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

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