A rectangular voltage pulse of height 9.00 V and width τ=0.200 ms, centered at the…

A rectangular voltage pulse of height \(9.00\,\mathrm{V}\) and width \(\tau=0.200\,\mathrm{ms}\), centered at the origin, is repeated in two different periodic trains. The first train \(x_1\) has period \(T_1=1.00\,\mathrm{ms}\); the second train \(x_2\) has period \(T_2=2.00\,\mathrm{ms}\). Explicitly, on a period symmetric about the origin, \[ x_i(t)=\begin{cases} 9.00\,\mathrm{V}, & |t|\le \tau/2,\\ 0, & \tau/2<|t|<T_i/2, \end{cases} \qquad i=1,2, \] each extended periodically with its own period \(T_i\). Time \(t\) is in seconds. Use the complex pair \[ a_k^{(i)}=\frac{1}{T_i}\int_{-T_i/2}^{T_i/2}x_i(t)\,e^{-\mathrm{j}k\omega_i t}\,\mathrm{d}t,\qquad \omega_i=\frac{2\pi}{T_i}, \] and \(\operatorname{sinc}(u)=\sin(\pi u)/(\pi u)\) with \(\operatorname{sinc}(0)=1\). (a) Derive a closed-form expression for \(a_k^{(i)}\) in terms of \(\tau\), \(T_i\), and \(k\), written with \(\operatorname{sinc}\), valid for \(i=1,2\) and every integer \(k\). Evaluate \(a_0^{(1)}\), \(a_{\pm 1}^{(1)}\), \(a_{\pm 5}^{(1)}\) and \(a_0^{(2)}\), \(a_{\pm 1}^{(2)}\), \(a_{\pm 2}^{(2)}\), \(a_{\pm 10}^{(2)}\) in volts. (b) Prove the exact relation \(a_{2\ell}^{(2)}=\frac12 a_\ell^{(1)}\) for every integer \(\ell\). Interpret the odd-index coefficients of \(x_2\) as additional spectral samples that appear when the period is doubled while the pulse shape is held fixed. (c) Using Parseval, compute the average of \(x_1^2\) over \(T_1\) and the average of \(x_2^2\) over \(T_2\) by piecewise time-domain integration (do not sum infinite series). Explain why the two averages are unequal even though each period contains an identical pulse, and compute their ratio.

Periodic repetition of a fixed pulse samples the pulse’s transform at the harmonic grid \(2\pi k/T\) and scales by \(1/T\). Lengthening \(T\) densifies the grid and reduces every sample.