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1. DESIGN AT 10GBPS USING MATLAB (SAMPLE ASSIGNMENT)
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RF Toolbox
This sample assignment shows how to use RF Toolbox to model a
differential high-speed backplane channel using rational functions. This
type of model is useful to signal integrity engineers, whose goal is to
reliably connect high-speed semiconductor devices with, for example,
multi-Gbps serial data streams across backplanes and printed circuit
boards.
Compared to traditional techniques such as linear interpolation, rational function fitting provides more insight into the
physical characteristics of a high-speed backplane. It provides a means, called model order reduction, of making a
trade-off between complexity and accuracy. For a given accuracy, rational functions are less complex than other
types of models such as FIR filters generated by IFFT techniques. In addition, rational function models inherently
constrain the phase to be zero on extrapolation to DC. Less physically-based methods require elaborate constraint
algorithms in order to force the extrapolated phase to zero at DC.

2. Figure 1: A differential high-speed backplane channel
Read the Single-Ended 4-Port S-Parameters and Convert Them to Differential 2-Port S-Parameters
Read a Touchstone data file, default.s4p, into an sparameters object. The parameters in this data file are the
50-ohm S-parameters of the single-ended 4-port passive circuit shown in Figure 1, given at 1496 frequencies ranging
from 50 MHz to 15 GHz. Then, get the single-ended 4-port S-parameters and use the matrix conversion
function s2sdd to convert them to differential 2-port S-parameters. Finally, plot the differential S11 parameter on a
Smith chart.
filename = 'default.s4p';
backplane = sparameters(filename);
data = backplane.Parameters;
freq = backplane.Frequencies;
z0 = backplane.Impedance;
Convert to 2-port differential S-parameters.
diffdata = s2sdd(data);
diffz0 = 2*z0;
% By default, |s2sdd| expects ports 1 & 3 to be inputs and ports 2 & 4 to
% be outputs. However if your data has ports 1 & 2 as inputs and ports 3 &
% 4 as outputs, then use 2 as the second input argument to |s2sdd| to
% specify this alternate port arrangement. For example,
% diffdata = s2sdd(data,2);
diffsparams = sparameters(diffdata,freq,diffz0)
fig1 = figure;
smith(diffsparams,1,1);

3. diffsparams =
sparameters: S-parameters object
NumPorts: 2
Frequencies: [1496x1 double]
Parameters: [2x2x1496 double]
Impedance: 100
rfparam(obj,i,j) returns S-parameter Sij
Compute the Transfer Function and Its Rational Function Object Representation
First, use the s2tf function to compute the differential transfer function. Then, use the rationalfit function to compute
the analytical form of the transfer function and store it in an rfmodel.rational object. The rationalfit function fits a
rational function object to the specified data over the specified frequencies. The run time depends on the computer,
the fitting tolerance, the number of data points, etc.
difftransfunc = s2tf(diffdata,diffz0,diffz0,diffz0);

4. delayfactor = 0.98; % Delay factor. Leave at the default of zero if your
% data does not have a well-defined principle delay
rationalfunc = rationalfit(freq,difftransfunc,'DelayFactor',delayfactor)
npoles = length(rationalfunc.A);
fprintf('The derived rational function contains %d poles.n', npoles);
rationalfunc =
rational with properties:
A: [25x1 double]
C: [25x1 double]
D: 0
Delay: 6.5982e-09
Name: 'Rational Function'
The derived rational function contains 25 poles.
Validate the Differential-Mode Frequency Response
Use the freqresp method of the rfmodel.rational object to get the frequency response of the rational function
object. Then, create a plot to compare the frequency response of the rational function object and that of the original
data. Note that detrended phase (i.e. phase after the principle delay is removed) is plotted in both cases.
freqsforresp = linspace(0, 20e9, 2000)';
resp = freqresp(rationalfunc,freqsforresp);
fig2 = figure;
subplot(2,1,1);
plot(freq*1.e-9,20*log10(abs(difftransfunc)),'r',freqsforresp*1.e-9, ...
20*log10(abs(resp)), 'b--', 'LineWidth', 2);
title(sprintf('Rational Fitting with %d poles',npoles),'fonts',12);
ylabel('Magnitude (decibels)'); xlabel('Frequency (GHz)');
legend('Original data', 'Fitting result');
subplot(2,1,2);
origangle=unwrap(angle(difftransfunc))*180/pi+360*freq*rationalfunc.Delay;
plotangle=unwrap(angle(resp))*180/pi+360*freqsforresp*rationalfunc.Delay;
plot(freq*1.e-9,origangle,'r', ...
freqsforresp*1.e-9,plotangle,'b--', 'LineWidth', 2);
ylabel('Detrended phase (deg.)'); xlabel('Frequency (GHz)');

5. legend('Original data', 'Fitting result');
Calculate and Plot the Differential Input and Output Signals of the High-Speed Backplane
Generate a random 2 Gbps pulse signal. Then, use the timeresp method of the rfmodel.rational object to
compute the response of the rational function object to the random pulse. Finally, plot the input and output signals of
the rational function model that represents the differential circuit.
datarate = 2*1e9; % Data rate: 2 Gbps
samplespersymb = 100;
pulsewidth = 1/datarate;
ts = pulsewidth/samplespersymb;
numsamples = 2^17;
numplotpoints = 10000;
t_in = double((1:numsamples)')*ts;
input = sign(randn(1, ceil(numsamples/samplespersymb)));
input = repmat(input, [samplespersymb, 1]);
input = input(:);
[output, t_out] = timeresp(rationalfunc,input,ts);
fig3 = figure;
subplot(2,1,1);
plot(t_in(1:numplotpoints)*1e9,input(1:numplotpoints),'LineWidth', 2);
title([num2str(datarate*1e-9), ' Gbps signal'], 'fonts', 12);
ylabel('Input signal'); xlabel('Time (ns)'); axis([-inf,inf,-1.5,1.5]);

6. subplot(2,1,2);
plot(t_out(1:numplotpoints)*1e9,output(1:numplotpoints),'LineWidth',2);
ylabel('Output signal'); xlabel('Time (ns)'); axis([-inf,inf,-1.5,1.5]);
Plot the Eye Diagram of the 2-Gbps Output Signal
Estimate and remove the delay from the output signal and create an eye diagram by using Communications System
Toolbox™ functions.
if ~isempty(which('commscope.eyediagram'))
if exist('eyedi', 'var'); close(eyedi); end;
eyedi = commscope.eyediagram('SamplingFrequency', 1./ts, ...
'SamplesPerSymbol', samplespersymb, 'OperationMode', 'Real Signal');
% Update the eye diagram object with the transmitted signal
estdelay = floor(rationalfunc.Delay/ts);
eyedi.update(output(estdelay+1:end));
end

7. if exist('eyedi', 'var'); close(eyedi); end;
close(fig1);
close(fig2);
close(fig3);