Mean square quantization error: Difference between revisions

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{{Short description|Figure of merit for the process of analog to digital conversion.}}

Mean square quantization error is a figure of merit for

{{no footnotes|date=August 2016}}

the process of analog to digital conversion.

{{Use dmy dates|date=June 2025}}

'''Mean square quantization error''' (MSQE) is a [[figure of merit]] for the process of [[Analog-to-digital converter|analog to digital conversion]].

In this conversion process, analog signals in a [[interval (mathematics)|continuous range]] of values are converted to a discrete set of values by comparing them with a sequence of thresholds.

As the input is varied, the input's value is recorded when

The quantization error of a signal is the difference between the original continuous value and its discretization, and the mean square quantization error (given some [[probability distribution]] on the input values) is the [[expected value]] of the square of the quantization errors.

the digital output changes.

For each digital output, the input's difference from ideal

Mathematically, suppose that the lower threshold for inputs that generate the quantized value <math>q_i</math> is <math>t_{i-1}</math>, that the upper threshold is <math>t_i</math>, that there are <math>k</math> levels of quantization, and that the [[probability density function]] for the input analog values is <math>p(x)</math>. Let <math>\hat x</math> denote the quantized value corresponding to an input <math>x</math>; that is, <math>\hat x</math> is the value <math>q_i</math> for which <math>t_i-1\le x<t_i</math>.

is normalized to the value of the least significant bit,

Then

then squared, summed, and normalized to the number of samples.

:<math>

\begin{align}

\operatorname{MSQE}&=\operatorname{E}[(x-\hat x)^2]\\

&=\int_{t_0}^{t_k} (x-\hat x)^2 p(x)\, dx\\

&= \sum_{i=1}^k \int_{t_{i-1}}^{t_i} (x-q_i)^2 p(x) \,dx.

\end{align}

</math>

==References==

*{{citation|title=Digital Image Processing: An Algorithm Approach|first=Madhuri A.|last=Joshi|edition=3rd|publisher=PHI Learning Pvt. Ltd.|year=2006|isbn=9788120329713|page=12|url=https://books.google.com/books?id=sWRXkyLinQ4C&pg=PA12}}.

*{{citation|title=Image and Video Compression for Multimedia Engineering: Fundamentals, Algorithms, and Standards|first1=Yun Q.|last1=Shi|first2=Huifang|last2=Sun|edition=2nd|publisher=CRC Press|year=2008|isbn=9781420007268|page=38|url=https://books.google.com/books?id=ztXLBQAAQBAJ&pg=PA38}}.

{{Reflist}}

[[Category:Statistical deviation and dispersion]]

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