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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 product code, there is a range of input values that produce the same product. That range is called a quantum (Q) and is equal to the Least Significant Bit (LSB). IQ can be calculated by dividing the range of the ADC by the number of steps in the staircase.

(1) Query = 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 the output is called the measurement error. Therefore, the measurement error can be between -1/2Q and +1/2Q.

This error can be considered as quantitative noise with RMS:

(2) v_qn = Q/sqrt(12)

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

We know that the power of the quantization noise is v_qn^2, but where is it concentrated or distributed in the frequency domain? A quantization error creates harmonics in the signal that amplify well above the Nyquist frequency. Due to the sampling step of the ADC, these harmonics are folded into the Nyquist band, pushing the total noise energy into the Nyquist band and the white spectrum (which is evenly distributed across all band frequencies). Some converters specialize in oversampling (sampling well above the Nyquist frequency) to spread the noise over a wide band and then digitally filter it. Thus, the power of the noise can be reduced.

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

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

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

To calculate the Signal-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)

In fact, the word:

SNR = 6.02N + 1.76 (dB)

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

For a detailed explanation of this topic, with some great looking numbers and formulas, click here.

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