## Signal Windowing Help

1. The problem statement, all variables and given/known data

$p[n]=x[n]w[n]$, where $w[n]$ is the rectangular window.

$x[n]=\sum_{k=-∞}^{∞} δ[n-k]$

$w[n]= 1,$ for $-M≤n≤M;$
$= 0$ otherwise

1. What is $X(e^{jw})$?
2. What is $P(e^{jw})$ when...
M=1?
M=10?
$j$ is the imaginary number. (it's the same as $i$.)

2. Relevant equations
N/A

3. The attempt at a solution

$X(e^{jw})=2\pi\sum_{k=-∞}^{∞}δ[n-2\pi k]$

I found $X(e^{jw}$ fairly easily, but I can't find $P(e^{jw})$ for either case of M. The answers from the book are below. Can someone tell me how I solve for this? I don't see why $X(e^{jw})$ is so different from $P(e^{jw})$

$P(e^{jw})=e^{jw}+1+e^{-jw}=1+2cos(\omega)$ when M=1
$P(e^{jw})=\frac{sin(21 \omega/2}{\omega/2}$ when M=1
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