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