1. Introduction
The academic researches of dithering technology have gained multitudinous remarkable attention in ADC area before 1951. It is widely acknowledged that developing innovative dithering signals are worthwhile and significative. And a large number of researchers have made a lot of efforts to concentrate on this attractive topic.
In [1], Goodall found using dithering technology on video pulse code modulation (PCM) to decrease the quantization effects. Then Robert did further research on developing contour effects with noise. He proposed that dithering added to an input signal of the ADC and then subtracting the dithering prior to its quantization and it is the earliest theory of adding or subtracting dithering [2]. Before 1960s, dithering had been widely used and researched. The processes of using Rober’s theory-add an analog noise prior the input signal of quantizer and then subtract the noise after the quantization have many different discoveries. Widrow clarified statistical independence of the quantization noise and the input signal can minimize losses of quantization [3]. Schuchman studied the effect of dithering on the quantization noise, and he gave a statistical independence condition of quantization noise and input signal [4]. Spang and Schultheiss found that dithering can change the frequency content of ADC’ noise. In the processes of dithering added, although the noises of specified frequency are decreased, the total errors are increased [5]. Blesser proposed the concept of non-subtractive dithering [6] [7]. In 1984, Vanderkooy and Lipshitz found that dithering effect can improve the quantization resolution [8]. They analyze the application of dithering in videos from theoretical and experimental. They applied the research results to the audio signal and proved that dithering can transform the distortion signals into small amplitude signals. In 1987, Blesser and Locanthi firstly discovered the narrow-band dithering [9]. By 1990s, Mahmound Fawzy Wagdy did much research on dithering technology and ADC theoretical analysis including dither can improve the noise of ADC and non-li- nearity of ADC transfer function [10]-[16]. A growing number of scholars make numerous contributions to improve the performance of ADC, but nobody proposes the optimum general principle for ADC judgments [17]-[23].
Dithering technology is extraordinarily efficient way to make totals errors can be independent with input, and applying dithering is the direct method to obtain these results. A model of non-subtractive dithering, the classical model quantization (CMQ), can be described in Figure 1.
The total errors of this system are defined as the differences between the system output and system input, and are denoted by
. Total errors, without dither added to input signal prior the quantization, can make independence of the input signals are not possible in theoretically because of the inherent nonlinearities of ADC. It is widely used that added dither decrease the relativity in terms of input signals and 
to improve the dynamic effects of ADC.
This paper discusses the quantization error and dithering. The typical dithering signals are simulated in this paper as follow. Using the simulation to illustrate the results that reconstructed dithering can improve the performance of ADC, compared with traditional typical dithering signal to some extent.
2. Mathodology
2.1. System Modeling
Dithering, straightforwardly, is a random “noise” process added to a signal prior to its quantization in order to decrease the quantization error and improve the
![]()
Figure 1. The classical model of quantization (CMQ).
performance of ADC. The dithering signal generation quantization system is shown in Figure 2. In each model, system inputs are denoted by x and system output by y. Thus, 
represents the dithering signal in CMQ. We use 
to distinguish the quantizer input from system input.
In this particular model, the value of DFT is used to reconstruct the dithering signal. x(n) are the sampled sequence of input signal, and xq(n) represent the quantization result with the dithering signal added zero. Meanwhile, x(n) subtract xq(n) are the value of v(n). The detailed parameters of this model are illustrated in Figure 2 as below.
2.2. A Condition for Total Error Moments Is Independent with Input
It has been shown by Wannamaker that total error induced by a non-subtractive dither (NSD) quantization system can be independent with input signal if and only if the dither’s characteristic function (CF) or CF (the Fourier transform of its probability density function (PDF) or PDF [24] [25]) obeys a certain condition. Wannamaker’s theorem [26] is as follow:
Theorem [26]: In an NSD quantization system, 
is independent of the distribution of the system input 
for 
if and only if the CF of the dithering satisfied as follow.
, 
and 
is nonnegative integer. where
, and 
is the step of quantization.
From Wannamaker [26], the statistical analysis of total error
, the difference between system output and system input, can be described as follows.
Now, we define a rectangular window function as
(1)
![]()
Figure 2. The dithering signal generation quantization system.
And, an impulse train functions can be defined as
(2)
The conditional PDF of total error ![]()
can be calculated as follow:
![]()
(3)
Then, taking the Fourier transform of (3), we find that the CF of ![]()
is given by
![]()
(4)
In this NSD system, we choose the input![]()
, since an arbitrary signal can be formed by a series of coefficient of Fourier Transform. Then, if the sine function meets the condition of Theorem, any other signals must be satisfied.
And the PDF of input is![]()
.
![]()
(5)
From Figure 2, there is a hysteresis in this system, and a system without dithering added is illustrated in Figure 3.
It is assumed that the interval of input ![]()
is big enough, and input ![]()
will be independent with![]()
. The quantization noise ![]()
cannot make independence of the input![]()
, but it is independent with input![]()
.
Then, the quantization error ![]()
can be used with ![]()
to reconstruct a di- thering based on DFT.
From Bernard Widrow [27], the CF of ![]()
can be calculated as follow.
![]()
(6)
From Figure 2, the CF of ![]()
multiply the CF of ![]()
may constitute the CF of dithering, since ![]()
is independence with![]()
.
![]()
(7)
According to the theorem [26], ![]()
is independent with any distributed input![]()
. And the conditional moments of total errors can be written as follows.
![]()
(8)
We take (7) substitute into (4).
![]()
(9)
From (8) and (9), the first moment of![]()
, or mean error of![]()
, is zero for all input signals, which indicate that the ADC can be referred to as a linearized model by use this reconstructed dither. However, the second moments of![]()
, or error variance of![]()
, is not zero, indicating that variance is dependent with input. This is sometimes referred to as noise modulation.
3. Simulation Analysis
In this simulation , the ADC is 4 bits. The number of the frequency, sample rate, amplitude and numbers of samples in one period for the input signal are respectively 1/40, 5, 1 and 200. The sampled signal through the input signal can avoid aliasing effect. And, this paper use NSD in this system.
The simulation result of traditional dither (Triangular-PDF Dither, Gaussian- PDF Dither and Uniform-PDF Dither) compared with the reconstructed dither illustrate in Figures 4(a)-(e).
From Figure 4(a), the quantization results of non-dither added loses lots of information compared to original input signal. And this result can decrease the accuracy and the distortion of the signal during its reconstruction Digital-to- Analog Converter (DAC).
From Figures 4(b)-(e), we can infer that the quantization results of different dither can keep the integrality of the input in some extent. From Figure 4(b) and Figure 4(c), the results curves are smoother than others. There are more signals burr for the result of Gaussian-PDF dither from Figure 4(d). It can be seen that more distortion for the result of triangular-PDF dither from Figure 4(e).
4. Conclusion
In this paper, different dithering have been discussed. And the simulation show that the dithering signal can decrease quantization error and improve the dynamic efficiency of ADC. Moreover, we can see that the effects of the reconstructed dithering are improved compared with others in the performance of ADC. And the accuracies of quantization results for the reconstructing dithering signal are enhanced. Therefore, the application of this reconstructed dithering based on DFT may be used in communication, image and video signal processing.
Acknowledgements
This work is fully supported by and the Innovation Group Build Plan of Sichuan (No. 2015TD0022), and the Talents Project of Sichuan University of Science and Engineering (No. 2014RC13), and the Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province (2017WZJ01), the Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informa- tionalization and Internet of Things (2017WZJ01), and Sichuan University of Science and Engineering talent introduction project (2017RCL11).