(Aktu Btech) Digital Signal Processing Important Unit-4 DFT and FFT

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Important Questions For Digital Signal Processing:
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Q1. Calculate the DFT of x(n) = cos an.

Ans. Given: x(n) = cos an

To Find: DFT. 

2. Putting n = 0, 1,………….N-1 

Q2. Find the 10-point DFT of the following sequence:

Ans. 1. For 10 point DFT, N= 10

Q3. Determine the 4-point discrete time sequence from its DFT X(k) = {4, 1 – j, -2, 1 + j}.

Ans. 1. IDFT is defined as

6. Therefore, the IDFT of the given DFT produces the following four point discrete time sequence.  

                                        x(n) = {1, 2, 0, 1}

Q4. Derive the relation between DFT and z-transform of a discrete time sequence x(n).

Ans. 1. The z-transform of x(n) is given as 

2. If X(z) is sampled at the N equally spaced points on the unit circle, these points will be

3. The R.H.S. of eq. (4.10.1) is DFT X(k), thus the relationship between DFT and z-transform

Q5. Compute the circular convolution of two discrete time sequences x₁ (n) = {1,2, 1,2} and x₂ (n) = {3, 2, 1, 4}.

Ans. Given: x₁ (n) = {1,2, 1,2}, x₂ (n) = {3, 2, 1, 4}

To Find: Circular convolution. 

Q6. Find the linear convolution using circular convolution of the following sequence:

                                             x(n) = {1,2, 1}, h(n) = {1, 2}.

Ans. Given: x(n) = {1,2, 1}, h(n) = {1, 2}

To Find: Linear convolution. 

3. Let us make length of x(n) and h(n) equal to 4 by adding zeros at end.  

5. Substituting the values of W₄ and x_n in eq. (4.17.1), we get

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