# Increasing SQNR by Increasing Bits Per Sample

December 3, 2019

Increasing the number of bits per sample will decrease the quantization noise and, therefore, increase SQNR. Theorem 3 quantifies this qualitative relationship for a full-scale 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 full-scale sinusoid input.

Let b be the number of bits used by a converter. Then:

(1) (2) #### Proof

(1) The power Ps of a sine waveform Asin(2𝜋𝑓 𝑡 + 𝜙) is A2/2

Proof: Let 𝜏 = 1/𝑓c  be one period of the sine waveform.

(3) (4) (5) by half-angle formula/squared identity (6) by half-angle 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 SQNRdB (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 2b>>1 (22) by property of logarithms (23)  ### Over-Sampling with Quantization Uniformly Distributed White Noise

Theorem 3 is a theoretical ideal. In practice, an N-bit 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 signal-to-noise-and-distortion ratio. Then:

(24) 