# 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) (6) (7) (2) For SQNR:

(8) (9) (10) (11) (12) (3) For SQNRdB:

(13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (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) 