A method for evaluating channels
Charles Moore, Avago Technologies
Adam Healey
Healey, LSI Corporation
100
00 Gb/s Backplane
ac p a e a
and
d Coppe
Copper S
Study
udy G
Group
oup
Singapore, March 2011
Supporters
• Brian Misek, Avago Technologies
• Rich
Ri h Mellitz,
M llit IIntel
t l
A method for evaluating channels
2
Background
• IEEE 802.3baTM-2010 introduced new methods for evaluating channels
– Pulse
P lse response amplit
amplitude,
de 85
85.8.3.3
833
– Integrated crosstalk noise (ICN), 85.10.7
• The influence of channel insertion loss deviation (ILD) and return loss
on link performance
f
was recognized by not rigorously quantified
f
A method for evaluating channels
3
Objectives
• Extend the ICN calculations to quantify the impact of channel ILD and
return loss
• For the study group, provide a method for estimating the suitability of
channels for use at some bit rate
– Justify
J if reach
h objectives
bj i
• For the (prospective) task force, provide a framework for the definition
of channel performance requirements
A method for evaluating channels
4
The basic method consists of 4 steps
1. Predict the available signal at the receiver given the channel and an
assumed transmitter
2. Compute an equivalent noise that represents the interference from
nearby channels (crosstalk) and residual inter-symbol interference
(ISI)
3. Apply an estimate of the additional SNR loss corresponding to “real
world” implementations of the receiver and transmitter
4. Compute the SNR and compare it to a target intended to provide a
symbol error ratio (SER) better than 10−12
This method is described and justified by text and equations. Equations
are derived in Appendix A.
Incidental
c de a values,
a ues, which
c may
ay cchanged
a ged without
ou cchanging
a g g the
e method,
e od, a
are
e
tabulated in Appendix B.
A method for evaluating channels
5
Signal measure
• The signal measure is the dibit amplitude which is the product of the
transmitted amplitude and the dibit gain
• Considers the influence of pre-cursor ISI from the first following bit
– Not correctable by the decision feedback equalizer
• Includes representations of the transmitter and receiver filters
The dibit gain is the peak of response to the
stimulus:
ti
l
+1
0.25
0.2
gdibit
0
1 UI
2 UI
Amplitude, V
0.15
0.1
0.05
0
−1
-0.05
-0.1
01
88
90
92
94
96
98
Time, UI
A method for evaluating channels
6
Sources of noise
Insertion loss deviation
Victim
transmitter
Ht
2
H ild
At2
=
1 + ( f ft )4
2
Receiver
Transmitter re-reflection
H rrtx
2
Hr =
2
1
1 + ( f f r )8
Receiver re-reflection
H rrrx
2
Transmitter-receiver re-reflection
H ft
2
A2fft
=
1 + ( f f ft ) 4
Far-end
aggressor
H nt
Near-end
aggressor
A method for evaluating channels
2
Ant2
=
1 + ( f f nt ) 4
H rrtxrx
2
Far-end crosstalk
Integrated crosstalk noise
10 − MDFEXTloss /10
Near-end crosstalk
10 − MDNEXTloss /10
7
Insertion loss deviation (ILD) noise
• Most cleanly designed channels with low reflections have a transfer
function which mayy be modeled as:
H = ex
log( H fit ) = x ≅ γ 0 + γ 1 f + γ 2 f + γ 4 f 2
γ i = α i + jβ i
• Since most channels will have this basic characteristic, it is reasonable
to expect that transmitters and receivers are designed to equalize it
• Deviations from the transfer function model will represent unexpected
perturbations that may
p
y be difficult to equalize
q
• The difference between the actual transfer function and the best fit to
the model may be considered to be error term whose power can be
added to the total noise
H ild = H − H fit
• The best fit is the one that minimizes Hild in the least mean squares
sense and this is a fit weighted by H
A method for evaluating channels
8
Best fit calculation
⎡ H ( f1 ) H ( f1 ) f1
⎢
