MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N dB," and Why You Should Care

Transcription

1 MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N dB," and Why You Should Care by Walt Kester REV. 0, INTRODUCTION You don't have to deal with ADCs or DACs for long before running across this often quoted formula for the theoretical signal-tonoise ratio (SNR) of a converter. Rather than blindly accepting it on face value, a fundamental knowledge of its origin is important, because the formula encompasses some subtleties which if not understood can lead to significant misinterpretation of both data sheet specifications and converter performance. Remember that this formula represents the theoretical performance of a perfect N-bit ADC. You can compare the actual ADC SNR with the theoretical SNR and get an idea of how the ADC stacks up. This tutorial first derives the theoretical quantization noise of an N-bit analog-to-digital converter (ADC). Once the rms quantization noise voltage is known, the theoretical signal-to-noise ratio (SNR) is computed. The effects of oversampling on the SNR are also analyzed. DERIVATION The maximum error an ideal converter makes when digitizing a signal is ±½ LSB as shown in the transfer function of an ideal N- bit ADC (Figure 1). The quantization error for any ac signal which spans more than a few LSBs can be approximated by an uncorrelated sawtooth waveform having a peak-to-peak amplitude of q, the weight of an LSB. Another way to view this approximation is that the actual quantization error is equally probable to occur at any point within the range ±½ q. Although this analysis is not precise, it is accurate enough for most applications. W. R. Bennett of Bell Laboratories analyzed the actual spectrum of quantization noise in his classic 1948 paper (Reference 1). With the simplifying assumptions previously mentioned, his detailed mathematical analysis simplifies to that of Figure 1. Other significant papers and books on converter noise followed Bennett's classic publication (References 2-6). Figure 1: Ideal N-bit ADC Quantization Noise

2 The quantization error as a function of time is shown in more detail in Figure 2. Again, a simple sawtooth waveform provides a sufficiently accurate model for analysis. The equation of the sawtooth error is given by The mean-square value of e(t) can be written: Performing the simple integration and simplifying, The root-mean-square quantization error is therefore Figure 2: Quantization Noise as a Function of Time The sawtooth error waveform produces harmonics which extend well past the Nyquist bandwidth of dc to f s /2. However, all these higher order harmonics must fold (alias) back into the Nyquist bandwidth and sum together to produce an rms noise equal to q/ 12. As Bennett points out (Reference 1), the quantization noise is approximately Gaussian and spread more or less uniformly over the Nyquist bandwidth dc to f s /2. The underlying assumption here is that the quantization noise is uncorrelated to the input signal. Under certain conditions where the sampling clock and the signal are harmonically related, the quantization noise becomes correlated, and the energy is concentrated in the harmonics of the signal however, the rms value remains approximately q/ 12. The theoretical signal-to-noise ratio can now be calculated assuming a full-scale input sinewave: The rms signal of the input signal is therefore The rms signal-to-noise ratio for an ideal N-bit converter is therefore

3 Bennett's paper shows that although the actual spectrum of the quantization noise is quite complex to analyze, the simplified analysis which leads to Eq. 9 is accurate enough for most purposes. However, it is important to emphasize again that the rms quantization noise is measured over the full Nyquist bandwidth, dc to f s /2. OVERSAMPLING AND UNDERSAMPLING In many applications, the actual signal of interest occupies a smaller bandwidth, BW, which is less than the Nyquist bandwidth (see Figure 3). If digital filtering is used to filter out noise components outside the bandwidth BW, then a correction factor (called process gain) must be included in the equation to account for the resulting increase in SNR as shown in Eq. 10. The process of sampling a signal at a rate which is greater than twice its bandwidth is referred to as oversampling. Oversampling in conjunction with quantization noise shaping and digital filtering are the key concepts in sigma-delta converters, although oversampling can be used with any ADC architecture. Figure 3: Quantization Noise Spectrum Showing Process Gain The significance of process gain can be seen from the following example. In many digital basestations or other wideband receivers the signal bandwidth is composed of many individual channels, and a single ADC is used to digitize the entire bandwidth. For instance, the analog cellular radio system (AMPS) in the U.S. consists of kHz wide channels, occupying a bandwidth of approximately 12.5 MHz. Assume a 65-MSPS sampling frequency, and that digital filtering is used to separate the individual 30-kHz channels. The process gain due to oversampling for these conditions is given by:

4 The process gain is added to the ADC SNR specification to yield the SNR in the 30-kHz bandwidth. In the above example, if the ADC SNR specification is 65 db (dc to f s /2), then it is increased to 95.3 db in the 30-kHz channel bandwidth (after appropriate digital filtering). Figure 4 shows an application which combines oversampling and undersampling. The signal of interest has a bandwidth BW and is centered around a carrier frequency f c. The sampling frequency can be much less than f c and is chosen such that the signal of interest is centered in its Nyquist zone. Analog and digital filtering removes the noise outside the signal bandwidth of interest, and therefore results in process gain per Eq. 10. Figure 4: Undersampling and Oversampling Combined Results in Process Gain CORRELATION BETWEEN QUANTIZATION NOISE AND INPUT SIGNAL YIELDS MISLEADING RESULTS Although the rms value of the noise is accurately approximated by q/ 12, its frequency domain content may be highly correlated to the ac-input signal under certain conditions. For instance, there is greater correlation for low amplitude periodic signals than for large amplitude random signals. Quite often, the assumption is made that the theoretical quantization noise appears as white noise, spread uniformly over the Nyquist bandwidth dc to f s /2. Unfortunately, this is not true in all cases. In the case of strong correlation, the quantization noise appears concentrated at the various harmonics of the input signal, just where you don't want them. In most practical applications, the input to the ADC is a band of frequencies (always summed with some unavoidable system noise), so the quantization noise tends to be random. In spectral analysis applications (or in performing FFTs on ADCs using spectrally pure sinewaves as inputs, however, the correlation between the quantization noise and the signal depends upon the ratio of the sampling frequency to the input signal. This is demonstrated in Figure 5, where the output of an ideal 12-bit ADC is analyzed using a 4096-point FFT. In the left-hand FFT plot (A), the ratio of the sampling frequency ( MSPS) to the input frequency (2.000 MHz) was chosen to be exactly 40, and the worst harmonic is about 77 db below the fundamental. The right hand diagram (B) shows the effects of slightly offsetting the input frequency to MHz, showing a relatively random noise spectrum, where the SFDR is now about 93 dbc

