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

**What is Noise Quantization?**

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 sequence of input values that produces the same output. That quantity is called a quantum (Q) and is synonymous with 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 ADC bits and the input range can be between 0 and V_ref.

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

This error can be calculated with RMS as a quantization noise:

(2) v_qn = Q/sqrt(12)

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

We know that the power of the quantized noise is v_qn^2, but where in the frequency domain is it concentrated or spread? The quantization error creates a harmonic in the signal that extends above the Nyquist frequency. Due to the ADC’s sampling step, these harmonics are shifted to the Nyquist band, putting the total audio power in the Nyquist band and with a fairly white spectrum (equally spread over all frequencies in the band). Some converters work specifically at oversampling (sampling above the Nyquist frequency) to spread the sound over a wider bandwidth and then digitally filter it. In this way, the power of the sound can be reduced.

**How does Signal-Noise Ratio (SNR) relate to the number of bits in a digital representation?**

Assuming an input sinusoidal with peak-to-peak amplitude V_ref, where V_ref is the reference voltage of an N-bit ADC (thus, taking 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) will result in

SNR = 6.02N + 1.76 (dB)

Actually, the term:

SNR = 6.02N + 1.76 (dB)

generalizes to any system that uses a digital representation. Therefore, a microprocessor that represents values in N-bits will have an SNR defined by the above formula.

For a detailed explanation of this topic, with nice numbers and formulas, click here.

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