Digital Signal Processing Reference
In-Depth Information
decoder based on Euclidean distances
decoder based on Hamming distances
transmitted sequence:
101 011
transmitted sequence:
101 011
natural mapping
8-PSK
natural mapping
8-PSK
noisy channel
noisy channel
received sequence:
110 011
received sequence:
110 011
Allowed subset
010
001
110 011
110 011
010
001
101 011
010 011
Allowed subset
011
000
011
000
8-PSK
distance: 2 bits
distance: 1 bit
111
111
100
100
d
min
distance metric not adapted
to modulation method
(natural mapping, 8-PSK)
101
110
101
110
modulation-aware decoder:
101 011
Hamming-based decoder: 010 011
Figure 2.4.
For 8-psk modulation with natural symbol mapping, a distance
metric other than Hamming must be used. Constellation points
'101' and '110' have the shortest Euclidean distance, but have a
Hamming distance of 2 bits.
As a consequence of the larger
Euclidean distance
between two op-
posing points in the 8-psk constellation, the probability of error (P
e
)
of the msb is obviously lower than for each of the other bits.
After transmission through a noisy channel, sequence '110011' is re-
ceived. A decoder which is aware of the way in which the encoded
data bits are modulated will correctly decide that '101011' has the
highest probability of being the originally transmitted sequence. The
Hamming distance based decoder would have chosen '010011'. This
example explains the importance of correct symbol mapping.
The bit-to-symbol mapper method for a Hamming-based decoder
should employ
Gray-coded mapping
, where the words of each adja-
cent constellation point only differ by 1 bit. Still, the Gray-coded sym-
bol mapper does not correctly reflect the Euclidean distance between
non-adjacent constellation points.
