Question

What is 4 point Discrete Fourier Transform of signal x(n) = (1, 2, 3, 4)
1. [10, 2j, -2, -2j]
2. [10, -2, -2, -2]
3. [10, -2+2j, -2, -2-2j]
4. [10j, -2, -2, -10j]

Answer 1

Correct Answer - Option 3 : [10, -2+2j, -2, -2-2j]

Concept:

The DFT of a sequence is given as:

\(\rm \left[ {\begin{array}{*{20}{c}} {X\left[ 0 \right]}\\ \rm {X\left[ 1 \right]}\\\rm {X\left[ 2 \right]}\\ \rm {X\left[ 3 \right]} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&1&1&1\\ \rm 1&\rm{ - j}&\rm{ - 1}&\rm j\\ 1&{ - 1}&1&{ - 1}\\ 1&\rm j&{ - 1}&\rm{ - j} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\rm x\left[0 \right]}\\ \rm{x\left[ 1 \right]}\\ \rm{x\left[ 2 \right]}\\ \rm{x\left[ 3 \right]} \end{array}} \right]\)

Analysis:

Substituting value we have

\(\rm\left[ {\begin{array}{*{20}{c}} {X\left[ 0 \right]}\\ \rm{X\left[ 1 \right]}\\ \rm{X\left[ 2 \right]}\\ \rm{X\left[ 3 \right]} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&1&1&1\\ 1&\rm{ - j}&{ - 1}&\rm j\\ 1&{ - 1}&1&{ - 1}\\ 1&\rm j&{ - 1}&\rm{ - j} \end{array}} \right]\left[ {\begin{array}{*{20}{r}} 1\\ 2\\ 3\\ { 4} \end{array}} \right] \)

\(= \left[ {\begin{array}{*{20}{r}} 10\\ \rm { 1-2j-3+4j}\\ 1-2+3-4\\ \rm{1+2j-3-4j} \end{array}} \right]\)

X[K] = [10, -2+2j, -2, -2-2j]

option(3) is the correct answer