digital comm v
DESCRIPTION
comunication digitalTRANSCRIPT
Digital Communication Modelb(n)
SinkSourceDecoder
s(m)b(n)Source Source
Coder
s(m)ChannelCoder
DigitalChannel
ChannelDecoder
c(l) c(l)
c = GbSource coder:
G =
�
���������
1 0 0 00 1 0 00 0 1 00 0 0 11 1 1 00 1 1 11 1 0 1
�
���������
Example: (7,4) Hamming code
dmin � 3
How to Correct Errors?
c = GbSource coder:
Channel decoder receives �c = c� e
s = H�c
G =
�
���������
1 0 0 00 1 0 00 0 1 00 0 0 11 1 1 00 1 1 11 1 0 1
�
���������
H =
�
�1 1 1 0 1 0 00 1 1 1 0 1 01 1 0 1 0 0 1
�
�
lower portion of G
identity
= H(c� e) = Hc�He
= HGb�He
= He
Decoder forms
How to Correct Errors?
c =
�
�b
- - - - -Glowerb
�
�c = Gb
H =�Glower I
�
G =�
IGlower
�
= 0Hc = HGb = [(Glowerb)� (Glowerb)]
Error Correction1. For each received “codeword”, form
2. If non-zero, find error sequence
3. Add error sequence to received “codeword”
4. Find data bits in corrected codeword
H�c = He
e He1000000 1010100000 1110010000 1100001000 0110000100 1000000010 0100000001 001
�c� e = (c� e)� e = c
H =
�
�1 1 1 0 1 0 00 1 1 1 0 1 01 1 0 1 0 0 1
�
�
Do we win?• Remember, to send data at the same rate, we must
scale the bit interval duration by the coding efficiency factor E = K/N
-5 0 5 10
10-8
10-6
10-4
10-2
100
Signal-to-Noise Ratio (dB)
Prob
abilit
y of
Blo
ck E
rror
Uncoded (K=4)
(7,4) Code
Hamming Codes• For a (N, K) code, number of single-bit errors = N
• Number of non-zero values for
• To correct any single-bit error, must have
• Maximally efficient when equality applies
N K E3 1 0.337 4 0.57
15 11 0.7331 26 0.8463 57 0.90
127 120 0.94
2N�K � 1 � N, N �K = number of coding bits
2N�K � 1 = N
H�c = 2N�K � 1
But…• The efficient Hamming code can correct all single-
bit errors
• When do double-bit errors become more likely than single-bit errors?
double-bit errors more likely whenpe
1� pe>
2N � 1
�N
1
�pe(1� pe)N�1
� �� �single-bit error probability
?<
�N
2
�p2
e(1� pe)N�2
� �� �double-bit error probability