Question is given like this
I did it like this here i have taken for one sided frequecny response
Method1
But in the solution they have calculated it with two sided filter
Method 2
Now clearly i am getting the different answer that is i am getting half of answer which is given in the solution of the book.
Now my doubt is which method is right and why? mathematical and theoretical reference if you provide will be helpful?
First of all, please post your questions using mathjax. Scanned images are unacceptable and may get your question closed before getting an answer.
Now, to your problem. The correct answer is yours. However, both you and "they" have computed the frequency response of the filter incorrectly. In particular, you have both ignored the periodicity of the discrete-time Fourier transform (DTFT). Your "solution" implies that the DTFT of $e^{i\pi n}$ is $\delta(\omega - \pi)$. This is not correct. It actually holds $$ \mathcal{F}(e^{i \pi n})=\sum_{k=-\infty}^{\infty}\delta(\omega - \pi - 2\pi k). $$
Using this formula you can now see that the correct frequency response is as shown in the figure (the dashed lines delineate the period $\pi<\omega<\pi$; the Fourier transform is periodic and extends over all $\omega \in \mathbb{R}$).
The other solution is based on expressing $(-1)^n$ as $\cos(\pi n)=\frac{1}{2} e^{i \pi n} + \frac{1}{2} e^{-i \pi n}$. I will leave it to you to verify that the DTFT of $\cos({\pi n})$ is also as shown above.
