Category: CSE348 Signals

In CSE 348 we assume a basic knowledge of complex numbers. So what we will do here is just a review rather than a complete…

DISCRETE PERIODIC FUNCTIONS AS FINITE DIMENSIONAL VECTORS As in continuous fourier series, we start with periodic discrete functions \(x[n]=x[n+N]\), where \(N\) is the period. We…

CONTINUOUS TIME FOURIER SERIES A CT fourier series is nothing but the expansion of a periodic function into a complex exponential basis. This is a…

SUM OF COMPLEX EXPONENTIALS Consider the geometric series $$\begin{eqnarray} D(u) &=& \sum_{n=-k}^{k} e^{-i u n} \\ &=& e^{-i u k}+e^{-i u (k-1)}+\ldots+e^{-i u (k-1)}+ e^{-i…

$$ \begin{eqnarray} x[n] = \frac{1}{2\pi} \int_{<2\pi>}X(\Omega)e^{i\Omega n}d\Omega\\\\ X(\Omega) = \sum_{k=-\infty}^{\infty}x[k]e^{-i\Omega k} \end{eqnarray}$$ DFT FOR PERIODIC CASE Consider a periodic signal $x[n]$, $x[n+N]=x[n]$. $$ \begin{eqnarray} X(\Omega)…

Theorem: $$ X(\omega)=\int_{t=-\infty}^{\infty}x(t) e^{-i \omega t} dt $$ $$ x(t)=\frac{1}{2\pi}\int_{\omega=-\infty}^{\infty}X(\omega) e^{i \omega t} d\omega $$ EXAMPLES Example 1: Find the FT of $x(t)=e^{-\alpha t}U(t)$ for…