DSP Applications and Student Projects - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 407

Modulation and AWGN for Soft Decision
In the soft decision decoding setup, the 1/0 output of the convolutional encoder is
mapped into an antipodal baseband signaling scheme (BPSK) by translating 0s to
-

1
on each convolutional encoder output symbol, where x is the encoder output symbol
and y is the output of the BPSK modulator.
AWGN is added to this modulated signal to create the effect of channel noise.
AWGN is a noise whose voltage distribution over time has characteristics that can
be described using a Gaussian distribution, that is, a bell curve. This voltage distri-
bution has zero mean and a standard deviation that is a function of the SNR of the
received signal. The standard deviation of this noise can be varied to obtain signals
with different SNRs at the decoder input.
A zero-mean Gaussian noise with standard deviation

1s and 1s to + 1s. This can be accomplished by performing the operation y = 2 x -

can be generated as
follows. In order to obtain Gaussian random numbers, we take advantage of the
relationships between uniform, Rayleigh, and Gaussian distributions. C only pro-
vides a uniform random number generator, rand( ) . Given a uniform random
variable U , a Rayleigh random variable R can be obtained using

s

( ( ) ) =◊ ( ( ) )

R =◊ 2 s

2

◊ ln 1 1 - U s 2 ◊ ln 1 1 - U

where

2 is the variance of the Rayleigh random variable. Given R and a second
uniform random variable V , a Gaussian random variable G can be obtained using

s

GR = cos V

Viterbi Decoding Algorithm
The Viterbi decoding algorithm uses the trellis diagram to perform the decoding.
The basic cycle repeated by the algorithm at each stage into the trellis is

1. Add : At each cycle of decoding, the branch metrics enumerating from the
nodes (states) of the previous stage are computed. These branch metrics are
added to the previously accumulated and saved path metrics.
2. Compare : The path metrics leading to each of the encoder's states are
compared.
3. Select : The highest-likelihood path (survivor) leading to each of the encoder's
states is selected, and the lower-likelihood paths are discarded.

A metric is a measure of the “distance” between what is received and all of the pos-
sible channel symbols that could have been received. The metrics for the soft deci-
sion and the basic Viterbi decoding techniques are computed using different
methods. For basic Viterbi decoding, the metric used is the Hamming distance, which
specifies the number of bits by which two symbols differ. For the soft decision tech-
nique, the metric used is the Euclidean distance between the signal points in a signal