# Estimate Standard Deviation of Quantization Noise of Complex-Valued Signal

Quantizing a complex signal to $p$ bits of precision can be modeled as a linear system that adds normally distributed noise with a standard deviation of ${\sigma }_{N}=\frac{{2}^{-p}}{\sqrt{6}}$ [1,2].

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

```p = 14; theoreticalQuantizationNoiseStandardDeviation = fixed.complexQuantizationNoiseStandardDeviation(p);```

The returned value is ${\sigma }_{N}=\frac{{2}^{-p}}{\sqrt{6}}$.

Create a complex signal with $n$ samples.

```rng('default'); n = 1e6; x = complex(rand(1,n),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 = 2.4902e-05 ```

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

`theoreticalQuantizationNoiseStandardDeviation`
```theoreticalQuantizationNoiseStandardDeviation = 2.4917e-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.