Keywords: ADC, Delta-Sigma, ENOB, SNR, SINAD, THD, signal-to-noise ratio, signal-to-noise and distortion ratio, total harmonic distortion, effective number of bits, analog-to-digital converter
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APPLICATION NOTE 6854

Abstract: SNR, SINAD, THD, and ENOB values are common measures of the dynamic performance of ADCs, and ENOB can be calculated directly from the known values of SNR and THD.

The dynamic performance of an analog-to-digital converter (ADC) is determined by the effective number of bits (ENOB). In this application note, we examine the relationship of ENOB with other dynamic characteristics of ADCs such as signal-to-noise ratio (SNR), signal-to-noise and distortion ratio (SINAD), and total harmonic distortion (THD). We also compare the theoretically calculated ENOB of the MAX11216 24-bit high performance delta-sigma ADC to the measured value obtained in the lab.

**Signal-to-Noise Ratio**

Signal-to-noise ratio (SNR) is the fundamental frequency signal power level (P_{S}) to the noise power level (P_{N}) ratio and is mathematically expressed in Equation 1.

SNR = 10log(P_{S}/P_{N}) = 10logP_{S} – 10logP_{N}. (EQ. 1)

The ideal theoretical SNR is calculated directly from the resolution (N bit) as SNR = (6.02 × N + 1.76)dB. However, for delta-sigma ADCs like the MAX11216, which features internal programmable low-pass digital filters, the SNR can be increased by adjusting the amount of filtering. Filter more by reducing the filter bandwidth for a higher SNR, and filter less by increasing the filter bandwidth for a higher data rate.

**Figure 1** shows a fast Fourier transform (FFT) of the MAX11216 with continuous mode, sinc filter, and buffer features. P_{S} is the fundamental frequency signal power level and P_{N} is the noise power level, resulting in an SNR of 110.4dB at a data rate of 8Ksps. FFT is an analog spectrum analyzer that measures the amplitude of the fundamental frequency and its various harmonics as well as the non-harmonic spurious and noise components of a digitized signal.

*Figure 1. MAX11216 FFT with SNR = 110.4dB, sample rate = 8Ksps, f _{IN} = 1KHz, V_{AVDD} = 3.6V, V_{AVSS} = 0V, V_{REF} = 3.6V, T_{A} = +25°C, and external clock = 8.192MHz.*

**Signal-to-Noise and Distortion Ratio**

Signal-to-noise and distortion ratio (SINAD) is the fundamental frequency signal power level (P_{S}) to the noise plus distortion power level (P_{N+D}) ratio. SINAD is mathematically expressed as in equation 2.

SINAD = 10log[P_{S}/(P_{N+D}] = 10logP_{S} – 10logP_{N+D}.(EQ. 2)

Distortion includes the harmonics and spurious as shown in **Figure 2** of a MAX11216 FFT with continuous mode, sinc filter, and buffer features and SINAD = 109.4dB at a data rate of 8Ksps.

*Figure 2. MAX11216 FFT with SINAD = 109.4dB, sample rate = 8Ksps, f _{IN} = 1KHz, V_{AVDD} = 3.6V, V_{AVSS} = 0V, V_{REF} = 3.6V, T_{A} = +25°C, and external clock = 8.192MHz.*

**Total Harmonic Distortion**

Total harmonic distortion (THD) is the ratio of the fundamental signal power level to the power of the sum of its harmonics excluding noise. Generally, the first five harmonics contribute to the most distortion. Hence, in calculating THD, only the first five harmonics are used as shown in Equation 3.

THD (dB) = 10log(P_{S}) – 10log(P_{2} + P_{3} + P_{4} + P_{5} + P_{6})(EQ. 3)

where P_{S} and P_{2} to P_{6} are in mW.

