Question

A signal x(t) has Fourier transform of $X\left(\omega \right)$. Which of the following is the Fourier transform of y(t) = x(-3t -4) in terms of $X\left(\omega \right)$ ?

• A. $\frac{1}{3}{e}^{-j\frac{4}{3}\omega }X\left(\frac{\omega }{3}\right)$
• B. $\frac{1}{3}{e}^{j\frac{4}{3}\omega }X\left(\frac{\omega }{3}\right)$
• C. $\frac{1}{3}{e}^{-j\frac{4}{3}\omega }X\left(\frac{-\omega }{3}\right)$
• D. $\frac{1}{3}{e}^{j\frac{4}{3}\omega }X\left(\frac{-\omega }{3}\right)$

Answer 1

The correct option is D $\frac{1}{3}{e}^{j\frac{4}{3}\omega }X\left(\frac{-\omega }{3}\right)$

$x\left(t\right)\stackrel{F.T}{⟷}X\left(\omega \right)$

$x(t-4)\stackrel{F.T}{⟷}{e}^{-j4\omega }X\left(\omega \right)$

$x(3t-4)\stackrel{F.T}{⟷}\frac{1}{3}{e}^{-j\frac{4}{3}\omega }X\left(\frac{\omega }{3}\right)$

$x(-3t-4)\stackrel{F.T}{⟷}\frac{1}{3}{e}^{j\frac{4}{3}\omega }X\left(\frac{-\omega }{3}\right)$