Quantization (signal processing)

A quantized signal curve in red. In gray the associated value-continuous course.

The quantization in the digital signal processing an image received by the digitization of analog signals and the compression is used by images and videos. The resulting deviation is called the quantization deviation .

An electronic component or function that performs this mapping of a value or a signal is called a quantizer .

General

In practical use, during quantization, a physical variable is often converted into an electrical signal within the scope of a measurement in accordance with a measurement principle for further processing and thus determined quantitatively. In the field of metrology, for example, a value with a limited number of digits is determined by reading a pointer position seen on the scale of an analog measuring device , which corresponds to a quantization. By writing down, the measured values can be stored in tables and processed numerically (e.g. calculation of the monthly average temperature from hourly individual measurements). Nowadays, such calculations are practically only carried out in computers , and the measured variable is quantized in an analog-digital converter . Various types of measurement errors occur here, such as linearity and quantization errors ; the latter can still cause quantization noise .

procedure

An analog signal can in principle be quantized continuously. In the context of digitization, however, the sampling takes place first , whereby it is converted into a time-discrete, value-continuous signal. This can be represented in signal theory by multiplying it with a Dirac comb . In practice, sample-and-hold circuits can be used for this purpose.

To discretize the values, the measurement range of the input variable is divided into a finite number of adjacent intervals and each is assigned a quantization level . The time-discrete signal is now converted to the individual levels via the quantization characteristic . The limits of quantization are formulated within the framework of the quantization theorem.

Often the quantized signal is then encoded , i. H. a unique number is assigned to each quantization level. In contrast to the actual quantization, this process is reversible. During the reconstruction, the values coded in this way are mapped back into values from the measurement range of the original signal.

The now time and value discrete signal is called a digital signal .

properties

How a quantizer maps the input signals in detail can be read from the quantization characteristic.

The resolution of the quantizer can be described by

the number of quantization levels ,
the word length, i. S. v. Number of bits that is at least necessary to represent the quantized values or
with a linear quantization also the size of the intervals.

The time interval between two successive sampling points is determined by the sampling rate .

Since a large amount of input values is mapped to a smaller amount during quantization , it is non-linear and also not reversible , because different input values can be mapped onto the same output value. However, quantization in equidistant steps is often called linear quantization in technician jargon (somewhat imprecisely) . In that terminology, there are non-linear quantization characteristics . Such non-linear quantization characteristics can be used to improve the dynamic range in the case of signals perceived as non-linear (e.g. audio signals), especially in the case of low resolutions .

In addition to sampling, quantization is a step in the digitization (analog-digital conversion) of analog signals. The most common and also the simplest type is scalar quantization. A scalar input value is mapped onto a scalar output value.

Procedure

The input value is rounded up or down to the nearest quantization level . Often the level with the smallest distance is mapped and thus has the lowest amount of quantization error. The quantization function is then:

{\ displaystyle Q (x) = \ operatorname {sgn} (x) \ cdot \ Delta \ cdot \ left \ lfloor {\ frac {\ left | x \ right |} {\ Delta}} + {\ frac {1} {2}} \ right \ rfloor}

With uniform quantization, the step size is a real constant with any value greater than 0 and indicates the length of the interval. ${\ displaystyle \ Delta}$

The step size is set to the value 1 for mapping to integer intervals . If the step size is sufficiently small in relation to the measuring range, the mean square deviation ( MSE ) according to the quantization is: ${\ displaystyle \ Delta}$

{\ displaystyle \ mathrm {MSE} = \ sigma ^ {2} \ approx {\ frac {\ Delta ^ {2}} {12}}}

which in this case is equal to the variance . In this context, the mean square deviation is also referred to as a quantization deviation. ${\ displaystyle \ sigma ^ {2}}$

Alternatively, the input value can also be rounded up or down, but then the mean error increases.

Lossy Compression

With lossy compression methods, the loss of information comes from the quantization of the input data. The attempt is made to remove "unimportant" information by encoding the signal with a partially reduced resolution.

Popular representatives for such compression techniques are MP3 , JPEG and MPEG .

Individual evidence

↑ BM Oliver, JR Pierce, Claude E. Shannon: The Philosophy of PCM . tape 36 . Proceedings of the IRE, November 1948, pp. 1324-1331 , doi : 10.1109 / JRPROC.1948.231941 .

Web links

The Scientist and Engineer's Guide to Digital Signal Processing. Retrieved April 10, 2013 .

literature

John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Pearson Education International, 2002, ISBN 0-13-095007-6 , Chapter 6.5, pp. 290 .
John G. Proakis, Dimitris G. Manolakis: Digital Signal Processing . 3. Edition. Prentice Hall, 1996, ISBN 0-13-394289-9 , Chapter 9.2, pp. 750 ff .

[Shannon1-1] BM Oliver, JR Pierce, Claude E. Shannon: The Philosophy of PCM . tape 36 . Proceedings of the IRE, November 1948, pp. 1324-1331 , doi : 10.1109 / JRPROC.1948.231941 .