Nondestructive Testing - Emergent Technologies

Nondestructive Testing - Emergent Technologies

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2007 Lecture 9a March 20, 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/ S. Mandayam/ ECOMMS/ECE Dept./Rowan University Plan Digital Communications Introduction Digital Communications Transceiver (CODEC/MODEM) Digital Baseband Communications Source Encoding Huffman Coding Error Control Coding Hamming Distance Error Detection Coding Parity Check Code Error Correction Coding Hamming Code S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics E le c tr ic a l C o m m u n ic a tio n S y s te m s

S ig n a ls D is c r e te S y s te m s C o n tin u o u s A n a lo g P r o b a b i li ty P o w e r & E n e r g y S ig n a ls AM S w itc h in g M o d u la to r E n v e lo p D e te c to r In fo r m a tio n C o n tin u o u s F o u rie r T r a n s fo rm D S B -S C P ro d u c t M o d u la to r C o h e re n t D e te c to r C o s ta s L o o p E n tr o p y D is c re te F o u rie r T ra n s fo rm SSB W e a v e r 's M e th o d P h a s in g M e th o d F re q u e n c y M e th o d C h a n n e l C a p a c i ty

B a s e b a n d a n d B a n d p a s s S i g n a ls F r e q u e n c y & P h a s e M o d u la tio n N a r ro w b a n d /W id e b a n d V C O & S lo p e D e te c to r PLL D ig ita l D ig ita l C o m m T r a n s c e iv e r B aseband CO DEC B andpass M O DEM S o u rc e E n c o d in g H u ffm a n c o d e s ASK PSK FSK E r ro r-c o n tro l E n c o d in g H a m m in g C o d e s BPSK S a m p lin g PAM Q PSK Q u a n tiz a tio n PCM M -a ry P S K

L in e E n c o d in g Q AM C o m p le x E n v e lo p e G a u s s ia n N o is e & S N R T im e D iv is io n M u x T 1 (D S 1 ) S ta n d a r d s R a n d o m V a r ia b le s N o is e C a lc u la tio n s P a c k e t S w itc h in g E th e r n e t IS O 7 -L a y e r P r o to c o l S. Mandayam/ ECOMMS/ECE Dept./Rowan University Digital Communications Some Milestones Claude Shannon, 1948 X.25 (Telephony) IEEE 802.3 (Ethernet) ARPANET, 1969 IEEE 802.5 (FDDI) ISO-OSI 7-layer Network Reference Model CDMA

GSM VOIP SIP protocols.com S. Mandayam/ ECOMMS/ECE Dept./Rowan University Digital Communications: Rationale Information Theory: What is the fundamental limit on the compression and refinement of information generated by the source? What is the fundamental limit on the transmission rate of information over a noisy channel? How do we approach these limits? S. Mandayam/ ECOMMS/ECE Dept./Rowan University Principle 1 0 1 0 Digital code Analog message 1 AM FM PM AM & PM 0 1

Digital message 1 1 0 0 modulate 0 Sinusoidal carrier S. Mandayam/ ECOMMS/ECE Dept./Rowan University Message 2 Message 3 Circuit Switching 1 2 3 S 1 2 2 3 S Sync bits Message 1 Message 2 Message 3 Packetizing Packet Switching H 1 H 3 H 3 H 2 H 1

H 2 H 3 H 1 H 2 Demultiplexer Message 1 Multiplexer Digital Communication Paradigms Message 1 Message 2 Message 3 Header bits Depacket -izing Message 1 Depacket -izing Message 2 Depacket -izing Message 3 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sampling Quantization

Source Encoder ADC Data Encryption Encoder Error Control Encoder Channel/ Line Encoder Modulator Analog i/p CODEC MODEM Audio Amp Reconstruction/ DAC Source Decoder Data Encryption Decoder

Error Control Decoder DEMUX Analog o/p Equalization / Decision Circuits Demodulator Multiple access channel Antialiasing Filter MUX Digital Communications Transceiver S. Mandayam/ ECOMMS/ECE Dept./Rowan University Source Encoding Analog Message A/D Converter Digital Source Source Source Encoder

Symbols (0/1) Source Entropy Why are we doing this? Encoded Symbols (0/1) Source-Coded Symbol Entropy S. Mandayam/ ECOMMS/ECE Dept./Rowan University Source Encoding Requirements Decrease Lav Unique decoding Instantaneous decoding S. Mandayam/ ECOMMS/ECE Dept./Rowan University Huffman Coding 2-Step Process Reduction List symbols in descending order of probability Reduce the two least probable symbols into one symbol equal to their combined probability Reorder in descending order of probability at each stage Repeat until only two symbols remain Splitting Assign 0 and 1 to the final two symbols remaining and work backwards Expand code at each split by appending a 0 or 1 to each code word

Example m(j) A P(j) 0.1 B 0.18 C 0.4 D 0.05 E 0.06 F 0.1 G 0.07 H 0.04 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sampling Quantization Source Encoder ADC

Data Encryption Encoder Error Control Encoder Channel/ Line Encoder Modulator Analog i/p CODEC MODEM Audio Amp Reconstruction/ DAC Source Decoder Data Encryption Decoder Error Control Decoder DEMUX

Analog o/p Equalization / Decision Circuits Demodulator Multiple access channel Antialiasing Filter MUX Digital Communications Transceiver S. Mandayam/ ECOMMS/ECE Dept./Rowan University Error Control Coding Error Detection (ARQ Technique) Error Correction (FEC Technique) Hamming Distance The number of locations (bits) at which two code words differ Theorem 1 A code with a Hamming distance of d >= t+1 can detect t errors in the received code word

Theorem 2 A code with a Hamming distance of 2t+1 <= d <= 2t+2 can detect and correct t errors in the received code word S. Mandayam/ ECOMMS/ECE Dept./Rowan University Error Control Codes Block Codes Convolutional Codes (memoryless) (with memory) Will not discuss! Principle Block Coder k information bits n encoded bits (n, k) systematic block code k n-k Information bits Parity bits

n-bit codeword Add Redundancy!! S. Mandayam/ ECOMMS/ECE Dept./Rowan University Parity Check Codes I7 I6 I5 I4 I3 I2 I1 P P is set such that the total no. of bits in the code word is even or odd S. Mandayam/ ECOMMS/ECE Dept./Rowan University Remediation for Detected Errors: ARQ Rx 1 2 ACK

Tx 1 3 3 NACK ACK 2 3 Error detected!!! S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code I4 I3 I2 I1 P3 P2 P1

Single error detection and correction code Hamming distance, d = 3 Fits into a general category of coding techniques called BCH codes Employs a Code Generator Matrix Syndrome Decoding Technique S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Encoding Parity bits I4 I3 I2 Code Generator I1 Info bits I3 I4 I2 I1 P3 P2

P1 C3 C2 C1 = C7 C6 C5 C4 Code word Decoding R7 R6 R5 R4 R3 R2 Received code word R1 Parity Check

s3 s2 s1 Error position indicator S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Code Generator Matrix 1 0 G 0 0 0 1 0 0 0 0 1 0 0 0 0 1 : : : : 1

0 1 1 1 1 0 1 1 1 1 0 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Encoder Info bits I4 I3 I2 Parity bits I1 P3 + + + P3 = I 4 I 2 I 1 P2 = I 4 I 3 I 1 P1 = I 4 I 3 I 2

P2 P1 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Syndrome Decoding Parity Check Matrix 1 0 1 1 : 1 0 0 H 1 1 0 1 : 0 1 0 1 1 1 0 : 0 0 1 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder Received code word R7 R6 R5 R4 R3 R2 Error Position Indicator R1 s3 s2 s1

+ + + s3 = R7 R5 R4 R3 s2 = R7 R6 R4 R2 s1 = R7 R6 R5 R1 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder Error position indicator (syndrome) s Error Position, e No error 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Summary

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