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: [cid:image002.png@01CE7195.91C42620] 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: [cid:image006.png@01CE7195.91C42620] 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: [cid:image009.png@01CE7195.91C42620] 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, ...}: [cid:image011.png@01CE7195.91C42620] (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] On Behalf Of Mike Steinberger Sent: Tuesday, June 25, 2013 9:29 AM To: 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 ________________________________ Confidentiality Notice. 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