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Lecture 2: FIR Filters

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Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters

Linear Phase FIR Filters

We have mentioned several times that FIR filters can have linear phase. Now we will show why. Suppose that $h[k] = h[n-k]$ (the coefficients are symmetric. In particular, consider the example
\begin{displaymath}h[k] = h[0] \delta[k] + h[1] \delta[k-1] + h[2] \delta[k-2] +
h[3]\delta[k-3] + h[4]\delta[k-4] + h[5]\delta[k-5]
\end{displaymath}

then
\begin{displaymath}\begin{aligned}
H(e^{j\Omega}) &= h[0] + h[1]e^{-j\Omega} + h...
...cos((3/2)\Omega) + 2h[2] \cos((1/2)\Omega)\right)
\end{aligned}\end{displaymath}

Then
\begin{displaymath}\arg H(e^{j\Omega}) = -5/2 \Omega,
\end{displaymath}

a linear function of phase.

We can pull a similar stunt with antisymmetry: $h[k] =-h[n-k]$ .

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 18). Lecture 2: FIR Filters. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node10.html. This work is licensed under a Creative Commons License Creative Commons License