## Decimation of a Band-Pass Signal and Its Inverse Operation

Decimation of a Band-Pass Signal. As was seen in the section entitled “Decimation,” if the input signal x(m) was low pass and band limited to [—n/M, n/M], the aliasing after decimation by a factor of M could be avoided [see Eq. (5)]. However, if a signal is split into M uniform frequency bands, at most one band will have its spectrum confined to [—n/M, n/M]. In fact, if a signal is split into M uniform real bands, one can say that band xk(n) will be confined to [—n(k + 1)/M, —nk/M] U [nk/M, n(k + 1)/M] [1] (see Fig. 11).

This implies that band k, k Ф 0 is not confined to [—n/M, n/M]. However, by examining Eq. (5) one can notice that aliasing is still avoided in this case. The only difference is that, after decimation, the spectrum contained in [—n(k + 1)/ M, —nk/M] is mapped to [0, n] if k is odd, and to [—n, 0] if k is even. Similarly, the spectrum contained in the interval [nk/M, n(k + 1)/M] is mapped to [—n, 0] if k is odd and to [0,

Critically Decimated М-Band Filter Banks. It is clear that if a signal x(m) is decomposed into M non-overlapping bandpass channels Bk, k = 0, . . ., and M — 1 such that UM=—o1 Bk = [—n, n], then it can be recovered from these M channels by just summing them up. However, as conjectured above, exact recovery of the original signal might not be possible if each channel is decimated by M. However, in the section entitled ‘‘Decimation of a Band-Pass Signal and Its Inverse Operation,’’ we examined a way to recover the band-pass channel from its subsampled version. All that is needed are interpolation operations followed by filters with passband [—n(k + 1)/M, —nk/M] U [nk/M, n(k + 1)/M] (see Fig. 13). This process of decomposing a signal and restoring it from the frequency bands is depicted in Fig. 14. We often refer to it as an M-band filter bank. The frequency bands uk(n) are called sub-bands. If the input signal can be recovered exactly from its sub-bands, it is called an M-band perfect reconstruction filter bank. Figure 15 details a perfect reconstruction filter bank for the 2-band case.

However, the filters required for the M-band perfect reconstruction filter bank described above are not realizable [see Eqs. (6) and (10)], that is, at best they can be only approximated (1). Therefore, in a first analysis, the original signal

Ho(z) |
j M |
uo(m) |
j M |
Go(z) |
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Hi(z) |
1 M |
ui(m) |
j M |
Gi(z) |
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x(n) |
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1 y(n) |
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HM – 1(z) |
m |
u M – |
1 M |
GM – 1(z) |
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