Statistical Signaling Analysis (SSA)

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Three fast time-domain system simulation techniques:

  • single-bit response method (SBR)
  • double-edge response method (DER)
  • multiple-edge response method (MER)

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Symmetric Rising and Falling Edges

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Single-Bit Response (SBR) Method

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Overlapping portions of a pulse response from neighboring bits are referred to as intersymbol interference (ISI). A received waveform is formed by superimposing, in time, the pulse responses of each bit in the sequence, as illustrated in Figure 9, assuming symmetric positive and negative pulses are transmitted for 1s and 0s

image-20240824193208821

To avoid spurious glitches between consecutive ones, rising and falling edge responses shall be symmetric. This is the limitation of SBR method.

Let \(p(t)\) be the SBR of the channel, \(t_s\) be the data sampling phase, \(T\) be the bit time, \(N_c\) is the number of UI in stored pulse response and \(b_m\) be the \(m\)th transmitted symbol. The voltage seen by the receiver's data sampler at the \(m\)th data sample is determined by \[ y_m = \sum_{k=m-N_c+1}^{m}b_kp(t_s+(m-k)T) \] where \(b_k \in [0, 1]\) and \(p(t) \ge 0\)

We always prepend \(Nc-1\) 0s in random bit stream for consistency.

image-20220429112902281

For computation convenient, the pulse need to be positive. For differential signal and amplitude \(V_{peak}\), the peak to peak is \(-V_{peak}\) to \(+V_{peak}\). After pulse added by \(V_{peak}\), peak to peak is \(0\) to \(+2V_{peak}\).

image-20220429154336080

image-20220429154423247

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hold on;

yy_sum = zeros(OSR*Ns, Ns);
for idxBit = 1:Ns
    bs_split = zeros(1, Ns+Nc-1);
    bs_split(idxBit) = bs(idxBit);
    yy = zeros(OSR, Ns);
    for ii = Nc:Nc+Ns-1
        bb = bs_split(ii:-1:ii-Nc+1);
        yy(:,ii-Nc+1) = sum(bb.*yrps, 2);
    end
    yy_cont2 = reshape(yy, [], 1);
    h = plot(yy_cont2);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
    yy_sum(:, idxBit) = yy_cont2;
end
yy_sum = sum(yy_sum, 2);    % merge
plot(yy_sum, 'k--');
plot(yy_cont, 'm-.');
grid on;
legend('sum', 'syn');
title('merge all single bit');
ylabel('mag');
xlabel('Time (\times Ts)');

The pulse response contain rising and falling edge. The 1 bit first rise from -1 to 1, then fall to -1; The 0 bit just do nothing for synthesized waveform with the help of falling edge of 1 bit.

The DC shift help deal with continuous 0 bits.

image-20220429174330324

another SBR example

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A = zeros(10,21);
n = [1:10];

% post cursor
for m = 1:3
    A(m, 11+n(m)) = 0.5;
    A(m, 11-n(m)) = 0.5;
end

% one
for m = 4:10
    A(m, 11) = 1;
end

% h0 or main cursor
h0 = zeros(1, 21);
h0(1, 1) = 0.5;
h0(1, 21) = 0.5;
out = h0;

for m = 1:10
    out = conv(out, A(m, :), "full");
end


stem(out)

143512636-0878e0fd-fe87-414c-9c73-52577eeb7593

143512677-ccefdf22-4e30-4e72-9220-bbe667671e79

Single Ended vs Differential Signaling

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Sai Goutham Sunkara, create a precursor in a single-bit response using RC filter [linkdin]

View image

Two poles is the minimum RC-filter order needed to produce a nonzero −1-UI precursor

The cursors are defined relative to the response peak, not to when the input arrived. A causal system can't respond before its input, but it can easily respond before its own peak. Any extra delay or smoothing that moves the peak more than 1 UI after the response begins will put signal at the −1 UI sample.

S-Parameter to Single Bit Response (SBR)

Mike Li, "S-Parameter to Single Bit Response (SBR) Transformation and Convergence Study" [https://ieee802.org/3/bj/public/may12/li_01_0512.pdf]

TODO 📅

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Double-Edge Response (DER) Method

To handle the more general cases, with asymmetric rising and falling edges, the system response can be constructed in terms of edge transitions instead of bit responses.

