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fixed.complexQuantizationNoiseStandardDeviation

Estimate standard deviation of quantization noise of complex-valued signal

Description

example

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

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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 σN=2-p6 [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 σN=2-p6.

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

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

Data Types: double

Output Arguments

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Noise standard deviation, returned as a scalar.

Tips

fixed.complexQuantizationNoiseStandardDeviation is used in these functions.

Algorithms

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

σq2=2qq/2q/2e2de=q26=22precisionBits6.

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

σq=2precisionBits6.

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.

Introduced in R2021b