[ibis-macro] Re: On impulse and step responses.
- From: David Banas <DBanas@xxxxxxxxxx>
- To: Mike Steinberger <msteinb@xxxxxxxxxx>
- Date: Tue, 25 Jun 2013 12:53:55 -0700
Hi Mike,
I’ve responded in-line, below, in order to preserve context.
Thanks,
-db
From: Mike Steinberger [mailto:msteinb@xxxxxxxxxx]
Sent: Tuesday, June 25, 2013 12:23 PM
To: David Banas
Cc: ibis-macro@xxxxxxxxxxxxx
Subject: Re: [ibis-macro] Re: On impulse and step responses.
Dave-
Even though "Kronecker Delta" has the word "delta" in it, it's not relevant to
this conversation.
The problem to be solved here is that we need a sampled time domain function
that expresses the transfer function of a linear, time invariant process in a
way that
1. Allows us to calculate the output of the process in response to a particular
sampled time domain input waveform
[David Banas] Isn’t this parroting my post, below? For instance, isn’t this
just a rewording of this excerpt from my text, below?:
That is, what input sequence to a discrete time, linear, shift invariant (LSI)
system will elicit a response from that system, which will then allow us to
predict the output of that system, in response to any arbitrary input, via
convolution?
2. Allows us to perform the same calculation and get a re-sampled version of
the same result if we change the sample interval and re-sample the input
waveform.
I submit that you can't solve this problem by ignoring the sample interval.
[David Banas] And you are correct. Each time Init() is called, we must
re-calculate the tap weights of our digital filter, according to the value we
find in the “sample_interval” parameter. I’m not sure how anything I said,
previously, would suggest that I was intending to ignore the sample interval.
What I claim is that there is no need for the EDA tool to scale the values in
the “impulse_matrix” parameter by the sample rate. At the time those values are
about to be processed by the filter, Init() has already adjusted the filter tap
weights to the new sample rate.
Your example assumes that the sample interval is equal to the symbol duration.
[David Banas] No, it doesn’t. Please show me, precisely, where it makes this
assumption.
That is, the delay between x(i-1) and x(i) is one symbol time.
[David Banas] I make no such assumption. The {xi} are intended as samples, not
bits. In fact, the notion of “bit time”, or “unit interval”, is nowhere to be
found in my previous post.
In some contexts, that's a perfectly fine problem to solve, but that doesn't
happen to be the problem we need to solve. We need waveforms which have been
uniformly sampled with a sample interval that is a small fraction of a symbol
duration.
[David Banas] I agree with you, and claim that nothing I have said, previously,
is inconsistent with this assertion.
You will find it instructive to re-derive the equations in your example in a
way that makes the sample interval an explicit variable.
[David Banas] Those equations aren’t so much “derived” as they are “quoted”
from a very well respected graduate level text on DSP. So, if you’re right and
they’re missing the sample rate as a necessary parameter, then DSP must be
being taught wrong in a fair number of American post-graduate courses. I
suspect that this is not the case.
I'm not going to derive those equations for you, but if you want to re-derive
them, I'm willing to check your math.
[David Banas] Thanks for the offer, but at this point I lack the confidence in
your understanding of the fundamental principles at hand, which would be
required, in order for me to refer to you as an authoritative reference on this
topic.
Respectfully yours,
Mike S.
On 06/25/2013 01:17 PM, David Banas wrote:
Hi Mike,
Thanks for the reply. Respectfully, you’re wrong.
This notion of enforcing an area of 1, when we move into the discrete time
domain, is very misguided.
As is noted in most DSP texts, the discrete time equivalent of the continuous
time Dirac delta function is the Kronecker delta sequence: {1, 0, 0, …}. And
the magnitude of the first element of that sequence is not scaled by the system
sampling rate.
https://en.wikipedia.org/wiki/Kronecker_delta#Digital_signal_processing
(Before you criticize me for relying upon Wikipedia for my understanding of DSP
fundamentals, let me assure you that I am posting the above only because it is
extremely convenient for all to view quickly, and that I have verified what it
says against my own graduate school DSP text: “Digital Signal Processing” by
Richard A. Roberts and Clifford T. Mullis.)
I believe we can understand why, by going back to the original motivation for
deriving the Dirac delta function and then following the same motivation for
deriving the Kronecker delta sequence. So, then, why was the Dirac delta
function useful? Well, because it produces output from a continuous time
system, which allows us to predict the output of that same system to any
arbitrary input, via continuous time convolution:
yt=τ=0∞h( 64;)∙x(t-τ)dτ
Where y(t) is the output of the system, in response to any arbitrary input,
x(t), and h(t) is the response of the system to the Dirac delta function as
input; the so called “impulse response” of the system. (I have assumed a causal
system.)
Now, what is the discrete time equivalent of the Dirac delta function? That is,
what input sequence to a discrete time, linear, shift invariant (LSI) system
will elicit a response from that system, which will then allow us to predict
the output of that system, in response to any arbitrary input, via convolution?