For example, if P_{S} = P_{1} = 1mW, P_{2} = 0.1nW, P_{3} = 0.01nW, P_{4} = 0.001nW, P_{5} = 0.0001nW, and P_{6} = 0.00001nW, then the THD is calculated as follows:

THD(dB) = 10log(1) – 10log[(0.1 + 0.01 + 0.001 + 0.0001 + 0.00001) × 10^{-6}]

THD(dB) = -69.5074dB

If the sixth harmonic of 0.00002nW was counted, the THD would increase to -69.5070dB, which is very insignificant.

**Figure 3** shows the MAX11216 with a THD of -116.3dB, an input frequency of 1KHz, a sample rate of 8Ksps, and continuous mode, sinc filter, and buffer features.

*Figure 3. MAX11216 FFT with THD = 116.3dB, sample rate = 8Ksps, f _{IN} = 1KHz, V_{AVDD} = 3.6V, V_{AVSS} = 0V, V_{REF} = 3.6V, T_{A} = +25°C, and external clock = 8.192MHz.*

The effective number of bits (ENOB) is the number of bits when both noise and distortion are considered and is mathematically expressed in Equation 4.

ENOB = (SINAD – 1.76)/6.02 (EQ. 4)

To express ENOB in terms of SNR and THD, see the following calculations:

- Use Equation 2 and Equation 1 as follows to determine Equation 5:

SINAD = 10log[P_{S}/(P_{N+D}] = 10logP_{S}– 10logP_{N+D}.

SNR = 10log (P_{S}/P_{N})

log (P_{S}/P_{N}) =^{SNR/10}

P_{S}/P_{N}= 10^{SNR/10}

P_{N}/P_{S}= 10^{-SNR/10}(EQ. 5)

- Equation 5 can be also expressed as follows:

P_{D}/P_{S}= 10^{-THD/10}(EQ. 6) - Add Equation 5 and Equation 6 to determine Equation 7.

(P_{N}+P_{D})/P_{S}= 10^{-SNR/10}+ 10^{-THD/10}

P_{S}/(P_{N+D}) = 1/(10^{-SNR/10}+ 10^{-THD/10}) = (10^{-SNR/10}+ 10^{-THD/10})^{-1}(EQ. 7) - Substitute Equation 7 into Equation 2.

SINAD = 10log(10^{-SNR/10}+ 10^{-THD/10})^{-1}

= -10log(10^{-SNR/10}+ 10^{-THD/10}) (EQ. 8) - Finally, substitute Equation 8 into Equation 4 to obtain the ENOB equation in terms of SNR and THD as follows:

ENOB = {[-10log(10^{-SNR/10}+ 10^{-THD/10})] – 1.76}/6.02 (EQ. 9)

Table 1 lists the simulated MAX11216 SNR values for different data rates with a sinc digital filter where V_{IN} = 0V, V_{AVDD} = 3.6V, V_{AVSS} = 0V, V_{REF} = 3.6V, and T_{A} = +25°C.

**Table 1. Simulated Continuous Mode SNR (dB) Versus Data Rate and PGA Gain with Sinc Filter**

Data Rate (sps) | Buffer | Gain = 1 | Gain = 8 | Gain = 128 |
---|---|---|---|---|

LN | LN | LN | ||

1.9 | 139.8 | 136.5 | 140.5 | 128 |

3.9 | 139.4 | 135.3 | 139.2 | 125.4 |

7.8 | 139.2 | 134.7 | 137.9 | 122.5 |

31.2 | 135.8 | 132.1 | 134.3 | 116.5 |

62.5 | 133.8 | 129.4 | 131.5 | 114 |

500 | 125.8 | 121.2 | 123.3 | 104.8 |

1000 | 123.1 | 118.5 | 120.4 | 101.8 |

4000 | 117.7 | 113.1 | 115.1 | 96.5 |

16000 | 114.6 | 109.9 | 112 | 93.4 |

64000 | 107.8 | 103.2 | 105.8 | 88.2 |

**Table 2** lists the calculated ENOB values using Equation 9 based on the SNR values where THD = 120dB (typical specification in the MAX11216 data sheet), V_{IN} = 0V, V_{AVDD} = 3.6V, V_{AVSS} = 0V, V_{REF} = 3.6V, and T_{A} = +25°C.

