Determine the number of data points used by the fft

Question

Problem # 1:
Given a sequence x(n) for 0≤n≤3, where x(0) = 1, x(1) = 1, x(2) = -1, and x(3) = 0, compute its DFT X(k).

(Use DFT formula, don't use MATLAB function)

Problem # 2:
Use inverse DFT and apply it on the Fourier components X(k) from problem 1 to get the original sequence back.
(Use IDFT formula, don't use MATLAB function)

Problem # 3:
Consider a digital sequence sampled at the rate of 20..000 Hz. If ‘J use the 8,000-point DFT to compute the spectrum, determine
a. the frequency resolution
b. the folding frequency in the spectrum.

Problem #4:
We use the DFT to compute the amplitude spectrum of a sampled data sequence with a sampling rate f_s= 2,000 Hz. It requires the frequency resolution to be less than 0.5 Hz. Determine the number of data points used by the FFT algorithm and actual frequency resolution in Hz, assuming that the data samples are available for selecting the number of data points.

Problem # 5:
Given the following sequence
and assuming that fs = 100 Hz, compute the amplitude spectrum, phase spectrum, and power spectrum.

Problem # 6:
Given the sequence x(n) = [2.2, 3.5, -3.2, 1.9, -2.6, 2], compute the windowed sequence xw using Hamming window function and calculate its amplitude and power spectra.

Problem # 7:
Calculate the spectrum of x(n) = [2.2, 3.5, -3.2, 1.9, -2.6, 2] using FFT algorithm.