Question

Question:

Assume that a complex multiplication takes 1 and that the amount of time to compute a DFT is determined by the amount of time it takes to perform all of the multiplications. The time required to compute a 1024-point DFT directly and using FFT is (approximately)

Option 1:

1s and 5ms

Option 2:

2s and 50ms

Option 3:

10s and 50ms

Option 4:

1s and 50ms

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Answer 1

Answer:

Answer: a

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

X(k)=∑N−1n=0x(n)e−j2πkn/N

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.

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Explanation:

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Answer 2

The time required to compute a 1024-point DFT directly and using FFT is 1s and 5ms

Step by step calculation:

GIVEN:

• Time required for 1 complex multiplication = 1 μs =  10⁻⁶ s
• Time required for DFT = time it takes to perform all of the multiplications

TO FIND:

• Time required to compute 1024-point DFT.

CALCULATIONS:

Direct calculation of N point DFT

Time required for complex multiplications =N²

For 1024-point DFT

Number of complex multiplications required  = 1024²

                                                                           = 1048576

Time required for 1 complex multiplication = 1 μs

Time required for 1048576 complex multiplication = 10⁻⁶ x 1048576

                                                                                    = 1.05 s

                                                                                    = 1s and 5ms

Final Answer- Option 1 1s and 5ms

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