(Solved): Problem 1. Fourier Transform of the Sin Signal Obtain the Fourier Transform X ( ) of signal 1 xt ...

Problem 1. Fourier Transform of the Sin Signal Obtain the Fourier Transform X ( ) ? of signal 1 xt t ( ) sin = ? by two different approaches 1a) 20 points: Using the Fourier Transform of the complex exponential { }1 1 2( ) j t F e ? = ? ?? ? ? 1b) 20 points: Using the Time Shift Theorem { } { } 0 0 ( ) () j t F xt t F xt e? ? ? = and the knowledge that: • { } 1 11 F t cos ( ) ( ) ? ?? ? ? ?? ? ? = ++ ? • sin cos( / 2) cos ( / 2 ) ? ?? ? ?? 1 1 tt t = ?= ? [ 1 1 ] 1c) 10 points: Obtain the expression for magnitude X ( ) ? and angle?X ( ) ? . Sketch X ( ) ? ?X ( ) ? . Discuss the significance of these plots in terms of harmonics.

 

Problem 1. Fourier Transform of the Sin Signal Obtain the Fourier Transform \( X(\omega) \) of signal \( x(t)=\sin \omega_{1} t \) by two different approaches 1a) 20 points: Using the Fourier Transform of the complex exponential \( F\left\{e^{j \omega_{1} t}\right\}=2 \pi \delta\left(\omega-\omega_{1}\right) \) 1b) 20 points: Using the Time Shift Theorem \( F\left\{x\left(t-t_{0}\right)\right\}=F\{x(t)\} e^{-j o t_{0}} \) and the knowledge that: - \( \quad F\left\{\cos \omega_{1} t\right\}=\pi \delta\left(\omega+\omega_{1}\right)+\pi \delta\left(\omega-\omega_{1}\right) \) - \( \sin \omega_{1} t=\cos \left(\omega_{1} t-\pi / 2\right)=\cos \left[\omega_{1}\left(t-\pi / 2 \omega_{1}\right)\right] \) 1c) 10 points: Obtain the expression for magnitude \( |X(\omega)| \) and angle \( \angle X(\omega) \). Sketch \( |X(\omega)| \) \( \angle X(\omega) \). Discuss the significance of these plots in terms of harmonics.