Calculate y(t) = x(1 – t) + x(– 1

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Direction: A continuous time signal X(t) has Fourier transform

X(jω) =	jω³
	1 + ω²

Calculate y(t) = x(1 – t) + x(– 1 – t)

1. – 2ω³ sin ω
  1 + ω²
2. j2ω³ cos ω
  1 + ω²
3. – j2ω³ cos ω
  1 + ω²
4. 2ω³ sin ω
  1 + ω²

Correct Option: C

If x(t) ←F.T.→ X(jω)
then
x(– t) ←F.T.→ X(– jω)
then
x(– t + 1) ←F.T.→ e^jω X(– jω)
then
x(– t – 1) ←F.T.→ e^{- jω} X(– jω)
Therefore,
x(1 – t) + x(– t – 1) ←F.T.→ e^{- jω} X(– jω) + e^jω X(– jω)
or

Y(jω) = e^–jω	(– jω)³	+	e^jω (– jω)³
	1 + (- ω)²		1 + (- ω)²

= – e^–jω	jω³	-	e^jω jω³
	1 + ω²		1 + ω²

=	– jω²	2		e^jω + e^–jω
	1 + ω²			2

=	– jω² 2 cos ω
	1 + ω²

=	– 2jω² cos ω
	1 + ω²

Hence, alternative (C) is the correct choice.