Systems Problems And Solutions Pdf: Signals And

\noindent\textbf13. Determine Fourier series coefficients for \(x(t)=2\cos(3\pi t)+\sin(4\pi t)\). \\ \textitAns: \(a_3=1, a_-3=1, a_4=-j/2, a_-4=j/2\), rest zero.

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

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\sectionLinear Time-Invariant (LTI) Systems and Convolution

\noindent\textbf14. Use differentiation in Z-domain: \(n a^n u[n] \leftrightarrow \fracaz^-1(1-az^-1)^2\). \noindent\textbf13

\subsection*Problem 10: Stability and Causality An LTI system has impulse response \(h(t) = e^\). Is it stable? Causal?

\sectionZ-Transform (Discrete-Time)

\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).