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)