DC Constraint

Chapter Contents:

Search Spectral Audio Signal Processing

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

The DTFT at frequency $\omega$ is given by

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

Using the symmetry,

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

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

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

Previous: Positive Window Sample Constraint
Next: Sidelobe Specification

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in