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When dealing with ADCs or DACs, you're bound to come across this oft-quoted formula for calculating a converter's theoretical signal-to-noise ratio (SNR). Rather than blindly trusting appearances, it is better to fundamentally understand where it comes from, as the formula contains subtleties that, if not delved into, can lead to misinterpretation of data sheet specifications and converter performance. Remember, this formula represents the theoretical performance of a perfect N-bit ADC. You can compare the actual SNR of an ADC to the theoretical SNR to see how it compares and differs.

overall architecture process

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An ideal converter digitizes a signal with a maximum error of ±½LSB, as shown in Figure 1 for the transfer function of an ideal N-bit ADC. For any ac signal spanning several LSBs, the quantization error can be approximated by an uncorrelated sawtooth waveform with peak-to-peak amplitude q (one LSB weight). Another way to look at this approximation is that the actual quantization error occurs at any point within ±½q with equal probability. While this analysis is not 100% accurate, it is accurate enough for most applications

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technical details

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In a classic 1948 paper by WR Bennett of Bell Laboratories, the actual spectrum of quantization noise was analyzed. Using the above simplifying assumptions, his detailed mathematical analysis can be simplified as shown in Figure 1. Following Bennett's classic paper, there are other important papers and books on converter noise. Figure 2 shows the quantization error versus input voltage in more detail. Again, a simple sawtooth waveform provides a sufficiently accurate model for analysis. The formula for calculating the sawtooth error is as follows:

Therefore, the root mean square quantization error is:

The sawtooth error waveform produces harmonics well beyond the Nyquist bandwidth from DC to fs/2, however, all of these higher order harmonics must fold back (alias) into the Nyquist bandwidth and add, yielding q/√ rms noise of 12. As Bennett pointed out, quantization noise is approximately Gaussian distributed almost uniformly over the Nyquist bandwidth from dc to fs/2. It is assumed here that the quantization noise is uncorrelated with the input signal. Under certain conditions, when the sampling clock, quantization noise versus time MT, and signal are harmonically correlated, the quantization noise will be correlated with the input signal, with energy concentrated in the harmonics of the signal, but still with an rms value of approximately q/√12. The theoretical signal-to-noise ratio can now be calculated with a full-scale input sine wave

The Bennett paper shows that although the actual spectrum of quantization noise is quite complex and difficult to analyze, the simplified analysis that leads to Equation 9 is accurate enough for most applications. However, it must be emphasized again that the rms quantization noise is measured over the full Nyquist bandwidth from dc to fs/2.

In many applications, the bandwidth BW occupied by the actual signal of interest is smaller than the Nyquist bandwidth (see Figure 3). If digital filtering is used to filter out noise components outside the bandwidth BW, a correction factor (called “processing gain”) must be included in the equation to reflect the resulting increase in SNR, as shown in Equation 10.

The process of sampling a signal at a rate that is more than twice the signal's bandwidth is called "oversampling." Oversampling, quantization noise shaping, and digital filtering are all important concepts in sigma-delta converters, but any ADC architecture can employ oversampling techniques

The meaning of processing gain can be illustrated by the following example. In many digital base stations or other wideband receivers, the signal bandwidth is made up of many independent channels, and an ADC digitizes the entire bandwidth. For example, the Analog Cellular Radio System (AMPS) in the United States consists of 416 channels of 30kHz bandwidth, occupying a bandwidth of about 12.5MHz. Assume a sampling rate of 65MSPS and use digital filtering to separate the individual 30kHz channels. Under these conditions, the processing gain resulting from oversampling is

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