Question

$\text{Consider the signal given below;}x\left[n\right]=[1,-2,\underset{\uparrow }{3},2,2]\text{If}X\left(e^{j\omega }\right)\text{is the discrete Fourier transform of x[n], then the value of}{\int }_{-\pi }^{\pi }X\left(e^{j\omega }\right)e^{-j\omega }d\omega \text{is\_\_\_\_\_\_\_\_}$

• A. -12.57

Answer 1

The correct option is A -12.57
$\text{We know that, from the definition of DTFT,}$

$X\left(e^{j\omega }\right)={\sum }_{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}$

$\text{where,}x\left[n\right]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X\left(e^{j\omega }\right)e^{j\omega n}d\omega$

$\text{by putting n = -1}$

$\therefore x[-1]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X\left(e^{j\omega }\right)e^{-j\omega }d\omega$

$\therefore {\int }_{-\pi }^{\pi }X\left(e^{j\omega }\right)e^{-j\omega }d\omega =2\pi x[-1]$

$=2\pi [-2]=-4\pi$

$=-12.57$