[SI-LIST] Re: Maximum frequency consideration for high-speed digital analysis of differential signals
- From: "Dr. Howard Johnson" <howie03@xxxxxxxxxx>
- To: <si-list@xxxxxxxxxxxxx>, <Cortex.Chen@xxxxxxxxxxxx>
- Date: Mon, 9 Jan 2006 13:48:45 -0800
Dear Rula Bakleh et.al.,
I'm going to begin by addressing the question from Cortex
about translation between different forms of risetime
measurement.
The relation between, say, 10-90% risetime and 20-80%
risetime depends on the exact shape of your signal.
The book High-Speed Digital Design: A Handbook of Black
Magic contains on page 401 a table showing seven types of
measurements, for three types of rising-edge shapes. (The
same table applies for falling edges.)
The seven columns of data in the table which I shall
reproduce below represent the following seven ways to
measure risetime:
(1) Take the derivative of your step waveform. You should
get a pulse. Compute the standard devication of that pulse.
(2) The 10-90% rise time, defined as 10-90% of the beginning
and ending values for the waveform under consideration (if
you are using end terminations these beginning and ending
values will NOT equal ground and VCC).
(3) The 20-80% rise time, defined as 20-80% of the beginning
and ending values for the waveform under consideration (same
note as above)
(4) Measure the value of the slope (dy/dx) at the 50% level
crossing. Take the signal swing dV, and divide it by the
slope at the 50% level crossing.
(5) Measure the peak value of the slope (dy/dx). This may
happen prior to the 50% level crossing. Take the signal
swing dV, and divide it by the maximum slope.
(6) Take the derivative of your step waveform. That forms
the impulse response. Do an FFT on this pulse. Find that
frequency where the high-frequency magnitude is rolled off
to a level 3dB below the DC value.
(7) Take the derivative of your step waveform. That forms
the impulse response of your system. Do an FFT on this
pulse. The RMS value is that cutoff frequency at which a
box-shaped frequency response would pass the same amount of
white noise energy as the filter formed by the impulse
response of your system.
The three waveforms considered in my table are:
(A) A one-pole R-C response.
(B) A two-pole, critically damped L-R-C response.
(C) A Gaussian response (i.e., the impulse response is a
gaussian pulse).
There is more detail about the pulse shapes in the book.
Here's the table (constant-width font required).
Tsig T10-90 T20-80 T50%slp Tmaxslp F3dB
FRMS
1-pole 1.00 .877 .553 .798 .399 .399 .626
2-pole 1.00 .947 .612 .900 .767 .363 .443
crit. damp
Gaussian 1.00 1.02 .672 1.00 1.00 .332 .354
>From this table you may extract ratios relevant to your
question. For example, the ratio of T10-90/T20-80 for a
1-pole response equals .877/.553 = 1.586 .
If the 20-80% rise time of a 1-pole system were tau seconds,
the 10-90% risetime would by tau*1.586, and the knee
frequency about 0.5 divided by that answer.
For a Gaussian rise time (more representative of high-speed
digital signals) the ratio T10-90/T20-80 equals 1.02/.672 =
1.518 .
For many digital systems the knee frequency (0.5/T10-90%) is
located near the 6dB rolloff point.
In high speed serial links, the eye at the end of the line
is all slopped over to a point where the rise time
practically equals the baud interval. For these systems the
"knee frequency" is therefore the same as
0.5/(baud_interval).
In such a system the sampling rate equals 1/(baud_interval),
so you could say the knee frequency is about
(1/2)*(sample_rate).
That is not a bad starting place for your thinking. You want
the channel response fairly flat, and the parasitics
manageably low, up to a frequency of at least 1/2 the sample
rate.
Don't get too hung up on the exact value of the knee
freuqency (or any of the other above measures of rise time
and bandwidth). The purpose of these measures is to give you
some idea of the importance of your parasitic elements.
For example, say you have a parasitic capacitance to ground
on the order of 1/2 pF. To evaluate the importance of that
element in your circuit you must know at what frequency the
component will be operated. The frequency that matters the
most is usually the highest freuqency of interest, the knee
frequency. Evaluate the impedance of your parasitic element
at that frequency (XC = 1/(2*pi*Fknee*C)).
Now sit back and compare XC to the impedance Z0 of your
system (likely 50 ohms, if the parasitic goes from one line
to ground, or 100 ohms if it does between two lines). The
value of XC always falls into one of three buckets.
If XC is orders of magnitude higher than Z0 then you know
that XC cannot hurt you. Ignore it!
On the other hand, if XC is lower than Z0, you don't need
any more analysis to tell you that your system is toast. No
matter what you do, nothing works with that much
capacitance.
If XC falls into the grey zone, from about 100 to 1000 ohms,
these are the problems we spend our days working on. If you
want to know what REALLY HAPPENS in this zone, use an
accurate tool. Use SPICE, or Laplace transforms, or Fourier
analysis, or differential equations, or build the circuit
and measure its performance.
The whole knee frequency concept is nothing more than a way
to focus your attention on one particular, very high
frequency of interest, so you can quickly weed out those
problems that are trivial, requiring no more effort, and
those that are devastating, requring something more than
analysis to overcome, leaving you to concentrate on those
vexing, intermediate little gotchas that suck up most of
your design effort.
Getting back to Rula's original question about channel
analysis, we can apply to his problem a quantitative rule of
thumb from the oscilloscope industry. When measuring a
system with bandwidth X, if the bandwidth of your scope
exceeds X by three times, then our scope vendors tell us the
limited bandwidth of the scope distorts the observed
measurments by only about five percent. (This a demonstrably
reasonable conclusion).
