SECTION 5 UNDERSAMPLING APPLICATIONS Fundamentals of Undersampling Increasing ADC SfdR and enob using External SHas Use of Dither Signals to Increase ADC Dynamic Range Effect of ADC Linearity and Resolution on SFDR and Noise in Digital Spectral Analysis Applications Future Trends in Undersampling adcs
1 SECTION 5 UNDERSAMPLING APPLICATIONS Fundamentals of Undersampling Increasing ADC SFDR and ENOB using External SHAs Use of Dither Signals to Increase ADC Dynamic Range Effect of ADC Linearity and Resolution on SFDR and Noise in Digital Spectral Analysis Applications Future Trends in Undersampling ADCs
sectiOn 5 UNDERSAMPLING APPLICATIONS Walt Kester An exciting new application for wideband, low distortion ADCs is called undersampling, harmonic sampling, bandpass sampling, or Super- Nyquist Sampling To understand these applications, it is necessary to review the basics of the sampling process The concept of discrete time and amplitude sampling of an analog signal is shown in Figure 5. 1. The continuous analog data must be sampled at discrete intervals, ts, which must be carefully chosen to insure an accurate representation of the original analog signal. It is clear that the more samples taken(faster sampling rates), the more accurate the digital representation, but if fewer samples are taken(lower sampling rates), a point is reached where critical information about the signal is Nyquist's Criteria given in Figure 5.2. Most textbooks state the Nyquist theorem 9 actually lost. This leads us to the statement of Shannons Information Theorem ar along the following lines: A signal must be sampled at a rate greater than twice its maximum frequency in order to ensure unambiguous data. The general assumption is that the signal has frequency components from de to some upper value, fa. The Nyquist Criteria thus requires sampling at a rate fs> 2fa in order to avoid overlapping aliased components For signals which do not extend to de, however, the minimum required sampling rate is a function of the bandwidth of the signal as well as its position in the frequency spectrum SAMPLING AN ANALOG SIGNAL *antiRat Figure 5.1
2 SECTION 5 UNDERSAMPLING APPLICATIONS Walt Kester An exciting new application for wideband, low distortion ADCs is called undersampling, harmonic sampling, bandpass sampling, or Super-Nyquist Sampling. To understand these applications, it is necessary to review the basics of the sampling process. The concept of discrete time and amplitude sampling of an analog signal is shown in Figure 5.1. The continuous analog data must be sampled at discrete intervals, ts , which must be carefully chosen to insure an accurate representation of the original analog signal. It is clear that the more samples taken (faster sampling rates), the more accurate the digital representation, but if fewer samples are taken (lower sampling rates), a point is reached where critical information about the signal is actually lost. This leads us to the statement of Shannon's Information Theorem and Nyquist's Criteria given in Figure 5.2. Most textbooks state the Nyquist theorem along the following lines: A signal must be sampled at a rate greater than twice its maximum frequency in order to ensure unambiguous data. The general assumption is that the signal has frequency components from dc to some upper value, fa. The Nyquist Criteria thus requires sampling at a rate fs > 2fa in order to avoid overlapping aliased components. For signals which do not extend to dc, however, the minimum required sampling rate is a function of the bandwidth of the signal as well as its position in the frequency spectrum. SAMPLING AN ANALOG SIGNAL Figure 5.1
SHANNONS INFORMATION THEOREM AND NYQUISTS CRITERIA Shannon: An Analog Signal with a Bandwidth of fa Must be Sampled at a Rate of f 2fa in Order to Avoid the Loss of Information The signal bandwidth may extend from DC to fa(Baseband Sampling)or from f, to f2, where fa=f2-f,(Undersampling, Bandpass Sampling, Harmonic Sampling, Super-Nyquist) Nyquist: If f <2fa, then a Phenomena Called Aliasing Will Occur. Aliasing is used to advantage in undersampling applications Figure 5.2 In order to understand the implications of aliasing in both the time and frequency domain, first consider the four cases of a time domain representation of a sampled sinewave signal shown in Figure 5.3. In Case 1, it is clear that an adequate number of samples have been taken to preserve the information about the sinewave In Case 2 of the figure, only four samples per cycle are taken; still an adequate number to preserve the information Case 3 represents the ambiguous limiting condition where fs=2fa. If the relationship between the sampling points and the sinewave were such that the sinewave was being sampled at precisely the zero crossings (rather than at the peaks, as shown in the illustration), then all information regarding the sinewave would be lost. Case 4 of Figure 5.3 represents the situation where fs<2fa, and th information obtained from the samples indicates a sinewave having a frequency which is lower than fs/2, i.e. the out-of-band signal is aliased into the Nyquist bandwidth between dc and f/2. As the sampling rate is further decreased, and the analog input frequency fa approaches the sampling frequency fs, the aliased signal approaches dc in the frequency spectrum
