fixed.complexQuantizationNoiseStandardDeviation

Estimate standard deviation of quantization noise of complex-valued signal

Since R2021b

Syntax

noiseStandardDeviation = fixed.complexQuantizationNoiseStandardDeviation(precisionBits)

Description

noiseStandardDeviation = fixed.complexQuantizationNoiseStandardDeviation(precisionBits) returns an estimate of the quantization noise standard deviation of a complex-valued signal with a quantization level q=2^{-precisionBits}, where precisionBits is the required number of bits of precision.

Examples

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 $${ϛ}_{\text{noise}}=\frac{2^{-p}}{\sqrt{6}}$$ [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 $${ϛ}_{\text{noise}}=\frac{2^{-p}}{\sqrt{6}}$$.

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.

Input Arguments

precisionBits — Required number of bits of precision

positive integer-valued scalar

Required number of bits of precision, specified as a positive integer-valued scalar.

Data Types: double

Output Arguments

noiseStandardDeviation — Noise standard deviation

scalar

Noise standard deviation, returned as a scalar.

Tips

fixed.complexQuantizationNoiseStandardDeviation is used in these functions.

• fixed.complexQRMatrixSolveFixedpointTypes

• fixed.complexQlessQRMatrixSolveFixedpointTypes

Algorithms

The variance of a complex-valued error sequence e(k) with quantization level q=2^{-precisionBits} [1][2] is

$${\sigma }_q^2=\frac{2}{q}{\displaystyle {\int }_{-q/2}^{q/2}{e}^{2}de=\frac{q^2}{6}=\frac{2^{-2precisionBits}}{6}}.$$

The standard deviation of a real error sequence e(k) is

$${\sigma }_q=\frac{2^{-precisionBits}}{\sqrt{6}}.$$

References

[1] Widrow, Bernard. "A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory." IRE Transactions on Circuit Theory 3, no. 4 (December 1956): 266-276.

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

Version History

Introduced in R2021b