5 and is limited by the spikes in the noise floor of the FFT. In both cases, the rms value of all the noise components is approximately q/ 12 (yielding a theoretical SNR of 74 db) but in the first case, the noise is concentrated at harmonics of the fundamental because of the correlation. Figure 5: Effect of Ratio of Sampling Clock to Input Frequency on Quantization Noise Frequency Spectrum for Ideal 12-bit ADC, 4096-Point FFT. (A) Correlated Noise, (B) Uncorrelated Noise Note that this variation in the apparent harmonic distortion of the ADC is an artifact of the sampling process caused by the correlation of the quantization error with the input frequency. In a practical ADC application, the quantization error generally appears as random noise because of the random nature of the wideband input signal and the additional fact that there is a usually a small amount of system noise which acts as a dither signal to further randomize the quantization error spectrum. It is important to understand the above point, because single-tone sinewave FFT testing of ADCs is one of the universally accepted methods of performance evaluation. In order to accurately measure the harmonic distortion of an ADC, steps must be taken to ensure that the test setup truly measures the ADC distortion, not the artifacts due to quantization noise correlation. This is done by properly choosing the frequency ratio and sometimes by summing a small amount of noise (dither) with the input signal. The exact same precautions apply to measuring DAC distortion with an analog spectrum analyzer. THEORETICAL NOISE FLOOR OF THE FFT OUTPUT Figure 6 shows the FFT output for an ideal 12-bit ADC. Note that the average value of the noise floor of the FFT is approximately 107 db below full-scale, but the theoretical SNR of a 12-bit ADC is 74 db. The FFT noise floor is not the SNR of the ADC, because the FFT acts like an analog spectrum analyzer with a bandwidth of f s /M, where M is the number of points in the FFT. The theoretical FFT noise floor is therefore 10log 10 (M/2) db below the quantization noise floor due to the processing gain of the FFT. In the case of an ideal 12-bit ADC with an SNR of 74 db, a 4096-point FFT would result in a processing gain of 10log 10 (4096/2) = 33 db, thereby resulting in an overall FFT noise floor of = 107 dbc. In fact, the FFT noise floor can be reduced even further by going to larger and larger FFTs; just as an analog spectrum analyzer's noise floor can be reduced by narrowing the bandwidth. When testing ADCs using FFTs, it is therefore important to ensure that the FFT size is large enough so that the distortion products can be distinguished from the FFT noise floor itself. Averaging a number of FFTs does not further reduce the noise floor, it simply reduces the variations between the individual noise spectral component amplitudes.

6 Figure 6: Noise Floor for an Ideal 12-bit ADC Using 4096-point FFT REFERENCES 1. W.R. Bennett, "Spectra of Quantized Signals," Bell System Technical Journal, Vol. 27, July 1948, p B.M. Oliver, J.R. Pierce, and C.E. Shannon, "The Philosophy of PCM," Proceedings IRE, Vol. 36, November 1948, p W.R. Bennett, "Noise in PCM Systems," Bell Labs Record, Vol. 26, December 1948, p H.S. Black and J.O. Edson, "Pulse Code Modulation," AIEE Transactions, Vol. 66, 1947, p H.S. Black, "Pulse Code Modulation," Bell Labs Record, Vol. 25, July 1947, p K.W. Cattermole, Principles of Pulse Code Modulation, American Elsevier Publishing Company,Inc., 1969, New York NY, ISBN Walt Kester, Analog-Digital Conversion, Analog Devices, 2004, ISBN , Chapter 2. Also Available as The Data Conversion Handbook, Elsevier/Newnes, 2005, ISBN , Chapter 2.

7 MT-002: What the Nyquist Criterion Means to Your Sampled Data System Design by Walt Kester REV. 0, INTRODUCTION A quick reading of Harry Nyquist's classic Bell System Technical Journal article of 1924 (Reference 1) does not reveal the true significance of the criterion which bears his name. Nyquist was working on the transmission of telegraph signals over a channel that was bandwidth limited. A thorough understanding of the modern interpretation of Nyquist's criterion is mandatory when dealing with sampled data systems. This tutorial explains in easy to understand terms how the Nyquist criterion applies to baseband sampling, undersampling, and oversampling applications. A block diagram of a typical real-time sampled data system is shown in Figure 1. Prior to the actual analog-to-digital conversion, the analog signal usually passes through some sort of signal conditioning circuitry which performs such functions as amplification, attenuation, and filtering. The lowpass/bandpass filter is required to remove unwanted signals outside the bandwidth of interest and prevent aliasing. Figure 1: Typical Sampled Data System The system shown in Figure 1 is a real-time system, i.e., the signal to the ADC is continuously sampled at a rate equal to f s, and the ADC presents a new sample to the DSP at this rate. In order to maintain real-time operation, the DSP must perform all its required computation within the sampling interval, 1/f s, and present an output sample to the DAC before arrival of the next sample from the ADC. An example of a typical DSP function would be a digital filter. Note that the DAC is required only if the DSP data must be converted back into an analog signal (as would be the case in a voiceband or audio application, for example). There are many applications where the signal remains entirely in digital format after the initial A/D conversion. Similarly, there are applications where the DSP is solely responsible for generating the signal to the