3 SHANNON’S INFORMATION THEOREM AND NYQUIST’S CRITERIA Shannon: An Analog Signal with a Bandwidth of fa Must be Sampled at a Rate of fs>2fa in Order to Avoid the Loss of Information. The signal bandwidth may extend from DC to fa (Baseband Sampling) or from f1 to f2 , where fa = f2 - f1 (Undersampling, Bandpass Sampling, Harmonic Sampling, Super-Nyquist) Nyquist: If fs<2fa , then a Phenomena Called Aliasing Will Occur. Aliasing is used to advantage in undersampling applications. Figure 5.2 In order to understand the implications of aliasing in both the time and frequency domain, first consider the four cases of a time domain representation of a sampled sinewave signal shown in Figure 5.3. In Case 1, it is clear that an adequate number of samples have been taken to preserve the information about the sinewave. In Case 2 of the figure, only four samples per cycle are taken; still an adequate number to preserve the information. Case 3 represents the ambiguous limiting condition where fs=2fa. If the relationship between the sampling points and the sinewave were such that the sinewave was being sampled at precisely the zero crossings (rather than at the peaks, as shown in the illustration), then all information regarding the sinewave would be lost. Case 4 of Figure 5.3 represents the situation where fs<2fa, and the information obtained from the samples indicates a sinewave having a frequency which is lower than fs /2, i.e. the out-of -band signal is aliased into the Nyquist bandwidth between dc and fs /2. As the sampling rate is further decreased, and the analog input frequency fa approaches the sampling frequency fs , the aliased signal approaches dc in the frequency spectrum
TIME DOMAIN EFFECTS OF ALIASING CASE 1: fs =8 fa CASE 2: fs =4 fa CASE 3: fs=2fa CASE 4: fs =1.3 fa Figure 5.3 The corresponding frequency domain representation of the above scenario is shown in Figure 5.4. Note that sampling the analog signal fa at a sampling rate fs actually produces two alias frequency components, one at fs+fa, and the other at fs-fa. The upper alias, fs+fa, seldom presents a problem, since it lies outside the Nyquist bandwidth. It is the lower alias component, fs-fa, which causes problems when the input signal exceeds the Nyquist bandwidth, f$/2
4 TIME DOMAIN EFFECTS OF ALIASING Figure 5.3 The corresponding frequency domain representation of the above scenario is shown in Figure 5.4. Note that sampling the analog signal fa at a sampling rate fs actually produces two alias frequency components, one at fs+fa, and the other at fs–fa. The upper alias, fs+fa, seldom presents a problem, since it lies outside the Nyquist bandwidth. It is the lower alias component, fs–fa, which causes problems when the input signal exceeds the Nyquist bandwidth, fs /2
FREQUENCY DOMAIN EFFECTS OF ALIASING ASED BANDWIDTH CASE t ' a',. REPEATS 2 CASE 3 a CASE 4 Figure 5.4 From Figure 5. 4, we make the extremely important observation that regardless of where the analog signal being sampled happens to lie in the frequency spectrum(as long as it does not lie on multiples of fs/2), the effects of sampling will cause either the actual signal or an aliased component to fall within the Nyquist bandwidth between dc and fs/2. Therefore, any signals which fall outside the bandwidth of interest, whether they be spurious tones or random noise, must be adequately filtered before sampling. If unfiltered, the sampling process will alias them back within the Nyquist bandwidth where they can corrupt the wanted signals Methods exist which use aliasing to our advantage in signal processing applications Figure 5.5 shows four cases where a signal having a 1MHz bandwidth is located in different portions of the frequency spectrum. The sampling frequency must be chosen such that there is no overlapping of the aliased components. In general, the sampling frequency must be at least twice the signal bandwidth, and the sampled signal must not cross an integer multiple of f/2
5 FREQUENCY DOMAIN EFFECTS OF ALIASING Figure 5.4 From Figure 5.4, we make the extremely important observation that regardless of where the analog signal being sampled happens to lie in the frequency spectrum (as long as it does not lie on multiples of fs/2), the effects of sampling will cause either the actual signal or an aliased component to fall within the Nyquist bandwidth between dc and fs/2. Therefore, any signals which fall outside the bandwidth of interest, whether they be spurious tones or random noise, must be adequately filtered before sampling. If unfiltered, the sampling process will alias them back within the Nyquist bandwidth where they can corrupt the wanted signals. Methods exist which use aliasing to our advantage in signal processing applications. Figure 5.5 shows four cases where a signal having a 1MHz bandwidth is located in different portions of the frequency spectrum. The sampling frequency must be chosen such that there is no overlapping of the aliased components. In general, the sampling frequency must be at least twice the signal bandwidth, and the sampled signal must not cross an integer multiple of fs /2