## 1 Introduction  
Fractional calculus (FC) generalizes the classical theory of differential calculus by including the concept of integrals and derivatives of real or complex order [1, 2]. Many systems in physics and engineering can be generalized in the light of the FC concepts. However, a number of applications still remain to be fully explored under the light of FC. The areas of signal processing, dynamical systems and control has received considerable attention and the results show that the adoption of FC is clearly a fruitful research strategy [3, 4, 5, 6].  
Sample and hold (S&H) systems are commonly applied in mix analog/digital systems, being the standard procedure in discrete time systems to adopt zero-order holds (S&H-0) [7, 8, 9, 10]. The devices, are simple to construct and 'hold' the signal sample during one time interval. They are a basic tool of the analog to digital (A/D) conversion. In the reverse operation (D/A) other types of S&H are used. For instance, the first-order hold (S&H-1) reconstructs signals as a piecewise linear approximations and may yield better fit at the price of a higher complexity. Other S&H are the fractional- and exponential- order holds, abbreviated as S&H-\(\beta\) and S&H-exp, with parameters \(0\leq\beta\leq 1\) and \(\tau>0\), respectively, that also improve the technique at the cost of a slight additional complexity [11, 12, 13, 14]. In fact, we can take advantage of the extra parameter \(\beta\) and \(\tau\) by tuning their values for a particular application. Nonetheless, we observe the introduction of high frequency artifacts, that may pose difficulties in some applications. We can reduce this kind of distortion by using a modification of some of the above S&H systems. This change consists in introducing a delay of one sampling interval until a new sample appears. Therefore, instead of an extrapolation, as above, we are doing an interpolation [10; 15]. Such procedures are very important in control [16; 17] and telecommunications [9; 18]. These topics are often neglected in practical applications. On one hand, the difference between theoretical and engineering implementations, due to physical restrictions, are overlooked. On the other hand, the possibilities for further improving the S&H systems, particularly when considering FC, remain to be explored.  
This paper starts by reviewing the fundamental concepts associated with the S&H systems and proposes one generalization in the perspective of FC. The classical and fractional S&H are then compared based on their frequency responses. For that purpose we adopt the Euclidean and Canberra distances [19]. The distinct realizations of S&H are then represented using the Hierarchical Clustering (HC) and multidimensional scaling (MDS) computational tools [20; 21; 22; 23]. Both schemes rely on constructing graphical representations of the items under study so that users can visualized their relationships. Items are 'leafs' of structures such as dendrogram or trees, for the HC, and as points in some abstract space, for the MDS. In both cases similar items are represented closely and vice-versa.  
Bearing these ideas in mind the manuscript is organized as follows. Section 2 introduces the fundamental concepts concerning S&H. Section 3 discusses real-world implementations of S&H systems and proposes a fractional-order generalization. Section 4 analyses the transfer functions of the S&H systems under discussion. Finally, Section 5 outlines the conclusions.