the resistance of MOS is not highly controlled -> \(R_T + Z_N\)
Peak power constraint of TX
FIR
Due to circuit limitation, circuit cannot have arbitrarily large
voltage on the output, i.e. a limited maximum swing. In order
to create the high frequency shape, the best we can do is lower DC
gain (low frequency gain < 1)
FIR is not increasing the amplitude on the edges
FIR is reducing the inner eye diagram
The maximum swing stays the same, \(\sum_i
|c_i|=1\)
Sharing termination keep a constant current through leg, which
improve TX speed in this way. On the other hand, the sharing termination
facilitate drain/source sharing technique in layout.
pull-up and pull-down
resistor
Original stacked structure
Pro's:
smaller static current when both pull up and pull down path is
on
Con's:
slowly switching due to parasitic capacitance behind pull-up and
pull-down resistor
with single shared linearization resistor
Pro's:
The parasitic capacitance behind the resistor still exists but is
now always driven high or low actively
Con's:
more static current
VM
Driver Equalization - differential ended termination
\[
V_o = D_{n+1}C_{-1}+D_nC_0+D_{n-1}C_{+1}
\]
where \(D_n \in \{-1, 1\}\)
\[
V_{\text{rx}} = V_{\text{dd}} \frac{(R_2-R_1)R_T}{R_1R_T+R_2R_T+R_1R_2}
\] With \(R_u=(L+M+N)R_T\)
\[\begin{align}
V_{\text{rxp}} &= \frac{1}{2} \cdot \frac{N}{L+M+N} \\
V_{\text{rxm}} &= \frac{1}{2} \cdot \frac{L+M}{L+M+N}
\end{align}\] So \[
V_{L}= \frac{1}{2}\cdot\frac{N-(L+M)}{L+M+N}
\] which is same with differential ended termination
Equation-2
\[\begin{align}
V_{\text{rxp}} &= \frac{1}{2} \\
V_{\text{rxm}} &= 0
\end{align}\] So \[
V_{M}= \frac{1}{2}
\] which is same with differential ended termination
Which can be simpified as \[\begin{align}
V_{\text{rx}} &= \frac{1}{2}(V_p - V_m) \\
&= \frac{1}{2}(\frac{2}{3}(2V_{\text{MSB}}+V_{\text{LSB}})-1) \\
&=\frac{1}{3}(2V_{\text{MSB}}+V_{\text{LSB}})-\frac{1}{2}
\end{align}\]
The above eqations demonstrate that the output \(V_{\text{rx}}\) is the linear sum of
MSB and LSB; LSB and
MSB have relative weight, i.e. 1 for LSB and
2 for MSB.
Assume pre cusor has \(L\) legs,
main cursor \(M\) legs and post cursor
\(N\) legs, which is same with the
convention in "Voltage-Mode Driver Equalization"
The number of legs connected with supply can expressed as \[
n_{up} = (1-d_{n+1})L + d_{n}M + (1-d_{n-1})N
\] Where \(d_n \in \{0, 1\}\),
or \[
n_{up} = \frac{1}{2}(-D_{n+1}+1)L + \frac{1}{2}(D_{n}+1)M +
\frac{1}{2}(-D_{n-1}+1)N
\] Where \(D_n \in \{-1,
+1\}\)
Then the number of legs connected with ground is \[
n_{dn}=L+M+N-n_{up}
\] where \(n_{up}+n_{dn}=L+M+N\)
Voltage resistor divider \[\begin{align}
V_o &=
\frac{\frac{R_{U}}{n_{dn}}}{\frac{R_U}{n_{dn}}+\frac{R_U}{n_{up}}} \\
&= \frac{1}{2}- \frac{1}{2}D_{n+1}\frac{L}{L+M+N}+
\frac{1}{2}D_{n}\frac{M}{L+M+N}-\frac{1}{2}D_{n-1}\frac{N}{L+M+N} \\
&= \frac{1}{2}-\frac{1}{2}D_{n+1}\cdot l+ \frac{1}{2}D_{n}\cdot
m-\frac{1}{2}D_{n-1}\cdot n
\end{align}\]
where \(l+m+n=1\)
\(V_{\text{MSB}}\) and \(V_{\text{LSB}}\) can be obtained
\[\begin{align}
V_{\text{MSB}} &= \frac{1}{2}-\frac{1}{2}D^{\text{MSB}}_{n+1}\cdot
l+ \frac{1}{2}D^{\text{MSB}}_{n}\cdot
m-\frac{1}{2}D^{\text{MSB}}_{n-1}\cdot n \\
V_{\text{LSB}} &= \frac{1}{2}-\frac{1}{2}D^{\text{LSB}}_{n+1}\cdot
l+ \frac{1}{2}D^{\text{LSB}}_{n}\cdot
m-\frac{1}{2}D^{\text{LSB}}_{n-1}\cdot n
\end{align}\]
Substitute the above equation into \(V_{\text{rx}}\), we obtain the relationship
between driver legs and FFE coefficients
After scaling, we obtain \[
V_{\text{rx}} = -l\cdot(2 \cdot
D^{\text{MSB}}_{n+1}+D^{\text{LSB}}_{n+1})+ m\cdot(2\cdot
D^{\text{MSB}}_{n}+D^{\text{LSB}}_{n}) - n \cdot(2\cdot
D^{\text{MSB}}_{n-1}+D^{\text{LSB}}_{n-1})
\] Where \(C_{-1} = l\), \(C_0 = m\) and \(C_{1}=n\), which is same with that of
NRZ
J. F. Bulzacchelli et al., "A 28-Gb/s 4-Tap FFE/15-Tap DFE Serial
Link Transceiver in 32-nm SOI CMOS Technology," in IEEE Journal of
Solid-State Circuits, vol. 47, no. 12, pp. 3232-3248, Dec. 2012, doi:
10.1109/JSSC.2012.2216414.
