Increasing SQNR by Increasing Bits Per Sample
December 3, 2019
Back to: Sampling & Reconstruction
Increasing the number of bits per sample will decrease the quantization noise and, therefore, increase SQNR. Theorem 3 quantifies this qualitative relationship for a fullscale sine waveform input. Take note that the SQNR increases approximately 6dB for every additional bit.
Theorem 3
Let SQNR be the signal to quantization noise ratio for a fullscale sinusoid input.
Let b be the number of bits used by a converter. Then:
(1)
(2)
Proof
(1) The power P_{s} of a sine waveform Asin(2๐๐cย ๐ก + ๐) is A2/2
Proof: Let ๐ = 1/๐cย be one period of the sine waveform.
(3)
(4)
(5) 
by halfangle formula/squared identity 
(6) 
by halfangle formula/squared identity 
(7)
(2) For SQNR:
(8) 
by definition of SQNR 
(9) 
by item (1) above and the proposition Var v(n) 
(10) 
from the lesson Quantization and Noise 
(11) 
from the lesson Quantization and Noise 
(12)
(3) For SQNRdB:
(13) 
by the definition of SQNR_{dB} 
(14) 
by proof (2) 
(15) 
by property of logarithms 
(16) 
by property of logarithms 
(17) 

(18) 
by property of logarithms 
(19) 
by property of logarithms 
(20) 
by property of logarithms 
(21) 
for 2^{b}>>1 
(22) 
by property of logarithms 
(23) 
OverSampling with Quantization Uniformly Distributed White Noise
Theorem 3 is a theoretical ideal. In practice, an Nbit converter has issues with inherent noise and doesn’t provide N bits of precision. Given this, system engineers want to know how many effective bits they can achieve. The b in Theorem 3 is equated in terms of SQNRdB to yield the following quantity called ENOB.
Let SINAD be the signaltonoiseanddistortion ratio. Then:
(24)