8 DAC. If a DAC is used, it must be followed by an analog anti-imaging filter to remove the image frequencies. Finally, there are slower speed industrial process control systems where sampling rates are much lower regardless of the system, the fundamentals of sampling theory still apply. There are two key concepts involved in the actual analog-to-digital and digital-to-analog conversion process: discrete time sampling and finite amplitude resolution due to quantization. This tutorial discusses discrete time sampling. THE NEED FOR A SAMPLE-AND-HOLD AMPLIFIER (SHA) FUNCTION The generalized block diagram of a sampled data system shown in Figure 1 assumes some type of ac signal at the input. It should be noted that this does not necessarily have to be so, as in the case of modern digital voltmeters (DVMs) or ADCs optimized for dc measurements, but for this discussion assume that the input signal has some upper frequency limit f a. Most ADCs today have a built-in sample-and-hold function, thereby allowing them to process ac signals. This type of ADC is referred to as a sampling ADC. However many early ADCs, such as Analog Devices' industry-standard AD574, were not of the sampling type, but simply encoders as shown in Figure 2. If the input signal to a SAR ADC (assuming no SHA function) changes by more than 1 LSB during the conversion time (8 µs in the example), the output data can have large errors, depending on the location of the code. Most ADC architectures are subject to this type of error some more, some less with the possible exception of flash converters having well-matched comparators. Figure 2: Input Frequency Limitations of Non-Sampling ADC (Encoder) Assume that the input signal to the encoder is a sinewave with a full-scale amplitude (q2 N /2), where q is the weight of 1 LSB. Taking the derivative: The maximum rate of change is therefore: Solving for f:

9 If N = 12, and 1 LSB change (dv = q) is allowed during the conversion time (dt = 8 µs), then the equation can be solved for f max, the maximum full-scale signal frequency that can be processed without error: f max = 9.7 Hz. This implies any input frequency greater than 9.7 Hz is subject to conversion errors, even though a sampling frequency of 100 ksps is possible with the 8-µs ADC (this allows an extra 2-µs interval for an external SHA to re-acquire the signal after coming out of the hold mode). To process ac signals, a sample-and-hold (SHA) function is added as shown in Figure 3. The ideal SHA is simply a switch driving a hold capacitor followed by a high input impedance buffer. The input impedance of the buffer must be high enough so that the capacitor is discharged by less than 1 LSB during the hold time. The SHA samples the signal in the sample mode, and holds the signal constant during the hold mode. The timing is adjusted so that the encoder performs the conversion during the hold time. A sampling ADC can therefore process fast signals the upper frequency limitation is determined by the SHA aperture jitter, bandwidth, distortion, etc., not the encoder. In the example shown, the sample-and-hold acquires the signal in 2 µs, the encoder converts the signal in 8 µs, yielding a total sampling period of 10 µs. This yields a sampling frequency of 100 ksps. and the capability of processing input frequencies up to 50 khz. It is important to understand a subtle difference between a true sample-and-hold amplifier (SHA) and a track-and-hold amplifier (T/H, or THA). Strictly speaking, the output of a sample-and-hold is not defined during the sample mode, however the output of a track-and-hold tracks the signal during the sample or track mode. In practice, the function is generally implemented as a trackand-hold, and the terms track-and-hold and sample-and-hold are often used interchangeably. The waveforms shown in Figure 3 are those associated with a track-and-hold. THE NYQUIST CRITERION Figure 3: Sample-and-Hold Function Required for Digitizing AC Signals A continuous analog signal is sampled at discrete intervals, t s = 1/f s, which must be carefully chosen to ensure an accurate

10 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. The mathematical basis of sampling was set forth by Harry Nyquist of Bell Telephone Laboratories in two classic papers published in 1924 and 1928, respectively. (See References 1 and 2 as well as Chapter 2 of Reference 6). Nyquist's original work was shortly supplemented by R. V. L. Hartley (Reference 3). These papers formed the basis for the PCM work to follow in the 1940s, and in 1948 Claude Shannon wrote his classic paper on communication theory (Reference 4). Simply stated, the Nyquist criterion requires that the sampling frequency be at least twice the highest frequency contained in the signal, or information about the signal will be lost. If the sampling frequency is less than twice the maximum analog signal frequency, a phenomenon known as aliasing will occur. In order to understand the implications of aliasing in both the time and frequency domain, first consider the case of a time domain representation of a single tone sinewave sampled as shown in Figure 4. In this example, the sampling frequency f s is not at least 2f a, but only slightly more than the analog input frequency f a the Nyquist criterion is violated. Notice that the pattern of the actual samples produces an aliased sinewave at a lower frequency equal to f s - f a. Figure 4: Aliasing in the Time Domain The corresponding frequency domain representation of this scenario is shown in Figure 5B. Now consider the case of a single frequency sinewave of frequency f a sampled at a frequency f s by an ideal impulse sampler (see Figure 5A). Also assume that f s > 2f a as shown. The frequency-domain output of the sampler shows aliases or images of the original signal around every multiple of f s, i.e. at frequencies equal to ± Kf s ± f a, K = 1, 2, 3, 4,...

