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Two-Channel Analysis Filter Bank

Module by: Ivan Selesnick

Consider a two-channel analysis filter bank (Figure 1).

Figure 1
Figure 1 (sec5fig1.png)

What is the transpose of this system? We can find it by finding first the matrix representation. To find the matrix representation, note that we already know the matrix representation of the two branches individually. s 0 = Q 0 x s 0 Q 0 x s 1 = Q 1 x s 1 Q 1 x So total system can be represented as a single matrix: s 0 s 1 = Q 0 x Q 1 x= Q 0 Q 1 x=Qx s 0 s 1 Q 0 x Q 1 x Q 0 Q 1 x Q x where Q= Q 0 Q 1 Q Q 0 Q 1 The transpose of the two-channel analysis filter bank is therefore: QT= Q 0 T Q 1 T Q Q 0 Q 1 and y=QT s 0 s 1 = Q 0 T Q 1 T s 0 s 1 = Q 0 T s 0 + Q 1 T s 1 y Q s 0 s 1 Q 0 Q 1 s 0 s 1 Q 0 s 0 Q 1 s 1 This corresponds to the diagram in Figure 2:

Figure 2
Figure 2 (sec5fig2.png)

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