# Discrete Fourier Transform Properties MCQ Quiz – Objective Question with Answer for Discrete Fourier Transform Properties

1. If x(n) is a finite duration sequence of length L, then the discrete Fourier transform X(k) of x(n) is given as ____________

A. $$\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}$$(L<N)(k=0,1,2…N-1)

B. $$\sum_{n=0}^{N-1}x(n)e^{j2πkn/N}$$(L<N)(k=0,1,2…N-1)

C. $$\sum_{n=0}^{N-1}x(n)e^{j2πkn/N}$$(L>N)(k=0,1,2…N-1)

D. $$\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}$$(L>N)(k=0,1,2…N-1)

If x(n) is a finite duration sequence of length L, then the Fourier transform of x(n) is given as

X(ω)=$$\sum_{n=0}^{L-1} x(n)e^{-jωn}$$

If we sample X(ω) at equally spaced frequencies ω=2πk/N, k=0,1,2…N-1 where N>L, the resultant samples are

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

2. If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is?

A. $$\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}$$

B. $$\sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}$$

C. $$\sum_{k=0}^{N-1}X(k)e^{j2πkn/N}$$

D. $$\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}$$

If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is given as

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

3. A finite duration sequence of length L is given as x(n)=1 for 0≤n≤L-1 = 0 otherwise, then what is the N point DFT of this sequence for N=L?

A. X(k) = L for k=0, 1, 2….L-1
B. X(k) = L for k=0 =0 for k=1,2….L-1
C. X(k) = L for k=0=1 for k=1,2….L-1
D. None of the mentioned

The Fourier transform of this sequence is

X(ω)=$$\sum_{n=0}^{L-1} x(n)e^{-jωn}=\sum_{n=0}^{L-1}e^{-jωn}$$

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

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

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

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}

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

The matrix W4 may be expressed as

W4=$$\begin{bmatrix}W_4^0& W_4^0& W_4^0& W_4^1\\W_4^0& W_4^0& W_4^2& W_4^3\\W_4^0& W_4^2& W_4^0& W_4^3\\W_4^4& W_4^6& W_4^6& W_4^9\end{bmatrix}=\begin{bmatrix}W_4^0& W_4^0& W_4^0& W_4^1\\W_4^0& W_4^0& W_4^2& W_4^3\\W_4^0& W_4^2& W_4^0& W_4^3\\W_4^0& W_4^2& W_4^2& W_4^1\end{bmatrix}$$

=$$\begin{bmatrix}1&1&1&1\\1&-j&-1&j\\1&-1&1&-1\\1&j&-1&-j\end{bmatrix}$$

Then X4=W4.x4=$$\begin{bmatrix}6\\ -2+2j\\ -2\\-2-2j\end{bmatrix}$$

8. If X(k) is the N point DFT of a sequence whose Fourier series coefficients are 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

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)= Nck

9. 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}

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.

10. If W4100=Wx200, then what is the value of x?

A. 2
B. 4
C. 8
D. 16