Question

Uma outra forma de representar a Transformada de Fourier Discreta é através da seguinte equação:

X left square bracket k right square bracket space equals space sum from n equals 0 to N minus 1 of x left square bracket n right square bracket W subscript N to the power of k n end exponent

Em que W subscript N space equals space e to the power of negative j fraction numerator 2 straight pi over denominator N end fraction end exponent. Para calcular todos os termos desta TFD é necessário realizar N² operações de adição e multiplicação, tal complexidade computacional não é aceitável em grandes quantidades de amostras, devido ao número exacerbado de operações. Para tanto, utiliza-se de um algoritmo computacional para viabilizar a determinação da TFD, chamado de Transformada Rápida de Fourier (FFT).

A sequência pode ser decomposta em duas sequências que contém as amostras pares e ímpares, a TFD será:

X left square bracket k right square bracket space equals space sum from n equals 0 to N over 2 minus 1 of x subscript 1 left square bracket n right square bracket W subscript N to the power of 2 k n end exponent plus W subscript N to the power of k sum from n equals 0 to N over 2 minus 1 of x subscript 2 left square bracket n right square bracket W subscript N to the power of 2 k n end exponentX left square bracket k right square bracket space equals X subscript 1 left square bracket k right square bracket plus W subscript N to the power of k X subscript 2 left square bracket k right square bracketX left square bracket k right square bracket space equals X subscript 1 open square brackets k minus N over 2 close square brackets plus W subscript N to the power of k X subscript 2 open square brackets k minus N over 2 close square brackets

Seja um sistema com 12 amostras, qual será a equação da oitava amostra?

Escolha uma:
a. X left square bracket 0 right square bracket equals X subscript 1 left square bracket 0 right square bracket space plus space W subscript 8 to the power of 0 X subscript 2 left square bracket 0 right square bracket.
b. X left square bracket 8 right square bracket equals X subscript 1 left square bracket 8 right square bracket space plus space W subscript 8 to the power of 12 X subscript 2 left square bracket 8 right square bracket.
c. X left square bracket 7 right square bracket space equals X subscript 1 open square brackets 1 close square brackets plus W subscript 12 to the power of 7 X subscript 2 open square brackets 1 close square brackets.
d. X left square bracket 7 right square bracket equals X subscript 1 left square bracket 7 right square bracket space plus space W subscript 12 to the power of 7 X subscript 2 left square bracket 7 right square bracket.
e. X left square bracket 9 right square bracket equals X subscript 1 left square bracket 5 right square bracket space plus space W subscript 8 to the power of 9 X subscript 2 left square bracket 5 right square bracket.

Answer 1

b.X[7] = X1[1]+ W12^7 X2[1].

Answer 2

A resposta correta é .X[7] = X1[1]+ W12^7 X2[1].