## 1 Introduction  
Mission- or safety-critical applications such as space, aerospace, nuclear, defense, electric power transmission and distribution, banking, financial, and industrial control and automation etc. usually employ some form of N-modular redundancy (NMR) for the hardware implementation to cope with unintended temporary faults and/or permanent failures of constituent circuits or systems [1, 2]. In an NMR hardware, (N - 1) identical copies of a function module are used along with the original function module; the function module may be a circuit or a system. Out of the N function modules, at the maximum, the temporary faults or permanent failures of (N - 1)/2 function modules are tolerated. In other words, the correct operation of at least (N + 1)/2 function modules are necessary. Supposing each of the N function modules produce K primary outputs, the corresponding outputs of each of the N function modules are combined separately using K N-input majority voters to generate the actual NMR hardware outputs [3]. A majority voter, as the name suggests, determines the output of an NMR implementation based on the Boolean majority. If N inputs are provided to a majority voter, the majority voter determines the primary output based on the identical values of (N + 1)/2 inputs. To implement NMR hardware, a total of N function modules and a requisite number of N-input majority voters, depending upon the number of outputs produced by the original function module, are required.  
Triple modular redundancy (TMR or 3 MR) is the minimum version of an NMR. In TMR, three function modules are used and the temporary fault or permanent failure of any one function module is tolerated. Thus, the correct operation of at least two function modules is necessary in TMR. A TMR implementation requires three-input majority voter(s). In fact, the three-input majority voter also finds use in majority and minority voted redundancy [4] and self-healing redundancy [5] schemes, which are relevant for mission- and safety-critical applications. In the literature, many designs for a three -input majority voter are discussed [6; 7; 8; 9; 10]. However, [6; 7; 8; 9] correspond to synchronous designs of the three-input majority voter, and [10] corresponds to a bundled-data asynchronous design which is not delay insensitive. To our knowledge, delay insensitive designs of the three-input majority voter have not been discussed thus far. In this context, this article presents quasi delay insensitive (QDI) designs of the three-input majority voter based on some QDI logic synthesis methods and analyzes which majority voters are preferable in terms of speed, power, and area. A QDI three input majority voter is necessary for implementing a QDI TMR circuit/system.  
QDI circuits form an attractive alternative to synchronous circuits because they are inherently modular and elastic due to being delay insensitive [11], able to cope with variations in process, voltage, and temperature [12], less susceptible to electromagnetic interference [13], low power consuming [14], more secure to attacks compared to synchronous circuits [15], and self-checking [16].  
The rest of the article is organized into five sections. Section 2 discusses the design preliminaries of QDI circuits. Section 3 describes the architecture of a QDI TMR implementation. Section 4 presents the QDI majority voter designs corresponding to some QDI logic synthesis methods utilizing delay insensitive dual rail data encoding and adhering to four-phase return-to-zero (RTZ) and four-phase return-to-one (RTO) handshake protocols. The simulation results obtained for the example QDI TMR implementations are given in Section 5, and finally, Section 6 concludes this article.