11 Figure 5: Analog Signal f a f s Using Ideal Sampler Has Images (Aliases) at ± Kf s ± f a, K = 1, 2, 3,... The Nyquist bandwidth is defined to be the frequency spectrum from dc to f s /2. The frequency spectrum is divided into an infinite number of Nyquist zones, each having a width equal to 0.5f s as shown. In practice, the ideal sampler is replaced by an ADC followed by an FFT processor. The FFT processor only provides an output from dc to f s /2, i.e., the signals or aliases which appear in the first Nyquist zone. Now consider the case of a signal which is outside the first Nyquist zone (Figure 5B). The signal frequency is only slightly less than the sampling frequency, corresponding to the condition shown in the time domain representation in Figure 4. Notice that even though the signal is outside the first Nyquist zone, its image (or alias), f s - f a, falls inside. Returning to Figure 5A, it is clear that if an unwanted signal appears at any of the image frequencies of f a, it will also occur at f a, thereby producing a spurious frequency component in the first Nyquist zone. This is similar to the analog mixing process and implies that some filtering ahead of the sampler (or ADC) is required to remove frequency components which are outside the Nyquist bandwidth, but whose aliased components fall inside it. The filter performance will depend on how close the out-of-band signal is to f s /2, and the amount of attenuation required. BASEBAND ANTIALIASING FILTERS Baseband sampling implies that the signal to be sampled lies in the first Nyquist zone. It is important to note that with no input filtering at the input of the ideal sampler, any frequency component (either signal or noise) that falls outside the Nyquist bandwidth in any Nyquist zone will be aliased back into the first Nyquist zone. For this reason, an antialiasing filter is used in almost all sampling ADC applications to remove these unwanted signals. Properly specifying the antialiasing filter is important. The first step is to know the characteristics of the signal being sampled. Assume that the highest frequency of interest is f a. The antialiasing filter passes signals from dc to f a while attenuating signals above f a. Assume that the corner frequency of the filter is chosen to be equal to f a. The effect of the finite transition from minimum to maximum attenuation on system dynamic range is illustrated in Figure 6A.

12 Figure 6: Oversampling Relaxes Requirements on Baseband Antialiasing Filter Assume that the input signal has full-scale components well above the maximum frequency of interest, f a. The diagram shows how full-scale frequency components above f s - f a are aliased back into the bandwidth dc to f a. These aliased components are indistinguishable from actual signals and therefore limit the dynamic range to the value on the diagram which is shown as DR. Some texts recommend specifying the antialiasing filter with respect to the Nyquist frequency, f s /2, but this assumes that the signal bandwidth of interest extends from dc to f s /2 which is rarely the case. In the example shown in Figure 6A, the aliased components between f a and f s /2 are not of interest and do not limit the dynamic range. The antialiasing filter transition band is therefore determined by the corner frequency f a, the stopband frequency f s - f a, and the desired stopband attenuation, DR. The required system dynamic range is chosen based on the requirement for signal fidelity. Filters become more complex as the transition band becomes sharper, all other things being equal. For instance, a Butterworth filter gives 6-dB attenuation per octave for each filter pole (as do all filters). Achieving 60-dB attenuation in a transition region between 1 MHz and 2 MHz (1 octave) requires a minimum of 10 poles not a trivial filter, and definitely a design challenge. Therefore, other filter types are generally more suited to applications where the requirement is for a sharp transition band and inband flatness coupled with linear phase response. Elliptic filters meet these criteria and are a popular choice. There are a number of companies which specialize in supplying custom analog filters. TTE is an example of such a company (Reference 5). As an example, the normalized response of the TTE, Inc., LE pole elliptic antialiasing filter is shown in Figure 7. Notice that this filter is specified to achieve at least 80 db attenuation between f c and 1.2f c. The corresponding passband ripple, return loss, delay, and phase response are also shown in Figure 7.

13 Figure 7: Characteristics of 11-Pole Elliptical Filter (TTE, Inc., LE1182-Series) From this discussion, we can see how the sharpness of the antialiasing transition band can be traded off against the ADC sampling frequency. Choosing a higher sampling rate (oversampling) reduces the requirement on transition band sharpness (hence, the filter complexity) at the expense of using a faster ADC and processing data at a faster rate. This is illustrated in Figure 6B which shows the effects of increasing the sampling frequency by a factor of K, while maintaining the same analog corner frequency, f a, and the same dynamic range, DR, requirement. The wider transition band (f a to Kf s - f a ) makes this filter easier to design than for the case of Figure 6A. The antialiasing filter design process is started by choosing an initial sampling rate of 2.5 to 4 times f a. Determine the filter specifications based on the required dynamic range and see if such a filter is realizable within the constraints of the system cost and performance. If not, consider a higher sampling rate which may require using a faster ADC. It should be mentioned that sigma-delta ADCs are inherently highly oversampled converters, and the resulting relaxation in the analog anti-aliasing filter requirements is therefore an added benefit of this architecture. The antialiasing filter requirements can also be relaxed somewhat if it is certain that there will never be a full-scale signal at the stopband frequency f s - f a. In many applications, it is improbable that full-scale signals will occur at this frequency. If the maximum signal at the frequency f s - f a will never exceed X db below full-scale, then the filter stopband attenuation requirement can be reduced by that same amount. The new requirement for stopband attenuation at f s - f a based on this knowledge of the signal is now only DR - X db. When making this type of assumption, be careful to treat any noise signals which may occur above the maximum signal frequency f a as unwanted signals which will also alias back into the signal bandwidth. UNDERSAMPLING (HARMONIC SAMPLING, BANDPASS SAMPLING, IF SAMPLING, DIRECT IF-TO- DIGITAL CONVERSION) Thus far we have considered the case of baseband sampling, i.e., all the signals of interest lie within the first Nyquist zone. Figure 8A shows such a case, where the band of sampled signals is limited to the first Nyquist zone, and images of the original band of frequencies appear in each of the other Nyquist zones. Consider the case shown in Figure 8B, where the sampled signal band lies entirely within the second Nyquist zone. The process of sampling a signal outside the first Nyquist zone is often referred to as undersampling, or harmonic sampling. Note that the image which falls in the first Nyquist zone contains all the information in the original signal, with the exception of its original location (the order of the frequency components within the spectrum is reversed, but this is easily corrected by re-ordering the output of the FFT).

