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System Diagram of the Running-Sum Filter Bank

Figure 9.15: DFT Filter Bank.

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

$\displaystyle y_k(n)$ $\displaystyle =$ $\displaystyle (h\ast x_k)(n) = \sum_{m=0}^{N-1}h(m)x_k(n-m)$  
  $\displaystyle =$ $\displaystyle (x_k\ast h)(n) = \sum_{m=n-(N-1)}^{n}x_k(m)h(n-m)$  
  $\displaystyle =$ $\displaystyle \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m }\hbox{\sc Shift}_{n,m}(\hbox{\sc Flip}(h))$  
  $\displaystyle =$ $\displaystyle \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m }
\protect$ (10.12)

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