H ( f 2 ) H ( f1 ) f1
F w= ⎢
⎢ M
M
⎢
⎢⎣ H ( f N ) H ( f N ) f N
⎡γ 0 ⎤
⎢γ ⎥
1
γ =⎢ ⎥
⎢γ 2 ⎥
⎢ ⎥
⎣γ 4 ⎦
H ( f1 ) f1
H ( f2 ) f2
M
H ( fN ) fN
H ( f1 ) f12 ⎤
⎥
H ( f 2 ) f 22 ⎥
⎥
M
⎥
2
H ( f N ) f N ⎥⎦
⎡ H ( f1 ) log( H ( f1 )) ⎤
⎢ H ( f ) log(
⎥
g(
H
(
f
2
2 ))
⎥
y=⎢
⎥
⎢
M
⎥
⎢
H
(
f
)
log(
H
(
f
))
N
N ⎦
⎣
γ lms = ( FwT Fw ) −1 FwT y
x lms = γ 0lms + γ 1lms
f + γ 2lms f + γ 4lms f 2
H fit = e x
lms
•T
To avoid
id undue
d iinfluence
fl
ffrom signals
i
l which
hi h contribute
t ib t littl
little noise,
i
th
the
range of the fit is limited such that fN < fmax
• The value of log(H( f )) is ambiguous but the form used here “unwraps”
the
h phase
h
– The magnitude may be fit independently from the phase by substituting the
magnitude of H( f ) for H( f ) in the expressions above to yield α lms
– The unwrapped phase may then be fit directly to yield β lms
A method for evaluating channels
9
ILD noise example
fci_cc_short_2to10
0
fci_cc_short_2to10
0.8
SDD21
Fit
0.7
-10
0.6
0.5
Amplitude, V
Magnitude, dB
B
-20
-30
-40
0.4
0.3
0.2
ILD
0.1
-50
-60
SDD21
Fit
ILD
0
5
10
15
Frequency, GHz
A method for evaluating channels
20
25
0
30
-0.1
25
30
35
40
45
50
55
60
Time, UI at 25.7813 GBd
65
70
75
10
Re-reflection interference (noise)
• Transmitter, receiver, and channel return loss influence the transfer
function of the assembled link
⎡ s11
S=⎢
⎣ s21
Transmitter
1
2
Channel
Γ1
Γ1
s12 ⎤
s22 ⎥⎦
Γ2
Transmitter re-reflection
s11
H rrtx = s11Γ1s21
s21
s21
s22
s21
Γ1
A method for evaluating channels
Receiver
s12
s21
Γ2
Γ2
Receiver re-reflection
H rrrx = s21Γ2 s22
Transmitter-receiver re-reflection
H rrtxrx = s21Γ2 s12 Γ1s21
11
Re-reflection noise example
fci_cc_short_15to7
0
RRTX
RRRX
RRTXRX
-10
0.06
0.04
Amplitude, V
Magnitude, dB
B
RRTX
RRRX
RRTXRX
SDD21
0.08
-20
-30
-40
0.02
0
-0.02
RRTXRX
-0.04
-0.06
-50
-0.08
-60
fci_cc_short_15to7
0.1
0
5
10
15
Frequency, GHz
A method for evaluating channels
20
25
30
-0.1
30
RRTX, RRRX
40
50
60
70
80
Time, UI at 25.7813 GBd
90
100
110
12
Calculating and combining noise
• Calculate the power of each noise term using the ICN integral
• Assume noise sources are statistically independent
2
2
2
σ ch2 = σ 2fx + σ nx2 + σ ild2 + σ rrtx
+ σ rrrx
+ σ rrtxrx
σ 2fx
σ 2fx = 2Δf ∑ W ft ( f n )10 − MDFEXT
loss
/ 10
n
σ nx2
σ nx2 = 2Δf ∑ Wnt ( f n )10 − MDNEXT
loss
/ 10
n
σ ild2
σ ild2 = 2Δf ∑ Wt ( f n ) H ild ( f n )
2
σ
= 2Δf ∑ Wt ( f n ) H rrtx ( f n )
2
2
σ rrtx
= 2Δf ∑ Wt ( f n ) H rrrx ( f n )
2
σ
2
rrtx
⎡
⎤⎡
⎤
Ant2
1
P( f n )
Wnt ( f n ) = ⎢
4 ⎥⎢
8⎥
⎣1 + ( f n f nt ) ⎦ ⎣1 + ( f n f r ) ⎦
n
2
σ rrrx
P( f ) =
n
2
σ rrtxrx
⎡
⎤⎡
⎤
At2
1
Wt ( f n ) = ⎢
P( f n )
4 ⎥⎢
8⎥
⎣1 + ( f n f t ) ⎦ ⎣1 + ( f n f r ) ⎦
⎤⎡
⎡
A2ft
⎤
1
W ft ( f n ) = ⎢
P( f n )
4 ⎥⎢
8⎥
⎣1 + ( f n f ft ) ⎦ ⎣1 + ( f n f r ) ⎦
n
2
rrtx
Weighting functions:
2
σ rrtxrx
= 2Δf ∑ Wt ( f n ) H rrtxrx ( f n )
2
1 ⎡ sin(π f f b ) ⎤
⎥
⎢
fb ⎣ π f fb ⎦
2
n
A method for evaluating channels
13
Implementation penalty
• Numerous other effects will limit the achievable SNR but are difficult to
predict without detailed simulation
• These effects include but are not limited to:
–
–
–
–
–
–
–
Transmitter jitter
Clock recovery error and jitter
Minimum slicer overdrive
Thermal noise and receiver noise figure
Baseline wander
Limitations on the accuracy of practical equalizers
“Real world” limitations on bandwidth,, linearity,
y, etc. of components
p
• Margin needs to built into the SNR estimate to allow for these effects
Amplitude penalty
Noise penalty
As = Adibit g ip
σ n2 = σ ch2 + σ ip2
A method for evaluating channels
14
Target SNR
• The SNR may now be computed as As / σn
• The SNR that is required for NRZ to provide a SER better than 10−12
may be
b calculated
l l d using
i the
h iinverse error ffunction
