Question

Use the Discrete Fourier transform (DFT) to determine the spectrum of the discrete-time signal x₁[n]=[1.0,-0.5,0.5,1.0] . The sequence values are zero for (n3). Use a 4point DFT. i.e. evaluate the DFT formula for N=4 X[k]=∑³ₙ₌₀ x[n]e−⁽²π/ᴺ⁾ᵏⁿ,k=0,⋯,3. The answer will be a sequence of four values. Determine the frequency resolution for a 4-point DFT.

Answer 1

To determine the spectrum of a discrete-time signal using the Discrete Fourier transform (DFT), we can evaluate the DFT formula for the given signal. In this case, the spectrum of the signal x₁[n] is [2, 0, -2, 0]. The frequency resolution for a 4-point DFT is 0.25.

Step-by-step explanation:

To determine the spectrum of the discrete-time signal x₁[n]=[1.0,-0.5,0.5,1.0] using the Discrete Fourier transform (DFT), we can evaluate the DFT formula for N=4.

X[k]=∑³ₙ₌₀ x[n]e−⁽²π/ᴺ⁾ᵏⁿ, k=0,⋯,3

Substituting the values of x[n], we have:

X[0] = 1.0e^(-2π/4*0) + (-0.5)e^(-2π/4*0) + 0.5e^(-2π/4*0) + 1.0e^(-2π/4*0) = 2

X[1] = 1.0e^(-2π/4*1) + (-0.5)e^(-2π/4*1) + 0.5e^(-2π/4*1) + 1.0e^(-2π/4*1) = 0

X[2] = 1.0e^(-2π/4*2) + (-0.5)e^(-2π/4*2) + 0.5e^(-2π/4*2) + 1.0e^(-2π/4*2) = -2

X[3] = 1.0e^(-2π/4*3) + (-0.5)e^(-2π/4*3) + 0.5e^(-2π/4*3) + 1.0e^(-2π/4*3) = 0

Therefore, the spectrum of the signal x₁[n] is [2, 0, -2, 0].

The frequency resolution for a 4-point DFT is given by Δf = 1/T, where T is the total duration of the signal. In this case, since the sequence values are zero for n>3, the total duration of the signal is 4. Therefore, the frequency resolution is Δf = 1/4 = 0.25.