Determine the fourier transform methods to determine the

Question

Problem 1: Determine the Fourier series coefficients of the following signals:

1. x₁[n] = 2 + cos(πn/3 + π/6) + sin(π/2)

2. x₂[n] = ∑^∞_k=-∞ [ (-1)^kδ(n - 2k)]

Problem 2: What time-domain signals have the following Fourier series/transform representations:

1. X₁[k] = cos(πk/3) + cos(πk/2 + π/3)

2. X₂(ω) = ∑²_k=-2 δ(ω - kπ/4) (|ω| ≤ π)

Problem 3: The FS representation of x[n] is given by X[k] = sin(kπ/3). Use Fourier series properties, without determining x[n], to determine the following DT signals:

1. v[n] = x[n + 1] ∗ x[n - 1] (*: convolution)

2. w[n] = sin(πn/2) x[n].

Problem 4: Determine the Fourier transform methods to determine the response of the following systems for the given input:

1. h₁[n] = sin(2πn/5)/πn and x₁[n] = sin(πn/3 + π/4) + cos(3πn/4)

2. h₂[n] = sin(πn/3) cos(πn/2)/πn and x₂[n] = sin(πn/3 + π/4) + cos(3πn/4)

3. h₃[n] = (1/2)^|n| and x₃[n] = (-1)ⁿ