C. Menolfi et al., "A 112Gb/S 2.6pJ/b 8-Tap FFE PAM-4 SST TX in 14nm
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J. Kim et al., "A 112Gb/s PAM-4 transmitter with 3-Tap FFE in 10nm
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[paper]
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Japan, 2025
RX Equalization Noise
Enhancement
Advanced Signal Integrity for High-Speed Digital Designs, S. H. Hall
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The periodic signal on VCTRL modulates the
VCO, giving rise to deterministic jitter
Timing Offsets Between Up and Dn Pulses
Mismatch Between Charge-Pump Current Sources
Incomplete Settling of Charge-Pump Currents
Finite Output Resistance of the Charge Pump
Up/Dn Timing Offset
If Dn pulse arrives \(\Delta T\)
after the Up pulse, the steady-state VCTRL will be slightly
lower than it would be without the \(\Delta T\) mismatch so as to return the
VCO's phase to match the reference clocks.
Vice versa, if If Up pulse arrives \(\Delta
T\) after the Dn pulse, the steady-state VCTRL will be slightly
higher than without \(\Delta
T\) mismatch
Current Sources Mismatch
Incomplete Settling
TODO 📅
W. Rhee, "Design of high-performance CMOS charge pumps in
phase-locked loops," 1999 IEEE International Symposium on Circuits
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University Press; 2024
2nd loop filter
PI (proportional - integral) Loop Filter
PFD Deadzone
Dead zone induced by incomplete settling of charge-pump
currents
This situation can be avoided by adding additional delay to the
AND gate in the PFD
For the sake of simplicity, \(V_{ctr}\) looks like a rectangular pulse
with an amplitude of \(I_{CP}R_1\) and
a duty ratio of (\(I_{leak}/I_{CP}\)),
whose first coefficient of Fourier series is
where \(I_\text{leak} \ll I_{CP}\)
is assumed
Then, the peak frequency deviation \(\Delta f\)\[
\Delta f = a_1 \cdot K_v = 2I_\text{leak}R_1 K_v
\] using narrowband FM approximation, we have \[
P_\text{spur} = 20\log\left(\frac{\Delta f}{2f_\text{ref}}\right) =
20\log\left(\frac{I_\text{leak}R_1 K_v}{f_\text{ref}}\right)
\]
W. Rhee, "Design of high-performance CMOS charge pumps in
phase-locked loops," 1999 IEEE International Symposium on Circuits
and Systems (ISCAS), Orlando, FL, USA, 1999, pp. 545-548 vol.2 [pdf]
—. Yu, Z., 2024. Phase-Locked Loops: System Perspectives and
Circuit Design Aspects. John Wiley & Sons
Lacaita, Andrea Leonardo, Salvatore Levantino, and Carlo Samori.
Integrated frequency synthesizers for wireless systems.
Cambridge University Press, 2007.
Hunting jitter is often referred to as
dithering jitter, the periodic time
error between data clock and input data, which exhibits a
limit-cycle behavior
BB PD
Youngdon Choi, Deog-Kyoon Jeong and W. Kim, "Jitter transfer analysis
of tracked oversampling techniques for multigigabit clock and data
recovery," in IEEE Transactions on Circuits and Systems II: Analog and
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Data Recovery
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Recovery in Serial Data Transmission Systems. [pdf]
—, Clock and Data Recovery for Serial Data Communications, focusing
on bang-bang CDR design methodology, ISSCC Short Course, February 2002.
[slides]
It's ternary, because early, late
and no transition
notice the transition density = 1 in
digital PLL
Linearing BB-PD
The effective PD gain is a function of the input jitter
pdf, it enables one to anticipate the effects of input jitter
on loop characteristics
BB Gain is the slope of average BB output \(\mu\), versus phase offset \(\phi\), i.e. \(\frac {\partial \mu}{\partial \phi}\),
BB only produces output for a transition and this de-rates the gain.
