DFT of a periodically repeated rectangular pulse 1

Find the DFT of x[n] = (u[n] − u[n − n $_x$ ])∗ δN $_0$ [n], 0 ≤ n $_x$ ≤N $_0$ using N $_0$ as the representation time.

Question

DFT of a periodically repeated rectangular pulse 1

Find the DFT of x[n] = (u[n] − u[n − n[latex]_x[/latex]])∗ δN[latex]_0[/latex] [n], 0 ≤ n[latex]_x[/latex] ≤N[latex]_0[/latex] using N[latex]_0[/latex] as the representation time.

Accepted Answer

Summing the finite-length geometric series,

[latex]\left(\mathrm{u}[n]-\mathrm{u}\left[n-n_x ight] ight) * \delta_{N_0}[n] \xleftrightarrow[N_0]{\mathcal{D} \mathcal{F} \mathcal{T}} \sum_{n=0}^{n_x-1} e^{-\jmath 2 \pi k n / N_0}[/latex]

[latex]\begin{aligned} & \left(\mathrm{u}[n]-\mathrm{u}\left[n-n_x ight] ight) * \delta_{N_0}[n] \xleftrightarrow[N_0]{\mathcal{D} \mathcal{F} \mathcal{T}} \frac{1-e^{-j 2 \pi k n_x / N_0}}{1-e^{j 2 \pi k n / N_0}}=\frac{e^{-j \pi k n_x / N_0}}{e^{-j \pi k / N_0}} \frac{e^{j \pi k n_x / N_0}-e^{-j \pi k n_x / N_0}}{e^{-j \pi k / N_0}-e^{-j \pi k / N_0}} \ & \left(\mathrm{u}[n]-\mathrm{u}\left[n-n_x ight] ight) * \delta_{N_0}[n] \xleftrightarrow[N_0]{\mathcal{D} \mathcal{F} \mathcal{T}} e^{-j \pi k\left(n_x-1 ight) / N_0} \frac{\sin \left(\pi k n_x / N_0 ight)}{\sin \left(\pi k / N_0 ight)}, 0 \leq n_x \leq N_0 \end{aligned}[/latex]

Question 7.1: DFT of a periodically repeated rectangular pulse 1 Find the ......

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