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The following M
atlab program explores a noisy system.
A sequence of four-level data is generated by calling the
pam.m routine. Noise is then added with
power specified by
p , the number of errors caused
by this amount of noise is calculated in
err .
b=[0.5 1 -0.6]; % define channelm=1000; s=sign(randn(1,m)); % binary source of length m
r=filter(b,1,s); % output of channeln=4; f=[0 1 0 0]'; % center spike initializationmu=.01; % algorithm stepsize
for i=n+1:m% iterate
rr=r(i:-1:i-n+1)'; % vector of received signal e=(f'*rr)*(1-(f'*rr)^2); % calculate error
f=f+mu*e*rr; % update equalizer coefficientsend
DMAequalizer.m find a DMA equalizer f for the channel b
(download file)
Typical outputs of
noisychan.m are shown in
[link] . Each plot shows the input sequence
(the four solid horizontal lines), the input plus the noise(the cloud of small dots), and the error between the
input and quantized output (the dark stars). Thus the dark stars thatare not at zero represent errors in transmission.
The noise
in the right-hand case is the maximum noise
allowable in the plausibility argument used to derive
[link] ,
which relates the average amplitudes of thesignal plus the noise to the number of levels in the signal.
For
(the same conditions as in
[link] (a)), the noise was chosen to be
independent and normally distributed with power
to ensure that
.
The middle plot used a noise with power
and the
left-hand plot had noise power
.
As can be seen from the plots, there were essentially no errorswhen using the smallest noise,
a handful of errors in the middle, and about
errors
when the power of the noise matched the Shannon capacity.Thus, this naive transmission of four-level data (i.e., with no
coding) has many more errors than the Shannon limit suggests.
Find the amplitudes of the -level (equally spaced) signal with unity power when
Use
noisychan.m to compare the noise performance of
two-level, four-level, and six-level transmissions.
Use
noisychan.m to compare the power requirements
for two-level, four-level, and six-level transmissions.Fix the noise power at
p=0.01 , and
find the error probability for four-level transmission.Experimentally find the power
that is required
to make the two-level and six-level transmissions have the sameprobability of error. Can you think of a way to calculate
this?
Consider the (asymmetric, nonuniformly spaced) alphabet consisting of the symbols .
noisychan.m to examine the noise performance of
this transmission by making a plot of the noise powerversus percentage of errors.
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