14 Figure 8: Undersampling and Frequency Translation Between Nyquist Zones Figure 8C shows the sampled signal restricted to the third Nyquist zone. Note that the image that falls into the first Nyquist zone has no spectral reversal. In fact, the sampled signal frequencies may lie in any unique Nyquist zone, and the image falling into the first Nyquist zone is still an accurate representation (with the exception of the spectral reversal which occurs when the signals are located in even Nyquist zones). At this point we can restate the Nyquist criterion as it applies to broadband signals: A signal of bandwidth BW must be sampled at a rate equal to or greater than twice its bandwidth (2BW) in order to preserve all the signal information. Notice that there is no mention of the absolute location of the band of sampled signals within the frequency spectrum relative to the sampling frequency. The only constraint is that the band of sampled signals be restricted to a single Nyquist zone, i.e., the signals must not overlap any multiple of f s /2 (this, in fact, is the primary function of the antialiasing filter). Sampling signals above the first Nyquist zone has become popular in communications, because the process is equivalent to analog demodulation. It is becoming common practice to sample IF signals directly and then use digital techniques to process the signal, thereby eliminating the need for an IF demodulator and filters. Clearly, however, as the IF frequencies become higher, the dynamic performance requirements on the ADC become more critical. The ADC input bandwidth and distortion performance must be adequate at the IF frequency, rather than only baseband. This presents a problem for most ADCs designed to only process signals in the first Nyquist zone an ADC suitable for undersampling applications must maintain dynamic performance into the higher order Nyquist zones. ANTIALIASING FILTERS IN UNDERSAMPLING APPLICATIONS Figure 9 shows a signal in the second Nyquist zone centered around a carrier frequency, f c, whose lower and upper frequencies are f 1 and f 2. The antialiasing filter is a bandpass filter. The desired dynamic range is DR, which defines the filter stopband attenuation. The upper transition band is f 2 to 2f s - f 2, and the lower is f 1 to f s - f 1. As in the case of baseband sampling, the antialiasing filter requirements can be relaxed by proportionally increasing the sampling frequency, but f c must also be changed so that it is always centered in the second Nyquist zone.

15 Figure 9: Antialiasing Filter for Undersampling Two key equations can be used to select the sampling frequency, f s, given the carrier frequency, f c, and the bandwidth of its signal, Δf. The first is the Nyquist criteria: The second equation ensures that f c is placed in the center of a Nyquist zone: where NZ = 1, 2, 3, 4,... and NZ corresponds to the Nyquist zone in which the carrier and its signal fall (see Figure 10). Figure 10: Centering an Undersampled Signal within a Nyquist Zone NZ is normally chosen to be as large as possible thereby allowing high IF frequencies. Regardless of the choice for NZ, the Nyquist criterion requires that f s > 2Δf. If NZ is chosen to be odd, then f c and its signal will fall in an odd Nyquist zone, and the image frequencies in the first Nyquist zone will not be reversed.

16 As an example, consider a 4-MHz wide signal centered around a carrier frequency of 71 MHz. The minimum required sampling frequency is therefore 8 MSPS. Solving Eq. 6 for NZ using f c = 71 MHz and f s = 8 MSPS yields NZ = However, NZ must be an integer, so we round to the next lowest integer, 18. Solving Eq. 6 again for f s yields f s = MSPS. The final values are therefore f s = MSPS, f c = 71 MHz, and NZ = 18. Now assume that we desire more margin for the antialiasing filter, and we select f s to be 10 MSPS. Solving Eq. 6 for NZ, using f c = 71 MHz and f s = 10 MSPS yields NZ = We round 14.7 to the next lowest integer, giving NZ = 14. Solving Eq. 6 again for f s yields f s = MSPS. The final values are therefore f s = MSPS, f c = 71 MHz, and NZ = 14. The above iterative process can also be carried out starting with f s and adjusting the carrier frequency to yield an integer number for NZ. SUMMARY This tutorial has covered the basics of the Nyquist criterion and the effects of aliasing in both the time and frequency domain. A working knowledge of the criterion was used to show how to adequately specify the antialiasing filter. Oversampling and undersampling examples were shown in relationship to modern applications in communications systems. REFERENCES 1. H. Nyquist, "Certain Factors Affecting Telegraph Speed," Bell System Technical Journal, Vol. 3, April 1924, pp H. Nyquist, Certain Topics in Telegraph Transmission Theory, A.I.E.E. Transactions, Vol. 47, April 1928, pp R.V.L. Hartley, "Transmission of Information," Bell System Technical Journal, Vol. 7, July 1928, pp C. E. Shannon, "A Mathematical Theory of Communication," Bell System Technical Journal, Vol. 27, July 1948, pp and October 1948, pp TTE, Inc., Olympic Blvd., Los Angeles, CA 90064, 6. Walt Kester, Analog-Digital Conversion, Analog Devices, 2004, ISBN , Chapter 2. Also Available as The Data Conversion Handbook, Elsevier/Newnes, 2005, ISBN , Chapter 2.

17 MT-003: Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so You Don't Get Lost in the Noise Floor by Walt Kester REV. 0, INTRODUCTION Six popular specifications for quantifying ADC dynamic performance are SINAD (signal-to-noise-and-distortion ratio), ENOB (effective number of bits), SNR (signal-to-noise ratio), THD (total harmonic distortion), THD + N (total harmonic distortion plus noise), and SFDR (spurious free dynamic range). Although most ADC manufacturers have adopted the same definitions for these specifications, some exceptions still exist. Because of their importance in comparing ADCs, it is important not only to understand exactly what is being specified, but the relationships between the specifications. There are a number of ways to quantify the distortion and noise of an ADC. All of them are based on an FFT analysis using a generalized test setup such as shown in Figure 1. Figure 1: Generalized Test Setup for FFT Analysis of ADC Output The spectral output of the FFT is a series of M/2 points in the frequency domain (M is the size of the FFT the number of samples stored in the buffer memory). The spacing between the points is f s /M, and the total frequency range covered is dc to f s /2, where f s is the sampling rate. The width of each frequency "bin" (sometimes called the resolution of the FFT) is f s /M. Figure 2 shows an FFT output for an ideal 12-bit ADC using the Analog Devices' ADIsimADC program. Note that the theoretical noise floor of the FFT is equal to the theoretical SNR plus the FFT process gain, 10 log(m/2). It is important to remember that the value for noise used in the SNR calculation is the noise that extends over the entire Nyquist bandwidth (dc to f s /2), but the FFT acts as a narrowband spectrum analyzer with a bandwidth of f s /M that sweeps over the spectrum. This has the effect of pushing the noise down by an amount equal to the process gain the same effect as narrowing the bandwidth of an analog spectrum analyzer. The FFT data shown in Figure 2 represents the average of 5 individual FFTs. Note that averaging a number of FFTs does not affect the average noise floor, it only acts to "smooth" the random variations in the amplitudes contained in each frequency bin.

