And Solutions Pdf — Signals And Systems Problems

\subsection*Problem 5: Fourier Transform of a Rectangular Pulse Compute the Fourier transform of \(x(t) = \textrect(t/T) = 1\) for \(|t| < T/2\), 0 otherwise.

\subsection*Problem 9: Nyquist Rate A signal \(x(t) = \textsinc(100t) + \textsinc^2(50t)\). Find the Nyquist sampling rate. 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 \(A^2/2\). \subsection*Solution The signal is periodic, so it has

\noindent\textbf15. Check: Input \(x(t-\tau)\) gives \(x(t-\tau)\cos t\), but for time-invariance we need \(x(t-\tau)\cos(t-\tau)\). \noindent\textbf15

\noindent\textbf15. Is \(y(t)=x(t)\cos(t)\) LTI? \textitAns: No, time-varying.

\noindent\textbf12. Using \(t^n e^-atu(t) \leftrightarrow \fracn!(s+a)^n+1\).