The discrete Fourier transform is given as X(k)=\(\sum_{n=0}^{N-1}e^{-j2πkn/N}\)

If N=L, then X(k)= L for k=0 =0 for k=1,2….L-1

4. The Nth root of unity WN is given as _________

A. ej2πN
B. e-j2πN
C. e-j2π/N
D. ej2π/N

Answer: C

We know that the Discrete Fourier transform of a signal

x(n) is given as X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1}x(n)W_N^{kn}\)

Thus we get Nth rot of unity WN=e-j2π/N

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

A. N2 complex multiplications and N(N-1) complex additions
B. N2 complex additions and N(N-1) complex multiplications
C. N2 complex multiplications and N(N+1) complex additions
D. N2 complex additions and N(N+1) complex multiplications

Answer: A

The formula for calculating N point DFT is given as

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

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.

6. Which of the following is digital signal processing is true?

A. WN*=\(\frac{1}{N} W_N^{-1}\)

B. WN-1=\(\frac{1}{N} W_N*\)

C. WN-1=WN*

D. None of the mentioned

Answer: B

If XN represents the N point DFT of the sequence xN in the matrix form, then we know that

XN=WN.xN

By pre-multiplying both sides by WN-1, we get
xN=WN-1.XN

But we know that the inverse DFT of XN is defined as

xN=\(\frac{1}{N} W_N*X_N\)

Thus by comparing the above two equations we get

WN-1 = \(\frac{1}{N} W_N*\)

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

A. {6,-2+2j-2,-2-2j}
B. {6,-2-2j,2,-2+2j}
C. {6,-2+2j,-2,-2-2j}
D. {6,-2-2j,-2,-2+2j}

Answer: C

The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property WNk+N/2=-WNk