Estimate Standard Deviation of Quantization Noise of Real-Valued Signal

Quantizing a real signal to $p$ bits of precision can be modeled as a linear system that adds normally distributed noise with a standard deviation of ${\sigma }_{N}=\frac{{2}^{-p}}{\sqrt{12}}$ [1,2].

Compute the theoretical quantization noise standard deviation with $p$ bits of precision using the fixed.realQuantizationNoiseStandardDeviation function.

p = 14;
theoreticalQuantizationNoiseStandardDeviation = fixed.realQuantizationNoiseStandardDeviation(p);

The returned value is ${\sigma }_{N}=\frac{{2}^{-p}}{\sqrt{12}}$.

Create a real signal with $n$ samples.

rng('default');
n = 1e6;
x = rand(1,n);

Quantize the signal with $p$ bits of precision.

wordLength = 16;
x_quantized = quantizenumeric(x,1,wordLength,p);

Compute the quantization noise by taking the difference between the quantized signal and the original signal.

quantizationNoise = x_quantized - x;

Compute the measured quantization noise standard deviation.

measuredQuantizationNoiseStandardDeviation = std(quantizationNoise)
measuredQuantizationNoiseStandardDeviation = 1.7607e-05

Compare the actual quantization noise standard deviation to the theoretical and see that they are close for large values of $n$.

theoreticalQuantizationNoiseStandardDeviation
theoreticalQuantizationNoiseStandardDeviation = 1.7619e-05

References

1. Bernard Widrow. “A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory”. In: IRE Transactions on Circuit Theory 3.4 (Dec. 1956), pp. 266–276.

2. Bernard Widrow and István Kollár. Quantization Noise – Roundoff Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge, UK: Cambridge University Press, 2008. 