Transition density = 0.5 for
random data
Input referred jitter from BB PD is
proportional to incoming jitter
BBPD gain simulation
L. Avallone, M. Mercandelli, A. Santiccioli, M. P. Kennedy, S.
Levantino and C. Samori, "A Comprehensive Phase Noise Analysis of
Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems I:
Regular Papers, vol. 68, no. 7, pp. 2775-2786, July 2021 [https://sci-hub.st/10.1109/TCSI.2021.3072344]
T. -K. Kuan and S. -I. Liu, "A Bang Bang Phase-Locked Loop Using
Automatic Loop Gain Control and Loop Latency Reduction Techniques," in
IEEE Journal of Solid-State Circuits, vol. 51, no. 4, pp. 821-831, April
2016 [https://sci-hub.st/10.1109/JSSC.2016.2519391]
coef_fit = np.polyfit(dt, et, 1) print(f'coef_fit: {coef_fit}')
x = np.linspace(-3.5, 3.5, 1000) y = coef_fit[0]*x + coef_fit[1]
plt.plot(dt, et, 'o') plt.plot(x, y, linewidth=2, linestyle='--')
# Calculate histogram counts and bin edges counts, bin_edges = np.histogram(dt, bins=100) # Find the maximum count max_count = counts.max() # Create weights to normalize the maximum height to 1 weights = np.ones_like(dt) / max_count plt.hist(dt, bins=100, weights=weights)
That is \[
P_{x_s x_s} (f)= \frac{1}{T_s}P_{xx}(f)
\] In going from discrete time to continuous
time, we must add a scale factor \(1/T\), the sample period
Y. Hu, T. Siriburanon and R. B. Staszewski, "Multirate Timestamp
Modeling for Ultralow-Jitter Frequency Synthesis: A Tutorial," in
IEEE Transactions on Circuits and Systems II: Express Briefs,
vol. 69, no. 7, pp. 3030-3036, July 2022
Daniel Boschen. GRCon24 - Quick Start on Control Loops with Python
Workshop [video, slides]
L. Avallone, M. Mercandelli, A. Santiccioli, M. P. Kennedy, S.
Levantino and C. Samori, "A Comprehensive Phase Noise Analysis of
Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems I:
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M. Zanuso, D. Tasca, S. Levantino, A. Donadel, C. Samori and A. L.
Lacaita, "Noise Analysis and Minimization in Bang-Bang Digital PLLs," in
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N. Da Dalt, "Linearized Analysis of a Digital Bang-Bang PLL and Its
Validity Limits Applied to Jitter Transfer and Jitter Generation," in
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Recovery in Serial Data Transmission Systems. [paper,slides]
hunting jitter
S. Jang, S. Kim, S. -H. Chu, G. -S. Jeong, Y. Kim and D. -K. Jeong,
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Bang–Bang Phase-Frequency Detection," in IEEE Transactions on Circuits
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[phd
thesis]
If \(m=0\)\[
\phi(t) \approx \frac{I_0C_0}{2q_\text{max}\Delta
\omega}\sin(\Delta\omega t)
\] If \(m\neq 0\) and \(m=n\)\[
\phi(t) \approx \frac{I_mC_m}{2q_\text{max}\Delta
\omega}\sin(\Delta\omega t)
\]
\(m\omega_0 +\Delta \omega \ge
0\)
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Corrections
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Given \(i(t) = I_m \cos[(m\omega_0 - \Delta
\omega)t]\) and \(m \ge 1\)
To compare the ring oscillator and VCO the total injected
charge to both should be the same
Tail filter
TODO 📅
P. Liu et al., "A 128Gb/s ADC/DAC Based PAM-4 Transceiver with
>45dB Reach in 3nm FinFET," 2025 Symposium on VLSI Technology and
Circuits (VLSI Technology and Circuits), Kyoto, Japan, 2025
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Oscillators," in IEEE Journal of Solid-State Circuits, vol. 52, no. 3,
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—, "25.3 A VCO with implicit common-mode resonance," 2015 IEEE
International Solid-State Circuits Conference - (ISSCC) Digest of
Technical Papers, San Francisco, CA, USA, 2015 [https://sci-hub.st/10.1109/ISSCC.2015.7063116]
Y. Hu, T. Siriburanon and R. B. Staszewski, "Intuitive Understanding
of Flicker Noise Reduction via Narrowing of Conduction Angle in
Voltage-Biased Oscillators," in IEEE Transactions on Circuits and
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XMODEL [pdf]
Since we have (ideally) no return current, the ground
reference becomes less important. The ground potential can
even be different at the sender and receiver or moving around within a
certain acceptable range. However, you need to be careful because
DC-coupled differential signaling (such as USB, RS-485, CAN) generally
requires a shared ground potential to ensure that the signals stay
within the interface's maximum and minimum allowable common-mode
voltage.
To understand the impact of the clock jitter on the performance of a
wireline system, the transfer functions of the PLL in the
transmitter side and the CDR loop in the receiver should
be taken into consideration
the minimum jitter occurs at the point
where the transmit PLL UGB is minimum and the
CDR UGB is maximized
the net rms jitter that impacts the performance of a wireline
transceiver is much lower than the rms jitter of the transmit PLL
the jitter requirements of the transmit PLL on the wireline system
is much more relaxed compared to the wireless transceiver
Low-Latency PCIe
TODO 📅
AC-coupling vs DC-coupling
TODO 📅
Linearity & Even-Order
Distortion
Odd-order distortion: symmetry
Even-Order Distortion: non-symmetry
(Effect of Mismatch)
B. Razavi, "Design considerations for direct-conversion receivers,"
in IEEE Transactions on Circuits and Systems II: Analog and Digital
Signal Processing, vol. 44, no. 6, pp. 428-435, June 1997 [http://www.seas.ucla.edu/brweb/papers/Journals/RTCAS97.pdf]
Corresponding to the three distinct voltage thresholds in the
PAM4 systems, it would need 12 slicers, 3
multiplexers, and one thermometer-to-binary decoder in
each deserialized data path, even if only one tap of the DFE is
unrolled
Look-Ahead Multiplexing DFE
The look-ahead multiplexing technique brings the key benefit that the
timing constraint can be significantly relaxed, as the iteration bound
is doubled at the expense of extra
hardware
"ISI cancellation" based equalization is conceptually more
straightforward but suffers from SNR penalty or error propagation
Jitter Amplification
by Passive Channels
Enhancing
Resolution with a \(\Delta \Sigma\)
Modulator
Sub-Resolution Time Averaging
\(\Delta \Sigma\) modulator
effectively dithers the LSB bit
between zero and one, such that you can get the effective
resolution of a much higher resolution DAC in the number of bits
Decimation
how they affect sampling phase
DLF's input bit-width can be reduced by decimating BBPD's
output. Decimation is typically performed by realizing either
majority voting (MV) or boxcar
filtering.
