A signal may be a complex function of time. The most important complex signal we will investigate is the complex exponential function \begin{eqnarray} f(t) =…
Comments closedCategory: 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…
Comments closedLinearity If $FT[x_1(t)] = X_1 (\omega)$ and $FT[x_2(t)] = X_2 (\omega)$, and if $a_1, a_2 \in \mathbb{R}$, then $FT[a_1x_1(t) + a_2x_2(t)] = a_1X_1 (\omega) +…
Comments closedVector Spaces Merriam-webster dictionary defines a vector as “….a quantity that has magnitude and direction and that is commonly represented by a directed line segment…
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DISCRETE TIME FOURIER TRANSFORMS Similarly to continuous-time fourier transformations, discrete-time fourier transforms is the expansion of a discrete signal $X[n]$ into a complex exponential basis…
Comments closedCONTINUOUS TIME FOURIER SERIES A CT fourier series is nothing but the expansion of continuous function $x(t)$ into the complex exponential basis. This is a…
Comments closedSUM OF COMPLEX EXPONENTIALS Consider the geometric series $$\begin{eqnarray} D(u) &=& \sum_{n=-k}^{k} e^{-i u n} \\ &=& e^{-i u k}+e^{-i u (k-1)}+\ldots+e^{-i u (k-1)}+ e^{-i…
Comments closed$$ \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)…
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