N-bit sampling AdC has an rms quantization noise of q/(sgrt 12)uniformly distributed within the Nyquist band of dC to fs/2(where q is the value of an lsB and fs is the sampling rate). Therefore, its SNR with a full-scale sinewave 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 ENOB= SNR-176dB 6.02dB SAMPLING ADC QUANTIZATION NOISE OUTPUT SIGNAL ADC RMS QUANTIZATION NOISE =gN 12 Figure 3.6 If we choose a much higher sampling rate the quantization noise is distributed over a wider bandwidth as shown in Figure 3.7. 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
6 N-bit sampling ADC has an rms quantization noise of q/(sqrt 12) uniformly distributed within the Nyquist band of DC to fs /2 (where q is the value of an LSB and fs is the sampling rate). Therefore, its SNR with a full-scale sinewave 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 ac tu al r eso lutio n ( o fte nk n o wnas its E f fe c tiv eNu m bero f B its or E N OB )will bede f ine d by ENOB SNR dB dB = - 176 6 02 . . . SAMPLING ADC QUANTIZATION NOISE Figure 3.6 If we choose a much higher sampling rate, the quantization noise is distributed over a wider bandwidth as shown in Figure 3.7. 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
OVERSAMPLING FOLLOWED BY DIGITAL FILTERING AND DECIMATION IMPROVES SNR AND ENOB fs ADC DECIMATOR QUANTIZATION NOISI REMOVED BY DIGITAL FILTER Figure 3.7 Since the bandwidth is reduced by the digital output filter, the output data rate may be lower than the original sampling rate 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 ten), 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 3.8) DECIMATION AFTER SAMPLING AT fs AND FILTERING, THE OUTPUT DATA RATE MAY BE REDUCED TO f/M WITHOUT LOSS OF INFORMATION FILTER PASSBAND 2 Figure 3.8
7 OVERSAMPLING FOLLOWED BY DIGITAL FILTERING AND DECIMATION IMPROVES SNR AND ENOB Figure 3.7 Since the bandwidth is reduced by the digital output filter, the output data rate may be lower than the original sampling rate 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 ten), 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 3.8). DECIMATION Figure 3.8
If we simply use over-sampling to improve resolution, we must over-sample by a factor of 2 2N to obtain an N-bit increase in resolution. The Sigma-Delta converte does not need such a high over-sampling ratio because it not only limits the signal passband, but also shapes the quantization noise so that most of it falls outside this passband If we take a l-bit adc (generally known as a comparator), drive it with the output of an integrator, and feed the integrator with an input signal summed with the output of a 1-bit DAC fed from the ADC output, we have a first-order Sigma-Delta modulator as shown in Figure 3.9. Add a digital low pass filter (LPF)and decimator at the digital output, and we have a Sigma-Delta ADC: the Sigma-Delta modulator shapes the quantization noise so that it lies above the passband of the digital output filter, and the eNob is therefore much larger than would otherwise be expected from the over-sampling ratio FIRST ORDER SIGMA-DELTA ADC 当∑ VREF Figure 3.9 B than one integration and summing stas Delta modulator, we can achieve higher orders of quantization noise shaping and even better ENOB for a given over-sampling ratio as is shown in figure 3. 10 for both a first and second-order Sigma-Delta modulator The block diagram for the second order Sigma-Delta modulator is shown in Figure 3. 11. Third, and higher, order Sigma- Delta ADCs were once thought to be potentially unstable at some values of input- recent analyses using finite rather than infinite gains in the comparator have shown that this is not necessarily so, but even if instability does start to occur, it is not important, since the DSP in the digital filter and decimator can be made to recognize incipient instability and react to prevent it
8 If we simply use over-sampling to improve resolution, we must over-sample by a factor of 2^2N to obtain an N-bit increase in resolution. The Sigma-Delta converter does not need such a high over-sampling ratio because it not only limits the signal passband, but also shapes the quantization noise so that most of it falls outside this passband. If we take a 1-bit ADC (generally known as a comparator), drive it with the output of an integrator, and feed the integrator with an input signal summed with the output of a 1-bit DAC fed from the ADC output, we have a first-order Sigma-Delta modulator as shown in Figure 3.9. Add a digital low pass filter (LPF) and decimator at the digital output, and we have a Sigma-Delta ADC: the Sigma-Delta modulator shapes the quantization noise so that it lies above the passband of the digital output