Question

A 4 x 4 DFT matrix is given by :

\(\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&1\\ 1&x \end{array}}&{\begin{array}{*{20}{c}} 1&1\\ { - 1}&y \end{array}}\\ {\begin{array}{*{20}{c}} 1&{ - 1}\\ 1&{ - j} \end{array}}&{\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&j \end{array}} \end{array}} \right]\)

(j² = -1)

Where values of x and y are_____ , ______ respectively.

1. 1, -1
2. -1, 1
3. -j, j
4. j, -j

Answer 1

Correct Answer - Option 4 : j, -j

The correct answer is option 4.

Definition of 4X4 DFT matrix  is, 

\(1/\sqrt{4}\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & W & W^2 & W^3\\ 1 & W^2 & W^4 & W^6\\ 1 & W^3 & W^6 & W^9 \end{bmatrix}\)

And the given example is,

\(\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&1\\ 1&x \end{array}}&{\begin{array}{*{20}{c}} 1&1\\ { - 1}&y \end{array}}\\ {\begin{array}{*{20}{c}} 1&{ - 1}\\ 1&{ - j} \end{array}}&{\begin{array}{*{20}{c}} 1&{ - 1}\\ { - 1}&j \end{array}} \end{array}} \right]\)

Given that j² = -1

compare the values of the definition matrix and given example,

W = x, x² = -1 and j² =-1  (W² = -1) then x=j

W³ = y, W3 ​= -j (i.e y=-j)

∴ Hence the correct answer is j, -j.