TABLE OF CONTENTS Perhaps the most popular example of M-band perfect reconstruction filter banks is given by the block transforms (see discrete transforms in Chapter 3). For instance, the discrete cosine transform (DCT) does essentially the same job as a filter bank: it divides a signal into several frequency components. The main difference is that, given a length- N signal, the DCT divides it into N frequency channels, whereas a filter bank divides it into M channels, M < N. However, in many applications, one wants to divide a length- N signal into J blocks, each having length M, and separately apply the transform to each one. This is done, for example, in the MPEG2 standard employed in digital video transmission (Le Gall, 1992; Whitaker, 1999). In it, instead of transmitting the individual pixels of a video sequence, each frame is first divided into 8 x 8 blocks. Then a DCT is applied to each block, and the DCT coefficients are transmitted instead. In this section, we show that such 'block transforms' are equivalent to M-band filter banks.
Consider a signal x( n), n = 0, ... , N - 1, divided into J blocks B j, with j = 0, ..., J - 1, each of size M. Block B j then consists of the signal x j( m), given by
| (9.71) |  |
for j = 0,...
