Help w/ exponential: H(w) = (1/2)(1 - e^{-jw}) = sin(w/2) e^{-jw/2} e^{j pi / 2}

superbrew

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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



\(\displaystyle y(n)\, =\, \dfrac{x(n)\, -\, x(n\, -\, 1)}{2}\)

\(\displaystyle Y(w)\, =\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, X(w)\)

\(\displaystyle \begin{align}H(w)\, &=\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, \\ \\ &=\, \left(\sin\left(\dfrac{w}{2}\right)\right)\, e^{-jw/2}\, e^{j\pi /2}\end{align}\)
 

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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



\(\displaystyle y(n)\, =\, \dfrac{x(n)\, -\, x(n\, -\, 1)}{2}\)

\(\displaystyle Y(w)\, =\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, X(w)\)

\(\displaystyle \begin{align}H(w)\, &=\, \dfrac{1}{2}\, \left(1\, -\, e^{-jw}\right)\, \\ \\ &=\, \left(\sin\left(\dfrac{w}{2}\right)\right)\, e^{-jw/2}\, e^{j\pi /2}\end{align}\)

Did you try to expand the last expression Euler's theorem and DeMoivre' s theorem?
 
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Thanks for replying.

I have not. It has been over 10 years since I have had to work with any of this stuff, and I am very rusty. I have looked into the 2 theorems that you posted, but I am not quite getting it.
 
Using Euler's formula, can you expand:

\(\displaystyle \displaystyle{e^{\frac{jw}{2}}}\)
 
\(\displaystyle \displaystyle{e^{\frac{jw}{2}}=cos{\frac{w}{2}}+jsin{\frac{w}{2}}}\) ?

Correct

Now expand (similarly):

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}}\)
 
Correct

Now expand (similarly):

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}}\)

I feel like I am close.....

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}=e^{\frac{jw}{2}}e^{\frac{pi}{2}}}\)

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}=jcos{\frac{w}{2}}-sin{\frac{w}{2}}}\)
 
I feel like I am close.....

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}=e^{\frac{jw}{2}}e^{\frac{pi}{2}}}\)

\(\displaystyle \displaystyle{e^{\left (\frac{jw}{2} \ + \ \frac{\pi}{2}\right )}=jcos{\frac{w}{2}}-sin{\frac{w}{2}}}\)
Yes you are close.

Now go back to the original problem and expand the RHS of (4).
 
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