Encoders
An encoder is a combinational circuit that performs the inverse operation of a decoder. If a device output code has fewer bits than the input code has, the device is usually called an encoder. e.g. 2ⁿ-to-n, priority encoders.

Decoders
A decoder is a multiple-input, multiple-output logic circuit that converts coded inputs into coded outputs, where the input and output codes are different; e.g. n-to-2n, BCD decoders

Noise Margin

Digital Logic Families
Logic families can be classified broadly according to the technologies they are built with. In earlier days we had vast number of these technologies, as you can see in the list below.

QUINE-McCLUSKEY MINIMIZATION

Quine-McCluskey minimization method uses the same theorem to produce the solution as the K-map method, namely X(Y+Y')=X

Minimization Technique

4-Variable K-Map

There are 16 cells in a 4-variable (W, X, Y, Z); K-map as shown in the figure below.

2-Variable K-Map

In any K-Map, each square represents a minterm. Adjacent squares always differ by just one literal (So that the unifying theorem may apply: X + X' = 1). 

Introduction
Simplification of Boolean functions is mainly used to reduce the gate count of a design. Less number of gates means less power consumption, sometimes the circuit works faster and also when number of gates is reduced, cost also comes down.

Multipliers
Multiplication is achieved by adding a list of shifted multiplicands according to the digits of the multiplier. An n-bit X n-bit multiplier can be realized in combinational circuitry by using an array of n-1 n-bit adders where each adder is shifted by one position. 

Subtracter
Subtracter circuits take two binary numbers as input and subtract one binary number input from the other binary number input. Similar to adders, it gives out two outputs, difference and borrow (carry-in the case of Adder). 

n-bit Carry Ripple Adder
An n-bit adder used to add two n-bit binary numbers can be built by connecting n full adders in series. Each full adder represents a bit position j (from 0 to n-1).

Arithmetic circuits are the ones which perform arithmetic operations like addition, subtraction, multiplication, division, parity calculation. Most of the time, designing these circuits is the same as designing muxers, encoders and decoders.

Floating Point Numbers
A real number or floating point number is a number which has both an integer and a fractional part. Examples for real real decimal numbers are 123.45, 0.1234, -0.12345, etc. Examples for real binary numbers are 1100.1100, 0.1001, -1.001, etc. In general, floating point numbers are expressed in exponential notation.

Error Detecting and Correction Codes
For reliable transmission and storage of digital data, error detection and correction is required. Below are a few examples of codes which permit error detection and error correction after detection.

Binary Codes
Binary codes are codes which are represented in binary system with modification from the original ones. Below we will be seeing the following: