Digital Logic, Arithmetic, and Conversions

1.The zero flag. This flag is set if a previous operation produces a value of zero.

2.The carry/overflow flag. This flag is set if there has been a carry or a borrow-out of the high-order bit of the operand.

3.The half-carry or digit-carry flag. This flag is set if there has been a carry or a bor- row-out of the low-order nibble of the operand.

Not all instructions affect all the flags. For example, loading a zero constant into a register may be said to produce a zero value; however, such an instruction may or may not affect the zero flag, according to the implementation on each particular device. More powerful and sophisticated microcontrollers sometimes implement other flags, such as flags to indicate a negative operand, an arithmetic overflow, or an interrupt.

4.0.2 Word Size

The word-size of a computer or a digital device refers to the number of bits used in storing data and in moving data in and out of the various machine units. In other words, a machine’s word-size is the native data unit for a particular architecture. In this manner we speak of the Pentium having a 32-bit word size or the PIC16x84 having an 8-bit word-size for data operations and 14-bit program words.

In the context of digital arithmetic and logic the data word-size determines the processing capabilities of each device. For example, a machine with an 8-bit word-size can perform unsigned addition of operands whose sum does not exceed the decimal value 255, since 255 is the largest unsigned integer that can be stored in eight bits. However, a machine with 16-bit words can perform unsigned additions up to a sum of 65,535 since it is the largest number that can be stored in 16 bits.

Therefore, the coding of numerical routines is determined by the word size of the machine or device. A device with 8-bit word-size requires multi-byte arithmetic to perform addition that exceeds a sum of 255, while a machine with a 16-bit word can do direct addition up to the sum 65,535. Considering that most popular microcontrollers have 8-bit word-sizes, we assume this limit in the arithmetic and logic algorithms and routines developed in this chapter.

4.1 Logical Instructions

The logical instructions include the Boolean operators, AND, OR, NOT, and XOR, as well as instructions to shift and rotate individual bits.

The logical instructions operate on a bit-by-bit basis; therefore, in the AND, OR, NOT, and XOR there is no interaction between bits. The action performed by the logical instructions is as follows:

1.AND, OR, and XOR logically combine each bit in the source operand with the corresponding bit in the destination operand. The result does not affect the neighboring bits.

2.The NOT operator inverts all bits in the destination operand.

These actions explain the term bitwise operation sometimes used to describe the instructions.

4.1.1 Logical AND

The AND instruction performs a bitwise logical AND of two operands. This determines that a bit in the result is set if and only if the corresponding bits are set in both operands. A frequent use of the AND operation is to clear one or more bits without affecting the remaining ones. This action is possible because ANDing with a 0 bit always clears the result bit and ANDing with a 1 bit preserves the original value of the first operand.

For example, if we have the binary coded decimal number 34 packed into a single byte, we can isolate the four low-order bits as follows:

The second operand, in this case 0FH, is called a mask. The AND operation preserves the 1-bits in the mask and clears the bits that are 0. Consequently, the mask 00000001B clears the seven high-order bits and preserves the original value of the low-order bit.

4.1.2 Logical OR

The OR operation performs the bitwise logical inclusive OR of two operands. After a logical OR, a bit in the result is set if one or both of the corresponding bits in the operands were set. A frequent use for the OR is to selectively set one or more bits. The action takes place because ORing with a 1-bit always sets the result bit, while ORing with a 0-bit preserves the original value in the first operand.

For example, to set the high-order bit (bit number 7) we can OR with a 1 bit, as follows:

The OR operation sets the bits that are 1 in the mask and preserves the bits that are masked 0.

4.1.3 Logical XOR

The XOR operator performs the bitwise logical exclusive OR of the two operands. Therefore, a bit in the result is set if the corresponding bits in the operands have opposite values. For this reason, XORing a value with itself always generates a zero result since all bits necessarily have the same value. On the other hand, XORing with a 1-bit inverts the value of the other operand, since 0 XOR 1 is 1 and 1 XOR 1 is 0. This toggling action of XORing with a 1 bit generates identical bitwise results as the NOT operation,

but by selecting the XOR mask, the programmer can control which bits of the operand are inverted and which are preserved.

In this manner it is possible to invert the four high-order bits of an operand by XORing with a mask that has these bits set. If the four low-order bits of the mask are clear, then the original values of the bits in the other operand are preserved in the result. For example:

In the previous example, the XOR operation inverts the bits that are 1 in the mask and preserves the bits that are masked 0. Consequently, the XOR mask 11110000B inverts the four high-order bits.

4.1.4 Logical NOT

In contrast with the other logical operators which require two operands, the NOT instruction acts on a single value. Its action is consistent with a Boolean NOT function, which converts all 1-bits to 0 and all 0-bits to 1. Arithmetically, the result is the one’s complement of the original value. This instruction can be useful in obtaining the two’s complement representation by performing the logical NOT and then adding one to the results.

4.2 Microcontroller Arithmetic

Microcontrollers are not designed for intensive numeric processing; therefore, they are not equipped with many arithmetic operators usually found in microprocessors. A typical mid-range microcontroller has instructions to add and subtract integers and perhaps to increment and decrement. Hardware multiplication is rarely available and even more so is division. Likewise, there is usually no hardware support for decimal and floating-point arithmetic. For this reason the microcontroller programmer is often challenged to provide most arithmetic and data processing operations in software.

In this discussion we assume a mid-range microcontroller, such as the PIC 16f8x. These devices contain primitives for adding and subtracting integers, shifting and rotating bits, incrementing and decrementing machine registers, some support for decimal operations and conversions, as well as the basic logic primitives AND, OR, XOR, and NOT. Multiplication and division operators, as well as floating-point operators, are not available in the mid-range devices.

4.2.1 Unsigned and Two’s Complement Arithmetic

In Chapter 3 we discussed the various representations for signed and unsigned binary and decimal numbers. Arithmetic operations of unsigned operands are the simplest. In this case we assume that the encoding always represents a positive number and that all bits relate to the number’s magnitude.

Unsigned arithmetic can be binary or decimal. In a machine with 8-bit words binary arithmetic on unsigned numbers use the entire range of the format. This is true even when the primitive operations are valid in two’s complement form; in fact, it is one of the great advantages of two’s complement representation. Table 4.1 shows a 4-bit binary in several numeric formats.

Table 4.1

Interpretations of 4-bit Binary Numbers

Assume a machine with a 4-bit word size and consider addition of two unsigned numbers:

Note, in the previous example, that if the encoding were in two’s complement form, the addition of the positive values 6 plus 7 would produce a result that overflows the capacity of the representation. In 4-bit two’s complement representation there is no way of encoding the value 13.

The question that arises is: in a device that performs two’s complement addition, must we always assume that the operands are in two’s complement form? The answer is: no. Signed addition of two’s complement operands and the unsigned addition of integer operands can be performed with identical processing and by the same electronic circuitry. It is the software that must take into account the encoding of the operands in order to interpret the results. For example, in the 4-binary digit device previously considered, the two’s complement addition of the values 6 and 7 produce an overflow, which can be detected by observing the change in the high-order bit (the sign bit) of the result. Therefore, in this case, the result of the addition operation is invalid. However, if the same decimal values represent unsigned operands, then the addition of 7 plus 6 produce the valid result 13. In either case the binary values of the operands, as well as the result, are the same.