### Polyphase Decomposition of Haar Example

Let's look at the polyphase representation for this example. Starting with the filter bank and its reconstruction (see Fig.11.17), the polyphase decomposition of is

(12.31) |

Thus, , and therefore

(12.32) |

We may derive polyphase synthesis filters as follows:

The polyphase representation of the filter bank and its reconstruction
can now be drawn as in Fig.11.18. Notice that the
reconstruction filter bank is formally the *transpose* of the analysis filter bank [263].
A filter bank that is inverted by its own transpose is said to be an
*orthogonal filter bank*, a subject to which we will return
§11.3.8.

Commuting the downsamplers (using the noble identities from
§11.2.5), we obtain Figure 11.19. Since
, this is simply the OLA form of an
*STFT filter bank* for
, with
, and rectangular
window
. That is, the DFT size, window length, and hop
size are all 2, and both the DFT and its inverse are simply
sum-and-difference operations.

**Next Section:**

Quadrature Mirror Filters (QMF)

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Haar Example