Note that deserialization is inherent to both
MV and boxcar filtering
Decimation is commonly employed to alleviate the high-speed
requirement. However, decimation increases loop-latency which causes
excessive dither jitter.
Decimation is basically, widen the data and slowing it down
Decimating by \(L\) means frequency
register only added once every \(L\)
UI, thus integral path gain reduced by \(L\) in linear model
proportional path gain is unchanged
CDR Linear Model
condition:
Linear model of the CDR is used in a frequency lock
condition and is approaching to achieve phase
lock
Using this model, the power spectral density (PSD) of jitter in the
recovered clock \(S_{out}(f)\) is \[
S_{out}(f)=|H_T(f)|^2S_{in}(f)+|H_G(f)|^2S_{VCO}(f)
\] Here, we assume \(\varphi_{in}\) and \(\varphi_{VCO}\) are uncorrelated as they
come from independent sources.
Using below notation \[\begin{align}
\omega_n^2=\frac{K_{PD}K_{VCO}}{C} \\
\xi=\frac{K_{PD}K_{VCO}}{2\omega_n^2}
\end{align}\]
We can rewrite transfer function as follows \[
H_T(s)=\frac{2\xi\omega_n s+\omega_n^2}{s^2+2\xi \omega_n s+\omega_n^2}
\]
The jitter transfer represents a low-pass filter
whose magnitude is around 1 (0 dB) for low jitter frequencies and drops
at 20 dB/decade for frequencies above \(\omega_n\)
the recovered clock track the low-frequency
jitter of the input data
the recovered clock DONT track the
high-frequency jitter of the input data
The recovered clock does not suffer from high-frequency jitter even
though the input signal may contain high-frequency jitter, which will
limit the CDR tolerance to high-frequency jitter.
Jitter Peaking in
Jitter Transfer Function
The peak, slightly larger than 1 (0dB) implies that jitter will be
amplified at some frequencies in the CDR, producing a
jitter amplitude in the recovered clock, and thus also in the recovered
data, that is slightly larger than the jitter amplitude
in the input data.
This is certainly undesirable, especially in applications such as
repeaters.
Jitter Generation
If the input data to the CDR is clean with no jitter, i.e., \(\varphi_{in}=0\), the jitter of the
recovered clock comes directly from the VCO jitter. The transfer
function that relates the VCO jitter to the recovered clock jitter is
known as jitter generation. \[
H_G(s)=\frac{\varphi_{out}}{\varphi_{VCO}}|_{\varphi_{in}=0}=\frac{s^2}{s^2+2\xi
\omega_n s+\omega_n^2}
\] Jitter generation is high-pass filter with
two zeros, at zero frequency, and two poles identical to those of the
jitter transfer function
Jitter Tolerance (JTOL)
To quantify jitter tolerance, we often apply a sinusoidal jitter of a
fixed frequency to the CDR input data and observe the BER of the CDR
The jitter tolerance curve DONT capture a CDR's true
tolerance to random jitter. Because we are applying
"sinusoidal" jitter, which is deterministic signal.
We can deal only with the jitter's amplitude and frequency instead of
the PSD of the jitter thanks to deterministic sinusoidal jitter signal.
\[
JTOL(f) = \left | \varphi_{in}(f) \right |_{\text{pp-max}} \quad
\text{for a fixed BER}
\] Where the subscript \(\text{pp-max}\) indicates the
maximum peak-to-peak amplitude. We can further expand
this equation as follows \[
JTOL(f)=\left| \frac{\varphi_{in}(f)}{\varphi_{e}(f)} \right| \cdot
|\varphi_e(f)|_\text{pp-max}
\]
Relative jitter, \(\varphi_e\) must
be less than 1UIpp for error-free operation
In an ideal CDR, the maximum peak-to-peak amplitude
of \(|\varphi_e(f)|\) is
1UI, i.e.,\(|\varphi_e(f)|_\text{pp-max}=1UI\)
Accordingly, jitter tolerance can be expressed in terms of the number
of UIs as \[
JTOL(f)=\left| \frac{\varphi_{in}(f)}{\varphi_{e}(f)} \right|\quad
\text{[UI]}
\] Given the linear CDR model, we can write \[
JTOL(f)=\left| 1+\frac{K_{PD}K_{VCO}H_{LF}(f)}{j2\pi f} \right|\quad
\text{[UI]}
\] Expand \(H_{LF}(f)\) for the
CDR, we can write \[
JTOL(f)=\left| 1-2\xi j \left(\frac{f_n}{f}\right) -
\left(\frac{f_n}{f}\right)^2 \right|\quad \text{[UI]}
\] At frequencies far below and above the natural frequency, the
jitter tolerance can be approximated by the following \[
JTOL(f) = \left\{ \begin{array}{cl}
\left(\frac{f_n}{f}\right)^2 & : \ f\ll f_n \\
1 & : \ f\gg f_n
\end{array} \right.
