# CS201 Lab: Design Sequential Adders & Subtractors

## Objectives

```
Investigate a 4-bit shift register and use it to store the results
Build sequential adder, subtractor, and adder-subtractor circuits to verify the suquential design procedures

```

```
A Half-adder is a combinational circuit that performs the addition of 2 bits.

Inputs: 	2 input bits to be added

Outputs: 	sum bit (least significant)
carry bit (most significant)

For example:

1 + 1 = 1 0 		c = 1;  s = 0

1 + 0 = 0 1		c = 0;  s = 1

```

```
A Full-adder is a combinational circuit that forms the arithmetic sum of 3 input bits.

Inputs: 	2 input bits to be added (x, y)
1 carry bit from previous lower significant position (z)

Outputs: 	sum bit (least significant)
carry bit (most significant)

For example:

1 + 1 + 1 = 1 1 	c = 1;  s = 1
1 + 0 + 1 = 1 0		c = 1;  s = 0

```

```
A binary adder is the circuit that generates the arithmetic sum of
two binary numbers of any length.

The subtraction of binary numbers can be done most conveniently
by means of complements of numbers.

Subtraction A - B can be done by taking the 2's complement of B and

The 2's complement can be obtained by taking the 1's complement and
adding 1 to the least significant pair of bits.

The 1's complement can be implemented with inverters and
a 1 can be added to the sum through the input carry.

The addition and subtraction operations can be combined into one common
circuit by including an exclusive-OR (XOR) gate with a full-adder.

A bit sequential adder-subtractor is shown in the following figure:

Input X is used to represent the bit of the binary number A.
Input Y is used to represent the bit of the binary number B.
The control signal Z controls the type of operation.

When Z = 0 the circuit is an adder, meanwhile, the D flip-flop
should be initialized to 0.
When Z = 1 the circuit becomes a subtractor. The D flip-flop
should be initialized to 1.

The XOR gate receives input of signal bit Z and the input of the operand bit Y.

When Z = 0, we have Y XOR Z = Y.
--------------------------------
The full adder receives the value of Y, the input carry is 0 because
D flip-flop was initialized to 0, and the circuit performs A plus B.

When Z = 1, we have Y XOR Z = Y' and the initial carry to 1.
------------------------------------------------------------
The Y input is complemented and a 1 is added through the initial input carry.
The circuit performs the operation: A plus the 2's complement of B.

For unsigned numbers A and B, this gives the A - B if A >= B or
2's complement of (B - A) if A < B.

For signed numbers, the result is A - B provided there is no overflow.

```

## Introducing a 4-bit Shift Register:

```
and shifting mode.

The shifted register "Shift Reg-4" can be obtained from "Simulation Logic.clf".

It demonstrates the following:

1.      How to use the 4 bit shift register.

2.      How to use the Load and Shift modes.

3.      How to use individual probes on each output to make
it easier to see what's happening in shift mode.

4.      Shifting direction is Q3<-Q2<-Q1<-Q0<-SI.

With LD set to 1:

The register is in shift left mode;
there is no shift right mode available.
The SI (shift input) line is used to set the value to be
shifted into the register. Clocking is positive edge triggered.

With LD set to 0:

The register is in parallel load mode.
Inputs D0 to D3 become outputs Q0 to Q3 on the next
rising clock pulse.

It can be used to store and diplay the output of your