Question

Let x(t) and $X\left(\omega \right)$ form a Fourier transform pair. The inverse Fourier transform of $X(-2\omega +1)$ is

• A. $e^{j\frac{t}{2}}x(-\frac{t}{2})$
• B. $\frac{1}{2}{e}^{j\frac{t}{2}}x(-\frac{t}{2})$
• C. 2$e^{j\frac{t}{2}}x(-\frac{t}{2})$
• D. $\frac{1}{2}{e}^{-j\frac{t}{2}}x(-\frac{t}{2})$

Answer 1

The correct option is B $\frac{1}{2}{e}^{j\frac{t}{2}}x(-\frac{t}{2})$

$x\left(t\right)⟷X\left(\omega \right)$

$e^{-jt}x\left(t\right)⟷X(\omega +1)$

$e^{j\frac{t}{2}}x(-t/2)⟷\frac{1}{|-\frac{1}{2}|}X(\frac{\omega }{(-\frac{1}{2})}+1)$

$e^{j\frac{t}{2}}x(-\frac{t}{2})⟷2X(-2\omega +1)$

$\frac{1}{2}{e}^{j\frac{t}{2}}x(-\frac{t}{2})⟷X(-2\omega +1)$