Question

Use the Discrete-time Fourier transform summation

[infinity]
X(eʲω) = ∑[infinity] x [n]e⁻ʲω to determine the spectrum of the discrete-time signal x₁ [n] = [1.0,−0.5,0.5,1.0].
ₙ₌−[infinity]
The sequence values are zero for n < 0 and n > 3

Answer 1

To find the spectrum of the discrete-time signal x₁[n] = [1.0, -0.5, 0.5, 1.0], the Discrete-time Fourier transform is applied to the given non-zero range, resulting in the frequency spectrum

Step-by-step explanation:

The question asks to determine the spectrum of the discrete-time signal x₁[n] = [1.0, -0.5, 0.5, 1.0] with the use of the Discrete-time Fourier transform . Given that x₁[n] is non-zero only for n ranging from 0 to 3, we can compute its Discrete-time Fourier transform by evaluating the summation of x₁[n]e⁻¹⁰⁷⁾⁾⁾⁷., where n is each point within that range.

Applying the formula, we get:
X(eˠᵗ) = x₁[0]e⁻ˠᵗ(0) + x₁[1]e⁻ˠᵗ(1) + x₁[2]e⁻ˠᵗ(2) + x₁[3]e⁻ˠᵗ(3)

= 1.0e⁻ˠᵗ(0) - 0.5e⁻ˠᵗ(1) + 0.5e⁻ˠᵗ(2) + 1.0e⁻ˠᵗ(3)

= 1 - 0.5e⁻ˠᵗ + 0.5e⁻2ˠᵗ + e⁻3ˠᵗ

This result is the frequency spectrum of the given discrete-time signal, showing how different frequency components are represented within the signal.