Quantization noise

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

A pH/Ion-meter with a resolution of only 0.1 mV is not sufficient because the ensuing quantization noise introduces an apparent deviation of at least 0.2%, and, more important in this particular case, these systematic effects lead to a bias that is strongly dependent on small shifts in 0. (See Fig. 4.24, left side.)... 

Horlick, G., Reduction of Quantization Effects by Time Averaging with Added Random Noise, Anal. Chem. 47, 1975, 352-354. 

This source of noise is not usually called noise in most technical contexts it is more commonly called error rather than noise, but that is just a label since it is a random contribution to the measured signal, it qualifies as noise just as much as any other noise source. So what is this mystery phenomenon It is the quantization noise introduced by the analog-to-digital (A/D) conversion process, and is engendered by the fact that for... 

The noise in the sensor signal is 0.5 digits. This corresponds to a 1-digit quantization noise of the A/D converter. At low concentration levels, the noise is equivalent to a CO-concentration variation of 0.03 ppm. Multiplying this value by three, the Emit of detection was assessed to be 0.1 ppm. The gas concentration resolution amounts... 

The spectral components are quantized and coded with the aim of keeping the noise, which is introduced by quantizing, below the masked threshold. Depending on the algorithm, this step is done in very different ways, from simple block companding to analysis-by-synthesis systems using additional noiseless compression. 

Especially in the case of low bit-rates (implying that a lot of quantization noise is introduced), the filter characteristics of the analysis and synthesis filters as determined by the prototype window / windowing function are a key factor for the performance of a coding system. 

In an analysis/synthesis filter bank, all quantization errors on the spectral components show up on the time domain output signal as the modulated signal multiplied by the synthesis window. Consequently, the error is smeared in time over the length of the synthesis window / prototype filter. As described above, this may lead to audible errors if premasking is not ensured. This pre-echo effect (a somewhat misleading name, a better word would be pre-noise) can be avoided if the filter bank is not static, but switched between different frequency/time resolutions for different blocks of the overlap/add. An example of this technique called adaptive window switching is described below. 

Non-uniform scalar quantization. While usually non-uniform scalar quantization is applied to reduce the mean squared quantization errors like in the well known MAX quantizer, another possibility is to implement some default noise shaping via the quantizer step size. This is explained using the example of the quantization formula for MPEG Layer 3 or MPEG-2 Advanced Audio Coding ... 

In this case, bigger values are quantized less accurately than smaller values thus implementing noise shaping by default. 

The bit allocation is derived from the SMR-values which have been calculated in the psychoacoustic model. This is done in an iterative fashion. The objective is to minimize the noise-to-mask ratio over every subband and the whole frame. In each iteration step the number of quantization levels is increased for the subband with the worst (maximum) noise-to-mask ratio. This is repeated until all available bits have been spent. 

After the anti-aliasing filter, the analog/digital converter (ADC) quantizes the continuous input into discrete levels. ADC technology has shown considerable improvement in recent years due to the development of oversampling and noise-shaping converters. However, a look at the previous technologies [Blesser, 1978] [Blesser and Kates, 1978][Fielder, 1989] will help appreciate the current state-of-the-art. 

Following the discussion in Bennett ( [Bennett, 1948]), we define the Signal to Noise Ratio (SNR) for a signal with zero mean and a quantization error with zero mean as follows first, we assume that the input is a sine wave. Next, we define the root mean square (RMS) value of the input as... 

The interpolation processor has a double buffered memory that permits updates during processing. Note that interpolation happens on every sample, thereby avoiding zipper noise due to coefficient quantization. This is an expensive strategy, however it always works. 

Roberts, 1976] Roberts, L. G. (1976). Picture Coding Using Pseudo-Random Noise. In Jayant, N. S., editor, Waveform Quantization and Coding, pages 145-154. IEEE Press. Reprinted from IEEE Trans, on Information Theory, vol. IT-8, Feb. 1962. 

Dual to prediction in time domain (with the result of flattening the spectrum of the residual), applying a filtering process to parts of the spectrum has been used to control the temporal shape of the quantization noise within the length of the window function of the transform [Herre and Johnston, 1996],... 

Noise allocation followed by scalar quantization and Huffman coding. In this method, no explicit bit allocation is performed. Instead, an amount of allowed noise equal to the estimated masked threshold is calculated for each scalefactor band. The scalefactors are used to perform a coloration of the quantization noise (i.e. they modify the quantization step size for all values within a scalefactor band) and are not the result of a normalization procedure. The quantized values are coded using Huffman coding. The whole process is normally controlled by one or more nested iteration loops. The technique is known as analysis-by-synthesis quantization control. It was first introduced for OCF [Brandenburg, 1987], PXFM [Johnston, 1989b] and ASPEC [Brandenburg et al., 1991], In a practical application, the following computation steps are performed in an iterative fashion ... 

The quantization noise is calculated by subtracting the reconstructed from the unquantized signal values and summing the energies per scalefactor band. 

Other coding tools in Layer 3 include a different (nonuniform) quantizer, analysis-by-synthesis control of the quantization noise and Huffman coding of the quantized values to increase the coding efficiency. All these have already been described earlier in this chapter. 

It is important to remember that this equation depends on the assumption that the quantizer is a fixed point, mid-tread converter with sufficient resolution so that the resulting quantization noise (enoise) is white. Furthermore, the input is assumed to be a full scale sinusoidal input. Clearly, few real world signals fit this description, however, it suffices for an upper bound. In reality, the RMS energy of the input is quite different due to the wide amplitude probability distribution function of real signals. One must also remember that the auditory system is not flat (see the chapter by Kates) and therefore SNR is at best an upper bound. 

Dither. Starting from Robert s pioneering paper [Roberts, 1976], the use of dither in audio was seriously analyzed by Vanderkooy and Lipshitz [Vanderkooy and Lipshitz, 1984], The basic idea is simple to whiten the quantization error, a random error signal is introduced. While the introduction of noise will make the signal noisier , it will also decorrelate the quantization error from the input signal (but not totally). Vanderkooy and Lipshitz also propose the use of triangular dither derived from the sum of two uniform random sources [Vanderkooy and Lipshitz, 1989],... 

Oversampling converters. Bennett s [Bennett, 1948] pioneering paper points out that the quantization noise is integrated over a frequency range. For a spectrally flat (i.e., white) signal the noise power is given by the following equation ... 

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