Convert Scattering Parameter to Impulse Response for SerDes System

This example covers two topics: the first focuses on the concatenation of three scattering parameters (S-Parameters) that represent a communication channel along with the analog components of a transmitter and of a receiver into a single 4-port S-Parameter, the second focuses on the conversion of this 4-port S-Parameter to an impulse response for use with SerDes Toolbox™. You will use the rationalfit function from the RF Toolbox™ to find the impulse response of the S-Parameter that represents a communication channel employing True and Complement differential signaling. For more information, see Modeling a High-Speed Backplane (4-Port S-Parameters to a Rational Function).

Configure Variables

The S-Parameter file must be a 4-port .s4p file. Normalize the transmitter (Tx) and receiver (Rx) impedances to 50 Ohms. The Datarate and SamplesPerSymb valuses must match the settings of the SerDes system and are set to 10 GHz and 32, respectively. The control parameters for the rationalfit function include delayStart, delayStep, delayStop, and nPoles. Adjust the resolution of the impulse response using pointsInImpulse.

filename = 'default.s4p';
TxR = 50;
TxC = 0.1e-12;
RxR = 50;
RxC = 0.2e-12;
datarate = 10*1e9;
samplespersymb = 32;

delayStart = 0.68;
delayStep = 0.01;
delayStop = 0.73;
nPoles = 120;
Rshort = 1.0e-6;
pointsInImpulse = 16384;
showPlots = true;

Create Differential S-Parameter Representation of Communication Channel

Calculate the symbol time and the sample interval. Import the S-Parameter data file and keep the original frequency content.

pulsewidth = 1/datarate;
ts = pulsewidth/samplespersymb;
origSparam = sparameters(filename); 
freq = origSparam.Frequencies;

Combine S-Parameter of Communication Channel with Analog Sections of Transmitter and Receiver

Create the analog Tx and Rx circuits as S-Parameters. Then combine the S-Paramater models of the Tx and Rx circuits with S-Parameter model of Communication Channel.

Convert Analog Model of Transmitter to S-Parameter

cktTx = circuit('txAnalog');
add(cktTx,[1 3],resistor(TxR,'R1'))
add(cktTx,[3 0],capacitor(TxC,'C1'))
add(cktTx,[2 4],resistor(TxR,'R2'))
add(cktTx,[4 0],capacitor(TxC,'C2'))
setports(cktTx,[1 0],[2 0],[3 0],[4 0])
StxAnalog = sparameters(cktTx,freq,50);

Configure Port Order for S-Parameter of Communication Channel

cktCh = circuit('ChannelSparam');
channel = nport(filename); % Create s-parameter circuit element
add(cktCh,[1 3 2 4],channel);
setports(cktCh,[1 0],[2 0],[3 0],[4 0])
Schannel = sparameters(cktCh,freq,50);

Convert Analog Model of Receiver to S-Parameter

cktRx = circuit('rxAnalog');
add(cktRx,[1 0],resistor(RxR,'R3'))
add(cktRx,[1 0],capacitor(RxC,'C3'))
add(cktRx,[2 0],resistor(RxR,'R4'))
add(cktRx,[2 0],capacitor(RxC,'C4'))
add(cktRx,[1 3],resistor(Rshort,'R5'))
add(cktRx,[2 4],resistor(Rshort,'R6'))
setports(cktRx,[1 0],[2 0],[3 0],[4 0])
SrxAnalog = sparameters(cktRx,freq,50);

Create Combined S-Parameter Object

Concatenate the S-Parameters using cascadesparams. Set values of data, freq and z0 for use by the rationalfit function. Use s2sdd to convert the data to a 4-port differential S-Parameter diffdata and set the port order to 1234. Note: RF Toolbox applies ports first from top to bottom on an S-parameter, then from left to right.

sparamWithAnalog = cascadesparams(StxAnalog,Schannel,SrxAnalog);

data = sparamWithAnalog.Parameters;
freq = sparamWithAnalog.Frequencies;
z0 = sparamWithAnalog.Impedance;

diffdata = s2sdd(data,2); 
diffz0 = 2*z0;
diffsparams = sparameters(diffdata,freq,diffz0);

Compute Analytical Form of Transfer Function with Rational Fit

Find the transfer function from the diffdata, reference impedance of the S-Parameters, source impedenance in the Tx analog model, and the load impedance in the Rx analog model.

zRef = diffz0;   % Reference impedance of S-Parameters
zSource = 1e-6;  % Short circuit the source impedance since it is included in the Tx Analog model.
zLoad = 1e6;     % Open circuit the load impedance since it is included in the Rx Analog model.
difftransfunc = s2tf(diffdata,zRef,zSource,zLoad);

Create the rationalfit of the diff S-Parameter. Sweep the delay factor and keep the best fit. Then plot the derived rationalfit result.

bestErr = 0;
bestDelay = 0;
bestRationalFit = rfmodel.rational();

for delaySweep = delayStart:delayStep:delayStop
    [rationalfunc,errdb] = rationalfit(freq,difftransfunc,-50,'DelayFactor',delaySweep,'IterationLimit',100,'NPoles',nPoles);
    if errdb < bestErr
        bestErr = errdb;
        bestDelay = delaySweep;
        bestRationalFit = rationalfunc;
    end
    fprintf('.'); 
end
......
rationalfunc = bestRationalFit;
npoles = length(rationalfunc.A);
fprintf('\nThe derived rational function achieved %f dB fit with %f delay and %d poles.\n',bestErr,bestDelay,npoles); 
The derived rational function achieved -69.958648 dB fit with 0.680000 delay and 120 poles.

Create the impulse response.

[imp,impt]=impulse(rationalfunc,ts,pointsInImpulse);

Evaluate Differential-Mode Frequency Response

Plot the magnitude and phase of the original transfer function and the output of rationalfit.

    freqsforresp = linspace(0,max(freq)*2,length(freq))';
    resp = freqresp(rationalfunc,freqsforresp);
    
    figure(11)
    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),'FontSize',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')

Convert Transfer Function to Impulse Response and Plot for Evaluation

Use the transfer function from rationalfit to find the impulse response using impulse.

[imp,impt]=impulse(rationalfunc,ts,pointsInImpulse);
plot(impt,imp);

See Also