\sectionZ-Transform (Discrete-Time)
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\subsection*Solution First term: \(e^-2tu(t) \leftrightarrow \frac1s+2\), \(\textRe(s) > -2\). \\ Second term: \(e^3tu(-t) \leftrightarrow -\frac1s-3\), \(\textRe(s) < 3\). \\ Thus \(X(s) = \frac1s+2 - \frac1s-3 = \frac-5(s+2)(s-3)\), ROC: \(-2 < \textRe(s) < 3\). signals and systems problems and solutions pdf
\subsection*Problem 9: Nyquist Rate A signal \(x(t) = \textsinc(100t) + \textsinc^2(50t)\). Find the Nyquist sampling rate.
\noindent\textbf13. Use Euler formulas and compare with exponential FS: \(x(t)=\sum a_k e^jk\omega_0 t\) with \(\omega_0=\pi\) (fundamental). \subsection*Problem 9: Nyquist Rate A signal \(x(t) =
\subsection*Solution Fundamental frequency \(\omega_0 = \pi\). \\ \(a_0 = \frac12\int_-0.5^0.5 1 dt = 0.5\). \\ \(a_n = \frac22\int_-0.5^0.5 \cos(n\pi t) dt = \frac2\sin(n\pi/2)n\pi\). \\ \(b_n=0\) (even symmetry). Hence \[ x(t) = 0.5 + \sum_n=1^\infty \frac2\sin(n\pi/2)n\pi \cos(n\pi t). \]
\section*Additional Problems (Brief Solutions) Use Euler formulas and compare with exponential FS:
\subsection*Problem 7: Region of Convergence Find the Laplace transform and ROC of \(x(t) = e^-2tu(t) + e^3tu(-t)\).