Test: Binary Codes - GATE MCQ

Binary Coded Decimal (BCD) is a type of binary encoding where each digit in a decimal number is represented in equivalent binary form.
In this code each decimal number is represented by 4-bit binary number.
Generally with the four bits we can represent sixteen numbers ranging from 0000 to 1111.
But in BCD only first ten of these are used (0000 to 1001) and remaining six combinations (1010 to 1111) are invalid.

Numbers greater than 9 are represented by digit by digit.
For instance, 962 in BCD can be represented by 12 bits which can be expressed as 1001 0110 0010.
Conclusion: option 4 is correct. One binary coded decimal requires 4 bits to store.

Concept:
BCD code (Binary coded decimal):
It is the 4-bit binary code representation of a decimal digit with each decimal digit replaced in the integer and fractional parts with its binary equivalent.
BCD Code uses four bits to represent the 10 decimal digits of 0 to 9 as shown in the below table.

Analysis
In BCD each decimal digit is represented by a 4-bit binary number.
The binary representation of 8 → 1000
The binary representation of 1 → 0001
(81)₁₀ = 10000001

A Binary Coded Decimal (BCD) is a type of binary representation for decimal values where each digit is represented by its 4-bit binary equivalent.
Binary-coded decimals are an easy way to represent decimal values, as each digit is represented by its own 4-bit binary sequence which only has 10 different combinations.
The following represents the 4-bit binary representation of decimal values: 

The BCD equivalent of 345 is 0011 0100 0101

Concept:

• 8421 is also known as BCD code
• BCD is a weighted code.
• In weighted codes, each successive digit from right to left represents weights equal to some specified value, and to get the equivalent decimal number to add the products of the weights by the corresponding binary digit.

The following represents the 4-bit binary representation of decimal values: 

• A number with an 'n' decimal digit will require 4k bits in BCD.
• Ex- Decimal 396 is represented in BCD with 12 bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit.
• As the decimal digits relating to 1010, 1011, 1100, 1101, 1110, and 1111 do not exist, these are invalid.

In BCD valid states are from 0 to 9. So 1100 is an invalid state.

Concept:
The procedure to convert a gray code to a binary one is as shown:

Given, B₃ B₂ B₁ B₀ = 0011
From the above circuit the binary equivalent =
A₃ = B₃ = 0
A₂ = B₃ ⊕ B₂ = 0 ⊕ 0 = 0
A₁ = 0 ⊕ B₁ = 0 ⊕ 1 = 1
A₀ = 1 ⊕ B₀ = 1 ⊕ 1 = 0
So, A₃ A₂ A₁ A₀ = 0010

In BCD to 7-segment Decoder, the outputs of a digital circuit are often displayed as decimal digits.
BCD to 7-segment decoder is a combinational circuit that converts a BCD number into signals that are required for the display of the value of that number on a seven-segment display.
The number of input lines is 4. 


Notes:
The decoder outputs are (a, b, c, d, e, f, g)
For the display of the digit 0, segments a, b, c, d, e, f will be lit as shown above.
A truth table can be formed for all digits from 0 to 9.

Concept:
The circuit for binary to gray code convertor is:

y₁ = x₁
y₂ = x₁ ⊕ x₂
y₃ = x₂ ⊕ x₃
y₄ = x₃ ⊕ x₄

Calculation:
Let the input to the circuit be 14
Binary value of 14 is 1110
∴ For an input 1110, we get the output as :
y₁ = 1
y₂ = 1 ⊕ 1 = 0
y₃ = 1 ⊕ 1 = 0
y₄ = 1 ⊕ 0 = 1
Hence the gray code for 14 is 1001

Concept:

• 8421 is also known as BCD code
• BCD is a weighted code.
• In weighted codes, each successive digit from right to left represents weights equal to some specified value, and to get the equivalent decimal number to add the products of the weights by the corresponding binary digit.

The following represents the 4-bit binary representation of decimal values: 

Application:
From the above Table:
4 = 0100
3 = 0011
43 = 0100 0011 in BCD

• A convolutional code is a type of error-correcting code. For generating a convolutional code, the information is passed sequentially through a linear finite-state shift register.
• An (n, k, L) convolutional code can be implemented with a k-input, n-output linear sequential circuit with input memory L.
• The constraint length is defined as K = n(L + 1). 
• Since each information bit remains in the encoder for up to (L + 1) time units, and during each time unit can affect any of the n encoder outputs.
• Therefore, K can be interpreted as the maximum number of encoder outputs that can be affected by a signal information bit. 

CONCEPT:
CRC stands for Cyclic Redundancy Check, is an error-detecting code used to detect errors in the frame received.
CRC is used to detect errors in the Data link layer.

CALCULATION:
STEP1: 
No. of bits in divisor - 4 bits, append 3 0's to data bits
Data bits after appending 0's - 100111001000
STEP2:
Binary division of 100111001000 with 1011:

STEP 3:
Append CRC 011 at last of data unit → 100111001011.
Hence, the correct answer is "option 2".