Compute the DFT of the following finite length sequences: i) x(n) = u(n) - u(n-N). Observe the result for N =1. ii) y[n] = cos (2 ⋅ 𝜋.𝑟.𝑛 𝑁 ). (Find the N-point DFT)

i) The Discrete Fourier Transform (DFT) of a sequence x(n) = u(n) - u(n-N) is given by the formula:

X(k) = Σ (from n=0 to N-1) x(n) * e^(-j2πn*k/N)

where u(n) is the unit step function. For the given sequence, x(n) = 1 for 0 <= n < N and x(n) = 0 otherwise. Therefore, the DFT becomes:

X(k) = Σ (from n=0 to N-1) e^(-j2πn*k/N)

This is a geometric series with N terms, the sum of which is given by:

X(k) = 1 - e^(-j2πk) / (1 - e^(-j2πk/N))

For N = 1, the DFT simplifies to X(k) = 1 for all k.

ii) The DFT of a sequence y[n] = cos(2πrn/N) is given by the formula:

Y(k) = Σ (from n=0 to N-1) y(n) * e^(-j2πn*k/N)

Substituting y(n) = cos(2πrn/N) into the formula gives:

Y(k) = Σ (from n=0 to N-1) cos(2πrn/N) * e^(-j2πn*k/N)

This is a more complex series to sum, and the result will depend on the specific values of r and N. However, it's important to note that the DFT of a cosine function will generally result in a pair of impulses at the frequencies corresponding to the cosine's frequency.