The DER method decomposes the input data pattern, in terms of rising and falling edge transitions. The system response can be calculated by superimposing the shifted versions of the rising and falling edge responses : \[ y_m = \sum_{k=m-N_c+1}^{m}(b_k-b_{k-1})s_k(t_s+(m-k)T) + y_{int} \] where

\[\begin{align} s_i(t) &= r(t) -V_{low} \quad \text{if} \: (b_i\gt b_{i-1}) \\ &= V_{high}-f(t) \quad \text{otherwise} \end{align}\]

\(r(t)\) and \(f(t)\) are the rising and falling edge responses,respectively. \(V_{high}\) and \(V_{low}\) are the steady state DC levels, in response to a constant stream of ones and zeros, respectively. \(y_{int}\) is the initial DC state (either \(V_{high}\) or \(V_{low}\) ).

We always prepend \(Nc\) 0s in random bit stream for consistency.

image-20220429191941805

der.drawio

image-20220430010336977

image-20220430013715680

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figure(1)
subplot(3, 1, 1)
plot(yrc);
hold on;
plot(yfc);
hold off;
legend('rising', 'falling')
grid on;
ylabel('mag');
xlabel('Time (\times Ts)');
title('step response');

subplot(3, 1, 2)
stem(bs, 'k'); grid on;
hold on;
stem((idxPreRspStart:idxPreRspEnd), bs(idxPreRspStart:idxPreRspEnd), "filled", 'r');
stem(idxPreRspStart, bs(idxPreRspStart), 'go');
stem(idxCurData, bs(idxCurData), "filled", 'm');
stem((idxPreRspStart:idxPreRspEnd)+0.5, 0.1.*bd(idxPreRspStart:idxPreRspEnd), 'bd-.');
hold off;
legend('', 'Nc bits', 'y_{int}', 'Current bit', 'Edge Transitions');
ylabel('mag');
xlabel('Time (\times UI)');
title('input stream');

subplot(3, 1, 3)
yy_cont = reshape(yy, [], 1);   % continuous version
plot(yy_cont); grid on;
title('continuous yout')
ylabel('mag');
xlabel('Time (\times Ts)');

figure(2)
hold on;
for idx = idxPreRspStart+1-Nc:idxCurData+32-Nc
    ys = yy(:, idx);
    tt = ((idx-1)*OSR+1:idx*OSR);
    h = plot(tt(:), ys(:), 'LineWidth',3);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
end
plot(yy_cont, 'm--', 'LineWidth',1);
hold off;
legend('syn')
ylabel('mag');
xlabel('Time (\times Ts)');
title('synthesize with step response');
grid on;

Cross Talk using Convolution

Charles Moore, Computing effect of cross talk using Convolution [https://grouper.ieee.org/groups/802/3/ap/public/channel_adhoc/moore_c1_0305.pdf]

Huang Chunxing, Statistical Eye Simulation Requirements [https://ibis.org/summits/oct06a/huang.pdf]

image-20260501222929928

Reference

T. C. Carusone, "Introduction to Digital I/O: Constraining I/O Power Consumption in High-Performance Systems," in IEEE Solid-State Circuits Magazine, vol. 7, no. 4, pp. 14-22, Fall 2015

—, ISSCC 2017 T6: Signal Integrity Analysis for Gb/s Links

Oh, Kyung Suk Dan, and Xing Chao Chuck Yuan. High-Speed Signaling: Jitter Modeling, Analysis, and Budgeting. Prentice Hall, 2011.

Ren, Jihong and Kyung Suk Oh. "Multiple Edge Responses for Fast and Accurate System Simulations." IEEE Transactions on Advanced Packaging 31 (2008) [https://sci-hub.jp/10.1109/TADVP.2008.2002201]

Shi, Rui. "Off-chip wire distribution and signal analysis." (2008). [pdf]

X. Chu, W. Guo, J. Wang, F. Wu, Y. Luo and Y. Li, "Fast and Accurate Estimation of Statistical Eye Diagram for Nonlinear High-Speed Links," in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 29, no. 7, pp. 1370-1378, July 2021, [https://sci-hub.ru/10.1109/TVLSI.2021.3082208]

Tingting Pang, DesignCon 2025: Fast BER Analysis Technique for Next Generation Chiplet Simultaneous Bi-Directional Transceiver