\]
the jitter tolerance at very high jitter frequencies
is limited to 1UIpp
1 2 3 4 5 6 7 8 9 10 11 12 13 14
clc; clear all;
f_fn = logspace(-1, 2, 60); for xi = [2, 1, 0.5, 0.2] jtol = abs(1- 1i*2*xi.*(1./f_fn)- (1./f_fn).^2); loglog(f_fn, jtol,LineWidth=2) disp(["min(JTpp)=", min(jtol),"@\xi=",xi]) hold on end grid on; xlabel("f/f_n") ylabel('JT_{pp}') legend('\xi=2', '\xi=1', '\xi=0.5', '\xi=0.2')
JTF, by jitter frequency, compares how much input signal jitter is
transferred to the output of a clock-recovery's PLL (recovered
clock)
Input signal jitter that is within the clock recovery PLL's loop
bandwidth results in jitter that is faithfully transferred (closed-loop
gain) to the clock recovery PLL's output signal. JTF in this situation
is approximately 1.
Input signal jitter that is outside the clock recovery PLL's loop
bandwidth results in decreasing jitter (open-loop gain) on the clock
recovery PLL's output, because the jitter is filtered out and no longer
reaches the PLL's VCO
Observed Jitter Transfer Function
Input Signal Versus Sampled Signal
OJTF compares how much input signal jitter is transferred to the
output of a receiver's decision making circuit as
effected by a clock recovery's PLL. As the recovered clock is the
reference for detecting the input signal
Input signal jitter that is within the clock
recovery PLL's loop bandwidth results in jitter on the recovered clock
which reduces the amount of jitter that can be detected. The input
signal and clock signal are closer in phase
Input signal jitter that is outside the clock
recovery PLL's loop bandwidth results in reduced jitter on the
recovered clock which increases the amount of jitter that can
be detected. The input signal and clock signal are more out of phase.
Jitter that is on both the input and clock signals can not detected or
is reduced
JTF and OJTF for 1st Order PLLs
The observed jitter is a complement to the PLL jitter transfer
response OJTF=1-JTF (Phase matters!)
OTJF gives the amount of jitter which is tracked and therefore not
observed at the output of the CDR as a function of the jitter rate
applied to the input.
The combination of the OJTF of a jitter measurement device and the
JTF of the clock generator under test gives the measured jitter as a
function of frequency.
For example, a clock generator with a type 1, 1st order PLL measured
with a jitter measurement device employing a golden PLL is \[
J_{\text{measured}} = \frac{\omega_1}{s+\omega_1}\frac{s}{s+\omega_2}
\]
Accurate measurement of the clock JTF requires that the OJTF cutoff
of the jitter measurement be significantly below that of the clock JTF
and that the measurement is compensated for the instrument's OJTF.
The overall response is a band pass filter because the clock JTF is
low pass and the jitter measurement device OJTF is high pass.
The compensation for the instrument OJTF is performed by measuring
the jitter of the reference clock at each jitter rate being tested and
comparing the reference jitter with the jitter
measured at the output of the DUT.
The lower the cutoff frequency of the jitter measurement device the
better the accuracy of the measurement will be.
The cutoff frequency is limited by several factors including the
phase noise of the DUT and measurement time.
Digital Sampling
Oscilloscope
How to analyze jitter:
TIE (Time Interval Error) track
histogram
FFT
TIE track provides a direct view of how the phase of
the clock evolves over time.
histogram provides valuable information about the
long term variations in the timing.
FFT allows jitter at specific rates to be measured
down to the femto-second range.
Maintaining the record length at a minimum of \(1/10\) of the inverse of the PLL loop
bandwidth minimizes the response error
reference
Dalt, Nicola Da and Ali Sheikholeslami. “Understanding Jitter and
Phase Noise: A Circuits and Systems Perspective.” (2018).
quantization noise is ~ bounded uniform distribution
Using unbounded Gaussian -> pessimistic BER prediction
AFE Nonlinearity
"total harmonic distortion" (THD) in
AFE
Relative to NRZ-based systems, PAM4 transceivers require more
stringent circuit linearity, equalizers which can implement multi-level
inter-symbol interference (ISI) cancellation, and improved
sensitivity
Because if it compresses, it turns out you have to use a much more
complicated feedback filter. As long as it behaves linearly,
the feedback filter itself can remain a linear FIR
Linearity can actually be a critical constraint in these signal
paths, and you really want to stay as linear as you can all the way up
until the point where you've canceled all of the ISI
Elad Alon, ISSCC 2014, "T6: Analog Front-End Design for Gb/s Wireline
Receivers"
BER with Quantization Noise
\[
\text{Var}(X) = E[X^2] - E[X]^2
\]
Impulse Response or Pulse
Response
TX FFE
TX FFE suffers from the peak power constraint, which in effect
attenuates the average power of the outgoing signal - the low-frequency
signal content has been attenuated down to the high-frequency level
H. Shakiba, D. Tonietto and A. Sheikholeslami, "High-Speed Wireline
Links-Part II: Optimization and Performance Assessment," in IEEE Open
Journal of the Solid-State Circuits Society, vol. 4, pp. 110-121, 2024
[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10579874]
R. Walker, “Designing Bang-Bang PLLs for Clock and Data Recovery in
Serial Data Transmission Systems,” in Phase-Locking in High-Performance
Systems, B. Razavi, Ed. New Jersey: IEEE Press, 2003, pp. 34-45. [http://www.omnisterra.com/walker/pdfs.papers/BBPLL.pdf]
P. K. Hanumolu, M. G. Kim, G. -y. Wei and U. -k. Moon, "A 1.6Gbps
Digital Clock and Data Recovery Circuit," IEEE Custom Integrated
Circuits Conference 2006, San Jose, CA, USA, 2006, pp. 603-606 [https://sci-hub.se/10.1109/CICC.2006.320829]
Da Dalt N. A design-oriented study of the nonlinear dynamics of
digital bang-bang PLLs. IEEE Transactions on Circuits and Systems I:
Regular Papers. 2005;52(1):21–31. [https://sci-hub.se/10.1109/TCSI.2004.840089]
Jang S, Kim S, Chu SH, Jeong GS, Kim Y, Jeong DK. An optimum loop
gain tracking all-digital PLL using autocorrelation of bang–bang phase
frequency detection. IEEE Transactions on Circuits and Systems II:
Express Briefs. 2015;62(9):836–840. [https://sci-hub.se/10.1109/TCSII.2015.2435691]
CDR Loop Latency
Denoting the CDR loop latency by \(\Delta
T\) , we note that the loop transmission is multiplied by \(exp(-s\Delta T)\simeq 1-s\Delta T\).The
resulting right-half-plane zero, \(f_z\) degrades the phase margin and must
remain about one decade beyond the BW\[
f_z\simeq \frac{1}{2\pi \Delta T}
\]
This assumption is true in practice since the bandwidth of the CDR
(few mega Hertz) is much smaller than the data rate (multi giga
bits/second).
Homayoun, Aliakbar and Behzad Razavi. “On the Stability of
Charge-Pump Phase-Locked Loops.” IEEE Transactions on Circuits and
Systems I: Regular Papers 63 (2016): 741-750.
N. Kuznetsov, A. Matveev, M. Yuldashev and R. Yuldashev, "Nonlinear
Analysis of Charge-Pump Phase-Locked Loop: The Hold-In and Pull-In
Ranges," in IEEE Transactions on Circuits and Systems I: Regular
Papers, vol. 68, no. 10, pp. 4049-4061, Oct. 2021
limit cycles imply self-sustained oscillators due nonlinear
nature
Ouzounov, S., Hegt, H., Van Roermund, A. (2007). SUB-HARMONIC
LIMIT-CYCLE SIGMA-DELTA MODULATION, APPLIED TO AD CONVERSION. In: Van
Roermund, A.H., Casier, H., Steyaert, M. (eds) Analog Circuit Design.
Springer, Dordrecht. [https://sci-hub.se/10.1007/1-4020-5186-7_6]
Digital CDR Category
DCO part is analogous so that it cannot be perfectly
modeled
Digital-to-phase converter is well-defined phase output, thus, very
good to model real situation
Z-domain modeling
The difference equation is \[
\phi[n] = \phi[n-1] + K_{DCO}V_C[n]\cdot T\cdot2\pi
\] z-transform is \[
\frac{\Phi(z)}{V_C(z)}=\frac{2\pi K_{DCO}T}{1-z^{-1}}
\]
where \(K_{DCO}\) : \(\Delta f\) (Hz/bit)
\(\Delta
\Sigma\)-dithering in DCO
Quantization noise
Here, \(\alpha_T\) is data
transition density
BBPD quantization noise
DAC quantization noise
M. -J. Park and J. Kim, "Pseudo-Linear Analysis of Bang-Bang
Controlled Timing Circuits," in IEEE Transactions on Circuits and
Systems I: Regular Papers, vol. 60, no. 6, pp. 1381-1394, June 2013 [https://sci-hub.st/10.1109/TCSI.2012.2220502]
Time to Digital Converter
(TDC)
Digital to Phase Converter
(DPC)
IIR low pass filter
simple approximation: \[
z = 1 + sT
\] bilinear-z transform \[
z =\frac{}{}
\]
FAQ
PLL vs. CDR
PLL
CDR
Clock edge periodic
Data edge random
Phase & Frequency detecting possible
Phase detecting possible , Frequency detecting impossible
PLL or FD(Frequency Detector) for frequency detecting in CDR
reference
J. Stonick. ISSCC 2011 "DPLL-Based Clock and Data Recovery" [slides,transcript]
P. Hanumolu. ISSCC 2015 "Clock and Data Recovery Architectures and
Circuits" [slides]
Amir Amirkhany. ISSCC 2019 "Basics of Clock and Data Recovery
Circuits"
Fulvio Spagna. INTEL, CICC2018, "Clock and Data Recovery Systems" [slides]
M. Perrott. 6.976 High Speed Communication Circuits and Systems
(lecture 21). Spring 2003. Massachusetts Institute of Technology: MIT
OpenCourseWare, [lec21.pdf]
Akihide Sai. ISSCC 2023, T5 "All Digital Plls From Fundamental
Concepts To Future Trends" [T5.pdf]
J. L. Sonntag and J. Stonick, "A Digital Clock and Data Recovery
Architecture for Multi-Gigabit/s Binary Links," in IEEE Journal of
Solid-State Circuits, vol. 41, no. 8, pp. 1867-1875, Aug. 2006 [https://sci-hub.se/10.1109/JSSC.2006.875292]
—, "A digital clock and data recovery architecture for
multi-gigabit/s binary links," Proceedings of the IEEE 2005 Custom
Integrated Circuits Conference, 2005.. [https://sci-hub.se/10.1109/CICC.2005.1568725]
P. Palestri et al., "Analytical Modeling of Jitter in
Bang-Bang CDR Circuits Featuring Phase Interpolation," in IEEE
Transactions on Very Large Scale Integration (VLSI) Systems, vol.
