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Which of the following is true regarding the number of computations required to compute an Npoint DFT?
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 N1 complex additions. So, in a total to perform Npoint DFT we perform N^{2} complex multiplications and N(N1) complex additions.
Which of the following is true regarding the number of computations required to compute DFT at any one value of ‘k’?
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 N1 complex additions. So, it requires 4N real multiplications and 4N2 real additions for any value of ‘k’ to compute DFT of the sequence.
Explanation: According to the symmetry property, we get W_{N}k+N/2=W_{N}^{k}.
The computation of XR(k) for a complex valued x(n) of N points requires:
Explanation: The expression for XR(k) is given as
So, from the equation we can tell that the computation of XR(k) requires 2N^{2} evaluations of trigonometric functions, 4N^{2} real multiplications and 4N(N1) real additions.
Divideandconquer approach is based on the decomposition of an Npoint DFT into successively smaller DFTs. This basic approach leads to FFT algorithms.
Explanation: T he development of computationally efficient algorithms for the DFT is made possible if we adopt a divideandconquer approach. This approach is based on the decomposition of an Npoint DFT into successively smaller DFTs. This basic approach leads to a family of computationally efficient algorithms known collectively as FFT algorithms.
If the arrangement is of the form 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, then which of the following mapping represents the above arrangement?
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.
If N=LM, then what is the value of W_{N}^{mqL}?
Explanation: We know that if N=LM, then W_{N}^{mqL}= W_{N/L}^{mq= WMmq.}
How many complex multiplications are performed in computing the Npoint DFT of a sequence using divideandconquer method if N=LM?
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)
How many complex additions are performed in computing the Npoint DFT of a sequence using divideandconquer method if N=LM?
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(M1) complex additions, second step do not require any additions and finally third step requires ML(L1) complex additions. So, Total number of complex additions= N(L+M2).
Which is the correct order of the following steps to be done in one of the algorithm of divide and conquer method?
1) Store the signal column wise
2) Compute the Mpoint DFT of each row
3) Multiply the resulting array by the phase factors WNlq.
4) Compute the Lpoint DFT of each column.
5) Read the result array row wise.
Explanation: According to one of the algorithm describing the divide and conquer method, if we store the signal in column wise, then compute the Mpoint DFT of each row and multiply the resulting array by the phase factors WNlq and then compute the Lpoint DFT of each column and read the result row wise.
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