- #1

DSRadin

- 12

- 1

**1. Given: The DTFT over the interval [itex] |ω|≤\pi, X\left ( e^{jω}\right )= cos\left ( \frac{ω}{2}\right ) [/itex]**

Find: [itex] x(n) [/itex]

b]3. Summary: My intuition tells me that a sinusoid in the frequency domain should pop out an impulse in the time domain.

BUT when running the synthesis equation where I normally end up with an orthogonality situation landing me a pair of delayed impulses, I end up with a delay of (n-1/2) and (n+1/2). As 'n' is a discrete time integer sample there is no data for (n-1/2) and (n+1/2) so I would expect the result to fall between samples and be zero.

Running all the way through the synthesis results in the following expression:

[itex] x(n) = \frac{1}{\pi}\left [ \frac{2sin\left ( \pi \left ( n+\frac{1}{2} \right ) \right )}{n+\frac{1}{2}} + \frac{2sin\left ( \pi \left ( n-\frac{1}{2} \right ) \right )}{n-\frac{1}{2}} \right ] [/itex]

Find: [itex] x(n) [/itex]

**2. Necessary Equations: IDTFT synthesis equation: [itex] x(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}X\left ( e^{jω} \right ) e^{j\omega n}d\omega[/itex]**

Euler's Identity: [itex] cos\left ( \omega \right ) = \frac{e^{j\omega n} + e^{-j\omega n}}{2}[/itex]

Euler's Identity: [itex] cos\left ( \omega \right ) = \frac{e^{j\omega n} + e^{-j\omega n}}{2}[/itex]

b]3. Summary: My intuition tells me that a sinusoid in the frequency domain should pop out an impulse in the time domain.

BUT when running the synthesis equation where I normally end up with an orthogonality situation landing me a pair of delayed impulses, I end up with a delay of (n-1/2) and (n+1/2). As 'n' is a discrete time integer sample there is no data for (n-1/2) and (n+1/2) so I would expect the result to fall between samples and be zero.

Running all the way through the synthesis results in the following expression:

[itex] x(n) = \frac{1}{\pi}\left [ \frac{2sin\left ( \pi \left ( n+\frac{1}{2} \right ) \right )}{n+\frac{1}{2}} + \frac{2sin\left ( \pi \left ( n-\frac{1}{2} \right ) \right )}{n-\frac{1}{2}} \right ] [/itex]

Which is great (I guess) - except that this expression results in a sinc function centered on zero.

Any guidance is appreciated!

I have a faint suspicion that the ω/2 would smear the time domain signal - but I don't quite have grasp enough of the theory to prove it.

Thanks for your help.

-DR