The GATE Grind

GATE 2019 EC – Question 38

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to $X[1]$ is shown in the figure. Let $W_6=\exp\left(-\frac{j2\pi}{6}\right)$. In the figure, what should be the values of the coefficients $a_1,a_2,a_3$ in terms of $W_6$ so that $X[1]$ is obtained correctly?

Diagram for GATE 2019 EC question 38
  1. $a_1=-1,a_2=W_6,a_3=W_6^2$
  2. $a_1=1,a_2=W_6^2,a_3=W_6$
  3. $a_1=1,a_2=W_6,a_3=W_6^2$
  4. $a_1=-1,a_2=W_6^2,a_3=W_6$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $a_1=1,a_2=W_6,a_3=W_6^2$

Explanation

For $X[1]$ the three 2-point sub-transforms are combined with twiddle factors $W_6^{0}$, $W_6^{1}$ and $W_6^{2}$ for the groups of samples $\{x[0],x[3]\}$, $\{x[1],x[4]\}$ and $\{x[2],x[5]\}$. So $a_1=1$, $a_2=W_6$ and $a_3=W_6^2$.