Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for whịch the signal-flow graph corresponding to {X}[1] is shown in the figure. Let W_{6}=\exp (-\frac{j 2 \pi}{6}) . 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?

Correct Answer is : a1=1,a2=W6,a3=W62

a_{1}=1, a_{2}=W_{6}^{2}, a_{3}=W_{6}

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

GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 52 of 65

Marks: +1, -0

Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for whịch the signal-flow graph corresponding to

{X}[1]

is shown in the figure. Let

W_{6}=\exp (-\;{j 2 π}/{6}) .

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?

Solution:

{x}( {k})=∑↙{ {n}=0}^{ {N}-1} {x}( {n}) {W}_{ {N}}^{ {kn}}

We are obtaining

{X}(1)

correctly

∴ {k}=1

∴ {x}(1)= {x}(0)+ {x}(1) {W}_{6}^{1}+ {x}(2) {w}_{6}^{2}+ {x}(3) {w}_{6}^{3}+ {x}(4) {w}_{6}^{4}+ {x}(5) {w}_{6}^{5}

We know that

w_{ {N}}^{k+\;{N}/{2}}=-W_{N}^{K}

∴ W_{6}^{3}=-W_{6}^{0}=-1

w_{6}^{4}=-W_{6}^{1}

w_{6}^{5}=-W_{6}^{2}

¿. comparing with given graph

a_{1}=1, a_{2}=W_{6,} \;\; a_{3}=W_{6}^{2}

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