16. 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.
17. Which of the following is true?
A. \(W_N^*=\frac{1}{N} W_{N^{-1}}\)
B. \(W_N-1=\frac{1}{N} W_{N^*}\)
C. \(W_N-1=W_{N^*}\)
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=1/N*XN
Thus by comparing the above two equations we get
WN-1=1/N WN*
18. 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
Then X4=W4.x4=\(\begin{bmatrix}6\\-2+2j\\-2\\-2-2j\end{bmatrix}\)
19. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
A. X(k)=Nck
B. X(k)=ck/N
C. X(k)=N/ck
D. None of the mentioned
Answer: A
The Fourier series coefficients are given by the expression
ck=\(\frac{1}{N} \sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} = \frac{1}{N}X(k)=> X(k)=Nc_k\)
20. 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: D
Given x(n)={0,1,2,3}
We know that the 4-point DFT of the above given sequence is given by the expression
X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} \)
In this case N=4
=>X(0)=6, X(1)=-2+2j, X(2)=-2, X(3)=-2-2j.
21. If W4100 = Wx200, then what is the value of x?
A. 2
B. 4
C. 8
D. 16
Answer: C
We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.
22. What are the values of z for which the value of X(z)=0?
A. Poles
B. Zeros
C. Solutions
D. None of the mentioned
Answer: B
For a rational z-transform X(z) to be zero, the numerator of X(z) is zero and the solutions of the numerator are called ‘zeros’ of X(z).
23. What are the values of z for which the value of X(z)=∞?
A. Poles
B. Zeros
C. Solutions
D. None of the mentioned
Answer: A
For a rational z-transform X(z) to be infinity, the denominator of X(z) is zero and the solutions of the denominator are called ‘poles’ of X(z).
24. If X(z) has M finite zeros and N finite poles, then which of the following condition is true?
A. |N-M| poles at origin(if N>M)
B. |N+M| zeros at origin(if N>M)
C. |N+M| poles at origin(if N>M)
D. |N-M| zeros at origin(if N>M)
Answer: D
If X(z) has M finite zeros and N finite poles, then X(z) can be rewritten as X(z)=z -M+N.X'(z).
So, if N>M then z has positive power. So, it has |N-M| zeros at origin.
25. If X(z) has M finite zeros and N finite poles, then which of the following condition is true?
A. |N-M| poles at origin(if N < M)
B. |N+M| zeros at origin(if N < M)
C. |N+M| poles at origin(if N < M)
D. |N-M| zeros at origin(if N < M)
Answer: A
If X(z) has M finite zeros and N finite poles, then X(z) can be rewritten as X(z)=z-M+N.X'(z).
So, if N < M then z has a negative power. So, it has |N-M| poles at the origin.
26. Which of the following signals have a pole-zero plot as shown below?
A. a.u(n)
B. u(an)
C. anu(n)
D. none of the mentioned
Answer: C
From the given pole-zero plot, the z-transform of the signal has one zero at z=0 and one pole at z=a.
So, we obtain X(z)=z/(z-A.
By applying inverse z-transform for X(z), we get
x(n)= anu(n).
