Originally Posted by dragonfyr
PLEASE do us all a favor and do a bit more research into what you imagine the ETC to be, as you severely pervert not only what it is, but apparently you have no clue as to its constituent parts or how it relates to other responses and to the Analytic as a whole.
As long as we are asking for favors, why don’t you do us a favor Dragon and read the research I provided *before* writing such remarks? Your first post yesterday was the same rehash of buzzwords from Localhost. Then, you go back and do a google search, find the paper from Don that I referenced, notice that what I said from it was right and come back and add another page to your post here. Why not have the courtesy of reading the paper I provided first? And why would you not be aware of such research when you are such a marketing machine for ETC? I mean who wants to be caught knowing less about ETC than me, Mr. Bob?
I didn’t want to respond yesterday because I wanted to find a simple way to explain these concepts so that at least one or two follow us in this discussion. I think I have found a way.
Let’s start at the top. ETC as a foundation uses the Impulse Response of the system. This is a method where we send a pulse through the system that in theory has infinite amplitude but lasts essentially zero amount of time. And then watch what the system does. That becomes the impulse response or IR for short. See the measurement graph from FHG I provided in my last post. Heyser took that response and said, let’s add in a second component to it and sum both of those and call them the Energy Time Curve or ETC. Mathematically this is what it looks like:
z(t) = IR(t) + j * Hilbert (IR(t))
No, don’t flip the page
. I will make sense out of this. For now think of this as the sum of two values. One is our original Impulse Response and the other, is the transformation of it using a mathematical method called the Hilbert Transform. The “J” in front of the second part lends to the terminology often used to describe complex numbers such as “imaginary” or “phase.” In this context though, they do not mean what we normally mean by those terms. Let’s put that aside for now.
We know our signal is an impulse. Another name for an impulse is a Dirac Delta. Folks smarter than us have solved the Hilbert transform of a Dirac Delta. It is 1/(pi * t). In other words, you multiply time by the number 3.1415926… and invert it and you have your second component. Here is a rough plot of that I created in Excel if you can’t visualize it:
Predictably we see that as the time approaches zero, the response goes to infinity (denominator becomes zero). But here is another thing that happens. If we give it negative time (e.g t= -0.2 seconds), it still gives us a value per the left part of the graph! We call this in mathematics and signal processing an acausal characteristic. The system is doing something before we tell it to do anything. It is like you taking a light and before you hook it up to a battery it lights up! Applied to our acoustic situation, it can tell you there is energy being emitted from a speaker before you give it a signal. You don’t see this as negative values on ETC because the math actually squares the Impulse and Hilbert transforms, adds them together and then takes the square root. So the result is always a positive value.
The error introduced by the Hilbert component per the negative part of the graph is the highest just before the impulse itself (i.e. t=0) so that is where you see them. As I noted, this translates into false amplitude computations in the valley right before the impulse peak and right at the peak. See the graph from Don’s paper in my last post.
Clearly no real system works this way. I won’t have any energy arriving at my ear before I play something. In contrast, the impulse response is causal and does represent how real systems work. As soon as you add the second Hilbert component to it to compute the ETC, you have created a response that cannot mathematically represent the way a real system works.
So we realize that the notion that we must add the Hilbert component in there or else we are not looking at the whole picture as claimed by both Dragon and Local goes out the window. If the addition invalidates the operation of a real system, it couldn’t have been something that was fundamentally there at the start.
Let’s now review if ETC actually represents the “energy” of the system. That is what the “E” in “ETC” stands for. Well, that is not it either. If we throw out the left hand part of the above graph and focus on the right with positive time values, we see that what is being added is the exponential decay times the Impulse Response. In other words, we are talking our signal and adding some of its own value back to it. That value keeps dropping with time. So if you have a single peak and nothing else, you would get that peak and then the graph would gently decline with time as the Hilbert transform value gets smaller and smaller.
Intuitively the above seems like how the energy dissipates in a system and that was Heyser’s hope. He got close but we can demonstrate mathematically that what he has done is an approximation and not a true computation of the energy in the system. And it is not just me saying it. Don says that and so do other experts. Speaking of other experts, the best I know when it comes to matters of signal processing as applied to audio are professors Vanderkooy and Lipshitz. What these two guys have forgotten about this topic is more than what all of us know about it
. You may have known them more for demonstrating the deficiencies in DSD audio (i.e. SACD) relative to PCM. Anyway, they performed an analysis of ETC and published it in the AES Journal. The title is another colorful one: ” Uses and Abuses of the Energy-Time Curve.”
Right away, you know this is not going to end well for folks who have a simplistic view of such concepts!
