Discrete Fourier Transform – Properties and Applications topics include: Discrete fourier transforms, their sampling and properties, linear filtering methods on DFT and their frequency analysis. The discrete Fourier transform (DFT) is a mathematical function that converts a sequence of equally-spaced samples of a function into a sequence of equally-spaced samples of the discrete-time Fourier transform. The DFT is a frequency domain representation of the original input sequence.  The DFT is used in engineering to convert a signal from the time domain into the frequency domain. This helps... Show more

Discrete Fourier Transform – Properties and Applications topics include: Discrete fourier transforms, their sampling and properties, linear filtering methods on DFT and their frequency analysis.

The discrete Fourier transform (DFT) is a mathematical function that converts a sequence of equally-spaced samples of a function into a sequence of equally-spaced samples of the discrete-time Fourier transform. The DFT is a frequency domain representation of the original input sequence. 

The DFT is used in engineering to convert a signal from the time domain into the frequency domain. This helps analyze the frequency components of the signal. The DFT is the core of many digital signal processing systems. 
The DFT has the following properties: Linearity, Time shifting, Time modulation, Multiplication. 
The basis functions of the DFT are a set of sine and cosine waves with unity amplitude.

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

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25 Questions

1. What is the circular convolution of the sequences X₁(n)={2,1,2,1} and x₂(n)={1,2,3,4}?

{16,16,14,14}{2,3,6,4}{14,16,14,16}{14,14,16,16}

2. What is the DFT of the four point sequence x(n)={0,1,2,3}?

{6,-2+2j-2,-2-2j}{6,-2+2j,-2,-2-2j}{6,-2-2j,2,-2+2j}{6,-2-2j,-2,-2+2j}

3. If W₄¹⁰⁰=W_x²⁰⁰, then what is the value of x?

82164

4. If X(k) is the N-point DFT of a sequence x(n), then circular time shift property is that N-point DFT of x((n-l))_N is X(k)e^-j2πkl/N.

MaybeDon't KnowFalseTrue

5. If the signal to be analyzed is an analog signal, we would pass it through an anti-aliasing filter with B as the bandwidth of the filtered signal and then the signal is sampled at a rate __________

Fs = 2BFs ≥ 2BFs ≤ 2BFs ≤ B

6. The N^th rot of unity W_N is given as ______________

ej2π/Ne-j2π/Ne-j2πNej2πN

7. In Overlap save method of long sequence filtering, what is the length of the input sequence block?

None of theseL+M-1L+ML+M+1

8. In which of the following methods, the output sequence is considered as shown in the below diagram?

None of theseOverlap add & save methodOverlap save methodOverlap add method

9. The condition with less number of samples L should be avoided.

MaybeDon't KnowTrueFalse

10. Which of the following is the disadvantage of Hanning window over rectangular window?

More side lobesNone of theseLess side lobesMore width of main lobe

11. What is the circular convolution of the sequences X₁(n)={2,1,2,1} and x₂(n)={1,2,3,4}, find using the DFT and IDFT concepts?

{14,14,16,16}{14,16,14,16}None of these{16,16,14,14}

12. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by c_k, then which of the following is true?

X(k)=NckNone of theseX(k)=ck/NX(k)=N/ck

13. Which of the following is true in case of Overlap add method?

M zeros are appended at first of each data blockM zeros are appended at last of each data blockM-1 zeros are appended at last of each data blockM-1 zeros are appended at first of each data block

14. If x(n) and X(k) are an N-point DFT pair, then X(k+N)=?

-X(k)X(-k)None of theseX(k)

15. Overlap add and Overlap save are the two methods for linear FIR filtering a long sequence on a block-by-block basis using DFT.

MaybeTrueFalseDon't Know

16. What is the DFT of the four point sequence x(n)={0,1,2,3}?

{6,-2+2j-2,-2-2j}{6,-2+2j,-2,-2-2j}{6,-2-2j,2,-2+2j}{6,-2-2j,-2,-2+2j}

17. What is the highest frequency that is contained in the sampled signal?

None of these2FsFsFs/2

18. If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is?

\(\sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\) \(\sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\)

19. The finite observation interval for the signal places a limit on the frequency resolution.

Don't KnowFalseMaybeTrue

20. What is the sequence y(n) that results from the use of four point DFTs if the impulse response is h(n)={1,2,3} and the input sequence x(n)={1,2,2,1}?

{1,4,9,11,8,3}{9,7,9,11}{9,9,7,11}{7,9,7,11}

21. If X(k) is the N-point DFT of a sequence x(n), then what is the DFT of x*(n)?

X*(k)X*(N-k)None of theseX(N-k)

22. In Overlap add method, what is the length of the input data block?

None of theseL-1L+1L

23. The first M-1 values of the output sequence in every step of Overlap save method of filtering of long sequence are discarded.

Don't KnowFalseMaybeTrue

24. What is the value of x(n)*h(n), 0≤n≤11 for the sequences x(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3} and h(n)={1,1,1} if we perform using overlap save fast convolution technique?

{1,3,3,1,1,3,5,2,2,2,3,6}{1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}{1,2,0,3,4,2,1,1,2,3,2,1,3}{1,3,3,-1,1,3,5,2,-2,2,3,6}

25. If {x(n)} is the signal to be analyzed, limiting the duration of the sequence to L samples, in the interval 0≤ n≤ L-1, is equivalent to multiplying {x(n)} by?

Rectangular windowHamming windowKaiser windowHanning window