**Table 2. Calculated Continuous Mode ENOB (Bits) Versus Data Rate and PGA Gain with Sinc Filter**

Data Rate (sps) | ENOB | |||
---|---|---|---|---|

Buffer | Gain = 1, Low Noise | Gain = 8, Low Noise | Gain = 128, Low Noise | |

1.9 | 19.6336811 | 19.62522357 | 19.63479484 | 19.53506307 |

3.9 | 19.6329602 | 19.62021353 | 19.63257437 | 19.45838575 |

7.8 | 19.63257437 | 19.61715616 | 19.62958982 | 19.3193091 |

31.2 | 19.62246601 | 19.59803115 | 19.61487866 | 18.79339941 |

62.5 | 19.6117322 | 19.56278581 | 19.59185023 | 18.4828525 |

500 | 19.4727366 | 19.23395095 | 19.3643726 | 17.09481505 |

1000 | 19.35361828 | 19.00585701 | 19.17360407 | 16.60710288 |

4000 | 18.92511435 | 18.36098144 | 18.62499223 | 15.73432624 |

16000 | 18.56137578 | 17.89619095 | 18.20615941 | 15.22101479 |

64000 | 17.57240744 | 16.83558105 | 17.25547296 | 14.35832751 |

**Table 3** lists the measured ENOB values.

**Table 3. Measured Continuous Mode ENOB (Bits) Versus Data Rate and PGA Gain with Sinc Filter**

Data Rate (sps) | Buffer | Gain = 1 | Gain = 8 | Gain = 128 |
---|---|---|---|---|

LN | LN | LN | ||

1.9 | 24.6 | 25.2 | 24.8 | 24.5 |

3.9 | 23.4 | 24.7 | 23.9 | 24.4 |

7.8 | 23.6 | 23.4 | 23.3 | 23.1 |

31.2 | 22.3 | 22.3 | 22.1 | 22 |

62.5 | 21.6 | 21.7 | 21.5 | 21.4 |

500 | 20.2 | 20.1 | 20.2 | 20 |

1000 | 19.7 | 19.6 | 19.5 | 19.3 |

4000 | 18.8 | 18.8 | 18.7 | 18.5 |

16000 | 18.3 | 18.8 | 18.5 | 18.6 |

64000 | 17.2 | 17.3 | 17.3 | 17.3 |

**Figure 4**, **Figure 5**, **Figure 6**, and **Figure 7** compare the measured and calculated ENOB values for the buffer, gain = 1, gain = 8, and gain = 128.

*Figure 4. MAX11216 calculated and measured ENOB with sinc filter, continuous mode, and buffer.*

*Figure 5. MAX11216 calculated and measured ENOB with sinc filter, continuous mode, and gain = 1.*

*Figure 6. MAX11216 calculated and measured ENOB with sinc filter, continuous mode, and gain = 8.*

*Figure 7. MAX11216 calculated and measured ENOB with sinc filter, continuous mode, and gain = 128*

Figure 8 shows the measured ENOB values versus the data rates for the buffer and various PGA gains.

*Figure 8. MAX11216 measured ENOB with the data rates for the buffer and various PGA gains.*

The SNR, SINAD, THD, and ENOB values are common measures of the dynamic performance of ADCs. The ENOB can be easily and accurately calculated based on the signal-to-noise ratio (SNR) and the total harmonic distortion (THD). The MAX11216 ENOB values obtained in the lab confirm that the measured data matches closely with the values calculated based on the ENOB equation. For delta-sigma ADCs with internal programmable digital filters, the measured data also confirms that increasing the amount of filtering increases the SNR and ENOB.