We have, for convolution in discrete time:
yi=k=0N-1hk∙xi-k
Where {yi} is the output sequence of the order-N system, in response to input
sequence, {xi}, and {hk} is the discrete time equivalent of the system impulse
response. That is, it is that sequence, which uniquely describes our LSI
system, such that we can use the equation, above, in order to find the output
sequence of the system, in response to any arbitrary input sequence.
Now, how do we find {hk} for some particular LSI system? Let’s take a simple,
2-tap FIR filter for example:
yi=b0∙xi+b1∙xi-1=k=01bk∙xi-k
And we find that, for a 2-tap FIR filter, {hk} = {bk}.
Now, we have to assume that we do not know {bk}, a priori. And the final
question that remains then is, “What sequence should we use as input to the
2-tap FIR filter, in order that it generate {bk} as its output?” When we find
the answer to this question, we will have found the discrete time equivalent of
the Dirac delta function.
Let’s try, first, the Kronecker delta sequence: {xk} = {1, 0, 0, …}:
yi=k=01bk∙xi-k=bi,i∈0,1;0 otherwise=b0,b1,0,0,…=bi(Ignoring trailing zeros.)
(since, in this case, xk = 0 for k /= 0).
And, so, we find that the discrete time equivalent to the Dirac delta function
is not the sequence, {<sample_rate>, 0, 0, …}, but simply, {1, 0, 0, …}.
If you’ve read this far, thanks! ;-)
-db
From: ibis-macro-bounce@xxxxxxxxxxxxx<mailto:ibis-macro-bounce@xxxxxxxxxxxxx>
[mailto:ibis-macro-bounce@xxxxxxxxxxxxx] On Behalf Of Mike Steinberger
Sent: Tuesday, June 25, 2013 9:29 AM
To: ibis-macro@xxxxxxxxxxxxx<mailto:ibis-macro@xxxxxxxxxxxxx>
Subject: [ibis-macro] Re: On impulse and step responses.
Dave-
I clarified that point for Greg Edlund on this e-mail reflector. Here's a
repeat of my response to him (minus the snippy comments):
If you want to use a narrow pulse of whatever shape, that's fine; however it is
essential that the pulse always has unit area (volts * seconds). Therefore, as
your pulse gets narrower and narrower, its amplitude has to get greater and
greater. In fact, the Dirac delta function has, by definition, unit area, in
that it's defined as the limit of your narrow pulse (with unit area) as the
width of the pulse goes to zero.
In the sampled data World, we don't actually take the width of the pulse to
zero. Rather, we leave it one sample wide, as being the narrowest pulse we can
generate in that domain. The sampled data equivalent of the (continuous time
domain) Dirac delta function therefore has a width of one sample and an
amplitude of one over the sample interval.
Hope this helps.
Mike S.
On 06/25/2013 11:04 AM, David Banas wrote:
Hi Fangyi,
Thanks for the reply.
Please, see below.
Thanks,
-db
From: fangyi_rao@xxxxxxxxxxx<mailto:fangyi_rao@xxxxxxxxxxx>
[mailto:fangyi_rao@xxxxxxxxxxx]
Sent: Thursday, June 20, 2013 8:18 AM
To: David Banas; ibis-macro@xxxxxxxxxxxxx<mailto:ibis-macro@xxxxxxxxxxxxx>
Subject: RE: On impulse and step responses.
David;
Step response has an unit of volt. Impulse response, which is the derivative of
step response by definition, has an unit of volt/sec.
[David Banas] If this discussion pertained to the continuous time domain, I
would agree with you, but it doesn’t. This discussion pertains to the discrete
time domain. (It has to, since we’re sending in a discrete set of samples,
taken at a uniform sampling interval, to Init().) And, in that domain, both
quantities must have the same unit, since we require:
uk=i=0khi
where {uk} is the “unit step response sequence” and {hi} the “unit pulse
response sequence” of the LSI discrete time system being discussed, and I have
taken the liberty of assuming we’re only interested in describing causal
systems. In our particular application, the most reasonable unit for these two
sequences seems to be “Volt”, which is why I’m very perplexed as to why several
of us seem to feel that “Volts/sec.” is the proper unit to be sending into
Init(). Does our current spec. call out the exact units to be used?
Also, please keep in mind that the Dirac delta function has an unit of 1/sec.
Regards,
Fangyi
From: ibis-macro-bounce@xxxxxxxxxxxxx<mailto:ibis-macro-bounce@xxxxxxxxxxxxx>
[mailto:ibis-macro-bounce@xxxxxxxxxxxxx] On Behalf Of David Banas
Sent: Thursday, June 20, 2013 7:50 AM
To: ibis-macro@xxxxxxxxxxxxx<mailto:ibis-macro@xxxxxxxxxxxxx>
Subject: [ibis-macro] On impulse and step responses.
Hi all,
In our work, we often take as a priori that the impulse response is the time
derivative of the step response. As I puzzle over this further, I realize that
I’m stumped by something very fundamental, which is this:
A quantity, which is the time derivative of some other quantity, cannot have
the same units as that other quantity. And, yet, when we
discuss/measure/simulate either a step response or an impulse response, we
expect to be talking about / measuring / viewing a voltage as a function of
time, in both cases! How can this be?
Thanks,
-db
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