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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)
Running
DMAequalizer.m
results in an equalizer
that is numerically similar to the equalizers of the previoustwo sections. Initializing with the “spike” at different
locations results in equalizers with different effectivedelays. The following exercises are intended to encourage
you to explore the DMA equalizer method.
Try the initialization
f=[0 0 0 0]'
in
DMAequalizer.m
. With this initialization,
can the algorithm open the eye? Tryincreasing
m
.
Try changing the stepsize
mu
.
What other nonzero initializations will work?
What happens in
DMAequalizer.m
when the stepsize
parameter
mu
is too large? What happens when it is too
small?
Add (uncorrelated, normally distributed) noise into
the simulation using the command
r=filter(b,1,s)+sd*randn(size(s))
.
What is the largest
sd
you can add, and still have no errors?
Does the initial value for
f
influence this number?
Try at least three initializations.
Modify
DMAequalizer.m
to generate a source sequence
from the alphabet
. For the default channel
[0.5 1 -0.6]
, find an equalizer that opens the eye.
Consider a DMA-like performance function . Show that the resulting gradient algorithm is
Hint: Assume that the derivative of the absolute value is the sign function.Implement the algorithm and compare its performance with the DMA of [link] in terms of
Consider a DMA-like performance function . What is the resulting gradient algorithm?Implement your algorithm and compare its performance with the DMA of [link] in terms of
f
,This section uses the M
atlab program
dae.m
which is
available on the website. The program demonstratessome of the properties of the least squares solution
to the equalization problem and its adaptive cousins:LMS, decision-directed LMS, and DMA.
Throughout these
simulations, other aspects of the system are assumedoptimal; thus, the downconversion is numerically perfect and
the synchronization algorithms are assumedto have attained their convergent values.
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