Question

5. A band-limited, periodic signal x(t) whose highest frequency is 25 Hz is sampled at 100 Hz over exactly one fundamental period to form the discrete-time signal x[n]. The samples are {x[0], x[1], x[2], x[3]} = {a, b, c, d}. Let one period of the DFT of those samples be {X[0], X[1], X[2], X[3]}. (a) What is the value of X[1] in terms of a, b, c, and d? (4 points) (b) What is the average value of x(t) in terms of a, b, c, and d? (4 points) (c) One of the numbers {X[0], X[1], X[2], X[3]} must be zero. Which one and why? (4 points) (d) Two of the numbers {X[0], X[1], X[2], X[3]} must be real. Which ones and why? (4 points) (e) Recall from class that the convolution of two real-valued signals of length N1 and N2 require N1 N2 real multiplications, while the computation of an N-point DFT (or IDFT) for N = 2L requires approximately 3NL/2 real multiplications. For the signal above (i.e., N1 = 4) and a filter with impulse response length N2 = 4, which method would be computationally simpler to compute the output of the filter: DFT or convolution? Comment on the result. (4 points)

          5. A band-limited, periodic signal x(t) whose highest frequency is 25 Hz is sampled at 100 Hz over exactly one fundamental period to form the discrete-time signal x[n]. The samples are
{x[0], x[1], x[2], x[3]} = {a, b, c, d}.
Let one period of the DFT of those samples be {X[0], X[1], X[2], X[3]}.
(a) What is the value of X[1] in terms of a, b, c, and d? (4 points)
(b) What is the average value of x(t) in terms of a, b, c, and d? (4 points)
(c) One of the numbers {X[0], X[1], X[2], X[3]} must be zero. Which one and why? (4 points)
(d) Two of the numbers {X[0], X[1], X[2], X[3]} must be real. Which ones and why? (4 points)
(e) Recall from class that the convolution of two real-valued signals of length N1 and N2 require N1 N2 real multiplications, while the computation of an N-point DFT (or IDFT) for N = 2L requires approximately 3NL/2 real multiplications. For the signal above (i.e., N1 = 4) and a filter with impulse response length N2 = 4, which method would be computationally simpler to compute the output of the filter: DFT or convolution? Comment on the result. (4 points)

5. A band-limited, periodic signal x(t) whose highest frequency is 25 Hz is sampled at 100 Hz over exactly one fundamental period to form the discrete-time signal x[n]. The samples are
x[0], x[1], x[2], x[3] = a, b, c, d.
Let one period of the DFT of those samples be X[0], X[1], X[2], X[3].
(a) What is the value of X[1] in terms of a, b, c, and d? (4 points)
(b) What is the average value of x(t) in terms of a, b, c, and d? (4 points)
(c) One of the numbers X[0], X[1], X[2], X[3] must be zero. Which one and why? (4 points)
(d) Two of the numbers X[0], X[1], X[2], X[3] must be real. Which ones and why? (4 points)
(e) Recall from class that the convolution of two real-valued signals of length N1 and N2 require N1 N2 real multiplications, while the computation of an N-point DFT (or IDFT) for N = 2L requires approximately 3NL/2 real multiplications. For the signal above (i.e., N1 = 4) and a filter with impulse response length N2 = 4, which method would be computationally simpler to compute the output of the filter: DFT or convolution? Comment on the result. (4 points)

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