Test: DFT Algorithm Computation - 1

Explanation: The formula for calculating N point DFT is given as

From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N² complex multiplications and N(N-1) complex additions.

Explanation: The formula for calculating N point DFT is given as

From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, it requires 4N real multiplications and 4N-2 real additions for any value of ‘k’ to compute DFT of the sequence.

Explanation: According to the symmetry property, we get W_Nk+N/2=-W_N^k.

Explanation: The expression for XR(k) is given as
So, from the equation we can tell that the computation of XR(k) requires 2N² evaluations of trigonometric functions, 4N² real multiplications and 4N(N-1) real additions.

Explanation: T he development of computationally efficient algorithms for the DFT is made possible if we adopt a divide-and-conquer approach. This approach is based on the decomposition of an N-point DFT into successively smaller DFTs. This basic approach leads to a family of computationally efficient algorithms known collectively as FFT algorithms.

Explanation: If we consider the mapping n=Ml+m, then it leads to an arrangement in which the first row consists of the first M elements of x(n), the second row consists of the next M elements of x(n), and so on.

Explanation: We know that if N=LM, then W_N^mqL= W_N/L^{mq= W_Mmq.}

Explanation: The expression for N point DFT is given as
The first step involves L DFTs, each of M points. Hence this step requires LM2 complex multiplications, second require LM and finally third requires ML2. So, Total complex multiplications= N(L+M+1)

Explanation: The expression for N point DFT is given as
The first step involves L DFTs, each of M points. Hence this step requires LM(M-1) complex additions, second step do not require any additions and finally third step requires ML(L-1) complex additions. So, Total number of complex additions= N(L+M-2).

Explanation: According to one of the algorithm describing the divide and conquer method, if we store the signal in column wise, then compute the M-point DFT of each row and multiply the resulting array by the phase factors WNlq and then compute the L-point DFT of each column and read the result row wise.