Oversampled Analog-to-Digital and Digital-to-Analog Converters:Filtering for Sigma-Delta Modulators

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Filtering for Sigma-Delta Modulators

In Sections 59.2 and 59.3, the discussion focused on the operation of the sigma-delta modulator core. While this core is the most unique aspect of sigma-delta data conversion, there are also filtering blocks that constitute an important part of sigma-delta A/D and D/A converters. In this section, the non- modulator components in baseband sigma-delta converters, namely the analog and digital filters, are described. First, the requirements of the analog anti-alias and reconstruction filters are described.

Second, typical architectures for the decimation and interpolation filters are discussed. While much of the design of these filters use standard techniques covered elsewhere in this volume, there are aspects of these filters that are specific to sigma-delta modulator applications.

Anti-Alias and Reconstruction Filters

The purpose of the anti-alias filter, shown in Figure 59.1 at the input of the sigma-delta A/D converter, is, as the name would indicate, to prevent aliasing. The sampling operation maps, or aliases, all frequencies into the range bounded by ± fS /2 [28]. Specifically, all signals within a baseband bandwidth of multiples of the sampling rate are mapped into the baseband. This is generally undesirable, so the anti-alias filter is designed to attenuate this aliasing to some tolerable level. One advantage of sigma-delta converters over Nyquist rate converters is that this anti-aliasing filter has a relatively wide transition region. As illustrated in Figure 59.27, the passband region for this filter is the signal bandwidth fB, while the stopband region for this filter is only within fB of the sampling rate. Thus, the transition region is 2(M-1)fB, and since M >> 1, the transition region is relatively wide. A wide transition region generally means a simple filter design. The precise nature of the anti-alias filter is application-dependent, and can be designed using any number of standard analog filter techniques [35].

The reconstruction filter, shown in Figure 59.2 at the output of the sigma-delta D/A converter, is also an analog filter. Its primary purpose is to remove unwanted out-of-band quantization noise. The extent to which this noise must be removed varies widely from system to system. If the analog output is to be applied to an element that is naturally bandlimited, such as a speaker, then very little attenuation may be necessary. In contrast, if the output is applied to additional analog circuitry, care must be taken lest the high-frequency noise distort and map itself into the baseband. Circuit techniques for this filter are addressed further in Section 59.5.5.

Decimation and Interpolation Filters

In general, the filter characteristics of the decimation filter, shown in Figure 59.1 at the output of the sigma-delta A/D converter, are much sharper than those of the anti-alias filter, that is, the transition region is narrower. The saving grace is that the filter is implemented digitally, and modern submicron processes have made complex digital filters economically feasible. Nonetheless, care must be taken or the filter architecture will become more computationally complex than is necessary.

The basic purpose of the decimation filter is to attenuate quantization noise and unwanted signals outside the baseband so that the output of the decimation filter can be down-sampled, or decimated, without significant aliasing. Normally, the most efficient means of accomplishing this is to apply a multirate filter architecture, such as that illustrated in Figure 59.28 [36,37]. The comb filter is a relatively

Oversampled Analog-to-Digital and Digital-to-Analog Converters-0058

Oversampled Analog-to-Digital and Digital-to-Analog Converters-0059

where R is the impulse response length of the comb filter. If R is set equal to the decimation ratio of the comb filter (the comb filter input rate divided by its output rate), then the filter zeros will occur at every point that would alias to DC [38,39]. If the filter order N is one more than the modulator order, then the comb filter will be adequate to attenuate the out-of-band quantization noise to the point where it does not adversely increase the baseband noise after decimation [40].

Following the comb filter is typically a series of one or more FIR filters. Since the sample rates of these FIR filters are much slower than the oversampled clock rate, each filter output can be computed over many clock cycles. Also, since the output of each filter is decimated, only the samples that will be output need to be calculated. These properties can be exploited to devise computationally efficient structures for decimation filtering [41].

In the example in Figure 59.28, the first FIR filter is decimating from 4´ to 2´ oversampling. Since the output of this filter is still oversampled, the transition region is relatively wide and the attenuation at midband need not be very high. Thus, an economical half-band filter (a filter in which every other coefficient is zero) can be used [37].

The final FIR filter is by far the most complex. It usually has to have a very sharp transition region, and for strict anti-alias performance it cannot be a halfband filter. In high-performance sigma-delta modulators, this filter is often in the range of 50–200 taps in length. Standard digital filter design techniques can be used to select that tap weights for this filter [28]. Since it is going to be a complex filter anyway, it can also be used to compensate for any frequency droop in the previous filter stages.

The interpolation filter, shown in Figure 59.2 at the input of the sigma-delta D/A converter, upsamples the input digital words to the oversampling rate. In many ways, this filter is the inverse of a decimation filter, typically comprising a complex upsampling FIR filter, optionally followed by one or more simple FIR filters, followed by an upsampling comb filter. The upsampling operation, without this filter, would produce images of the baseband spectrum at multiples of the baseband frequency. The purpose of the interpolation is to attenuate these images to a tolerable level. What constitutes tolerable is very much a system-dependent criterion. Design techniques for the interpolation filter parallel those of the decimation filter discussed above.

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