| Line 12: | Line 12: | ||
\end{align}</math> | \end{align}</math> | ||
| − | b) | + | b) Using Euler Formula, we have |
| + | |||
| + | <math>\begin{align} | ||
| + | x[n] &= e^{\frac{j\pi n}{3}}(\frac{ e^{\frac{j\pi n}{6}} + e^{-\frac{j\pi n}{6}} }{2}) \\ | ||
| + | &= \frac{1}{2}e^{\frac{j\pi n}{2}} + \frac{1}{2}e^{\frac{j\pi n}{12}} | ||
| + | \end{align}</math> | ||
| + | |||
| + | Observing that <math>x[n]</math> has fundamental period <math>N=12</math>. Using IDFT, we have | ||
| + | |||
| + | <math>\begin{align} | ||
| + | x[n] &= \frac{1}{N}\sum_{n=0}^{N-1}e^{\frac{j2\pi nk}{N}} \\ | ||
| + | \frac{1}{2}e^{\frac{j\pi n}{2}} + \frac{1}{2}e^{\frac{j\pi n}{12}} &= \frac{1}{12}\sum_{n=0}^{11}e^{\frac{j2\pi nk}{12}} | ||
| + | \end{align}</math> | ||
| + | |||
| + | For <math>k=0,1,...,N-1</math> | ||
| + | |||
| + | |||
---- | ---- | ||
==Question 2== | ==Question 2== | ||
Revision as of 14:27, 13 October 2011
Homework 4, ECE438, Fall 2011, Prof. Boutin
Question 1
a) For $ k=0,1,...,N-1 $
$ \begin{align} X_N(k) &= \sum_{k=0}^{N-1}x[n]e^{-\frac{j2\pi nk}{N}} \\ &= x[0]e^{-\frac{j2\pi 0\cdot k}{N}} \\ &= 1 \end{align} $
b) Using Euler Formula, we have
$ \begin{align} x[n] &= e^{\frac{j\pi n}{3}}(\frac{ e^{\frac{j\pi n}{6}} + e^{-\frac{j\pi n}{6}} }{2}) \\ &= \frac{1}{2}e^{\frac{j\pi n}{2}} + \frac{1}{2}e^{\frac{j\pi n}{12}} \end{align} $
Observing that $ x[n] $ has fundamental period $ N=12 $. Using IDFT, we have
$ \begin{align} x[n] &= \frac{1}{N}\sum_{n=0}^{N-1}e^{\frac{j2\pi nk}{N}} \\ \frac{1}{2}e^{\frac{j\pi n}{2}} + \frac{1}{2}e^{\frac{j\pi n}{12}} &= \frac{1}{12}\sum_{n=0}^{11}e^{\frac{j2\pi nk}{12}} \end{align} $
For $ k=0,1,...,N-1 $
Question 2
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