Question 7.2: DFT of a periodically repeated rectangular pulse 2 Find the ......
DFT of a periodically repeated rectangular pulse 2
Find the DFT of x[n] = (u[n − n_0 ] − u[n − n_1]) ∗ δ_{N0} [n], 0≤ n_1 − n_0 ≤ N_0.
From Example 7.1 we already know the DFT pair
If we apply the time-shifting property
\mathrm{x}\left[n-n_y\right] \xleftrightarrow[N]{\mathcal{D F} \mathcal{T}} \mathrm{X}[k] e^{-j 2 \pi k n_y / N}to this result we have
Now, let n_0 = n_y and let n_1 = n_y + n_x .
Consider the special case in which n_0 + n_1 = 1. Then
\mathrm{u}\left[n-n_0\right]-\mathrm{u}\left[n-n_{\mathrm{l}}\right] * \delta_{N_0}[n] \xleftrightarrow[N_0]{\mathcal{D F} \mathcal{T}} \frac{\sin \left(\pi k\left(n_1-n_0\right) / N_0\right)}{\sin \left(\pi k / N_0\right)}, n_0+n_1=This is the case of a rectangular pulse of width n_1 − n_0 = 2n_1 −1, centered at n = 0. This is analogous to a continuous-time, periodically repeated pulse of the form
T_0 rect( t/ w) ∗δ_{T_0} (t).Compare their harmonic functions.
The harmonic function of T_0 \operatorname{rect}(t / w) * \delta_{T_0}(t) is a sinc function. Although it may not yet be obvious, the harmonic function of u[n − n_0 ] − u[n − n_1] is a periodically repeated sinc function.