Before we fine DFT first let we check it out the what is DFT.

Find the DFT of the following finite equation sequence of length L.

Find the DFT of the following finite equation sequence of length L.

x(n) = A for 0≤ n ≤ L-1

= 0 otherwise

We have

N-1

X(K) = ∑ x(n) . e

^{–j2πkn/N}^{ }n=0

N-1

X(K) = ∑ A . e

^{–j2πkn/N}^{ }n=0

N-1

X(K) = ∑ x(n) .( e

^{–j2πk/N })^{n}^{ }

We have standard summation formula

N2

X(K) = ∑ a

^{K }= a^{N1 }- a^{N2+1 }/ 1- a^{ }K=N1

here N1=0, N2 = L-1 and a = e

^{–j2πk/N}
N-1

X(K) = ∑ A [ ( e

^{–j2πk/N })^{0}- .( e^{–j2πk/N })^{L-1+1 }/1- e^{–j2πk/N ]}^{ }n=0

^{ }N-1

X(K) = ∑ A [ ( 1 - .( e

^{–j2πkL/N })^{ }/ 1- e^{–j2πk/N }^{ }n=0