I have an example problem that I am working on, but I am not sure how it got from the second to last step to the last step. Can someone demonstrate for me the steps, or point me in the right direction? Thanks
\(\displaystyle y(n)\, =\, \dfrac{x(n)\, -\, x(n\, -\, 1)}{2}\)
\(\displaystyle Y(w)\, =\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, X(w)\)
\(\displaystyle \begin{align}H(w)\, &=\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, \\ \\ &=\, \left(\sin\left(\dfrac{w}{2}\right)\right)\, e^{-jw/2}\, e^{j\pi /2}\end{align}\)
\(\displaystyle y(n)\, =\, \dfrac{x(n)\, -\, x(n\, -\, 1)}{2}\)
\(\displaystyle Y(w)\, =\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, X(w)\)
\(\displaystyle \begin{align}H(w)\, &=\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, \\ \\ &=\, \left(\sin\left(\dfrac{w}{2}\right)\right)\, e^{-jw/2}\, e^{j\pi /2}\end{align}\)
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