\sectionFourier Series
\sectionLinear Time-Invariant (LTI) Systems and Convolution signals and systems problems and solutions pdf
\noindent\textbf14. Z-transform of \(x[n]=n(1/3)^n u[n]\). \textitAns: \(\frac(1/3)z^-1(1-(1/3)z^-1)^2\), \(|z|>1/3\). signals and systems problems and solutions pdf
\subsection*Problem 3: Convolution Integral Given \(x(t) = e^-tu(t)\) and \(h(t) = u(t) - u(t-2)\), compute \(y(t) = x(t) * h(t)\). 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 \(A^2/2\).
\sectionLaplace Transform