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## What Is Quantization Noise and How to Use It to Calculate the SNR of a Digital Representation?

**What is quantization noise?**

When an ADC converts a continuous signal into a discrete digital representation, the transfer function is like a ladder. For each output code, there is a range of input values that produce the same output. That range is called the quantum (Q) and is equal to the least significant bit (LSB). Q can be calculated by dividing the range of the ADC by the number of steps in the ladder.

(1) Q = V_ref / 2^N.

In the above equation, N is the number of bits of the ADC and the input range can be between 0 and V_ref.

The difference between input and output is called quantization error. Therefore, the quantization error can be between -1/2Q and +1/2Q.

This error can be considered as the quantization noise with RMS:

(2) v_qn = Q/sqrt(12)

**What is the frequency spectrum of quantization noise?**

We know the quantization noise power is v_qn^2, but where is it concentrated or spread in the frequency domain? Quantization error creates harmonics in the signal that extend above the Nyquist frequency. Due to the ADC’s sampling stage, these harmonics are folded into the Nyquist band, pushing the total noise power into the Nyquist band and creating a nearly white spectrum (spread evenly over all frequencies in the band). Some converters specifically employ oversampling (sampling above the Nyquist frequency) to spread the noise over a wider band and then digitally filter it. Thus, the noise power can be reduced.

**How is signal-to-noise ratio (SNR) related to the number of bits in a digital representation?**

Assuming the input is sinusoidal with peak-to-peak amplitude V_ref , where V_ref is the reference voltage of the N-bit ADC (therefore, occupying the full-scale of the ADC), is its RMS value.

(3) V_rms = 2^NQ / (2*sqrt(2))

To calculate the signal-to-noise ratio, we divide the RMS of the input signal V_rms by the RMS of the quantization noise v_qn:

(4) SNR = 20log(V_rms / v_qn)

Substituting equations (2) and (3) into (4) will lead to

SNR = 6.02N + 1.76 (dB)

Actually, the word:

SNR = 6.02N + 1.76 (dB)

Generalizes to any system using digital representation. So, a microprocessor representing values with N bits will have the SNR defined by the formula above.

For a detailed explanation of this topic, including statistics and good-looking formulas, click here.

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