30-11-2012, 06:11 PM
Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications: A Tutorial
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INTRODUCTION
In recent years there has been tremendous progress in
the multirate processing of digital signals. Unlike the single-
rate system, the sample spacing in a multirate system
can vary from point to point [I], [2]. This often results in more
efficient processing of signals because the sampling rates
at various internal points can be kept as small as possible.
Unfortunately, thisalso results in the introduction of a new
type of error, i.e., aliasing, which should somehow be canceled
eventually.
The basic building blocks in a multirate digital signal processing
(DSP) system are decimators and interpolators. In
1981, an excellent tutorial article on decimation and interpolation
appeared in [3]. Subsequent to this a text on the
subject of multirate systems has also been published by the
same authors [4]. Since then, a number of new developments
have taken place in the area, particularly in multirate
SOME APPLICATIONOSF MULTIRASTYSETE MS
We shall now review a number of important applications
of multirate filters and filter banks, with pointers to the literature
for details, examples, and demonstrations. In section
IX, several unconventional applications are also outlined.
Applications in the design of transmultiplexers (which
are devices for conversion between frequency division
multiplexing (FDM) and time-division multiplexing (TDM))
are not discussed here in detail, primarily because of the
excellent treatment already available in [13]. Also see [I41
for the correspondence between transmultiplexers and
analysiskynthesis filter banks. The input to a TDM-to-FDM
converter is a signal y(n), which is the time-multiplexed version
of M signals y&), 0 5 k 5 M - 1. Given y(n), the components
yk(n) can easily be separated out by use of a commutator
switch [4], [13]. TheseM signals are then modulated
using distinct carrier frequencies. The carrier frequencies
uk, 0 5 k 5 M - 1 are chosen so that there is sufficient spectral
gap between the messages. A sum of these M signals
(which is the FDM signal) is then transmitted through the
channel. The total channel bandwidth is therefore required
to exceed the sum of signal bandwidths because of the safeguard
gap between adjacent spectra. The gap enables one
to obtain perfect recovery of the multiplexed signals yk(n)
at a future point.
Analog Voice Privacy Systems
These systems [29] are intended to communicate speech
over standard analogtelephone links,whileat thesametime
ensuring voice privacy. The main idea is to split a signal x(n)
into M subband signals xk(n) and then divide each subband
signal into segments in the time domain. These segments
of subband signals are then permuted and recombined into
a single encrypted signal y(n), which can then be transmitted
(after DIA conversion). For example, if there are five
subbands and twenty-five time segments in each subband,
then there are 125! possible permutations, and unless an
eavesdropper has the key for decryption, he will be unable
to perform a pleasant job of eavesdropping. The aims of the
designer of such a privacy system are: the encrypted message
should be unintelligible, decryption without a key
should be very difficult, and the decrypted signal should
VAIDYANATHAN: MULTIRATE DIGITAL FILTERS 61
be of good quality retaining naturalness and voice characteristics.
These features have indeed been achieved by
Cox et al. [29].
THET WO-CHANNEQLM F BANK
Consider the filter bank system of Fig. 13, which is a maximally
decimated analysiskynthesis system (or a QMF bank)
[4], [46]-[50]. The applications of this system were outlined
in greater detail in section Ill. The QMF bank is an LTV system
(because of decimators and interpolators). In this section
we shall study the aliasing and imaging effects created
by the decimators and interpolators, emphasizing the M =
2 case. The filters Ho(z) and H,(z) are low-pass and high-pass,
respectively (Fig. 12(b)). The overlapping responses of Fig.
12(b) ensures that no portion of X(e9 is severely attenuated.
However, this overlap also means that the filters do not
bandlimit the subband signals xo(n) and x,(n) to a sufficient
extent, which results in aliasing when xo(n) and x,(n) are
decimated. We should choose the synthesis filters Fo(z) and
Fl(z) to cancel this aliasing, as is explained next.