Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

DC Constraint

The DTFT at frequency $ \omega$ is given by

$\displaystyle W\left(\omega \right)=\sum _{n=-L}^{L}w\left(n\right)e^{-j\omega n}.$ (4.69)

By zero-phase symmetry,
$\displaystyle W\left(\omega \right)$ $\displaystyle =$ $\displaystyle h\left(0\right)+2\sum _{n=1}^{L}h\left(n\right)\cos \left(n\omega \right)$  
  $\displaystyle =$ $\displaystyle \left[\begin{array}{cccc}
1 & 2\cos \left(\omega \right) & \cdots & 2\cos \left(L\omega \right)\end{array}\right]\left[\begin{array}{c}
h\left(0\right)\\
h\left(1\right)\\
\vdots \\
h\left(L\right)\end{array}\right]$  
  $\displaystyle =$ $\displaystyle d\left(\omega \right)^{T}h.$  

So $ W\left(0\right)=1$ can be expressed as

$\displaystyle \zbox {d\left(0\right)^{T}h=1.}$ (4.70)


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA