Solution to Question 1 of HW8
Solution to Question 2 of HW8
a)
$ x[n]=6\delta[n]+5 \delta[n-1]+4 \delta[n-2]+3 \delta[n-3]+2 \delta[n-4]+\delta[n-5]\,\! $
Using the 6-point DFT formula,
$ \begin{align} X[k] &=\sum_{n=0}^{5}\left(6\delta[n]+5 \delta[n-1]+4 \delta[n-2]+3 \delta[n-3]+2 \delta[n-4]+\delta[n-5]\right)e^{-j\frac{2\pi}{6}kn} \\ &= 6 + 5e^{-j\frac{2\pi}{6}k} + 4e^{-j\frac{2\pi}{6}2k} + 3e^{-j\frac{2\pi}{6}3k} + 2e^{-j\frac{2\pi}{6}4k} + e^{-j\frac{2\pi}{6}5k} \\ \end{align} $
b)
We use the 6-point inverse-DFT formula to obtain $ y_6[n] $
$ y_6[n]=\frac{1}{6}\sum_{k=0}^{5} W_6^{-2k} X[k] e^{j\frac{2\pi}{6}nk} = \frac{1}{6}\sum_{k=0}^{5} X[k] e^{j\frac{2\pi}{6}(n+2)k} \quad \text{where} \;\; W_N=e^{-j\frac{2\pi}{N}} $
If you compare this with the 6-point inverse-DFT of $ X[k] $
$ x_6[n]=\frac{1}{6}\sum_{k=0}^{5}X[k] e^{j\frac{2\pi}{6}nk} $
then, you will notice that $ y_6[n]=x_6[(n+2)\text{mod}6] $. Thus, it becomes
$ y_6[n]=4\delta[n]+3 \delta[n-1]+2\delta[n-2]+\delta[n-3]+6 \delta[n-4]+5\delta[n-5]\,\! $
(Producting $ W^{-2k} $ to $ X[k] $ yields circular-shifting to the left by 2 in the periodic discrete-time signal)
c)
$ h[n]=\delta[n]+\delta[n-1]+\delta[n-2]\,\! $
computing the circular convolution with $ x[n] $ and $ h[n] $,
$ \begin{align} y[n] =& x[n]\circledast_6 h[n] \\ =& \quad \{\quad 6,\quad 5,\quad 4,\quad 3,\quad 2,\quad 1\} \\ & +\! \{\quad 1,\quad 6,\quad 5,\quad 4,\quad 3,\quad 2\} \\ & +\! \{\quad 2,\quad 1,\quad 6,\quad 5,\quad 4,\quad 3\} \\ =& \quad \{\quad 9,\;\;12,\;\;\!15,\;\;12,\quad 9,\quad 6\} \\ =& 9\delta[n]+12\delta[n-1]+15\delta[n-2]+12\delta[n-3]+9\delta[n-4]+6\delta[n-5] \\ \end{align} $
d)
In order for the periodic repetition (with period N) of the usual convolution between x[n] and h[n] to be the same with the N-point circular convolution,
$ N \geq L+M-1 $ where L is the length of x[n] and M is the length of h[n].
Therefore, $ N\geq8 $.
Solution to Question 3 of HW8
a)
$ \begin{align} X(\omega)&=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} \\ &=\sum_{n=-\infty}^{\infty}(2\delta [n]+6\delta [n-1]-\delta [n-2])e^{-j\omega n} \\ &=2+6e^{-j\omega}-e^{-j2\omega} \end{align} $
$ \begin{align} Y(\omega)&=\sum_{n=-\infty}^{\infty}y[n]e^{-j\omega n} \\ &=\sum_{n=-\infty}^{\infty}x[-n]e^{-j\omega n} \\ &=\sum_{n=-\infty}^{\infty}(2\delta [-n]+6\delta [-n-1]-\delta [-n-2])e^{-j\omega n} \\ &=2+6e^{j\omega}-e^{j2\omega} \end{align} $
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