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Electrical Communications Systems ECE.09.331 Spring 2008. Lecture 11a April 8, 2008. Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/ecomms/. Plan. Digital Baseband Communications Error Control Coding Hamming Distance

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electrical communications systems ece 09 331 spring 2008

Electrical Communications SystemsECE.09.331Spring 2008

Lecture 11aApril 8, 2008

Shreekanth Mandayam

ECE Department

Rowan University

http://engineering.rowan.edu/~shreek/spring08/ecomms/

slide2
Plan
  • Digital Baseband Communications
    • Error Control Coding
      • Hamming Distance
      • Error Detection Coding
        • Parity Check Code
      • Error Correction Coding
        • Hamming Code
  • Pulse Amplitude Modulation
      • Sampling
      • Quantization
    • TDM Standards
      • T1 (DS-1) Line
      • North American TDMA Hierarchy
digital communications transceiver
Digital Communications Transceiver

Anti-

aliasing

Filter

Error

Control

Encoder

Data

Encryption

Encoder

Channel/

Line

Encoder

Source

Encoder

Sampling

Quantization

Modulator

MUX

ADC

Analog i/p

CODEC

MODEM

Multiple access channel

Analog o/p

Error

Control

Decoder

Data

Encryption

Decoder

Source

Decoder

Audio

Amp

Reconstruction/

DAC

Equalization /

Decision

Circuits

Demod-ulator

DEMUX

error control coding
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

error control codes

Block

Coder

n

encoded

bits

k

information

bits

k

n-k

Information bits

Parity bits

Add

Redundancy!!

n-bit codeword

Error Control Codes

Block Codes

(memoryless)

Convolutional Codes

(with memory)

Principle

Will not discuss!

(n, k) systematic block code

parity check codes

I6

I4

I7

I5

I3

I2

I1

P

Parity Check Codes

P is set such that the total no. of bits in the code word is even or odd

remediation for detected errors arq

1

Rx

2

3

3

NACK

ACK

ACK

1

2

3

Tx

Error detected!!!

Remediation for Detected Errors: ARQ
7 4 hamming code

I3

I1

I4

I2

P3

P2

P1

(7, 4) Hamming Code
  • 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
7 4 hamming code1

I3

I3

I1

I1

I4

I4

I2

I2

P3

s3

s2

P2

P1

s1

R6

C6

R4

C4

C7

R7

C5

R5

C3

R3

C2

R2

R1

C1

(7, 4) Hamming Code

Encoding

Parity bits

Code

Generator

=

Info bits

Code word

Decoding

Parity

Check

Received code word

Error position

indicator

7 4 hamming code2
(7, 4) Hamming Code

Code Generator Matrix

hamming encoder

I3

I1

I4

I2

P3

P2

P1

+

+

+

Hamming Encoder

Info bits

Parity bits

P3 = I4 I2 I1

P2 = I4 I3 I1

P1 = I4 I3 I2

syndrome decoding
Syndrome Decoding

Parity Check Matrix

hamming decoder

s3

s2

s1

R6

R4

R7

R5

R3

R2

R1

Hamming Decoder

Received code word

Error Position Indicator

+

+

+

s3 = R7 R5 R4  R3

s2 = R7 R6 R4  R2

s1 = R7 R6 R5  R1

hamming decoder1
Hamming Decoder

Error position indicator

(syndrome)

s

Error Position, e

No error

digital communications transceiver1
Digital Communications Transceiver

Anti-

aliasing

Filter

Error

Control

Encoder

Data

Encryption

Encoder

Channel/

Line

Encoder

Source

Encoder

Sampling

Quantization

Modulator

MUX

ADC

Analog i/p

CODEC

MODEM

Multiple access channel

Analog o/p

Error

Control

Decoder

Data

Encryption

Decoder

Source

Decoder

Audio

Amp

Reconstruction/

DAC

Equalization /

Decision

Circuits

Demod-ulator

DEMUX

pulse code modulation pcm
Pulse Code Modulation (PCM)
  • Standard for all digital communications

Analog

Signal

Sample &

Hold

Digital

Bitstream

Quantize

Encode

PAM

PCM

pulse amplitude modulation flat top sampling

8

Analog

PAM

6

4

2

0

-2

-4

-6

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

time

Pulse Amplitude Modulation: Flat-top Sampling

Matlab Demo: pam.m

quantization
Quantization

Uniform Non-Uniform

m-Law A-Law

Mid-rise Mid-tread

Matlab Demos:

quantization.m

pcm.m

  • Mid-rise
  • Mid-tread
  • Mid-rise
  • Mid-tread

Matlab Demos:

companding.m

pcm_mucompand.m

pcm_mucompand2.m

pcm_acompand.m

pcm_acompand2.m

digital communications transceiver2
Digital Communications Transceiver

Anti-

aliasing

Filter

Error

Control

Encoder

Data

Encryption

Encoder

Channel/

Line

Encoder

Source

Encoder

Sampling

Quantization

Modulator

MUX

ADC

Analog i/p

CODEC

MODEM

Multiple access channel

Analog o/p

Error

Control

Decoder

Data

Encryption

Decoder

Source

Decoder

Audio

Amp

Reconstruction/

DAC

Equalization /

Decision

Circuits

Demod-ulator

DEMUX

time division multiplexing

Decode

Time Division Multiplexing

Message 1

Quantize

&

Encode

Message 2

Channel

Message N

Message 1

LPF

Message 2

LPF

Message N

LPF

t1 line tdm format

Channel 1

Channel 24

Channel 2

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

7

7

7

8

8

8

T1-Line TDM Format

1 Frame = (24 Channels + F) = 125 ms

F

193 bits

Frame Sync Bit

Even Frames: 0 0 1 1 1 0

Odd Frames: 1 0 1 0 1 0

Every Sixth Frame

LSB (8th bit) is a Signaling bit

Bipolar R-Z Line Encoding

north american tdma hierarchy

24

6

1

7

1

1

1

2

4

1

5th

M

U

X

3rd

M

U

X

2nd

M

U

X

4th

M

U

X

DS-2

6.312 Mbs

96 VF

DS-3

44.736 Mbs

672 VF

DS-4

274.176 Mbs

4032 VF

DS-1

1.44 Mbs

24 VF

North American TDMA Hierarchy

64 kbs

1 VF

DS-0

1st

M

U

X

DS-5

560.160 Mbs

8064 VF

  • Figure 3-40, p. 207 in Couch
  • Table 3-8, p. 208 in Couch
  • Table 3-9, p. 209 in Couch