filter, and the ENOB is therefore much larger than would otherwise be expected from the over-sampling ratio. FIRST ORDER SIGMA-DELTA ADC Figure 3.9 By using more than one integration and summing stage in the Sigma-Delta modulator, we can achieve higher orders of quantization noise shaping and even better ENOB for a given over-sampling ratio as is shown in Figure 3.10 for both a first and second-order Sigma-Delta modulator. The block diagram for the secondorder Sigma-Delta modulator is shown in Figure 3.11. Third, and higher, order Sigma-Delta ADCs were once thought to be potentially unstable at some values of input - recent analyses using finite rather than infinite gains in the comparator have shown that this is not necessarily so, but even if instability does start to occur, it is not important, since the DSP in the digital filter and decimator can be made to recognize incipient instability and react to prevent it
SIGMA-DELTA MODULATORS SHAPE QUANTIZATION NOISE DIGITAl 一 FILTER igure 3. 10 SECOND-ORDER SIGMA-DELTA ADC ))1 Figure 3.11 and the amount of over-sampling necessary to achieve a particular SNR odulator Figure 3.12 shows the relationship between the order of the Sigma-Delta
9 SIGMA-DELTA MODULATORS SHAPE QUANTIZATION NOISE Figure 3.10 SECOND-ORDER SIGMA-DELTA ADC Figure 3.11 Figure 3.12 shows the relationship between the order of the Sigma-Delta modulator and the amount of over-sampling necessary to achieve a particular SNR
SNR VERSUS OVERSAMPLING RATIO FOR FIRST. SECOND, AND THIRD-ORDER LOOPS 120 THIRD-ORDER LOOP. SNR dB) 15dB OCTAVE FIRST-ORDER LOOP 2nd ORDER LOOPS DO NOT OBEY LINEAR MODEL OVERSAMPLING RATIO, K Figure 3. 12 The Sigma-Delta ADCs that we have described so far contain integrators, which are low pass filters, whose passband extends from DC. Thus, their quantization noise is pushed up in frequency. At present, all commercially available Sigma-Delta ADCs are of this type(although some which are intended for use in audio or telecommunications applications contain bandpass rather than lowpass digital filters to eliminate any system dC offsets). Sigma-Delta DCs are available with resolutions up to 24-bits for DC measurement applications(AD7710, AD7711 AD7712, AD7713, AD7714), and with resolutions of 18-bits for high quality digita audio applications(AD1879) But there is no particular reason why the filters of the Sigma-Delta modulator should be LPFs, except that traditionally adCs have been thought of as being baseband devices, and that integrators are somewhat easier to construct than bandpass filters. If we replace the integrators in a Sigma-Delta ADC with bandpass filters(BPFs), the quantization noise is moved up and down in frequency to leave a virtually noise-free region in the pass-band(see Reference 1). If the digital filter is then programmed to have its pass band in this region, we have a Sigma-Delta ADC with a bandpass, rather than a low pass characteristic(see Figure 3. 13). Although studies of this architecture are in their infancy, such ADCs would seem to be ideally suited for use in digital radio receivers, medical ultrasound, and a number of other application
1 0 SNR VERSUS OVERSAMPLING RATIO FOR FIRST, SECOND, AND THIRD-ORDER LOOPS Figure 3.12 The Sigma-Delta ADCs that we have described so far contain integrators, which are low pass filters, whose passband extends from DC. Thus, their quantization noise is pushed up in frequency. At present, all commercially available Sigma-Delta ADCs are of this type (although some which are intended for use in audio or telecommunications applications contain bandpass rather than lowpass digital filters to eliminate any system DC offsets). Sigma-Delta ADCs are available with resolutions up to 24-bits for DC measurement applications (AD7710, AD7711, AD7712, AD7713, AD7714), and with resolutions of 18-bits for high quality digital audio applications (AD1879). But there is no particular reason why the filters of the Sigma-Delta modulator should be LPFs, except that traditionally ADCs have been thought of as being baseband devices, and that integrators are somewhat easier to construct than bandpass filters. If we replace the integrators in a Sigma-Delta ADC with bandpass filters (BPFs), the quantization noise is moved up and down in frequency to leave a virtually noise-free region in the pass-band (see Reference 1). If the digital filter is then programmed to have its pass-band in this region, we have a Sigma-Delta ADC with a bandpass, rather than a low pass characteristic (see Figure 3.13). Although studies of this architecture are in their infancy, such ADCs would seem to be ideally suited for use in digital radio receivers, medical ultrasound, and a number of other applications