Question

Use the Discrete-time Fourier transform summation X(e^jω) = ∑[n=−[infinity] to [infinity]] x[n]e^−jinω 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 n<0 and n> [missing value].

Answer 1

In this case, the spectrum of the discrete-time signal x₁[n] is X(e^jω) = 1.0e^(-j0ω) - 0.5e^(-j1ω) + 0.5e^(-j2ω) + 1.0e^(-j3ω)

Step-by-step explanation:

AI-generated answer

To determine the spectrum of the discrete-time signal x₁[n] = [1.0, -0.5, 0.5, 1.0], we can use the given Discrete-time Fourier transform summation formula:

X(e^jω) = ∑[n=-∞ to ∞] x[n]e^(-jinω)

Let's substitute the values of x₁[n] into the formula and evaluate the summation step-by-step:

X(e^jω) = x[0]e^(-j0ω) + x[1]e^(-j1ω) + x[2]e^(-j2ω) + x[3]e^(-j3ω)

1. Evaluating x[0]:

X(e^jω) = 1.0e^(-j0ω)

2. Evaluating x[1]:

X(e^jω) = 1.0e^(-j0ω) - 0.5e^(-j1ω)

3. Evaluating x[2]:

X(e^jω) = 1.0e^(-j0ω) - 0.5e^(-j1ω) + 0.5e^(-j2ω)

4. Evaluating x[3]:

X(e^jω) = 1.0e^(-j0ω) - 0.5e^(-j1ω) + 0.5e^(-j2ω) + 1.0e^(-j3ω)

Now, we have the expression for X(e^jω), which represents the spectrum of the discrete-time signal x₁[n].

It is important to note that the missing value for n > [missing value] limits the range of the summation in the given formula. To provide a complete and accurate spectrum, the missing value needs to be specified. Once that value is provided, the summation can be evaluated accordingly.