over the non-zero range of the signal.

Step-by-step explanation:

The question asks about finding the Discrete-Time Fourier Transform of the given signal xa, which is a non-periodic discrete-time signal. The signal is defined as Vn = {O,5n; 0 ≤ n ≤ N0⁻¹, 0; otherwise}. The formula for the Discrete-Time Fourier Transform is X(e⁻ᵇΩ) = ∑ from n = -∞ to ∞ of xne⁻ᵇΩn. To calculate the transform, we only need to consider the range where the signal is not zero.

Performing the analysis only for the non-zero part of the signal, we simplify the transform as follows:

1. Identify the non-zero portion of the signal, which is 5n for 0 ≤ n ≤ N0⁻¹.
2. Substitute the defined range of n into the Discrete-Time Fourier Transform formula. This gives us a finite sum to compute: X(e⁻ᵇΩ) = ∑ from n = 0 to N0⁻¹ of 5ne⁻ᵇΩn.
3. Perform the sum to obtain the Discrete-Time Fourier Transform of the given signal within its non-zero range.