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This document contains a problem set on digital communications from the Department of Electrical Engineering at the University of Isfahan. The problem set covers topics including source and channel coding, prefix codes, the Kraft-McMillan inequality, linear block codes, repetition codes, and error detection/correction. It contains 4 problems asking students to identify prefix codes, apply concepts like the Kraft-McMillan inequality, find encoding tables and error correction capabilities of codes, and evaluate syndromes for single and double error patterns.
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![University of Isfahan - Department of Electrical Engineering
Problem 3
Consider a (6,3) linear block code defined by the generator matrix
G =
1 0 0 1 1 0
0 1 0 0 1 1
0 0 1 1 0 1
(a) Determine if the code is a Hamming code. Find the parity check matrix
H of the code in systematic form.
(b) Find the encoding table for the linear block code.
(c) What is the minimum distance dmin of the code. How many errors can
the code detect. How many errors can the code correct.
(d) Suppose c = [ 1 1 1 0 0 0 ] is sent and r = [ 1 1 1 0 0 1 ] is received.
Show how the code can correct this error.
Problem 4
Consider the (5, 1) repetition code:
G = 1 1 1 1 1
H =
1 0 0 0 1
0 1 0 0 1
0 0 1 1 1
0 0 0 1 1
Evaluate the syndrome s for the following error patterns:
(a) All possible single-error patterns.
(b) All possible double-error patterns
Explain why syndromes for all single-error and double-error patterns are
distinct.
2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur like single-bit and burst errors. It explains that redundancy is needed to detect or correct errors. Various coding techniques are described, including block coding, linear block codes, cyclic codes, and checksums. Specific codes like parity-check codes, Hamming codes, and cyclic redundancy checks are explained in detail with examples. The document emphasizes that error detection codes can only detect certain error types, while error correction codes can correct errors.
Digital electronics(EC8392) unit- 1-Sesha Vidhya S/ ASP/ECE/RMKCET
Number systems, Number conversion,Logic Gates,Boolean Theorem and Laws,Boolean Simplification,NAND,NOR Implementation,K-MAP simplification and Tabulation Method
Chapter 10
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
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Digital electronics(EC8392) unit- 1-Sesha Vidhya S/ ASP/ECE/RMKCET
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PS3
- 1. University of Isfahan - Department of Electrical Engineering Digital Communications Due: 94/8/19 Problem Set 3: Source and Channel Coding Instructor: M. F. Sabahi TA: I. Taghavi Problem 1 Consider the four codes listed below: Symbol Code I Code II Code III Code IV s0 0 0 0 00 s1 10 01 01 01 s2 110 001 011 10 s3 1110 0010 110 110 s4 1111 0011 111 111 (a) Two of these four codes are prefix codes1. Identify them, and construct their individual decision trees. (b) Apply the Kraft-McMillan inequality to codes I, II, III, and IV. Discuss your results in light of those obtained in part (a). Problem 2 A discrete memoryless source and its source code is described in the following table: Symbol S0 S1 S2 S3 S4 S5 S6 Probability 1/4 1/4 1/8 1/8 1/8 1/16 1/16 Code word 10 11 001 010 011 0000 0001 Explain why the source code has an efficiency of 100 percent. 1 A prefix code is defined as a code in which no code word is the prefix of any other code word. 1
- 2. University of Isfahan - Department of Electrical Engineering Problem 3 Consider a (6,3) linear block code defined by the generator matrix G = 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 (a) Determine if the code is a Hamming code. Find the parity check matrix H of the code in systematic form. (b) Find the encoding table for the linear block code. (c) What is the minimum distance dmin of the code. How many errors can the code detect. How many errors can the code correct. (d) Suppose c = [ 1 1 1 0 0 0 ] is sent and r = [ 1 1 1 0 0 1 ] is received. Show how the code can correct this error. Problem 4 Consider the (5, 1) repetition code: G = 1 1 1 1 1 H = 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 0 1 1 Evaluate the syndrome s for the following error patterns: (a) All possible single-error patterns. (b) All possible double-error patterns Explain why syndromes for all single-error and double-error patterns are distinct. 2