Signals And Systems Problems And Solutions Pdf May 2026
\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 \(A^2/2\).
\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 \[ x(t) = 0.5 + \sum_n=1^\infty \frac2\sin(n\pi/2)n\pi \cos(n\pi t). \]
\noindent\textbf12. Using \(t^n e^-atu(t) \leftrightarrow \fracn!(s+a)^n+1\). signals and systems problems and solutions pdf
\subsection*Problem 7: Region of Convergence Find the Laplace transform and ROC of \(x(t) = e^-2tu(t) + e^3tu(-t)\).
\subsection*Problem 10: Stability and Causality An LTI system has impulse response \(h(t) = e^\). Is it stable? Causal? \subsection*Solution The signal is periodic, so it has
\subsection*Problem 3: Convolution Integral Given \(x(t) = e^-tu(t)\) and \(h(t) = u(t) - u(t-2)\), compute \(y(t) = x(t) * h(t)\).
\noindent\textbf11. \(x[n]=\delta[n]+\delta[n-1]+\delta[n-2]\), \(h[n]=\delta[n]+\delta[n-1]\). Convolution gives \(y[n]=\delta[n]+2\delta[n-1]+2\delta[n-2]+\delta[n-3]\). \\ \(b_n=0\) (even symmetry)
\sectionFourier Series
