DFT Efficient Computation - Fast Fourier Transform Algorithms topics include:  Computation of discrete fourier transforms and fast fourier transforms, various approaches to their computation which include filtering and quantization and applications of FFT algorithms. The Fast Fourier Transform (FFT) is a computationally efficient method for calculating the Discrete Fourier Transform (DFT). It is a key tool in digital signal processing applications and is used as a benchmark for evaluating digital signal processor (DSP) performance.  The FFT algorithm is more efficient than a direct DFT... Show more

DFT Efficient Computation - Fast Fourier Transform Algorithms topics include:  Computation of discrete fourier transforms and fast fourier transforms, various approaches to their computation which include filtering and quantization and applications of FFT algorithms.

The Fast Fourier Transform (FFT) is a computationally efficient method for calculating the Discrete Fourier Transform (DFT). It is a key tool in digital signal processing applications and is used as a benchmark for evaluating digital signal processor (DSP) performance. 

The FFT algorithm is more efficient than a direct DFT computation. A direct DFT computation requires O(N^2) operations, while the FFT algorithm requires O(N log N) operations. This is a significant speedup for large data sets. 
For example, when N = 512, the direct DFT computational complexity is proportional to N2 = 262144, whereas the FFT computational complexity is proportional to Nlog2N = 2048. This means FFT is 32 times faster than DFT. 
The FFT algorithm was developed by Cooley-Tukey in 1965. The idea came about during a meeting of President Kennedy's Science Advisory Committee. The committee was discussing how to detect nuclear tests by the Soviet Union by setting up sensors around the country. An FFT algorithm was needed to analyze the output of these sensors. 

Related Test: Digital Signal Processing Practice Test: Discrete Fourier Transform - Properties and Applications

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Digital Signal Processing Practice Test: DFT Efficient Computation - Fast Fourier Transform Algorithms

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25 Questions

1. Every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.

Don't KnowTrueMaybeFalse

2. What is the total number of quantization errors in the computation of single point DFT of a sequence of length N?

2N4N12N8N

3. The computation of XR(k) for a complex valued x(n) of N points requires _____________

4N(N-1) real additionsAll of the mentioned2N2 evaluations of trigonometric functions4N2 real multiplications

4. If the desired number of values of the DFT is less than log₂N, a direct computation of the desired values is more efficient than FFT algorithm.

MaybeDon't KnowTrueFalse

5. If X(k) is the N/2 point DFT of the sequence x(n), then what is the value of X(k+N/2)?

F1(k)-WNk F2(k)F1(k)+WNk F2(k)F1(k)+F2(k)None of these

6. The following butterfly diagram is used in the computation of __________

All of the mentionedDecimation-in-frequency FFTNone of theseDecimation-in-time FFT

7. Which of the following is true regarding the number of computations required to compute an N-point DFT?

N2 complex additions and N(N+1) complex multiplicationsN2 complex multiplications and N(N-1) complex additionsN2 complex multiplications and N(N+1) complex additionsN2 complex additions and N(N-1) complex multiplications

8. How many complex additions are required to be performed in linear filtering of a sequence using FFT algorithm?

(N/2)logNNlog2N2Nlog2N(N/2)log2N

9. If g(n) is a real valued sequence of 2N points and x₁(n)=g(2n) and x₂(n)=g(2n+1), then what is the value of G(k), k=N,N-1,…2N-1?

X1(k)+W2kX2(k)X1(k)-W2kX2(k)X1(k)-W2kNX2(k)X1(k)+W2kNX2(k)

10. The total number of complex multiplications required to compute N point DFT by radix-2 FFT is?

(N/2)log2NNone of these(N/2)logNNlog2N

11. Divide-and-conquer approach is based on the decomposition of an N-point DFT into successively smaller DFTs. This basic approach leads to FFT algorithms.

FalseTrueDon't KnowMaybe

12. How many multiplications are required to calculate X(k) by chirp-z transform if x(n) is of length N?

N+1N-1NNone of these

13. How many complex multiplications are need to be performed to calculate chirp z-transform?(M=N+L-1)

Mlog2(M-1)log2M(M-1)log2MMlog2M

14. For a decimation-in-time FFT algorithm, which of the following is true?

Input is shuffled and output is in orderInput is in order and output is shuffledBoth input and output are shuffledBoth input and output are in order

15. How many complex multiplication are required per output data point?

[(N/2)logN]/L[Nlog22N]/LNone of these[(N/2)log2N]/L

16. What is the model that has been adopt for characterizing round of errors in multiplication?

Additive white noise modelSubtractive white noise modelMultiplicative white noise modelNone of these

17. What is the signal-to-noise ratio?

σX2-σq2σX2+σq2σX2.σq2σX2/σq2

18. If we store the signal row wise and compute the L point DFT at each column, the resulting array must be multiplied by which of the following factors?

WNpmWNlqWNpq

19. What is the real part of the N point DFT XR(k) of a complex valued sequence x(n)?

\(\sum_{n=0}^{N-1} [x_R (n) cos⁡\frac{2πkn}{N} + x_I (n) sin⁡\frac{2πkn}{N}]\) None of these\(\sum_{n=0}^{N-1} [x_R (n) cos⁡\frac{2πkn}{N} – x_I (n) sin⁡\frac{2πkn}{N}]\) \(\sum_{n=0}^{N-1} [x_R (n) sin⁡\frac{2πkn}{N} + x_I (n) cos⁡\frac{2πkn}{N}]\)

20. If x₁(n) and x₂(n) are two real valued sequences of length N, and let x(n) be a complex valued sequence defined as x(n)=x₁(n)+jx₂(n), 0≤ n≤ N-1, then what is the value of x₂(n)?

\(\frac{x(n)+x*(n)}{2j}\) \(\frac{x(n)-x*(n)}{2j}\) \(\frac{x(n)+x*(n)}{2}\) \(\frac{x(n)-x*(n)}{2}\)

21. What is the range in which the quantization errors due to rounding off are uniformly distributed as random variables if Δ=2^-b?

(0,Δ)(-Δ/2,Δ/2)(-Δ,0)None of these

22. What is the value of the variance of quantization error in FFT algorithm, compared to that of direct computation?

EqualGreaterCannot be comparedLess

23. What is the variance of the output DFT coefficients |X(k)|?

\(\frac{1}{2N}\) \(\frac{1}{3N}\) \(\frac{1}{N}\) \(\frac{1}{4N}\)

24. Decimation-in frequency FFT algorithm is used to compute H(k).

MaybeDon't KnowFalseTrue

25. Which is the correct order of the following steps to be done in one of the algorithm of divide and conquer method?
i) Store the signal column wise
ii) Compute the M-point DFT of each row
iii) Multiply the resulting array by the phase factors W_N^lq.
iv) Compute the L-point DFT of each column.
v) Read the result array row wise.

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