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The Hebrew University in Jerusalem
Givat Ram Campus
Autumn Semester-2010
Modern Radio Communications
Course # 67732
Professor I. Kalet
13 October 2010
Block Codes
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BLOCK CODES
In this file, we will try to give a very short introduction to Block
Coding. First, all codes have the following properties listed below.
Some of the terms, e.g., Eb,inf/N0, will be more clear after this week’s
lesson.
 Codes improve detectability performance, i.e., for
the same transmitted power and the same information
bit rate (or same Eb,inf/N0), the Pr{} is decreased.
 Coding (Usually) Increases bandwidth
Coding introduces some delay
 Complex implementation
(especially the decoder)
P
Rb, info
CELLULAR
PHONE
Eb, info=P Tb,info=P/Rb,info
P =Eb, info Rb, info
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BLOCK CODES
The names of some typical block codes
 HAMMING
 GOLAY
 BCH
 REED-SOLOMON
 CONCATENATED CODES
(inner and outer codes)
The description, given in this file, is a very brief
description of a simple block code. The important
pages are pages 4-7, and 8-11.
4
5
Simple example of an (n,k) Block Code
This (7,4) Hamming code is capable of
detecting and correcting one error!!
HAMMING CODE -(7,4)
Information Bits in Block
(7,4)
Total Number of Bits in Block
There are (7-4) = 3 parity or check bits in the block-These do not
carry any new information and are redundant coding bits.
In general
 (n,k) code
 (n-k) parity bits
Coding usually increases the bandwidth
 UNCODED
-Rinfo=2400 bps
 CODED- After Coding we transmit
-R trans=2400(7/4)=4200 bps
The bandwidth increases by a factor of 7/4!
6
Block Coding plus BPSK Modulation
N0/2 watts/Hz
Info
Bits
Coded
Bits
Block
Coder
BPSK
MOD
Demodulator
Block
Decoder
Soft-Decision Decoding
Hard-Decision Decoding
Demodulator
cos 2f0t
T

0
7
Hamming (7,4) Code
Information
Bits
a3=1
a5=0
a6=1
a7=1
Transmitted
(Coded) Bits
HAMMING
CODER
(7, 4)
a1, a2 and a4 are
the redundant
bits, created by
the equations
below
a1
a2
a3
a4
a5
a6
a7
Eb,info bit
a7
a6
a5
a3
1
1
0
1
Info
Bits
Eb,transmitted= (4/7) Eb,info bit [(4/7)=-2.43 dB)]
a7
a6
a5
a4
a3
a2
a1
1
1
0
0
1
1
0
Pb,tr
Transmitted Bit
ENCODER
a1=a3  a5  a7=0
a2=a3  a6  a7=1
a4=a5  a6  a7=0
(Eb,trans)
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Binary Representation of Place Numbers in
Block
4
2
1
an (n)
0
0
1
1
0
1
0
2
0
1
1
3
1
0
0
4
1
0
1
5
1
1
0
6
1
1
1
7
9
DECODER
 Will detect and correct one error.
(can detect two (or more) errors but
cannot correct them!)
Demodulated Signal-After Receiver
including one error
1 0 0 0 1 1 0
a4
a2 a1
Once again (at the receiver) we generate the parity
bits, a1R, a2R, and a4R based on the received
information bits.
1 0 0 1 1 0 0
a4R
a2R a1R
Then, we do modulo 2 additions between the received parity bits,a1 ,
a2 and a4 , and the parity bits a1R , a2R and a4R , which we generated
at the receiver based on the received information bits.
a1R=0
a2R=0
a4R=1



a1= 0
a2= 1
a4= 0
=
=
=
0
1
1
1 1 0  is equivalent to “6”
There is an error in place number 6!
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What is the Pr{} ?
Prblock{}=Pr{two or more errors
in block}
Prblock{}=1-Pr{no errors or
one error}
7
=1-[(1-Pb,tr) +( )Pb,tr(1-Pb,tr)
7
1
Pb,tr =Probability of error of a
transmitted bit
The Question is“Have “things” improved??"
6
11
PROBABILITY OF ERROR
HAMMING (7,4) CODE
Eb/N0,trans,dB Eb/N0,info,dB
CODED
CODED
Pbit,trans
CODED
Pblock
CODED
P dec-info
CODED
+1.43 dB
10-1
.15
.08
+.5 dB
+4.3
6.73
10-2
2*10-3
10-3
+6.8
+6.8
9.23
10-3
2.1*10-5
10-5
+9.6
+8.4
10.83
10-4
2.2*10-7
10-7
+11.2
-1 dB
Eb/N0,info,dB
UNCODED
If we convert (4/7) into dB we get -2.43 dB
 Eb/N0, info, dB=Eb/N0, trans, dB + 2.43 dB
 Pdec=Pr{error in decoded bits} (1/2) Pblock
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BLOCK CODE PERFORMANCE
In general, the minimum Hamming distance, min dH,
determines the Pr{}
min dH=2t+1
The letter “t” represents the number of bit errors that
we can detect and correct.
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Block Codes
Coding Performance
B. Sklar, Digital Communications-Fundamentals and ApplicationsPrentice –Hall, New York, 1988
Hamming (7,4), t=1
Code
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Soft Decision versus Hard Decision
Detected Bits
1 0 0 0 1 1 0
cos 2f0t
T

0
The BER performance of soft-decision decoding is
about two and a half dB better than that of harddecision decoding (@ 10-5 ) but requires a lot of DSP.
However, the Viterbi Algorithm makes soft-decoding
possible for convolutional codes.
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SOFT DECISION IS ABOUT 2.5 DB
BETTER THAN HARD DECISION AT
Probability of Error =10-5
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Summary
The description given in this file is a very brief
description of a simple block code.The important
pages are pages 4-7, and 8-11.