# John Errington's Data Conversion Website

# Quantization Error and Signal - to - Noise Ratio calculations

The signal to noise ratio of a quantized signal is 2+6*(no of bits), as shown in the following table.

## Resolution and Signal to Noise Ratio for signals coded as n bits |
|||||

bits, n |
levels, 2n | Weighting of LSB, 2-n | SNR, dB | ||

1 | 2 | 0.5 | 8 | ||

2 | 4 | 0.25 | 14 | ||

3 | 8 | 0.125 | 20 | ||

4 | 16 | 0.0625 | 26 | ||

5 | 32 | 0.03125 | 32 | ||

6 | 64 | 0.01563 | 38 | ||

7 | 128 | 0.00781 | 44 | ||

8 | 256 | 0.00391 | 50 | ||

9 | 512 | 0.00195 | 56 | ||

10 | 1024 | 0.00098 | 62 | ||

11 | 2048 | 0.00048 | 68 | ||

12 | 4096 | 0.00024 | 74 | ||

13 | 8192 | 0.00012 | 80 | ||

14 | 16384 | 0.00006 | 86 | ||

15 | 32768 | 0.00003 | 92 | ||

16 | 65536 | 0.00001 | 98 |

These values are for a signal matched to the full-scale range of the converter. If a signal with a range of 5V is measured by an 8 bit ADC with a range of 10V then only 7 bits are effectively in use, and a signal to noise ratio of 44 rather than 50 will apply.

#### Proof:

Suppose that the instantaneous value of the input voltage is measured by an ADC with a Full Scale Range of Vfs volts, and a resolution of n bits. The real value can change through a range of q = Vfs / 2n volts without a change in measured value occurring.

The value of the measured signal is Vm = Vs - e, where

Vm is the measured value,

Vs is the actual value, and

e is the error.

The maximum value of error in the measured signal is

emax = (1/2)(Vfs / 2n) or emax = q/2 since q = Vfs / 2n

The RMS value of quantization error voltage is

whence

The Signal to Noise Ratio (SNR) is defined as

It is normally quoted on a logarithmic scale, in deciBels ( dB ).

or | ||

The RMS signal voltage is then | ||

The error, or quantization noise signal is | ||

Thus the signal - to - noise ratio in dB. is | ||

since Vfs = 2n q, then | ||

which simplifies to |

N.B. This equation is true only if the input signal is exactly matched to the Full Scale Range of the converter. For signals whose amplitude is less than the FSR the Signal - to - Noise Ratio will be reduced.

**Download a .pdf file of the analysis of quantization error and signal to noise ratio**