The conversion of a signal from analog to digital generally comprises two processes: sampling, which converts a signal (a function of time) into a discrete-time signal (a sequence of real numbers); and quantization of the amplitude values, a nonlinear process that assigns a binary code to each signal sample [1]. In quantization designs, under certain conditions (e.g., Quantization Theorem I and II in [2]), from a point of view of moments, the quantization error can be modeled as an additive random process, independent of the input signal and distributed uniformly [2]. This linear behavior is highly desirable in many circumstances [2]- [5].

However, if the input has magnitude values comparable to the quantizer step, the error signal e cannot be modeled as an additive random process due to its strongly dependence on the input [6]. The most common approach to solve it has been the use of Dithered quantizer because it satisfies the QT II, thus controlling the statistical properties of the quantization error and their relationship to the input signal of the system by adding a random signal, called the Dither signal, to the input signal before quantization [3], [7]. It is widely employed, for example in audio and image processing applications to reduce perceptual artifacts [8]; in ADCs to improve the resolution [9], [10] and security [11]; in sigma-delta digital modulators [12]-[14]; in Watermark to hide data [15] in Light detection [16,17]; and in finite frames [18]-[20]. Recently, Dither quantization has been used in the quantization of random linear mapping outputs, particularly in compressed sensing [21]-[23] and in spatial Dithering for graph signal to find statistically good sampling sets [24]. Despite the benefits of the Dither quantizer, one of its disadvantages is computational complexity to generate random processes (Dither signals) subject to an arbitrary joint probability distribution [7] because, in practice, they are commonly generated as a combination of n uniform independent, identically distributed, random variables. Sanyal and Sun [25] proposed a new method for adding Dither signals to vector quantizers, which achieves a better mismatch shaping performance with low hardware usage compared to existing Dither techniques. Akyol and Rose [5] developed a constrained Dither quantizer that outperforms the conventional Dither quantizer with similar complexity. In [26], it was demonstrated how the resolution of a uniform quantizer increases by using a deterministic Dither signal, and in [27], the authors designed an alternative Dithering scheme that replaced the Dithering addition by an orthogonal transformation.

Given this limitation, a new framework called Quantum Signal Processing (QSP) [28] that is aimed at developing new or modifying existing signal processing algorithms by borrowingfrom the principles of quantum mechanics and some of its interesting axioms and constraints, presents a probabilistic quantizer that has a direct relationship with the Dither quantizer and can generate a Dither signal with an arbitrary joint probability distribution using only a uniform random variable as input. Therefore, it reduces the computational complexity; however, it needs a mapping function with specific characteristics in time and frequency domain.

This paper shows that the positive B-splines satisfy the conditions to be a mapping for the uniform probabilistic quantizer and establishes a relation between the order of B-spline and the rendering of conditional moments of the error. One advantage of using B-splines is that they can be explicitly characterized, thus, their analysis is easier by eliminating convergence issues. Finally, we verify the result through numerical simulations and show that the proposed approach performs on par with Dither quantizer, but only requires the generation of a random variable.

In value amplitude quantization, a binary code is assigned to each signal sample, then the quantization process performs a mapping from the analog world to the digital one. It is necessary to acknowledge the inevitable degradation suffered by the signal, which is characteristic of the quantization process. More precisely, the quantizer input is a sequence that must belong to the space of sequences of finite energy (ℓ²) to bring stability to the quantification process. For each input value x _n , there will be a corresponding quantized value given by y _n = Q(x _n ), where the quantizer Q selects one of the quantization levels according to a defined rule, and the difference between each level is equal to Δ. In the operation of the QSP quantizer, the scalar quantizer defined as:

It was taken as a base, where the operator [. ] returns the largest integer, which is less than or equal to its argument. The Δ step is commonly referred to as the least significant bit (LSB) since a change in the input signal of one step at the most will generate a change in the LSB of the binary code of the output [2]. The transfer function of the scalar quantizer is shown in Figure 1.

Fig. 1

To observe the performance of the probabilistic quantizer, first a 1 Khz sinusoidal, shown in Figure 8, is used as input; and the centered B-spline of order 2 (see Figure 5a) is used as mapping ℎ in the quantizer. The discrete random variable 𝑤 ∈ℤ is generated with probabilities 𝑝_𝑘=ℎ(𝑥−𝑘𝛥), and the quantized output y is given by 𝑤Δ. Note that this relation is input dependent. To generate 𝑤, we use the inversion method [35], which consists in obtaining the partial sums 𝑠_𝑘=Σ𝑝𝑖𝑖≤𝑘 and generating only one uniform [0,1] random variable 𝑣. The sum can be seen as a monotone transformation, then 𝑝(𝑠_𝑘≤𝑣≤𝑠𝑘+1)=𝑠_𝑘+1−𝑠_𝑘=𝑝𝑘. To obtain the output, we must generate a realization of 𝑣, and then quantize the input equal to 𝑘𝛥, where 𝑘 is obtained from 𝑠_𝑘≤𝑣≤𝑠_𝑘+1. This can be done with any input and the code for all simulations is available online (GitHub repository https://github.com/jphoyos/bspline_mapping).

Fig. 8

As the order the spline is 2, only the first and second moment of the error are independent of the input (Figure 7). Hence, vestiges of the input signal are visible in the error signal Figure 8(c), as in the output signal Figure 8(b). The power spectral density shown in Figure 8(d) presents no distortion and suggests that the error is spectrally white. Our results are the same as those obtained by [7], but in our mappings we only needed to generate one random variable to do the quantization with the B-spline mapping and obtained the same performance. Also, to analyze the Rate-Distortion

where 𝒫(𝑋,𝐷) is the set of 𝑃_𝑌|𝑋 that satisfy 𝔼[𝑑(𝑋,𝑌)]≤𝐷, we obtained the upper bounds on 𝑅(𝐷) by evaluating the mutual information 𝐼(𝑋,𝑌) for a specific 𝑌 and distortion 𝐷. Figure 9 presents the rate-distortion of the proposed mappings in the probabilistic quantizer and the standard Dither quantizer for a standard unit variance scalar Gaussian source. We used the mean square error distortion as measure 𝑑 for each quantizer, guaranteeing that the distribution is on the interval [−𝛥/2,𝛥/2].

The probabilistic quantizer with B-spline of order 0 obtains the same performance as the traditional Dither quantizer with uniform distribution, while the B-spline of order 1 and 3 obtain a rate equal to a Dither quantizer with Gaussian distribution. Conversely, with the fractional order, the worst performance is obtained due to the anti-symmetric distribution. Then, the best rate was obtained by the B-spline of order 3 and the standard Dither quantizer with a Gaussian distribution, which renders all moments of the error independent of the system input; unfortunately, achieving it is complex because it would be necessary to generate and add theoretically infinite independent random variables [6]. Using a probabilistic quantizer with B-splines of order p≥3, the same rate is obtained and such complexity disappears since only one random variable is required to generate the distribution, rendering the p moments of the quantization error.

Fig. 9

This paper proves that the centered positive B-splines fulfill all the necessary conditions to be mappings in a probabilistic quantizer. We show that the use of B-splines as mappings brings mathematical advantages because they can be explicitly characterized, thus facilitating their analysis as well as that of the conditional moments of the error. This explicit characterization eliminates convergence issues that may arise with other mapping techniques. Furthermore, we established a meaningful relationship between the order of B-splines and the rendering of conditional moments of the error. This finding provides valuable insights to optimize the quantization process and achieve higher performance. Additionally, it is noteworthy that the implementation of B-splines as mappings requires generating a single random variable, which simplifies the overall system complexity. Finally, through extensive numerical experimentation, we validated the effectiveness of the probabilistic quantizer utilizing B-splines. Our experimental results have shown that this approach achieves performance on par with the NSD quantizer, a well-established benchmark. This result reinforces the practical feasibility and usefulness of employing B-splines in the quantization process.