i
• SNR2 is approximately 7.03
multi level modulation (L-PAM):
(L PAM):
• For a fixed differential output voltage, multi-level
• yields a reduction in the spacing between adjacent levels
• yields somewhat smaller RMS interference amplitude
• allows more bits per symbol hence lower symbol rates
Symbol rate
f b = R log 2 ( L)
Target SNR
SNRL = SNR2
3
L2 − 1
• Note that inner signal levels have a higher propensity for error and in
the limit of large L this effectively doubles the SER
• For an SER of 10−12, this could increase the target SNR by 0
0.1
1 dB
A method for evaluating channels
15
Comparison to Salz SNR
• Salz SNR represents an upper bound on the performance of decision
feedback equalizers
– Readily computed from the channel insertion loss to crosstalk ratio (ICR)
– However, practical equalizers will not perform as well
• Salz
S
S
SNR
is not very sensitive to pre-cursor ISI,
S ILD, and re-reflection
f
– These can be significant error terms in this analysis
– 10GBASE-KR ICR limit assumes a 3 dB SNR penalty due to ILD (and rereflection?)
– When considered in detail using this method, the penalty may actually be
much greater
• Salz SNR should also be adjusted by an implementation penalty
– Implementation penalty is explicitly included by this method
A method for evaluating channels
16
Conclusions
• Link operation at less than a given symbol error ratio can be estimated
from a determination of the SNR
• The SNR can be derived from characteristics which are represented by
collections of scattering parameters
• Reasonable suggestions for these calculations have been shown
• The authors recommend:
– The Study Group use this method, including reasonable refinements, as part
of the process to determine channel reach objectives
– Should a Task Force be formed, use this method (and refinements) as the
framework for defining channel performance requirements
A method for evaluating channels
17
Appendix A
Derivation of formulae
Dibit amplitude
The signal measure is dibit amplitude which is the product of the peakto peak transmitted amplitude and the dibit gain.
to-peak
gain
Adibit = At g dibit
A.1
A dibit is a signal which consists of a unit positive pulse 1 unit interval
(UI) long followed by a unit negative pulse 1 UI long. The dibit gain is the
peak of the response
p
p
of a system
y
to a dibit. In this case,, the system
y
consists of the transmitter low pass filter, which accounts for the
transmitter rise time, the channel, and the receiver low pass filter. Dibit
gain can be calculated with a time domain simulator or in the frequency
domain by performing the integral:
∞
⎡ sin 2 (π f f b ) ⎤
j 2 π fτ
df
dibit (τ ) = 2∫ ⎢2 j
⎥ H t ( f ) SDD 21( f ) H r ( f ) e
π f
⎦
0 ⎣
A method for evaluating channels
A.2
19
Dibit amplitude, continued
If the transmitter and receiver low pass filters are real, i.e. zero phase,
then the peak of the dibit will occur near:
⎡
⎤
γ1
+ γ 2 + 2γ 4 (0.38 f b )⎥ + 0.5
τ e = imag ⎢
⎢⎣ 2 0.38 f b
⎥⎦
A.3
The coefficients γ i are derived as shown on slide 9. The units of τe are
unit intervals.
When computed this way, the dibit gain will usually be accurate to better
than 0.5 dB. For better accuracy, values at τe ± 0.2 UI can be computed
and the peak found by quadratic fitting.
A method for evaluating channels
20
Insertion loss deviation (ILD) noise
The objective is to find the set of coefficients γ lms that minimize the mean
squared error between H and e x.
ε n = H ( f n ) − e x( f
x lms = γ 0lms + γ 1lms
n)
A.4
f + γ 2lms f + γ 4lms f 2
A5
A.5
Begin by linearizing e x around γ lms.
e ≅e
x
x lms
+ ∑ (γ i − γ
i
lms
i
∂e x
)
∂γ i
A.6
γ lms
E
Expansion
i off Equation
E
ti A.6
A 6 yields
i ld Equation
E
ti A.7
A 7 and
d Equation
E
ti A.8.
