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Lecture 5: Ideal and Practical Filters

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Schedule :: Fourier Series Formulas :: Nonperiodic Functions :: Useful Functions :: Simple Functions :: Fourier Transforms :: Properties :: System Analysis :: Signal Distortion :: Filters :: Parseval's Formulas

The ideal low-pass filter allows all signals below some cutoff of $W$ rad/sec to pass through undistorted, while completely cutting off all other frequencies. This is sometimes said to be the ideal "brick wall'' filter. If $f(t)$ is strictly bandlimited to $W$ rad/sec, then the output is simply delayed:

\begin{displaymath}y(t) = f(t-t_d) \end{displaymath}

The ideal filter has

\begin{displaymath}\vert H(\omega)\vert = \rect(\omega/2W) \quad\quad \angle H(\omega) =
e^{-j\omega t_d}

The impulse response of the filter is then

\begin{displaymath}h(t) = \Fc^{-1}\left[\rect(\omega/(2W)) e^{-j\omega t_d}\right] =
\frac{W}{\pi} \sinc(W(t-t_d))

Plot $h(t)$ . Observe that it is noncausal , and hence physically nonrealizable. We cannot build an ideal lowpass filter, or an ideal bandpass either, for that matter. We could get pretty close, but creating a system with a large $t_d$ , so that most of the sinc function would be included. One approach to determine the effect of this truncation of the impulse response

\begin{displaymath}\hhat(t) = h(t)u(t) \end{displaymath}

is to use the properties of F.T.

\begin{displaymath}\Hhat(\omega) = \frac{1}{2\pi}H(\omega)*U(\omega) = \frac{1}{2\pi}
H(\omega)*(\pi \delta(\omega) + \frac{1}{j\omega})\end{displaymath}

The effect is to smear the response by the convolution.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Lecture 5: Ideal and Practical Filters. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: This work is licensed under a Creative Commons License Creative Commons License