29, no. 7, pp. 1392-1401, July 2021 [https://sci-hub.se/10.1109/TVLSI.2021.3068450]
Rhee, W. (2020). Phase-locked frequency generation and clocking :
architectures and circuits for modern wireless and wireline
systems. The Institution of Engineering and Technology
M.H. Perrott, Y. Huang, R.T. Baird, B.W. Garlepp, D. Pastorello, E.T.
King, Q. Yu, D.B. Kasha, P. Steiner, L. Zhang, J. Hein, B. Del Signore,
"A 2.5 Gb/s Multi-Rate 0.25μm CMOS Clock and Data Recovery Circuit
Utilizing a Hybrid Analog/Digital Loop Filter and All-Digital
Referenceless Frequency Acquisition," IEEE J. Solid-State Circuits, vol.
41, Dec. 2006, pp. 2930-2944 [https://cppsim.com/Publications/JNL/perrott_jssc06.pdf]
—, et al., "Modeling of ADC-Based Serial Link Receivers With
Embedded and Digital Equalization," in IEEE Transactions on
Components, Packaging and Manufacturing Technology, vol. 9, no. 3,
pp. 536-548, March 2019 [https://sci-hub.se/10.1109/TCPMT.2018.2853080]
K. Zheng, "System-Driven Circuit Design for ADC-Based Wireline Data
Links", Ph.D. Dissertation, Stanford University, 2018 [https://purl.stanford.edu/hw458fp0168]
S. Cai, A. Shafik, S. Kiran, E. Z. Tabasy, S. Hoyos and S. Palermo,
"Statistical modeling of metastability in ADC-based serial I/O
receivers," 2014 IEEE 23rd Conference on Electrical Performance of
Electronic Packaging and Systems [pdf]
We typically use the two spectra, \(S_{\phi
n}(f)\) and \(S_{out}(f)\),
interchangeably, but we must resolve these inconsistencies.
voltage spectrum is called Lorentzian
spectrum
The periodic signal \(x(t)\) can be
expanded in Fourier series as:
Assume that the signal is subject to excess phase noise,
which is modeled by adding a time-dependent noise
component \(\alpha(t)\). The noisy
signal can be written \(x(t+\alpha(t))\), the added excess phase
\(\phi(t)=
\frac{\alpha(t)}{\omega_0}\)
The autocorrelation of the noisy signal is by definition:
The autocorrelation averaged over time results in:
By taking the Fourier transform of the autocorrelation, the spectrum
of the signal \(x(t + \alpha(t))\) can
be expressed as
It is also interesting to note how the integral in Equation
9.80 around each harmonic is equal to the power of the harmonic
itself \(|X_n|^2\)
The integral \(S_x(f)\) around
harmonic is \[\begin{align}
P_{x,n} &= \int_{f=-\infty}^{\infty}
|X_n|^2\frac{\omega_0^2n^2c}{\frac{1}{4}\omega_0^4n^4c^2+(\omega
+n\omega_0)^2}df \\
&= |X_n|^2\int_{\Delta
f=-\infty}^{\infty}\frac{2\beta}{\beta^2+(2\pi\cdot\Delta f)^2}d\Delta f
\\
&= |X_n|^2\frac{1}{\pi}\arctan(\frac{2\pi \Delta
f}{\beta})|_{-\infty}^{\infty} \\
&= |X_n|^2
\end{align}\]
The phase noise does not affect the total power in the
signal, it only affects its distribution
Without phase noise, \(S_v(f)\) is
a series of impulse functions at the harmonics of \(f_o\).
