ADCs for DSP, part 2

Sigma-delta analog-digital converters (Σ-Δ ADCs) have been known for nearly 30 years, but only recently has the technology (high density digital VLSI) existed to manufacture them as inexpensive monolithic integrated circuits. They are now used in many applications where a low cost, low bandwidth, low power, high resolution ADC is required.

There have been innumerable descriptions of the architecture and theory of Σ-Δ ADCs, but most commence with a maze of integrals and deteriorate from there. In the Applications Department at Analog Devices, we frequently encounter engineers who do not understand the theory of operation of Σ-Δ ADCs and are convinced, from study of a typical published article, that it is too complex to easily comprehend.

There is nothing particularly difficult to understand about Σ-Δ ADCs, as long as one avoids the detailed mathematics, and this section has been written in an attempt to clarify the subject. A Σ-Δ ADC contains very simple analog electronics (a comparator, voltage reference, a switch, and one or more integrators and analog summing circuits), and quite complex digital computational circuitry. This circuitry consists of a digital signal processor (DSP) that acts as a filter (generally, but not invariably, a low-pass filter). It is not necessary to know precisely how the filter works to appreciate what it does. To understand how a Σ-Δ ADC works, familiarity with the concepts of oversampling, quantization noise shaping, digital filtering, and decimation is required.

Figure 3-10: Sigma-Delta ADCs.

Let us consider the technique of oversampling with an analysis in the frequency domain. Where a dc conversion has a quantization error of up to 1/2 LSB, a sampled data system has quantization noise. A perfect classical N-bit sampling ADC has an RMS quantization noise of q/√12 uniformly distributed within the Nyquist band of DC to f_s/2 (where q is the value of an LSB and f_s is the sampling rate) as shown in Figure 3-11A. Therefore, its SNR with a full-scale sine wave input will be (6.02N + 1.76) dB. If the ADC is less than perfect, and its noise is greater than its theoretical minimum quantization noise, then its effective resolution will be less than N-bits. Its actual resolution (often known as its effective number of bits or ENOB) will be defined by

If we choose a much higher sampling rate, Kf_s (see Figure 3-11B), the rms quantization noise remains q/√12, but the noise is now distributed over a wider bandwidth dc to Kf_s/2. If we then apply a digital low-pass filter (LPF) to the output, we remove much of the quantization noise, but do not affect the wanted signal—so the ENOB is improved. We have accomplished a high resolution A/D conversion with a low resolution ADC. The factor K is generally referred to as the oversampling ratio. It should be noted at this point that oversampling has an added benefit in that it relaxes the requirements on the analog antialiasing filter.

Figure 3-11: Oversampling, Digital Filtering, Noise Shaping, and Decimation.

Since the bandwidth is reduced by the digital output filter, the output data rate may be lower than the original sampling rate (Kf_s) and still satisfy the Nyquist criterion. This may be achieved by passing every Mth result to the output and discarding the remainder. The process is known as "decimation" by a factor of M. Despite the origins of the term (decem is Latin for 10), M can have any integer value, provided that the output data rate is more than twice the signal bandwidth. Decimation does not cause any loss of information (see Figure 3-11B).

If we simply use oversampling to improve resolution, we must oversample by a factor of 2^2N to obtain an N-bit increase in resolution. The Σ-Δ converter does not need such a high oversampling ratio because it not only limits the signal pass band, but also shapes the quantization noise so that most of it falls outside this pass band as shown in Figure 3-11C.

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