**by W Kester · Cited by 87 — formula for the theoretical signal-to-noise ratio (SNR) of a converter. Rather than blindly accepting it on face value, a fundamental knowledge of its **

**103 KB – 7 Pages**

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MT-001TUTORIAL Taking the Mystery out of the Infamous Formula, “SNR = 6.02N + 1.76dB,” and Why You Should Care by Walt Kester INTRODUCTION 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 rati o (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 converte r performance. Remember that this formula represents the theoretical performa nce 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. QUANTIZATION NOISE MODEL 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 (F igure 1). The quantization error for any ac signal which spans more than a few LSBs can be appr oximated by an uncorrela ted sawtooth waveform having a peak-to-peak amplitude of q, the wei ght of an LSB. Another way to view this approximation is that the actual quantization erro r 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. ANALOG INPUT DIGITAL OUTPUT ERROR(INPUT ŒOUTPUT) q = 1 LSB Figure 1: Ideal N-bit ADC Quantization Noise Rev.A, 10/08, WK Page 1 of 7

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MT-001W. R. Bennett of Bell Laboratories analyzed th e actual spectrum of qua ntization noise in his classic 1948 paper (Reference 1). With the simp lifying assumptions previously mentioned, his detailed mathematical analysis simplifies to that of Figure 1. Other significant papers and books on converter noise followed Bennett’s cl assic publication (References 2-6). The quantization error as a function of time is s hown in more detail in Figure 2. Again, a simple sawtooth waveform provides a sufficiently accur ate model for analysis. The equation of the sawtooth error is given by e(t) = st, Œq/2s < t < +q/2s. Eq. 1 The mean-square value of e(t) can be written: s2/q s2/q 22dt)st( qs)t( e. Eq. 2 Performing the simple integration and simplifying, 12q)t(e 22. Eq. 3 The root-mean-square quantiz ation error is therefore 12q)t(enoiseonquantizati rms2 . Eq. 4 t+q2+q2Œq2Œq2SLOPE = s +q2s+q2sŒq2sŒq2se(t) 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 fs/2. However, all these higher order harmonics must fold (alias) back into the Nyquist bandwidth and sum together to pr oduce 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 Page 2 of 7
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MT-001here is that the quantization noise is uncorrelated to the input signal. Under certain conditions where the sampling clock and the signal are ha rmonically related, the quantization noise becomes correlated, and the energy is concentrated in the harmonics of the signalŠhowever, the rms value remains approximately q/ 12. The theoretical signal-to-noise ratio can now be calculated assuming a full-scale input sinewave: ).ft2sin( 22q )t(v SinewaveFSInput N Eq. 5 The rms value of the input signal is therefore 22 2q inputFSofvaluerms N. Eq. 6 The rms signal-to-noise ratio for an ideal N-bit converter is therefore noiseonquantizati ofvaluerms inputFSofvaluerms log20SNR 10 Eq. 7 23log202log20 12/q 22/2q log20SNR 10N10N10 Eq. 8 SNR = 6.02N + 1.76dB, over the dc to f s/2 bandwidth. Eq. 9 Bennett's paper shows that although the actual spectrum of th e 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 emphasi ze again that the rms quantization noise is measured over the full Nyquist bandwidth, dc to f s/2. FREQUENCY SPETRUM OF QUANTIZATION NOISE 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. BW2flog10dB76.1N02.6SNR s10 , over the bandwidth BW. 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 qua ntization noise shaping and digital filtering are the key concepts in sigma-delta conv erters, although oversampling can be used with any ADC architecture. Page 3 of 7
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MT-001 fs2RMS VALUE = q12q = 1 LSB MEASURED OVER DC TO fs2q / 12 fs/ 2NOISE SPECTRAL DENSITYSNR = 6.02N + 1.76dB + 10log 10FOR FS SINEWAVE fs2ŁBW BWProcess Gain 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. cons ists of 416 30-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: dB3.30 10302 1065 log10 BW2 flog10GainocessPr 3610s10. Eq. 11 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 remove s the noise outside the signal bandwidth of interest, and therefore results in process gain per Eq. 10. Page 4 of 7
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MT-001ZONE3 0.5f sfs1.5f s2fs2.5f s3fs3.5f sBWZONE1 SNR = 6.02N + 1.76dB + 10log 10FOR FS SINEWAVE fs2ŁBW Process Gain fc Figure 4: Undersamp ling and Oversampling Combined Results in Process Gain CORRELATION BETWEEN QUANTIZATION NOISE AND INPUT SIGNAL YIELDS MISLEADING RESULTS T 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-inpu t signal under certain conditions. For instance, there is greater correlation for low amplitude pe riodic 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 signa l, just where you don't want them. Although the rms value of the noise is accurately approximated by q/ 12, its frequency domain content may be highly correlated to the ac-inpu t signal under certain conditions. For instance, there is greater correlation for low amplitude pe riodic 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 signa l, 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 qua ntization noise tends to be random. In spectral analysis applications (or in performing FFTs on ADCs using sp ectrally 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. In most practical applications, the input to the ADC is a band of frequencies (always summed with some unavoidable system noise), so the qua ntization noise tends to be random. In spectral analysis applications (or in performing FFTs on ADCs using sp ectrally 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 (80.000 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 2.111 MHz, showing a relatively random noise spectrum, where the SFDR is now about 93 dBc 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. 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 (80.000 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 2.111 MHz, showing a relatively random noise spectrum, where the SFDR is now about 93 dBc 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. Page 5 of 7
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MT-001 fS= 80.000 MSPS, f IN= 2.000 MHz fS= 80.000 MSPS, f IN= 2.111 MHz 0510 152025 303540 FREQUENCY -MHz 0510152025303540 FREQUENCY -MHz 0>“50>“100 >“150>“200 0>“50>“100 >“150 >“200(A) CORRELATED NOISE (B) UNCORRELATED NOISE SFDR = 77 dBc SFDR = 93 dBc SNR = 74 dBc SNR = 74 dBc Figure 5: Effect of Ratio of Sa mpling Clock to Input Frequency on Quantization Noise Frequency Spectru m 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 correl ation of the quantization error with the input frequency. In a practical ADC application, the quantization error generally app ears 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, becau se single-tone sinewave FFT testing of ADCs is one of the universally accepted methods of pe rformance evaluation. In order to accurately measure the harmonic distortion of an ADC, steps mu st 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 ex act same precautions apply to measuring DAC distortion with an analog spectrum analyzer. SNR, PROCESS GAIN, AND FFT NOISE FLOOR RELATIONSHIPS 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 10log10(4096/2) = 33 dB, thereby resulting in an overall FFT noise floor of 74 + 33 = 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 anal yzer’s noise floor can be reduced by narrowing Page 6 of 7

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