System Diagram of the Running-Sum Filter Bank

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**Figure 9.15:** DFT Filter Bank.
$\begin{figure}\input fig/BPFB.pstex_t \end{figure}$

Figure 9.15 shows the system diagram of the complete -channel filter bank constructed using length FIR running-sum lowpass filters. The th channel computes:

$\displaystyle y_k(n)$	$\displaystyle =$	$\displaystyle (h\ast x_k)(n) = \sum_{m=0}^{N-1}h(m)x_k(n-m)$
	$\displaystyle =$	$\displaystyle (x_k\ast h)(n) = \sum_{m=n-(N-1)}^{n}x_k(m)h(n-m)$
	$\displaystyle =$	$\displaystyle \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m }\hbox{\sc Shift}_{n,m}(\hbox{\sc Flip}(h))$
	$\displaystyle =$	$\displaystyle \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m } \protect$	(10.4)

written by 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.

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