Quantization error

We shall establish an upper bound on error probability for the best code of rate B and block length N on this channel by considering a decoder that quantizes the output before decoding. Theorem 4-10 already provides a bound on error probability for such a quantized channel. We then find the limit of this bound as the quantization becomes infinitely fine. 

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 ). 

Since chemical reactions usually show significant nonadiabaticity, there are naturally quantitative errors in the predictions of the vibrationally adiabatic model. Furthermore, there are ambiguities about how to apply the theory such as the optimal choice of coordinate system. Nevertheless, this simple picture seems to capture the essence of the resonance trapping mechanism for many systems. We also point out that the recent work of Truhlar and co-workers24,34 has demonstrated that the reaction dynamics is largely controlled by the quantized bottleneck states at the barrier maxima in a much more quantitative manner than expected. 

There is a shortest reasonable exposure time that is related to the frequency of the heartbeat quartz and the adjustment of the voltage-to-frequency converters. Below the reasonable exposure interval quantization errors become a problem and the measured value will be chosen from a small number of possible steps. 

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... 

Finally, for the PT problem, dynamical friction effects have been examined for a model for a phenol-amine acid-base reaction in methyl chloride solvent [12]. With the quantization of the proton and the O-N vibration, the problem can be reduced to a one-dimensional solvent coordinate problem, similar to the ET case. Again, GH theory is found to agree with the MD results to within the error bars of the computer simulation. 

Figure 3. Watermark embedding using Costa s seheme with a scalar component codebook (SCS). The watermark letter e is embedded after dithered uniform scalar quantization of x and the addition of the scaled quantization error as watermark Wr,.

One of the first applications of CMD to a realistic and important system was to study the quantum dynamical effects in water. It was found that, even at 300 K, the quantum effects are remarkably large. This finding, in turn, led us to have to reparameterize the flexible water model (called the SPC/F2 model) in order to obtain good agreement with a variety of experimental properties for the neat liquid. An example of the large quantum effects in water can be seen in Fig. 3 in which the mean-spared displacement correlation function, ( x(t) - x(0) 2) is plotted. (These are new results which are better converged than those in Ref 34.) Shown are the quantum CMD and the classical MD results for the SPC/F2 model. The mean-squared displacement for the quantized version of the model is 4.0 X 10- m s-f while the classical value is 4.0 x 10 m s-f The error in these numbers is about 15%. These results suggest that quantum effects increase the diffusivity of liquid water by a factor of two. 

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 ... 

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... 

If coder-generated artifacts are spread in time in a way that they precede a time domain transition of the signal (e.g. a triangle attack), the resulting audible artifact is called pre-echo . Since coders based on filter banks always cause a spread in time (in most cases longer than 4 ms) of the quantization error, pre-echoes are a common problem to audio coding systems. 

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],... 

An example of a typical sequence of onset times is shown in Figure 9.12a. Implied in the figure is that in general there can be more than one onset time per analysis frame. Although any one of the onset times can be used, in the face of computational errors due to discrete Fourier transform (DFT) quantization effects, it is best to choose the onset time which is nearest the center of the frame, since then the resulting phase errors will be minimized. This procedure determines a relative onset time, which is in contrast... 

While the image is resolved into picture elements conversion of the brightness values into numerical values ( quantization ) also is required Figure 12 shows how the continuous density values black, through grey, to white are quantized on a scale from 0 to 255 since the values in each different range are converted on the same basis there may be some variation from the true basis this is known as the quantization error , and the calculation reported is the one that limits this error to a minimum.3... 

Although the usual quantization is mathematically convenient, it completely ignores uncertainties induced by unavoidable measurement errors around the boundaries between the individual intervals. This is highly unrealistic. Quantization forced by the limited resolution of the measuring instrument involved can be made more realistic by replacing the crisp intervals with fuzzy intervals or fuzzy numbers. This is illustrated for our example in Fig. 4b. Fuzzy sets are in this example fuzzy numbers expressed by the shown triangular membership functions, which express the linguistic... 

Measurements invariably involve errors and uncertainties. Only a few of these are due to mistakes on the part of the experimenter. More commonly, errors are caused by faulty calibrations or standardizations or random variations and uncertainties in results. Frequent calibrations, standardizations, and analyses of known samples can sometimes be used to lessen all but the random errors and uncertainties. In the limit, however, measurement errors are an inherent part of the quantized world in which we live. Because of this, it is impossible to peiform a chemical analysis that is totally free of errors or uncertainties. We can only hope to minimize errors and estimate their size with acceptable accuracy. In this and the next two chapters, we explore the nature of experimental errors and their effects on the results of chemical analyses. 

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