18 Figure 2: FFT Output for an Ideal 12-Bit ADC, Input = 2.111MHz, f s = 82MSPS, Average of 5 FFTs, M = 8192, Data Generated from ADIsimADC The FFT output can be used like an analog spectrum analyzer to measure the amplitude of the various harmonics and noise components of a digitized signal. The harmonics of the input signal can be distinguished from other distortion products by their location in the frequency spectrum. Figure 3 shows a 7-MHz input signal sampled at 20 MSPS and the location of the first 9 harmonics. Aliased harmonics of f a fall at frequencies equal to ±Kf s ± nf a, where n is the order of the harmonic, and K = 0, 1, 2, 3,... The second and third harmonics are generally the only ones specified on a data sheet because they tend to be the largest, although some data sheets may specify the value of the worst harmonic. Harmonic distortion is normally specified in dbc (decibels below carrier), although in audio applications it may be specified as a percentage. It is the ratio of the rms signal to the rms value of the harmonic in question. Harmonic distortion is generally specified with an input signal near full-scale (generally 0.5 to 1 db below full-scale to prevent clipping), but it can be specified at any level. For signals much lower than full-scale, other distortion products due to the differential nonlinearity (DNL) of the converter not direct harmonics may limit performance. Figure 3: Location of Distortion Products: Input Signal = 7 MHz, Sampling Rate = 20 MSPS Total harmonic distortion (THD) is the ratio of the rms value of the fundamental signal to the mean value of the root-sum-square

19 of its harmonics (generally, only the first 5 harmonics are significant). THD of an ADC is also generally specified with the input signal close to full-scale, although it can be specified at any level. Total harmonic distortion plus noise (THD + N) is the ratio of the rms value of the fundamental signal to the mean value of the root-sum-square of its harmonics plus all noise components (excluding dc). The bandwidth over which the noise is measured must be specified. In the case of an FFT, the bandwidth is dc to f s /2. (If the bandwidth of the measurement is dc to f s /2 (the Nyquist bandwidth), THD + N is equal to SINAD see below). Be warned, however, that in audio applications the measurement bandwidth may not necessarily be the Nyquist bandwidth. Spurious free dynamic range (SFDR) is the ratio of the rms value of the signal to the rms value of the worst spurious signal regardless of where it falls in the frequency spectrum. The worst spur may or may not be a harmonic of the original signal. SFDR is an important specification in communications systems because it represents the smallest value of signal that can be distinguished from a large interfering signal (blocker). SFDR can be specified with respect to full-scale (dbfs) or with respect to the actual signal amplitude (dbc). The definition of SFDR is shown graphically in Figure 4. Figure 4: Spurious Free Dynamic Range (SFDR) The Analog Devices' ADIsimADC ADC modeling program allows various high performance ADCs to be evaluated at various operating frequencies, levels, and sampling rates. The models yield an accurate representation of actual performance, and a typical FFT output for the AD bit, 80-MSPS ADC is shown in Figure 5. Note that the input frequency is MHz and is aliased back to MHz by the sampling process. The output also displays the locations of the first five harmonics. In this case, all the harmonics are aliases. The program also calculates and tabulates the important performance parameters as shown in the left-hand data column. Figure 5: AD Bit, 80MSPS ADC fin = MHz, fs = 80MSPS,

20 Average of 5 FFTs, M = 8192, Data Generated from ADIsimADC SIGNAL-TO-NOISE-AND-DISTORTION RATIO (SINAD), SIGNAL-TO-NOISE RATIO (SNR), AND EFFECTIVE NUMBER OF BITS (ENOB) SINAD and SNR deserve careful attention, because there is still some variation between ADC manufacturers as to their precise meaning. Signal-to-Noise-and-Distortion (SINAD, or S/(N + D) is the ratio of the rms signal amplitude to the mean value of the root-sum-square (rss) of all other spectral components, including harmonics, but excluding dc. SINAD is a good indication of the overall dynamic performance of an ADC because it includes all components which make up noise and distortion. SINAD is often plotted for various input amplitudes and frequencies. For a given input frequency and amplitude, SINAD is equal to THD + N, provided the bandwidth for the noise measurement is the same for both (the Nyquist bandwidth). A typical plot for the AD bit, 65-MSPS ADC is shown in Figure 6. Figure 6: AD bit, 65-MSPS ADC SINAD and ENOB for Various Input Full-Scale Spans (Range) The SINAD plot shows that the ac performance of the ADC degrades due to high-frequency distortion and is usually plotted for frequencies well above the Nyquist frequency so that performance in undersampling applications can be evaluated. SINAD plots such as these are very useful in evaluating the dynamic performance of ADCs. SINAD is often converted to effective-number-ofbits (ENOB) using the relationship for the theoretical SNR of an ideal N-bit ADC: SNR = 6.02N db. The equation is solved for N, and the value of SINAD is substituted for SNR: Note that Equation 1 assumes a full-scale input signal. If the signal level is reduced, the value of SINAD decreases, and the ENOB decreases. It is necessary to add a correction factor for calculating ENOB at reduced signal amplitudes as shown in Equation 2: The correction factor essentially "normalizes" the ENOB value to full-scale regardless of the actual signal amplitude. Signal-to-noise ratio (SNR, or sometimes called SNR-without-harmonics) is calculated from the FFT data the same as SINAD, except that the signal harmonics are excluded from the calculation, leaving only the noise terms. In practice, it is only necessary

