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- What Is Quantization Noise and How to Use It to Calculate the SNR of a Digital Representation?
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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 staircase. For each output code, there is a range of input values that produce the same output. This range is called the quantum (Q) and is equivalent to the least significant bit (LSB). Q can be calculated by dividing the range of the ADC by the number of steps.

(1) Q = V_ref / 2^N.

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

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

This error can be thought of as RMS quantization noise:

(2) v_qn = Q/sqrt(12)

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

We know that the power of quantization noise is v_qn^2, but where in the frequency domain does it concentrate or disperse? The quantization error creates harmonics in the signal that extend well above the Nyquist frequency. Because of the ADC’s sampling rate, these harmonics are folded into the Nyquist band, pushing all the noise power into the Nyquist band and approximately the white spectrum (distributed equally across all bands). Some converters specifically operate in oversampling (sampling well above the Nyquist rate) to spread the noise over a wider band and then digitally filter it. In this way, the noise power can be reduced.

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

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

(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) leads to

SNR = 6.02 N + 1.76 (dB)

In fact, this term is:

SNR = 6.02 N + 1.76 (dB)

generalizes to any system using a digital representation. Thus, a microprocessor representing values with N bits has an SNR defined by the above formula.

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