A8
e x ≅ e x + (γ 0 − γ 0lms )e x + (γ 1 − γ 1lms )e x
lms
lms
lms
f + (γ 2 − γ 2lms )e x f + (γ 4 − γ 4lms )e x f 2
e x ≅ e x + (γ 0 + γ 1 f + γ 2 f + γ 4 f 2 )e x − (γ 0lms + γ 1lms
lms
A method for evaluating channels
lms
lms
f + γ 2lms f + γ 4lms f 2 )e x
lms
lms
A.7
A.8
21
Insertion loss deviation (ILD) noise, continued
If the model is a good representation of the channel over the frequency
range of interest and the error is small,
small it can be assumed that:
H ≅e
x lms
l ( H ) ≅ x lms
log(
A.9
A 10
A.10
Substituting Equations A.9 and A.10 into Equation A.8:
e x ≅ H ( f ) + (γ 0 + γ 1 f + γ 2 f + γ 4 f 2 ) H ( f ) − log( H ( f )) H ( f )
A.11
Substituting A.11 into the expression for the fit error Equation A.4 yields:
ε n ≅ log( H ( f n )) H ( f n ) − (γ 0 + γ 1 f n + γ 2 f n + γ 4 f n2 ) H ( f n )
A.12
Equation A.12
A 12 is the basis for determining the best fit coefficients γ lms as
shown on slide 9.
A method for evaluating channels
22
Re-reflection noise
So far, the channel transfer function has been assumed to be SDD21
which is measured under the condition of the transmitter and receiver
are terminated by the reference impedance. When terminated with a
transmitter with output reflection coefficient Γ1 and a receiver with input
reflection coefficient Γ2, the transfer function from port 1 to port 2
becomes:
H12 =
s21
1 − (Γ1s11 + Γ2 s22 + Γ1Γ2 s21s12 − Γ1Γ2 s11s22 )
⎡ s11
S=⎢
⎣ s21
Transmitter
A method for evaluating channels
1
Γ1
A 13
A.13
s12 ⎤
s22 ⎥⎦
Channel
2
Γ2
Receiver
23
Re-reflection noise, continued
Given that 1/(1−x) is approximately 1+x when |x| << 1, the transfer
function is approximately:
H12 ≅ s21 (1 + Γ1s11 + Γ2 s22 + Γ1Γ2 s21s12 − Γ1Γ2 s11s22 )
A.14
Under this assumption, the magnitude of the denominator is close to 1
so these reflection terms have negligible effect on the dibit gain. Note
that the last term in Equation A.14
A 14 is the product of the of second and
third terms, which are assumed to be small, and may be safely ignored.
In addition, knowing that s21 = s21 yields:
3
H12 ≅ s21 + s11s21Γ1 + s21s22 Γ2 + s21
Γ1Γ2
A.15
The last three terms are called the transmitter re-reflection,
re-reflection receiver rereflection, and transmitter-receiver re-reflection respectively. An intuitive
interpretation of these terms is given on slide 11.
A method for evaluating channels
24
Re-reflection noise computation
It would be possible to compute the true transfer function if all of the Sparameters for the channel and the complex reflection coefficients of the
transmitter and receiver were known. In general, the provider of the
channel will only have access to limits on the magnitude of the reflection
coefficients.
coefficients
A method for evaluating channels
25
Appendix B
Parameters for calculations
General parameters
Parameter
Symbol
Value
Symbol
y
rate,, GHz
fb
Variable
Victim differential output amplitude, mV peak
At
400
Victim transmitter 3 dB bandwidth, GHz
ft
0.55 x fb
Far-end disturber differential output amplitude, mV peak
Aft
400
Far-end disturber 3 dB bandwidth, GHz
fft
0.55 x fb
N
Near-end
d di
disturber
t b differential
diff
ti l output
t t amplitude,
lit d mV
V peak
k
Ant
600
Near-end disturber 3 dB bandwidth, GHz
fnt
1.00 x fb
Receiver 3 dB bandwidth
bandwidth, GHz
fr
0 75 x fb
0.75
Maximum frequency for transfer function fit, GHz
fmax
0.75 x fb
Implementation amplitude penalty, V/V
gip
0.667
Implementation noise penalty, mV
σip
0.024 At g dibit
A method for evaluating channels
27
Transmitter and receiver reflection coefficients
Parameter
Symbol
Value
Transmitter DC reflection coefficient, V/V
g01
0.161
Transmitter return loss reference frequency, GHz
f1
1.25 x fb
Receiver DC reflection coefficient, V/V
g02
0.161
Receiver return loss reference frequency, GHz
f2
1.25 x fb
Γ1 =
2
g + ( f f1 )
1 + ( f f1 ) 2
2
01
-2
2
B.1
-4
Γ2
2
2
g 02
+ ( f f2 )2
=
1 + ( f f 2 )2
B.2
Magnitude,, dB
-6
-8
8
-10
-12
-14
-16
A method for evaluating channels
0
0.2
0.4
0.6
Frequency/(Symbol rate)
0.8
1
28