[ibis-macro] Re: On impulse and step responses.
- From: "Walter Katz" <wkatz@xxxxxxxxxx>
- To: "'David Banas'" <DBanas@xxxxxxxxxx>, "'Todd Westerhoff'" <twesterh@xxxxxxxxxx>
- Date: Wed, 26 Jun 2013 14:13:09 -0400 (EDT)
David,
The following is extracted from:
http://en.wikipedia.org/wiki/Integral
To paraphrase: The formal mathematic definition of an integral of f(t)dt
from A to B is the limit of the sum of sum(f(ti)*(ti+1-ti)) as the mesh of
(ti+1-ti) becomes smaller. This is a fundamental concept in the Theory of
Continuous Function.
Walter
Riemann integral[edit
<http://en.wikipedia.org/w/index.php?title=Integral&action=edit§ion=9> ]
Main article: Riemann integral
<http://en.wikipedia.org/wiki/Riemann_integral>
<http://en.wikipedia.org/wiki/File:Integral_Riemann_sum.png>
http://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Integral_Riemann_sum.png/220px-Integral_Riemann_sum.png
<http://en.wikipedia.org/wiki/File:Integral_Riemann_sum.png>
http://bits.wikimedia.org/static-1.22wmf7/skins/common/images/magnify-clip.png
Integral approached as Riemann sum based on tagged partition, with irregular
sampling positions and widths (max in red). True value is 3.76; estimate is
3.648.
The Riemann integral is defined in terms of Riemann sums
<http://en.wikipedia.org/wiki/Riemann_sum> of functions with respect to
tagged partitions of an interval. Let [a,b] be a closed interval
<http://en.wikipedia.org/wiki/Interval_(mathematics)> of the real line;
then a tagged partition of [a,b] is a finite sequence
a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le
x_n = b . \,\!
<http://en.wikipedia.org/wiki/File:Riemann_sum_convergence.png>
http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/250px-Riemann_sum_convergence.png
<http://en.wikipedia.org/wiki/File:Riemann_sum_convergence.png>
http://bits.wikimedia.org/static-1.22wmf7/skins/common/images/magnify-clip.png
Riemann sums converging as intervals halve, whether sampled at ■ right, ■
minimum, ■ maximum, or ■ left.
This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed
by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi].
A Riemann sum of a function f with respect to such a tagged partition is
defined as
\sum_{i=1}^{n} f(t_i) \Delta_i ;
thus each term of the sum is the area of a rectangle with height equal to
the function value at the distinguished point of the given sub-interval, and
width the same as the sub-interval width. Let Δi = xi−xi−1 be the width of
sub-interval i; then the mesh of such a tagged partition is the width of the
largest sub-interval formed by the partition, maxi=1…n Δi. The Riemann
integral of a function f over the interval [a,b] is equal to S if:
For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b]
with mesh less than δ, we have
\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.
When the chosen tags give the maximum (respectively, minimum) value of each
interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum
<http://en.wikipedia.org/wiki/Darboux_integral> , suggesting the close
connection between the Riemann integral and the Darboux integral
<http://en.wikipedia.org/wiki/Darboux_integral> .
From: David Banas [mailto:DBanas@xxxxxxxxxx]
Sent: Wednesday, June 26, 2013 2:06 PM
To: Walter Katz; 'Todd Westerhoff'
Cc: ibis-macro@xxxxxxxxxxxxx
Subject: RE: [ibis-macro] Re: On impulse and step responses.
Hi Walter,
The concept of “area under” a discrete sequence of values is meaningless.
-db
From: Walter Katz [mailto:wkatz@xxxxxxxxxx]
Sent: Wednesday, June 26, 2013 10:31 AM
To: David Banas; 'Todd Westerhoff'
Cc: ibis-macro@xxxxxxxxxxxxx
Subject: RE: [ibis-macro] Re: On impulse and step responses.
David,
I believe the definition of a Dirac Delta Function is that the area under it
is 1. The area under a continuous curve is the integral of f(t)*dt. For a
discrete representation of this integral becomes the sum(f(ti)*(ti+1-ti)).
Note that sample_rate is 1/(ti+1-ti). Thus the area under {1, 0, 0, …} is
1/sample_rate while the area under {<sample_rate>, 0, 0, …} is 1.
Walter
From: ibis-macro-bounce@xxxxxxxxxxxxx
[mailto:ibis-macro-bounce@xxxxxxxxxxxxx] On Behalf Of David Banas
Sent: Wednesday, June 26, 2013 1:07 PM
To: Todd Westerhoff
Cc: ibis-macro@xxxxxxxxxxxxx
Subject: [ibis-macro] Re: On impulse and step responses.
Hi Todd,
There are two things being discussed, which is probably adding to the
confusion.
Firstly, I’m challenging Mike’s belief that the discrete time equivalent to
the Dirac delta is the sequence, {<sample_rate>, 0, 0, …}. I believe the
discrete time equivalent to the Dirac delta is the sequence, {1, 0, 0, …}. I
think this one has probably been exhausted, unfortunately without
resolution.
Secondly, the current spec. fails to name the units, which are to be assumed
for the values passed into Init(), via the impulse_matrix parameter. This is
the issue, which is less academic, more practical, and more worthy of the
committee’s time. Here is the current relevant language, excerpted from IBIS
v5.1:
“impulse_matrix” points to a memory location where the collection of channel
voltage impulse responses, … The algorithmic model is expected to modify the
impulse responses in place by applying a filtering behavior, for example, an
equalization function, if modeled in the AMI_Init function. …
(Note that my omissions of any original text are indicated by ellipses, and
any emphasis is entirely mine.)
Now, the language “impulse response” is vague in signal processing parlance,
as it can refer to either:
1. The continuous time “impulse response function”, or
2. The discrete time “unit pulse response sequence”,
both of which are more precise concepts.
In deciding which of the two interpretations, above, to accept, one notes
the use of the language, “voltage” (i.e. – NOT “volts/sec.”), as a
qualifying preface to the term, “impulse response.” Therefore, one could
defensibly argue that `2’ should be assumed, since the continuous time
impulse response function must have units of “Volts/sec.”, as has been
pointed out now numerous times in this discussion.
Further confidence in choice ‘2’ is gained, by noting that ours is a
necessarily discrete time application. (It takes place entirely within the
state space of a digital computer, and the interface between the model and
the rest of the system is a discrete sequence of numbers.)
Finally, the language, “in place by applying a filtering behavior,” suggests
that the model should expect to be receiving values with units most natural
to direct digital filter application to the unmodified input, which would be
“Volts”.
-db
From: Todd Westerhoff [mailto:twesterh@xxxxxxxxxx]
Sent: Wednesday, June 26, 2013 7:39 AM
To: David Banas
Cc: ibis-macro@xxxxxxxxxxxxx
Subject: On impulse and step responses.
Dave,
Can I ask you to confirm your initial question on this subject? I want to
make sure I understand what we’re saying before we get too deep into the
math.
It seems to me that you’ve called into question the mathematics behind
impulse responses being created for current IBIS-AMI models, and therefore
how IBIS-AMI models must be written to process those impulse responses. If
there is indeed a problem with the math, it follows that the current
standard would need to be either updated or extended, depending on how the
details play out.
Is that what you’re saying?
Todd.
Todd Westerhoff
VP, Software Products
Signal Integrity Software Inc. • www.sisoft.com
6 Clock Tower Place • Suite 250 • Maynard, MA 01754
(978) 461-0449 x24 • twesterh@xxxxxxxxxx
“I want to live like that”
-Sidewalk Prophets
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