DSP Applications and Student Projects - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 431

10.23.1 Multirate Filter

With multirate processing, a filter can be realized with fewer coefficients than with
an equivalent single-rate approach. Possible applications include a controlled noise
source and background noise synthesis.

Introduction
Multirate processing uses more than one sampling frequency to perform a desired
processing operation. The two basic operations are decimation , which is a sampling-
rate reduction, and interpolation , which is a sampling-rate increase. Decimation
techniques have been used in filtering. Multirate decimators can reduce the com-
putational requirements of the filter. Interpolation can be used to obtain a sampling-
rate increase. For example, a sampling-rate increase by a factor of K can be achieved
by padding K 1 zeros between pairs of consecutive input samples x i and x i +1 .We

can also obtain a noninteger sampling-rate increase or decrease by cascading the
decimation process with the interpolation process. For example, if a net sampling-
rate increase of 1.5 is desired, we would interpolate by a factor of 3, padding
(adding) two zeros between each input sample, and then decimate with the inter-
polated input samples shifted by 2 before each calculation. Decimating or interpo-
lating over several stages generally results in better efficiency [60-67].

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Design Considerations
A binary random signal is fed into a bank of filters that are used to shape the output
spectrum. The functional block diagram of the multirate filter is shown in Figure
10.60. The frequency range is divided into 10 octave bands, with each band 3 -octave
controllable. The control of each octave band is achieved with three filters. The co-
efficients of these filters are combined to yield a composite filter with one set of
coefficients for each octave. Only three unique sets of filter coefficients (low, middle,
and high) are required, because the center frequency and the bandwidth are pro-
portional to the sampling frequency. Each of the 3 -octave filters has a bandwidth of
approximately 23% of its center frequency, a stopband rejection of greater than
45 dB, with an amplitude that can be controlled individually. This control provides
the capability of shaping an output pseudorandom noise spectrum. The sampling
rate of the output is chosen to be 16,384 Hz. Forty-one coefficients are used for the
highest 3 -octave filter to achieve these requirements. The middle 3 -octave filter coef-
ficients were used as BP41.cof in Chapter 4.
In order to meet the filter specifications in each region with a constant sampling
rate, the number of filter coefficients must be doubled from one octave filter to the
next lower one. As a result, the lowest-octave filter would require 41 ¥ 2 9 coeffi-

cients. With 10 filters ranging from 41 to 41 2 9 coefficients, the computational

requirements would be considerable. To reduce these computational requirements,
a multirate approach is used, as shown in Figure 10.60.

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