## 1 Introduction  
Digital signal processing (DSP) has been widely adopted in the Internet of Things (IoT), sensor systems, communication systems, digital image processing, and artificial intelligence (AI). Mobile handheld electronic devices, such as mobile phones, tablets, and notebooks, involve new technology and applications such as three-dimensional computer graphics and real-time systems in DSP. These require circuits for complex arithmetic computations such as multiplication, division, square root, squaring, and powering, which entail additional hardware costs and longer latency. To reduce the hardware costs and transmission delays, recent studies have developed novel methods to replace the complex arithmetic computations, such as the CORDIC algorithm [1], the table-based algorithm using rectangular multipliers [2], and the logarithmic number system (LNS) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] to handle arithmetic computations. The CORDIC algorithm [1] uses an iterative method, and is not suitable for three-dimensional real-time DSP because of the limitation of operation speed. The table-based algorithm using rectangular multipliers [2] requires quite a large hardware memory for storage. The logarithmic number system (LNS) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] reduces the implementation area and transmission delay for arithmetic computations in DSP. LNS-based arithmetic computation can simplify the complex operations by transforming multiplication to addition, division to subtraction, square root to right shifts, squaring to left shifts, poweringto continuous addition, and reciprocals to complementing. Table 1 shows the LNS-based operations. The LNS consists of three main units: the logarithmic converter, simple arithmetic unit (SAU), and antilogarithmic converter, as shown in Figure 1. The logarithmic converter is used to convert two binary inputs (e.g., \(a\) and \(b\) in Table 1) into the logarithmic system format (\(\log_{2}\)\(a\) and \(\log_{2}\)\(b\)). The SAU uses the shifter, complement, adder, or subtractor operations to perform a simple mathematical computation in the logarithmic system format. The antilogarithmic function is used to convert the final arithmetic computation result of the logarithmic system and simple arithmetic unit into a fixed-point binary output. The present study focuses on antilogarithmic converters with efficient error-area-delay products (eADPs) using the fractional-bit compensation schemes. In recent years, many schemes have been proposed for antilogarithmic conversion systems, such as the straight-line method [3], look-up table method [7; 15], shift-and-add method [5; 11], bit correction scheme [13], and constant compensation scheme [16]. In 1962, [PERSON] [3] first used an approximation method to approximate the antilogarithmic converter, which adopted \(out=1+m\) to approximate an \(out=2^{m}\) curve. Although this method produces considerably large antilogarithmic conversion, it is simple and uses little hardware area. [PERSON] et al. [7; 15] used the look-up read-only memory (ROM) method to \begin{table}
\begin{tabular}{c c c} \hline \hline
**Arithmetic Computation** & **Arithmetic Operation** & **LNS-Based Operation** \\ \hline Multiplication & \(out=a\cdot b\) & \(A+B\) \\ \hline Division & \(out=a/b\) & \(A-B\) \\ \hline Square root & \(out=\sqrt{a}\) & \(A>>1\) \\ \hline Squaring & \(out=a^{2}\) & \(A<<1\) \\ \hline Powering & \(out=a^{b}\) & \(\underbrace{A+A+.....+A}_{b}\) \\ \hline Reciprocal & \(out=1/a\) & \(-X\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LNS-based operations for arithmetic computations. Figure 1: Logarithmic number system. implement the antilogarithmic converter. This method obtains highly accurate antilogarithmic conversion values from the SAU and logarithmic converter. However, the look-up table scheme demands additional hardware area. [PERSON] and [PERSON] [5] and [PERSON] and [PERSON] [11] used the shift-and-add scheme to approximate an \(out=2^{m}\) curve.