With phase noise, the impulse functions spread, becoming fatter and
shorter but retaining the same total power
Phase
perturbed by a stationary noise with Gaussian PDF
If keep \(\phi_{rms}\) in \(R_x(\tau)\), i.e. \[
R_x(\tau)=\frac{A^2}{2}e^{-\phi_{rms}^2}\cos(2\pi f_0
\tau)e^{R_\phi(\tau)}\approx \frac{A^2}{2}e^{-\phi_{rms}^2}\cos(2\pi f_0
\tau)(1+R_\phi(\tau))
\] The PSD of the signal is \[
S_x(f) = \mathcal{F} \{ R_x(\tau) \} =
\frac{P_c}{2}e^{-\phi_{rms}^2}\left[S_\phi(f+f_0)+S_\phi(f-f_0)\right] +
\frac{P_c}{2}e^{-\phi_{rms}^2}\left[\delta(f+f_0)+\delta(f-f_0)\right]
\] ❗❗above Eq isn't consistent with stationary
white noise process - the following section
Phase
perturbed by a stationary WHITE noise process
Assuming that the delay line is noiseless
Expanding the cosine function we get \[\begin{align}
R_y(t,\tau) &= \frac{A^2}{2}\left\{\cos(2\pi
f_0\tau)E[\cos(\phi(t)-\phi(t-\tau))] - \sin(2\pi
f_0\tau)E[\sin(\phi(t)-\phi(t-\tau))]\right\} \\
&+ \frac{A^2}{2}\left\{\cos(4\pi
f_0(t+\tau/2-T_D))E[\cos(\phi(t)+\phi(t-\tau))] - \sin(4\pi
f_0(t+\tau/2-T_D))E[\sin(\phi(t)+\phi(t-\tau))] \right\}
\end{align}\]
where, both the process \(\phi(t)-\phi(t-\tau)\) and \(\phi(t)+\phi(t-\tau)\) are independent of
time \(t\), i.e. \(E[\cos(\phi(t)+\phi(t-\tau))] =
m_{\cos+}(\tau)\), \(E[\cos(\phi(t)-\phi(t-\tau))] =
m_{\cos-}(\tau)\), \(E[\sin(\phi(t)+\phi(t-\tau))] =
m_{\sin+}(\tau)\) and \(E[\sin(\phi(t)-\phi(t-\tau))] =
m_{\sin-}(\tau)\)
The second term in the above expression is periodic in \(t\) and to estimate its PSD, we compute the
time-averaged autocorrelation function\[
R_y(\tau) = \frac{A^2}{2}\left\{\cos(2\pi f_0\tau)m_{\cos-}(\tau) -
\sin(2\pi f_0\tau)m_{\sin-}(\tau)\right\}
\]
After nontrivial derivation
Phase perturbed by a Weiner
process
The phase process \(\phi(t)\) is
also gaussian but with an increasing variance which
grows linearly with time\(t\)
The spectrum of \(y(t)\) is
determined by the asymptotic behavior of \(R_y(t,\tau)\) as \(t\to \infty\)
❗❗ \(\lim_{t\to\infty}R_y(t,\tau)\) rather than
time-averaged autocorrelation function of cyclostationary
process, ref. Demir's paper
We define \(\zeta(t,
\tau)=\phi(t)+\phi(t-\tau) = \phi(t)-\phi(t-\tau) +
2\phi(t-\tau)\), the expected value of \(\zeta(t,\tau)\) is 0, the variance is \(\sigma_{\zeta}^2=(k\sigma)^2(\tau +
4(t-\tau))=(k\sigma)^2(4t-3\tau)\)\[
E[\cos(\zeta(t,\tau))]=\frac{1}{\sqrt{2\pi
\sigma_{\zeta}^2}}\int_{-\infty}^{\infty}e^{-\zeta^2/2\sigma_{\zeta}^2}\cos(\zeta)d\zeta
= e^{-\sigma_{\zeta}^2/2}=e^{-(k\sigma)^2(4t-\tau)}
\] i.e., \(\lim _{t\to \infty}
E[\cos(\zeta(t,\tau))] = \lim_{t\to \infty}e^{-(k\sigma)^2(4t-\tau)} =
0\)
For \(E[\sin(\zeta(t,\tau))]\), we
have \[
E[\sin(\zeta(t,\tau))] = \frac{1}{\sqrt{2\pi
\sigma_{\zeta}^2}}\int_{-\infty}^{\infty}e^{-\zeta^2/2\sigma_{\zeta}^2}\sin(\zeta)d\zeta
\] i.e., \(E[\sin(\zeta(t,\tau))]\) is odd
function, therefore \(E[\sin(\zeta(t,\tau))]=0\)
A. Hajimiri and T. H. Lee, "A general theory of phase noise in
electrical oscillators," in IEEE Journal of Solid-State
Circuits, vol. 33, no. 2, pp. 179-194, Feb. 1998 [paper],
[slides]
A. Demir, A. Mehrotra and J. Roychowdhury, "Phase noise in
oscillators: a unifying theory and numerical methods for
characterization," in IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications, vol. 47, no. 5, pp. 655-674,
May 2000 [https://sci-hub.se/10.1109/81.847872]
Godone, A. & Micalizio, Salvatore & Levi, Filippo. (2008). RF
spectrum of a carrier with a random phase modulation of arbitrary slope.
[https://sci-hub.se/10.1088/0026-1394/45/3/008]
Bae, Woorham; Jeong, Deog-Kyoon: 'Analysis and Design of CMOS
Clocking Circuits for Low Phase Noise' (Materials, Circuits and Devices,
2020)
\[
V_{\text{rx}} = V_{\text{dd}} \frac{(R_2-R_1)R_T}{R_1R_T+R_2R_T+R_1R_2}
\] With \(R_u=(L+M+N)R_T\)
\[\begin{align}
V_{\text{rx}} &= \frac{1}{2}(V_p - V_m) \\
&= \frac{1}{2}(\frac{2}{3}(2V_{\text{MSB}}+V_{\text{LSB}})-1) \\
&=\frac{1}{3}(2V_{\text{MSB}}+V_{\text{LSB}})-\frac{1}{2}
\end{align}\]
