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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 $W_{6}=exp\left(-\:\dfrac{j2\pi}{6}\right).$ 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?

  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. $a_{1}=-1,a_{2}=W_{6}^{2},a_{3}=W_{6}$
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