21 to exclude the first 5 harmonics, since they dominate. The SNR plot will degrade at high input frequencies, but generally not as rapidly as SINAD because of the exclusion of the harmonic terms. A few ADC data sheets somewhat loosely refer to SINAD as SNR, so you must be careful when interpreting these specifications and understand exactly what the manufacturer means. THE MATHEMATICAL RELATIONSHIPS BETWEEN SINAD, SNR, AND THD There is a mathematical relationship between SINAD, SNR, and THD (assuming all are measured with the same input signal amplitude and frequency. In the following equations, SNR, THD, and SINAD are expressed in db, and are derived from the actual numerical ratios S/N, S/D, and S/(N+D) as shown below: Eq. 3, Eq. 4, and Eq. 5 can be solved for the numerical ratios N/S, D/S, and (N+D)/S as follows: Because the denominators of Eq. 6, Eq. 7, and Eq. 8 are all equal to S, the root sum square of N/S and D/S is equal to (N+D)/S as follows: Therefore, S/(N+D) must equal: and hence,

22 Eq. 12 gives us SINAD as a function of SNR and THD. Similarly, if we know SINAD and THD, we can solve for SNR as follows: Similarly, if we know SINAD and SNR, we can solve for THD as follows: Equations 12, 13, and 14 are implemented in an easy to use design tool on the Analog Devices' website. It is important to emphasize again that these relationships hold true only if the input frequency and amplitude are equal for all three measurements. SUMMARY Because SINAD, SNR, ENOB, THD, THD + N, and SFDR are common measures of ADC dynamic performance, a complete understanding of them in the context of the manufacturers' data sheet is critical. This tutorial has defined the quantities and derived the mathematical relationship between SINAD, SNR, and THD. REFERENCES 1. Walt Kester, Analog-Digital Conversion, Analog Devices, 2004, ISBN , Chapter 2. Also Available as The Data Conversion Handbook Elsevier/Newnes, 2005, ISBN , Chapter 2.

23 1/18/200 MT-004: The Good, the Bad, and the Ugly Aspects of ADC Input Noise Is No Noise Good Noise? by Walt Kester REV. 0, INTRODUCTION All analog-to-digital converters (ADCs) have a certain amount of "input-referred noise" modeled as a noise source connected in series with the input of a noise-free ADC. Input-referred noise is not to be confused with quantization noise which only occurs when an ADC is processing an ac signal. In most cases, less input noise is better, however there are some instances where input noise can actually be helpful in achieving higher resolution. This probably doesn't make sense right now, so you will just have to read further into this tutorial to find out how SOME noise can be GOOD noise. INPUT-REFERRED NOISE (CODE TRANSITION NOISE) Practical ADCs deviate from ideal ADCs in many ways. Input-referred noise is certainly a departure from the ideal, and its effect on the overall ADC transfer function is shown in Figure 1. As the analog input voltage is increased, the "ideal" ADC (shown in Figure 1A) maintains a constant output code until the transition region is reached, at which point the output code instantly jumps to the next value and remains there until the next transition region is reached. A theoretically perfect ADC has zero "code transition" noise, and a transition region width equal to zero. A practical ADC has certain amount of code transition noise, and therefore a transition region width that depends on the amount of input-referred noise present (shown in Figure 1B). Figure 1B shows a situation where the width of the code transition noise is approximately one least significant bit (LSB) peak-to-peak. Figure 1: Code Transition Noise (Input-referred Noise) and its Effect on ADC Transfer Function All ADC internal circuits produce a certain amount of rms noise due to resistor noise and "kt/c" noise. This noise is present even for dc-input signals, and accounts for the code transition noise. Today, code transition noise is most often referred to as inputreferred noise rather than code transition noise. The input-referred noise is most often characterized by examining the histogram of a number of output samples when the input to the ADC is a dc value. The output of most high speed or high resolution ADCs is a distribution of codes, centered around the nominal value of the dc input (see Figure 2). To measure its value, the input of the ADC is either grounded or connected to a heavily decoupled voltage source, and a large number of output samples are collected and plotted as a histogram (sometimes referred to as a grounded-input histogram). Since the noise is approximately Gaussian, the standard deviation of the histogram, σ, can be calculated, corresponding to the effective input rms noise. Reference 1 provides a detailed description of how to calculate the value of σ from the histogram data. It is common practice to express this rms noise in terms of LSBs rms, although it can be expressed as an rms voltage referenced to the ADC full-scale input range.

