Quantization theorem

The Quantisierungstheorem provides in the context of signal theory in the quantization , this is the conversion of a continuous-value signal into a discrete-value signal, an indication of the error-free reconstructability of the original continuous-value signal. It represents the counterpart to the Nyquist-Shannon sampling theorem , which describes the limits for error-free reconstruction in the time domain during sampling . The quantization theorem was formulated in 1961 by Bernard Widrow with the help of the Fourier transformation of distribution densities and can be assigned to the field of statistical signal processing.

description

Zone scanning and spectral display

An amplitude-continuous and band-limited signal with a distribution density function , as shown in the figure on the right, is converted by the quantizer Q into an amplitude-discrete signal with the distribution density function . The continuous distribution function is converted into a discrete distribution density function by integration over the individual quantization intervals with the width Q, limited in the sketch by the dashed areas . The two associated frequency spectra and the distribution density functions, which are formed by the Fourier transformation and the discrete Fourier transformation , are shown by way of example in the sketch on the right with a red curve. Due to the discrete nature of the distribution density function , the associated spectrum has a periodic continuation with the quantization frequency . ${\displaystyle x}$ ${\displaystyle p_{X}(x)}$ ${\displaystyle y}$ ${\displaystyle p_{Y}(y)}$ ${\displaystyle p_{X}(x)}$ ${\displaystyle p_{Y}(y)}$ ${\displaystyle P_{X}(u)}$ ${\displaystyle P_{Y}(u)}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle p_{Y}(y)}$ ${\displaystyle P_{Y}(u)}$ ${\displaystyle u_{0}}$

The quantization theorem now states that if the quantization frequency is : ${\displaystyle u_{0}}$

${\displaystyle u_{0}={\frac {2\pi }{Q}}}$

twice as large as the highest frequency component in , the individual frequency components of the time-discrete distribution density function do not overlap. This case is shown in the figure below on the right. Only then is it possible to reconstruct the value-continuous distribution density function from the quantized distribution density function . ${\displaystyle P_{X}(u)}$ ${\displaystyle P_{Y}(u)}$ ${\displaystyle p_{X}(x)}$ ${\displaystyle p_{Y}(y)}$

If the double quantization frequency is less than double the frequency component in , there is a spectral overlap in the distribution of the discrete distribution density function and an error-free mapping to the value-continuous distribution density function is not possible. ${\displaystyle P_{X}(u)}$

swell

1. ↑ Bernhard Widrow: Statistical analysis of amplitude-quantized sampled-data systems . In: Transactions of the American Institute of Electrical Engineers , Part II: Applications and Industry . tape 79 , no. 6 , 1961, pp. 555-568 , doi : 10.1109 / TAI.1961.6371702 ( PDF ).

literature

• Bernard Widrow, István Kollár: Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications . Cambridge University Press, 2008, ISBN 978-0-521-88671-0 .