# 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 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 šš of a sine waveform Asin(2ššcĀ š” + š) is A2/2

Proof: Let š = 1/šcĀ be one period of the sine waveform.

(3)

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(6)

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(2) For SQNR:

(8)

(9)

(10)

(11)

(12)

(3) For SQNRdB:

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

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(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)