You can apply the scope bandwidth rule to his system. The
results depend on what you are trying to predict. If you
just want to model the eye amplitude at the center of the
eye, then the fastest risetime you care much about is a full
baud interval in length (200 ps). The highest frequency you
care much about in that case equals half the baud rate, or
2.5 GHz, so you can expect an S-parameter channel
characterization taken to 7.5 GHz should correctly model the
most important aspects of the channel, including the
amplitude of the signal at the center of the eye, to within
about 5% error. Going out to ten GHz would not be
unreasonable. Much beyond that, in my opinion, for simple
eye-opening analysis, is not necessary.
If, on the other hand, you wish to model the exact shape of
the rising edge right out of the transmitter, an interval
much shorter than 200 ps, you would need a correspondingly
higher bandwidth.
Your thoughts?
Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
tel +1 509-997-0505, howie03@xxxxxxxxxx
http:\\sigcon.com -- High-Speed Digital Design seminars,
publications and films
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx
[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of
Cortex.Chen@xxxxxxxxxxxx
Sent: Sunday, January 08, 2006 8:16 PM
To: rula.bakleh@xxxxxxxxxx
Cc: tom@xxxxxxxxxxxxx; si-list@xxxxxxxxxxxxx;
esayre3@xxxxxxxx
Subject: [SI-LIST] Re: Maximum frequency consideration for
high-speed
digital analysis of differential signals
Rula,
Maybe you could focus on knee frequency (0.5/Tr, Fknee)
instead of BW/f3dB.
It means that the behavior of a circuit at frequencies above
Fknee hardly affects digital performance.
In detail, refer to the section 1.1 of High-Speed Digital
Design - A Handbook of Black Magic (Dr. Howard Johnson's
Book@93).
I also have a question for this equation, 0.5/Tr.
It seems to be extracted from the 10% to 90% of Tr.
Who can tell me the knee frequency for 20% to 80% of Tr?
Regards,
Cortex
-----Original Message-----
From: Ed Sayre III [mailto:esayre3@xxxxxxxx]
Sent: Friday, January 06, 2006 11:48 PM
To: Cortex Chen (³¯¥ÃªN)
Cc: tom@xxxxxxxxxxxxx; rula.bakleh@xxxxxxxxxx;
si-list@xxxxxxxxxxxxx
Subject: Re: [SI-LIST] Re: Maximum frequency consideration
for high-speed digital analysis of differential signals
Cortex,
The value of 0.35 in your calculation is base on the
pole location of
the 1st order system. How are you deriving the 0.22 value?.
I am very
curious to understand the thinking behind this.
Regards
-Ed
At 10:25 AM 1/6/2006 +0800, Cortex.Chen@xxxxxxxxxxxx wrote:
>0.35/Tr ---- 10% to 90%
>0.22/Tr ---- 20% to 80%
>
>Regards,
>
>Cortex
>________________________________
>
>From: si-list-bounce@xxxxxxxxxxxxx ¢DN2z Tom Dagostino
>Sent: 2006/1/6 [?P¡¦A?-] ?W?E 10:03
>To: rula.bakleh@xxxxxxxxxx; si-list@xxxxxxxxxxxxx
>Subject: [SI-LIST] Re: Maximum frequency consideration for
high-speed
>digital analysis of differential signals
>
>
>
>I'm not sure how you are computing your bandwidth. If you
have a 25 psec
>risetime (assuming 10-90%) your -3dB bandwidth is
0.35/25psec or 14 GHz.
>
>When analyzing bandwidth requirements for systems like this
most people talk
>about passing the 3rd or 5th harmonic of the clock. In
your case that is
>7.5 or 12.5 GHz. Some analysis of your system requirements
is in order.
>
>Tom Dagostino
>Teraspeed(R) Labs
>13610 SW Harness Lane
>Beaverton, OR 97008
>503-430-1065
>tom@xxxxxxxxxxxxx
>www.teraspeed.com
>
>Teraspeed Consulting Group LLC
>121 North River Drive
>Narragansett, RI 02882
>401-284-1827
>
>-----Original Message-----
>From: si-list-bounce@xxxxxxxxxxxxx
>[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of
rula.bakleh@xxxxxxxxxx
>Sent: Thursday, January 05, 2006 2:19 PM
>To: si-list@xxxxxxxxxxxxx
>Subject: [SI-LIST] Maximum frequency consideration for
high-speed
>digital analysis of differential signals
>
>
>Hi Everyone,
>
>
>I'm designing a differential signal for the Rocket I/O
interface. My signal
>speed is 5 Gbit/sec, operating frequency is 2.5 GHz, and
fastest edge rate
>is 25 psec or 40 GHz.
>
>
>
>In regards to performing channel analysis, what is the
highest frequency
>content to consider when designing this particular
differential signal
>serial link? If I am measuring or simulating S parameters
for various
>portions of this link what is the highest frequency of
interest that I need
>to consider as a rule of thumb and why?
>
>
>
>Is it the operating frequency only, multiples/harmonics of
the operating
>freq, edge rate frequency (f=1/trise), multiples/harmonics
of the edge rate
>freq, or some fraction of the edge rate? I'm finding
conflicting
>information: some people say that it's the full edge rate,
others say 0.35
>or 0.5 of the edge rate, and yet others say multiples of
the edge frequency
>should be taken into account for analysis and design. I'm
hoping that
>somebody could shed some light on this topic for me.
>
>
>
>Thank You,
>
>Rula Bakleh
>
>
>
>
>
>
>
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