This document describes modeling and simulating a differential high-speed backplane channel using rational functions in MATLAB. It reads in S-parameter data for a 4-port single-ended backplane channel, converts it to a 2-port differential model, fits a rational function to the differential transfer function, validates the frequency response fit, simulates the channel response to a 2 Gbps random pulse signal, and plots the input, output and eye diagram. The rational function model contains 25 poles.

1. DESIGN OF 10 GBPS USING MATLAB
Our online Tutors are available 24*7 to provide Help with Design of 10 gbps
Homework/Assignment or a long term Graduate/Undergraduate Design of 10 gbps Project.
Our Tutors being experienced and proficient in Design of 10 gbps ensure to provide high
quality Design of 10 gbps Homework Help. Upload your Design of 10 gbps Assignment at
‘Submit Your Assignment’ button or email it to info@assignmentpedia.com. You can use our
‘Live Chat’ option to schedule an Online Tutoring session with our Design of 10 gbps Tutors.
HIGH-SPEED BACKPLANE
This example 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.
Figure 1: A differential high-speed backplane channel

2. 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);
diffsparams =
sparameters: S-parameters object
NumPorts: 2
Frequencies: [1496x1 double]
Parameters: [2x2x1496 double]
Impedance: 100
rfparam(obj,i,j) returns S-parameter Sij

3. 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);
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

4. 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)');
legend('Original data', 'Fitting result');

5. 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]);
subplot(2,1,2);

6. 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);
visit us at www.assignmentpedia.com or email us at info@assignmentpedia.com or call us at +1 520 8371215