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:

$$y_k(n) = (h\ast x_k)(n) = \sum_{m=0}^{N-1}h(m)x_k(n-m)$$

$$= (x_k\ast h)(n) = \sum_{m=n-(N-1)}^{n}x_k(m)h(n-m)$$

$$= \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m }\hbox{\sc Shift}_{n,m}(\hbox{\sc Flip}(h))$$

$$= \sum_{m=n-(N-1)}^{n}x(m)e^{-j\omega_k m } \protect$$ (10.12)