ADC SNR Gain

ADC SNR Gain

Why the ADC SNR Formula is 6.02N + 1.76 dB

The full-scale quantization noise model assumes:

• The quantization step is Δ = 2VFS / 2N
• Quantization noise is uniformly distributed from -Δ/2 to +Δ/2

1. Quantization Noise Power

A uniform noise distribution over width Δ has variance:

σ² = Δ² / 12

2. Signal Power (Full-Scale Sine Wave)

The RMS value of a sine wave with peak amplitude VFS is:

V_rms = V_FS / √2

Signal power:

P_s = V_FS² / 2

3. SNR Calculation

Using SNR = Ps / Pq, and substituting the expressions for signal and noise:

SNR = 3 · 2^2N

4. Convert SNR to dB

SNR_dB = 10log(3) + 20N log(2)

Using constants:

• log₁₀(2) = 0.3010 → 20 log(2) = 6.02 dB
• 10 log(3) = 4.77 dB

After RMS correction for sine vs. square wave (−3.01 dB), the final expression becomes:

SNR_ADC = 6.02N + 1.76 dB

This is the standard “ideal ADC SNR” formula.

 

1. ADC SNR Gain (Resolution-Based SNR Gain)

Increasing the ADC resolution increases SNR. Using:

SNR_ADC = 6.02N + 1.76 dB

A 1-bit increase in resolution gives:

ΔSNR = 6.02 dB per bit

 

Every extra ADC bit = +6.02 dB SNR improvement

Example:
12-bit → 14-bit = +2 bits → +12.04 dB SNR

 

2. Sampling SNR Gain (Oversampling Gain)

When sampling faster than Nyquist, quantization noise spreads over a larger bandwidth, while the signal stays within its own bandwidth.

Oversampling Ratio (OSR)

OSR = f_s / (2B)

SNR improvement due to oversampling:

SNR gain = 10 log₁₀(OSR)

For quantization noise, doubling the sampling rate gives:

SNR gain = 3 dB per 2× oversampling

Convert Oversampling to ENOB

ΔENOB = (1/2) log₂(OSR)

4× oversampling = +6 dB SNR = +1 extra ENOB 

16× oversampling = +12 dB SNR = +2 extra ENOB

 

Summary Table

Improvement Method	Rule	SNR Gain	ENOB Gain
Increase ADC resolution	+1 bit	+6.02 dB	+1 bit
Oversampling (OSR)	2× oversampling	+3 dB	+0.5 bits
	4× oversampling	+6 dB	+1 bit
	16× oversampling	+12 dB	+2 bits

ENOB (Effective Number of Bits) Basics

Let's go through an example where we start with a fixed bit resolution for an ADC (say, 12 bits), and then after some processing (such as noise reduction), we calculate the ENOB (Effective Number of Bits). This will help illustrate how processing can improve the effective resolution of a system.

Scenario

1. Fixed Bit Resolution of ADC: You have an ADC with a nominal resolution of 12 bits. This means the ADC can theoretically distinguish 2^12 = 4096 levels of the input signal. However, due to noise in the system, the actual precision is lower than this.

2. Initial SNR (before processing): Before applying any processing (like noise reduction), the Signal-to-Noise Ratio (SNR) is measured to be 45 dB. This is the signal quality you start with.

3. Processing (Noise Reduction): After applying noise reduction or other signal enhancement techniques (e.g., filtering, averaging), the SNR improves to 55 dB.

Step 1: Initial ENOB (Before Processing)

First, let’s calculate the ENOB before processing, based on the initial SNR of 45 dB. We use the formula for ENOB:

ENOB = (SNR - 1.76) / 6.02

For the initial SNR = 45 dB:

ENOB_before = (45 - 1.76) / 6.02 = 43.24 / 6.02 = 7.19 bits

Interpretation: The 12-bit ADC has an SNR of 45 dB, which means the effective resolution is only 7.19 bits. This means, due to noise, the ADC is only able to provide the precision of a system that could distinguish between 2^7.19 ≈ 150 levels (instead of the ideal 4096 levels).

Step 2: ENOB After Processing (Improved SNR)

Now, let’s calculate the ENOB after processing, when the SNR improves to 55 dB.

ENOB_after = (55 - 1.76) / 6.02 = 53.24 / 6.02 = 8.86 bits

Interpretation: After applying processing that improves the SNR to 55 dB, the effective resolution increases to 8.86 bits. This means the ADC is now effectively distinguishing between 2^8.86 ≈ 250 levels—an improvement from the 150 levels at 45 dB SNR.

Step 3: Comparing the Results

Let’s summarize the difference before and after processing:

• Before processing: With an SNR of 45 dB, the ENOB is 7.19 bits.
• After processing: With an improved SNR of 55 dB, the ENOB increases to 8.86 bits.

Conclusion: This demonstrates the impact of noise reduction or processing: the effective resolution (ENOB) has increased, making the system more accurate. Even though the ADC has a fixed resolution of 12 bits, noise was reducing the actual usable resolution. By improving the SNR through processing, we’ve boosted the effective precision of the ADC.

Summary

Fixed Bit Resolution (Nominal): 12 bits

Initial SNR: 45 dB → ENOB = 7.19 bits

After Processing (Improved SNR): 55 dB → ENOB = 8.86 bits

Further Reading

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