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