19-11-2012, 04:16 PM
Multirate digital signal processing
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In multirate digital signal processing the sampling rate of a signal is changed in or-
der to increase the e±ciency of various signal processing operations. Decimation, or
down-sampling, reduces the sampling rate, whereas expansion, or up-sampling, fol-
lowed by interpolation increases the sampling rate. Some applications of multirate
signal processing are:
² Up-sampling, i.e., increasing the sampling frequency, before D/A conversion in
order to relax the requirements of the analog lowpass antialiasing ¯lter. This
technique is used in audio CD, where the sampling frequency 44.1 kHz is increased
fourfold to 176.4 kHz before D/A conversion.
² Various systems in digital audio signal processing often operate at di®erent sam-
pling rates. The connection of such systems requires a conversion of sampling
rate.
² Decomposition of a signal intoM components containing various frequency bands.
If the original signal is sampled at the sampling frequency fs (with a frequency
band of width fs=2, or half the sampling frequency), every component then con-
tains a frequency band of width 1
2fs=M only, and can be represented using the
sampling rate fs=M. This allows for e±cient parallel signal processing with pro-
cessors operating at lower sampling rates. The technique is also applied to data
compression in subband coding, for example in speech processing, where the vari-
ous frequency band components are represented with di®erent word lengths.
² In the implementation of high-performance ¯ltering operations, where a very nar-
row transition band is required. The requirement of narrow transition bands leads
to very high ¯lter orders. However, by decomposing the signal into a number of
subbands containing the passband, stopband and transition bands, each compo-
nent can be processed at a lower rate, and the transition band will be less narrow.
Hence the required ¯lter complexity may be reduced signi¯cantly.
The theory of multirate signal processing is quite extensive, and not entirely trivial.
Here we will discuss some of the main ideas only.
Sampling rate conversion by non-integer factors.
Sampling rate conversion by a non-integer factor F may be achieved by the use of
expansion and decimation operations. If the conversion factor can be expressed as a
rational number, i.e., the ratio of two integers, F = L=M, then the obvious way to
achieve the conversion is to apply expansion by the factor L followed by lowpass ¯ltering
(to extract the low-frequency component of the expanded signal) and decimation with
the factor M. A problem with this direct approach occurs when the integers L and
M are large, in which case there will be very high requirements on the anti-aliasing
¯lters. In practice this problem is avoided in multirate signal processing by performing
the sampling rate conversions in several stages with small integer factors (for example,
M = L = 2) at each stage.
Analysis and synthesis ¯lter banks
A basic operation in multirate signal processing is to decompose a signal into a number
of subband components, which can be processed at a lower rate corresponding to the
bandwidth of the frequency bands. Recall from equation (2.13) that down-sampling
mixes frequency components in the original signal by aliasing and frequency folding.
Therefore, the signal should be ¯ltered before decimation. Figure 2.3 shows the decom-
position of a signal into two subband components. The purpose of the ¯lters H1 and
H2 is to extract the low- and high-frequency components of the signal x before deci-
mation. For example, the component xD1 may contain the low-frequency components
of x, and the component xD2 may contain the low-frequency components of x. The set
of ¯lters shown in Figure 2.3 is called an analysis ¯lter bank. If required, the signals
may be decimated further into narrower subband components, cf. Figure 2.4. If H1 is a
low-pass ¯lter and H2 a high-pass ¯lter, xD1 is the low-frequency component consisting
of the subband [0; ¼=2], whereas xD21 and xD22 contain the subbands [¼=2; 3¼=4] and
[3¼=4; ¼], respectively.
Subband decomposition
In order to present the basic techniques involved in decomposing a signal into sub-
band components, let's consider a simple case where a signal is decomposed into two
components: a low-frequency component and a high-frequency component. See Figure
2.3. The purpose of the ¯lters H1 and H2 is to extract the low- and high-frequency
components of the signal x. For perfect signal decomposition, H1 should be an ideal
low-pass ¯lter with the passband [0; ¼=2], and H2 should be an ideal high-pass ¯lter
with the passband [¼=2; ¼], cf. Figure 2.6. Filters having the characteristics shown in
Figure 2.6 are called brickwall ¯lters.
In order to gain insight into the subband decomposition problem, suppose for the
moment that it were possible to design ideal brickwall ¯lters with the responses shown
in Figure 2.6. As the outputs of the brickwall ¯lters are composed of frequencies in
bands having widths ¼=2, they can be exactly represented using only half of the original
sampling rate. More precisely, it follows from the properties of the brickwall ¯lters H1
and H2 and the relation (2.13) that the down-sampled signals xD1 and xD2 in Figure
2.3 have the Fourier-transforms