# Taking the Mystery out of the Infamous Formula, "SNR = 6.02N dB," and Why You Should Care. by Walt Kester

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1 ITRODUCTIO Taking the Mystery out of the Infamous Formula, "SR = dB," and Why You Should Care by Walt Kester MT-001 TUTORIAL You don't have to deal with ADCs or DACs for long before running across this often quoted formula for the theoretical signal-to-noise ratio (SR) 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 -bit ADC. You can compare the actual ADC SR with the theoretical SR and get an idea of how the ADC stacks up. This tutorial first derives the theoretical quantization noise of an -bit analog-to-digital converter (ADC). Once the rms quantization noise voltage is known, the theoretical signal-to-noise ratio (SR) is computed. The effects of oversampling on the SR are also analyzed. QUATIZATIO OISE MODEL The maximum error an ideal converter makes when digitizing a signal is ±½ LSB as shown in the transfer function of an ideal -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. DIGITAL OUTPUT AALOG IPUT ERROR (IPUT OUTPUT) q = 1 LSB Figure 1: Ideal -bit ADC Quantization oise Rev.A, 10/08, WK Page 1 of 7

2 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 -6). The quantization error as a function of time is shown in more detail in Figure. 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 e(t) = st, q/s < t < +q/s. Eq. 1 s = + q / s e (t) (st) dt. Eq. q q / s q e (t) =. Eq. 3 1 q rms quantization noise = e (t) =. Eq q e(t) SLOPE = s q t q s +q s Figure : Quantization oise as a Function of Time The sawtooth error waveform produces harmonics which extend well past the yquist bandwidth of dc to /. However, all these higher order harmonics must fold (alias) back into the yquist bandwidth and sum together to produce an rms noise equal to q/ 1. As Bennett points out (Reference 1), the quantization noise is approximately Gaussian and spread more or less uniformly over the yquist bandwidth dc to /. The underlying assumption Page of 7

3 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/ 1. The theoretical signal-to-noise ratio can now be calculated assuming a full-scale input sinewave: The rms value of the input signal is therefore q Input FS Sinewave = v(t) = sin(πft). Eq. 5 rms value of q FS input =. Eq. 6 The rms signal-to-noise ratio for an ideal -bit converter is therefore rms value of FS input SR = 0log10 Eq. 7 rms value of quantization noise q / 3 SR = 0log10 = 0log10 + 0log10 Eq. 8 q / 1 SR = dB, over the dc to / bandwidth. Eq. 9 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 yquist bandwidth, dc to /. FREQUECY SPETRUM OF QUATIZATIO OISE In many applications, the actual signal of interest occupies a smaller bandwidth, BW, which is less than the yquist 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 SR as shown in Eq. 10. f SR = db + 10log s 10, over the bandwidth BW. Eq. 10 BW The process oampling 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. Page 3 of 7

4 OISE SPECTRAL DESITY RMS VALUE = q 1 q = 1 LSB MEASURED OVER DC TO q / 1 / BW SR = dB + 10log 10 BW FOR FS SIEWAVE Process Gain Figure 3: Quantization oise 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 1.5 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: f Process Gain 10log s = 10 = 10log10 = 30.3 db. Eq. 11 BW The process gain is added to the ADC SR specification to yield the SR in the 30-kHz bandwidth. In the above example, if the ADC SR specification is 65 db (dc to /), 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 yquist zone. Analog and digital filtering removes the noise outside the signal bandwidth of interest, and therefore results in process gain per Eq Page 4 of 7

7 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. (db) 0 ADC FULLSCALE 0 = 1-BITS M = dB = dB RMS QUATIZATIO OISE LEVEL M 33dB = 10log 10( ) FFT OISE FLOOR 74dB 107dB 10 BI SPACIG = 4096 REFERECES Figure 6: oise Floor for an Ideal 1-bit ADC Using 4096-point FFT 1. W. R. Bennett, "Spectra of Quantized Signals," Bell System Technical Journal, Vol. 7, July 1948, pp B. M. Oliver, J. R. Pierce, and C. E. Shannon, "The Philosophy of PCM," Proceedings IRE, Vol. 36, ovember 1948, pp W. R. Bennett, "oise in PCM Systems," Bell Labs Record, Vol. 6, December 1948, pp H. S. Black and J. O. Edson, "Pulse Code Modulation," AIEE Transactions, Vol. 66, 1947, pp H. S. Black, "Pulse Code Modulation," Bell Labs Record, Vol. 5, July 1947, pp K. W. Cattermole, Principles of Pulse Code Modulation, American Elsevier Publishing Company, Inc., 1969, ew York Y, ISB Walt Kester, Analog-Digital Conversion, Analog Devices, 004, ISB , Chapter. Also Available as The Data Conversion Handbook, Elsevier/ewnes, 005, ISB , Chapter. 8. Hank Zumbahlen, Basic Linear Design, Analog Devices, 006, ISB: Also available as Linear Circuit Design Handbook, Elsevier-ewnes, 008, ISB-10: , ISB-13: Chapter 5. Copyright 009, Analog Devices, Inc. All rights reserved. Analog Devices assumes no responsibility for customer product design or the use or application of customers products or for any infringements of patents or rights of others which may result from Analog Devices assistance. All trademarks and logos are property of their respective holders. Information furnished by Analog Devices applications and development tools engineers is believed to be accurate and reliable, however no responsibility is assumed by Analog Devices regarding technical accuracy and topicality of the content provided in Analog Devices Tutorials. Page 7 of 7

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