Here is the abstract:"Two basic aspects of the familiar Heyser energy-time curve (ETC) as applied to electroacoustic system measurements are addressed, 1) its inherently acausal nature, which means that it should not be interpreted literally as representing the energy flow of a physical system, and, more importantly, 2) the way in which the appearance of an actual ETC is affected by the detailed nature of the data from which it is computed, and especially by any frequency-domain window that is used in its computation. Theoretical and experimental data are used to illustrate the variety of behavior that can occur, and show how the processing can either enhance or falsify the measurement.”
We clearly see confirmation of what I explained in that the system is not causal and hence cannot represent the energy despite what the acronym in ETC stands for.
In their article, they also make this comment:"Later Heyser preferred to think of the ETC as being not just the real envelope e(t), but rather the full complex analytic signal z(t) itself. We shall use this latter terminology in this paper, although the points that we wish to make do not depend upon this choice.”
So we see that the use of the terms used by Heyser to describe the system as having a “complex” and “real” components are just that: *his* terminology. The notion by both Dragon and Local then that this refers to us having an AC and DC component is confusion on their part. The added component to the Impulse Response by Heyser in the form of Hilbert transform is not giving us some magic representation of AC energy of a signal. If you doubt this, read the above statements again on how this is not an energy computation.
The professors make another excellent point regarding band-limiting of ETC which is a defense Dragon is putting forward. That is a notion that says let’s filter the signal that we run through the system to the audio bands we are interested in. Well, that is easier said than done when it comes to signal processing. Let’s say you want to see frequencies from 2Khz to 10 Khz. Non-intuitively if you precisely cut off the signal at 2 and 10 Khz, what results is actually a distorted ETC response! Nature doesn’t like to have things have sharp discontinuities where you go from nothing to something. The solution is the so called windowing functions. This takes those sharp edges of spectrum and smears them on either end. The choice of the “window” function determines its shape. Fancy term for the error is "spectrum leakage."
Problem with those window functions is that they all have pros and cons with respect to the impact on what is being measured. If you are performing frequency response measurements you can play with these and immediately see their effect. They do not usually get in the way of analyzing the signal. In the computation of ETC however, such window functions are often hidden and even if they are there, users have no idea what they really do. As our dear professors write on issues with ETC, there is peril in sticking one’s head under the sand there:”The indiscriminate use of spectral-weighting windows (commonly the Hamming window) in the computation of the ETC, which can frequently falsify the system’s true ETC in a way that may not be apparent to the user.”
They go on to say,”We believe that drastic spectral windowing should only be used in specialized cases where its effect is known and understood”
Hence the reason when I talked about band-limitting ETCs that it needs to be in expert hands. Here is their summary of ETC analysis:”We have seen that the customary energy-time curve, or ETC, cannot represent the energy flow of a system, because it is not causal and does not contain the relevant conjugate variables. The discrete processing in a sampled-data system gives rise to ETC that are similar to the continuous time descriptions, but due to the nature of the processing, time aliasing occurs and alternate-bin ripples often appear in the envelop.”
Let me now address the political spin by Dragon that I am describing all of this as to put ETC in bad light. That is very far from the truth. This discussion started with the erroneous comment by Localhost that FHG researchers should not have used the Impulse response in order to measure the amplitude of the reflection and then reduce it by 10 dB relative to the main signal. It was claimed that such display must be wrong because it ignored the “AC’ component. As I have shown, there is no “AC” signal and further, if what you want is the amplitude of the reflection, the impulse response gives you that and since it is a causal computation, it does represent the real life aspect of our system (room) under test.
ETC attempts to take the Impulse response and account for how the signal may decay over time in a loudspeaker or a room. It comes close to that but as a matter of theory and mathematics, it is a different animal than true computation of the system energy. That does not mean it is not useful however. ETC gives you better visibility into peaks by taking away the “grass” that is in the Impulse Response. In that manner, it aids in analysis of the system by a human looking at the graph. And that is a good thing. As with anything you need to understand what your tool measures and doesn’t measure.
My main beef with ETC is not any of this but the fact that what it tells you is devoid of psychoacoustics. Psychoacoustics involves the knowledge of frequency spectrum and how it relates to us hearing them. Such data is absent and misleading at times in ETC. This is why you don’t want to use it. Not because of accuracy errors.
It is ironic however that our two champions of ETC don’t seem to appreciate its true nature, mathematics and limitations. I can’t blame them. Who likes math? Almost no one.
But if you are going to go after someone on a forum by throwing these words around, you better know what they really mean. It is damn dangerous to assume that your counterpart is going to be scared off the buzzwords and let you win the argument by default. Some people can walk toe to toe with you and then some on such topics
Edit: finished the one sentence I had left hanging
.Edited by amirm - 7/2/12 at 12:30pm