Arithmetic Circuits in Digital Systems

Chapter 6: Arithmetic Circuits

• Arithmetic circuits perform arithmetic operations such as addition, subtraction, multiplication, and division.
• They use logic gates to process binary numbers.
• Common types include adders, subtractors, and multipliers.
• Arithmetic circuits are essential in computers and calculators for numerical calculations.

Importance of Arithmetic Circuits

• They perform basic mathematical functions in digital systems ranging from calculators to microprocessors.
• In computers, they enable the CPU to carry out tasks like addition, subtraction, and multiplication.
• They lay the groundwork for complex digital operations, including arithmetic logic units.
• They facilitate efficient handling of numerical data in various applications like financial software and gaming.
• Essential in embedded systems like control systems and robotics for processing inputs and performing calculations.
• They are used in digital communication systems for error detection and correction, as well as signal processing tasks.

Binary Addition

• Binary addition is similar to decimal addition but uses only 0 and 1.
• Adding one-bit numbers results in four combinations: 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (carry 1).
• Adding larger numbers requires adding each pair of bits with a carry-in (Cin).

Half Adder and Full Adder Circuit

• The half adder adds two bits (A and B) from the 1s column, generating a sum (S) and a carry-out (Co).
• The full adder adds three bits (Cin, A, and B) from the 2s, 4s, and 8s columns, producing a sum (S) and a carry-out (Cout).
• Full adder operation takes into account the carry from previous lower-order bits.

Parallel Adder Circuit

• Parallel adders are used for adding multiple bits simultaneously.
• Constructing a 4-bit parallel adder involves using one half adder and three full adder circuits.
• The half adder adds the first column (A1+B1) and generates S1 and Co.
• Full adders are utilized to add the 2s, 4s, and 8s columns.
• The carry-out (Co) from one adder stage is connected to the carry-in (Cin) of the next adder stage.
• The 7483 is a commercial 4-bit full adder with four full adders connected internally, requires the Cin to be connected to ground (0) for 4-bit addition.
• An 8-bit parallel adder is created using two 7483 ICs and connects the carry-out (Cout) of the first IC to the carry-in (Cin) of the second IC.

Signed Numbers

• Binary numbers can represent positive or negative values.
• Signed numbers are commonly represented using sign-magnitude or 2's complement.
• In sign-magnitude, a number consists of a magnitude and a sign.
• Zero signifies a positive number while One signifies a negative number.
• An n-bit sign-magnitude number represents integers ranging from -(2^(n-1)-1) to +(2^(n-1)-1).
• A 4 bits sign-magnitude number represents number from -7 to +7.
• Sign-magnitude has two representations of zero: -0 and +0.

Two's Complement Number

• The complement number system makes binary addition and subtraction faster and easier.
• It uses complementation processes, and can be efficiently performed on binary numbers.
• An n-bit, 2's complement represents integers from -2^(n-1) to +2^(n-1)-1.
• A 4 bits complement numbers, represents values from -8 to +7.
• A positive number is represented similar to sign-magnitude numbers with a 0 sign bit: i.e. +6 = 0110.
• A negative number is obtained by complementing each bit of the equivalent positive number including the sign bit, then adding 1 to the result.
• Any carry that occurs out of the Most Significant Bit (MSB) is ignored.

2's Complement Number Examples:

• For the number +30, magnitude is 11110, therefore the 2’s complement for +30 is 011110 (with sign bit)
• For the number -30, magnitude is 11110, therefore the 2’s complement for -30 is 100010
• For the number +30 using 8 bits the 2’s complement is 00011110
• For the number -30 using 8 bits the 2’s complement is 11100010

Determining the Decimal Value of 2's Complement Numbers

• 00100010 is positive, with a decimal value of +34.
• 10100010 is negative. Complement and add 1, which equals +(-94).
• 11111111 is negative number, complement and add 1 to give decimal value of -1

Limits

• Available bits in computer's arithmetic unit limits range represented in the machine.
• Numbers falling outside handled range can't be handled by a machines.
• 2's complement systems represents in -2^(n-1) ≤ N ≤ 2^(n-1) – 1 (With n bit available).
• If result falls outside limits range, overflow occurs.

Two’s Complement Number Example Ranges:

• An 8-bit number ranges from -128 (10000000) to +127 (01111111).
• A 16-bit number ranges from -32,768 to +32,767.
• A 32-bit number ranges from -2,147,483,648 to +2,147,483,647.

Two's Complement Arithmetic

• Addition and subtraction use the 2's complement system
• Includes the cases: A+B, A-B, -A+B, -A-B.

2's Complement: Case 1: A+B Two positive numbers:

• Both A and B are positive, use to 2's complement. Overflow:
• Cases occur that operation produces result outside number system’s range.
• Incorrect result occurs adding numbers within system but larger than the largest number.
• Addition overflow occurs when the sign of the sum differs from signs of both addends.

Two's Complement: Case 2: -A-B Two negative numbers:

• –A and –B represented as 2's complement and performed: (-Α) + (-Β)

Two's Complement: Case 3: Both 2's Complement - A-B

• Positive number unchanged, negative represented in 2's complement form.

Two's Complement: Case 4: Both 2's Complement - -A+B

• The negative number is put in 2’s complement, positive one remains unchanged

Subtractor Circuit

• 4-bit parallel adder is modified to creates a subtractor circuit.
• 2's complement system performs: S = A-B = A + B + 1 .
• Two’s complement for B obtained doing complement inverts each bit, then add 1 to Least Significant Bit (LSB).
• In a 4-bit subtractor the 2’s complement formed with Inverters adding Cinput- 1 to Last Significant Bit(LSB), with the Most Significant Bit (MSB) using A4/B4/S4 for signed bits and ignore the Cout

Subtractor Circuit Limitation

• Can only accept values between -7 (1001) and +7(0111), limited values -7 to +7
• Arithmetic operation results an overflow if output beyond value accepted.

8-bit Subtractor Circuit

• Connect two 4-bit subtractor circuits.
• Then the Cout connected to the Cin of preceding subtractor.
• Values accepted are -127 (10000001), +127(0111111), result value between the numbers too

Adder-Subtractor Circuit

• Combines adder and subtractor, 4-bit circuits identical.

• Control used as addition, an extra input to circuits

• Control input is 0, then it performs a addition with: A + B + 0, Cout =0

• Control input set to 1, then it performs: A + B + 1, Cout = 1

• Circuit has control input, performs Subtraction / A + B + 1 = 0 if Cout

• Control 1 then XOR inverter performs Subtraction

Explore the role of arithmetic circuits in digital communication and CPUs. Understand binary and decimal addition differences. Learn about half-adders and full-adders in constructing complex digital operations.