Discrete Fourier Transform Interview Questions and Answers:

1. What is zero padding? What are its uses?

Ans. Let the sequence x(n) has a length L. If we want to find the N-point DFT (N > L) of the sequence x(n), we have to add (N — L) zeros to the sequence x(n). This is known as zero padding. The uses of padding a sequence with zeros are

• We can get “better display” of the frequency spectrum.
• With zero padding, the DFT can be used in linear filtering.

2. Define discrete Fourier series?

Ans. Consider a sequence x_p(n) with a period of N samples so that x_p(n) = x_p(n + lN); Then discrete Fourier series of the sequence x_p(n) is defined as

3. What do you understand by periodic convolution?

Ans. Let x_1p(n) and x_2p(n) be two periodic sequences each with period N with

If

then the periodic sequence x_3p(n) with Fourier series coefficients X_3p(k) can be obtained by periodic convolution, defined as

The convolution in the form of Eq. (2) is known as periodic convolution, as the sequences in Eq. (2) are all periodic with period N, and the summation is over one period.

4. Define circular convolution?

Ans. Let x₁(n) and x₂(n) are finite duration sequences both of length N with DFTs X₁(k) and X₂(k). If X₃(k) = X₁(k)X₂(k), then the sequence x₃(n) can be obtained by circular convolution, defined as

5. Distinguish between linear and circular convolution of two sequences

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6. How will you obtain linear convolution from circular convolution?

Ans. Consider two finite duration sequences x(n) and h(n) of duration L samples and M samples respectively. The linear convolution of these two sequences produces an output sequence of duration L + M — 1 samples, whereas, the circular convolution of x(n) and h(n) give N samples where N = Max(L, M). In order to obtain the number of samples in circular convolution equal to L + M — 1, both x(n) and h(n) must be appended with appropriate number of zero valued samples. In other words, by increasing the length of the sequences x(n) and h(n) to L + M — 1 points and then circularly convolving the resulting sequences we obtain the same results as that of linear convolution.

7. What is meant by sectioned convolution?

Ans. If the data sequence x(n) is of long duration, it is very difficult to obtain the output sequence y(n) due to limited memory of a digital computer. Therefore, the data sequence is divided up into smaller sections. These sections are processed separately one at a time combined later to get the output.

8. What are the two methods used for the sectional convolution?

Ans. The two methods used for the sectional convolution are (1) the overlap-add method and (2) overlap-save method.

9. Write briefly about overlap-save method?

Ans. In the method the data sequence is divided into N-point sections x_i(n). Each section contains the last M —1 data points of the previous section followed by L new data points to form a data sequence of length N = L+ M —1. If we take N-point circular convolution of x_i(n) with h(n), the first M – 1 points will not agree with the linear convolution of x_i(n) and h(n) because of aliasing; the remaining (N — M + 1) points however will agree with the linear convolution. Hence we discard the first (M — 1) points of filter section x_i(n) N h(n). This process is repeated for all sections and the filtered sections are abutted together.

10. Write briefly about overlap-add method?

Ans. In this method the size of the input data block x_i(n) is L. To each data block we append M – 1 zeros and perform N-point (N = L+ M —1) circular convolution of x_i(n) with h(n). Since each data block is terminated with M — 1 zeros, the last M —1 points from each output block must be overlapped and added to first M —1 points of the succeeding block. Hence, this method is called overlap-add method.

11. State the difference between (i) overlap-save method (ii) overlap-add method.

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12. Distinguish between DFT and DTFT.

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13. Distinguish between Fourier series and Fourier transform.

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