The correct answer is $\boxed{{a_1} = 1,{a_2} = {W_6},{a_3} = W_6^2}$.
The decimation-in-time FFT algorithm is a recursive algorithm that can be used to compute the discrete Fourier transform (DFT) of a sequence. The algorithm works by dividing the sequence into smaller subsequences, which are then transformed using the DFT. The results of the DFT of the subsequences are then combined to compute the DFT of the original sequence.
The signal-flow graph shown in the figure is for a six-point decimation-in-time FFT algorithm. The coefficients $a_1$, $a_2$, and $a_3$ are the twiddle factors that are used to multiply the input sequence before it is transformed. The twiddle factors are chosen so that the DFT of the input sequence is computed correctly.
The twiddle factors for a six-point decimation-in-time FFT algorithm are given by
$$W_6 = \exp \left( { – \frac{{j2\pi }}{6}} \right)$$
$$W_6^2 = \exp \left( { – \frac{{j4\pi }}{6}} \right) = – W_6$$
$$W_6^3 = \exp \left( { – \frac{{j6\pi }}{6}} \right) = – W_6^2 = W_6$$
Therefore, the values of the coefficients $a_1$, $a_2$, and $a_3$ in terms of $W_6$ are
$$a_1 = 1$$
$$a_2 = W_6$$
$$a_3 = W_6^2$$