Leading Zero Counter - Digital System Design

In a binary number, leading zeros are the zero digits in the most significant positions of data, up to the position in which the first one is present. For a binary number $x = 0000100011010010$ the leading leading zero count is 4. Leading zero counter is a very important combinational circuit in designing the floating-point architectures to do the normalization operation.

Here an architecture to count maximum 15 leading zeros is presented. A simple modular architecture is presented in [1]. In [1], authors designed higher order leading zero counter using 4-bit leading zero counters. In a binary number ( $x$ ), the leading zero count ( $q$ ) varies from 0 to 3. If all the bits are zero then it generates a signal ( $a$ ) to indicate that the number is zero. The truth table for 4-bit leading zero counter (LZC-4) is shown below in Table 1.

Table 1: Truth Table for 4-bit Leading Zero Counter

Using the K-map optimization technique the Boolean expression for the outputs of LZC-4 can be obtained and these are

The architecture of the LZC-4 is shown in Figure 1 . It outputs two bits $\{q_1,q_0\}$ and also output an signal $a$ which indicates that all the bits are zero.

Figure 1: A basic architecture for 4-bit Leading Zero Counter

Higher order leading zero counters can be designed using the basic LZC-4 blocks. The four bits are together called as nibble. A 16-bit binary number has four nibbles. The outputs $a_i$ and $z_i$ is generated by the $i^{th}$ nibble from the MSB side. Lets consider to design a LZC-8 block to count 7 leading zeros. In this case, the count will vary from 0 to 7. If $a_0=0$ , the zero count value depends on $z_0$ and if $a_0=1$ then $z_1$ decides the overall count value. The truth table for 16-bit leading zero counter (LZC-16) is shown in Table 2.

Table 2: Truth Table for 16-bit Leading Zero Counter

The LZC-16 is designed using the basic LZC-4 block. The upper bits of the counter are evaluated by a block which takes the inputs $a_i$ from the LZC-4 blocks. This block is called as leading zero encoder (LZE-4) which is shown in Figure 2. The logical expressions for the outputs of this blocks are decided by the Table 2 . Using K-map the logical expressions are defined as

Figure 2: A basic architecture for 4-bit Leading Zero Encoder

Finally the overall architecture of LZC-16 is shown in Figure 3 . There are four LZC-4 blocks used here and one LZE block is used. The output of the LZE-4 block which are the upper bits of the counter selects the lower bits trough a MUX.

Figure 3: A basic architecture for 16-bit Leading Zero Counter

[1]. Nebojša Z. Milenković and Vladimir V. Stanković and Miljana Lj. Milić, “MODULAR DESIGN OF FAST LEADING ZEROS COUNTING CIRCUIT”, Journal of ELECTRICAL ENGINEERING, VOL. 66, NO. 6, 2015, 329–333

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