24 Figure 2: Effect of Input-Referred Noise on ADC "Grounded Input" Histogram for ADC with Small Amount of DNL Although the inherent differential nonlinearity (DNL) of the ADC may cause some minor deviations from an ideal Gaussian distribution (some DNL is shown in Figure 2, for instance), it should be at least approximately Gaussian. If there is significant DNL, the value of s should be calculated for several different dc input voltages, and the results averaged. If the code distribution has large and distinct peaks and valleys, for instance this indicates either a poorly designed ADC or, more likely, a bad PC board layout, poor grounding techniques, or improper power supply decoupling (see Figure 3). Another indication of trouble is when the width of the distribution changes drastically as the dc input is swept over the ADC input voltage range. Figure 3: Grounded Input Histogram for Poorly Designed ADC and/or Poor Layout, Grounding, or Decoupling NOISE-FREE (FLICKER-FREE) CODE RESOLUTION The noise-free code resolution of an ADC is the number of bits beyond which it is impossible to distinctly resolve individual codes. The cause is the effective input noise (or input-referred noise) associated with all ADCs and described above. This noise can be expressed as an rms quantity, usually having the units of LSBs rms. Multiplying by a factor of 6.6 converts the rms noise

25 into peak-to-peak noise (expressed in LSBs peak-to-peak). The total range (or span) of an N-bit ADC is 2 N LSBs. The total number of noise-free counts is therefore equal to: The number of noise-free counts can be converted into noise-free code resolution by taking the base-2 logarithm as follows: The noise-free code resolution specification is generally associated with high-resolution sigma-delta measurement ADCs. It is most often a function sampling rate, digital filter bandwidth, and programmable gain amplifier (PGA) gain. Figure 4 shows a typical table taken from the AD7730 sigma-delta measurement ADC. Figure 4: Noise-Free Code Resolution for the AD7730 Sigma-Delta ADC Note that for an output data rate of 50 Hz and an input range of ±10 mv, the noise-free code resolution is 16.5 bits (80,000 noise-free counts). The settling time under these conditions is 460 ms, making this ADC an ideal candidate for a precision weigh scale application. Data such as this is available on most data sheets for high resolution sigma-delta ADCs suitable for precision measurement applications. The ratio of the FS range to the rms input noise (rather than peak-to-peak noise) is sometimes used to calculate resolution. In this case, the term effective resolution is used. Note that under identical conditions, effective resolution is larger than noise-free code resolution by log 2 (6.6), or approximately 2.7 bits. Some manufacturers prefer to specify effective resolution rather than noise-free code resolution because it results in a higher number of bits the user should check the data sheet closely to make sure which is actually specified. INCREASING ADC "RESOLUTION" AND REDUCING NOISE BY DIGITAL AVERAGING The effects of input-referred noise can be reduced by digital averaging. Consider a 16-bit ADC which has 15 noise-free bits at a sampling rate of 100 ksps. Averaging two samples per output sample reduces the effective sampling rate to 50 ksps and increases the SNR by 3 db, and the noise-free bits to Averaging four samples per output sample reduces the sampling rate to 25 ksps, increases the SNR by 6 db, and increases the noise-free bits to 16. In fact, if we average sixteen samples per output, the output sampling rate is reduced to 6.25 ksps, the SNR increases by

26 another 6 db, and the noise-free bits increase to 17. The arithmetic in the averaging must be carried out to the larger number of significant bits in order to take advantage of the extra "resolution." The averaging process also helps "smooth" out the DNL errors in the ADC transfer function. This can be illustrated for the simple case where the ADC has a missing code at quantization level "k". Even though code "k" is missing because of the large DNL error, the average of the two adjacent codes, k 1 and k + 1 is equal to k. This technique can therefore be used effectively to increase the dynamic range of the ADC, at the expense of overall output sampling rate and extra digital hardware. It should be noted, however, that averaging will not correct the inherent integral nonlinearity of the ADC. Now, consider the case of an ADC that has extremely low input-referred noise, and the histogram shows a solid code all the time. What will digital averaging do for this ADC? This answer is simple it will do nothing! No matter how many samples we average, we will still get the same answer. However, as soon as we add enough noise to the input signal so that there is more than one code in the histogram, the averaging method starts working again. Some small amount of noise is therefore good (at least with respect to the averaging method), but the more noise present at the input, the more averaging is required to achieve the same resolution. DON'T CONFUSE EFFECTIVE NUMBER OF BITS (ENOB) WITH "EFFECTIVE RESOLUTION" OR "NOISE-FREE CODE RESOLUTION" Because of the similarity of the terms, effective number of bits and effective resolution are often assumed to be equal. This is not the case. Effective number of bits (ENOB) is derived from an FFT analysis of the ADC output when the ADC is stimulated with a full-scale sinewave input signal. The rss value of all noise and distortion terms is computed, and the ratio of the signal to the noise and distortion is defined as SINAD, or S/(N+D). The theoretical SNR of a perfect N-bit ADC is given by: The calculated value of SINAD for the ADC is substituted for SNR in Eq. 5, and the equation solved for N, yielding the ENOB: The noise and distortion used to calculate SINAD and ENOB include not only the input-referred noise but also the quantization noise and the distortion terms. SINAD and ENOB are used to measure the dynamic performance of an ADC, while effective resolution and noise-free code resolution are used to measure the noise of the ADC under dc input conditions where there is no quantization noise. USING NOISE DITHER TO INCREASE ADC SPURIOUS FREE DYNAMIC RANGE There are two fundamental limitations to maximizing SFDR in a high speed ADC. The first is the distortion produced by the frontend amplifier and the sample-and-hold circuit. The second is that produced by non-linearity in the actual transfer function of the encoder portion of the ADC. The key to high SFDR is to minimize the non-linearity of each. There is nothing that can be done externally to the ADC to significantly reduce the inherent distortion caused by the ADC front end. However, the differential nonlinearity in the ADC encoder transfer function can be reduced by the proper use of dither (external noise which is summed with the analog input signal to the ADC). Dithering improves ADC SFDR under certain conditions (References 2-5). For example, even in a perfect ADC, there is some correlation between the quantization noise and the input signal. This can reduce the SFDR of the ADC, especially if the input signal is an exact sub-multiple of the sampling frequency. Summing broadband noise (about 1/2 LSB rms in amplitude) with the input signal tends to randomize the quantization noise and minimize this effect (see Figure 5A). In most systems, however, there is enough noise riding on top of the signal so that adding additional dither noise is not required. The input-referred noise of the ADC may also be enough to produce the same effect. Increasing the wideband rms noise level beyond approximately one LSB will proportionally reduce the ADC SNR and results in no additional improvement.