CH 8, 4PP
$y[k] = \sum_{n=0}^{5} (7 - 3w[n])e^{-j\frac{2\pi kn}{6}}$
And,
$y[k] = \sum_{n=0}^{5} (7)e^{-j\frac{2\pi kn}{6}} - 3w[n]$
From orthogonal property, from equation (8.7) provided in the textbook,
$\sum_{k=0}^{N-1} a^k = \frac{1 - a^N}{1 - a}$
Form the orthogonal property, the DFT of the signal $y[k]$:
$Y[k] = 7\left(\frac{1 - e^{-j\frac{2\pi k}{6}}}{1 - e^{-j\frac{2\pi k}{6}}}\right) - 3W[k]$
To calculate the DFT of $y[n]$. Firstly, calculate $w[n]$ by taking Inverse
discrete Fourier transform $w[k]$. Then calculate DFT of $y[n]$ by substitute
the value of $w[n]$.
Enter the following code in MATLAB to show the output of the signal.
w=[1,0,j,0,-j,0];
w[n] = ifft(w);
y=7-3w[n];
Y[k]=39,0,-3j,0,+3j,0
The following is the output of the system $Y[k]$:
$Y[k] = 39, 0, -3j, 0, +3j, 0$
Therefore, the 6-point DFT $Y[k]$ is equal to $39, 0, -3j, 0, +3j, 0$.
