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- Software receiver design
- Software receiver design
- Linear equalization
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.
Use
DMAequalizer.m to find an equalizer that can open the eye
for the channel
b=[1 1 -0.8 -.3 1 1] .
- What equalizer length
n is needed?
- What initializations for
f did you use?
- How does the converged answer compare with the designs in
Exercises
[link] ,
[link] , and
[link] ?
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.
- What equalizer length
n is needed?
- What is an appropriate value of
?
- What initializations for
f did you use?
- Is this a fundamentally easier or more difficult
task than when equalizing a binary source?
- How does the answer compare with the designs in
Exercises
[link] ,
[link] , and
[link] ?
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
- speed of convergence,
- number of errors in a noisy environment
(recall
[link] ), and
- ease of initialization.
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
- speed of convergence of the equalizer
coefficients
f ,
- number of errors in a noisy environment
(recall
[link] ), and
- ease of initialization.
Examples and observations
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.
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Source:
OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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