Let an input 𝑥[𝑛] having discrete time Fourier transform

𝑋(𝑒^𝑗𝛺) = 1 − 𝑒^−𝑗𝛺 + 2𝑒^−3𝑗𝛺 be passed through an LTI system. The frequency response of the LTI system is \(\rm H(e^{j\Omega)}=1-\frac{1}{2}e^{-j2\Omega}\). The output 𝑦[𝑛] of the system is

The correct option is 3

Given:  𝑦[𝑛]  = x[n] ∗ h[n] 

We know that the convolution in one domain results in convolution in another domain.

Y(𝑒𝑗𝛺) =𝑋(𝑒𝑗𝛺) H(𝑒𝑗𝛺) 

= (1 − 𝑒−𝑗𝛺 + 2𝑒−3𝑗𝛺)(\(1-\frac{1}{2}e^{-j2\Omega}\))

=1 - 𝑒−𝑗𝛺 + 2.5𝑒−3𝑗𝛺 - 0.5𝑒−2𝑗𝛺 - 𝑒−5𝑗𝛺

Taking IDFT we have:

y[n] =\(\rm \delta[n]-\delta[n-1]-\frac{1}{2}\delta[n-2]+\frac{5}{2}\delta[n-3]-\delta[n-5]\)