Signals And Systems Problems And Solutions Pdf «2024-2026»

\section*Additional Problems (Brief Solutions)

\sectionSystem Properties

\noindent\textbf12. Find Laplace transform of \(t e^-2tu(t)\). \textitAns: \(1/(s+2)^2\), ROC \(\textRe(s)>-2\). signals and systems problems and solutions pdf

\subsection*Solution The signal is periodic, so it has infinite energy but finite average power. \[ P = \lim_T\to\infty \frac1T \int_-T/2^T/2 |x(t)|^2 dt = \frac1T_0 \int_0^T_0 A^2 \cos^2(2\pi f_0 t + \theta) dt \] Using \(\cos^2(\cdot) = \frac1+\cos(2\cdot)2\), the integral of the cosine term over one period is zero: \[ P = \fracA^2T_0 \int_0^T_0 \frac12 dt = \fracA^22. \] Hence \(x(t)\) is a power signal with power \(A^2/2\). -2\). \subsection*Solution The signal is periodic