Nice to see where your misunderstanding has come from Elias, but unfortunately you have things back to front. The response to a sinusoid, just like the response to any input, is determined by the frequency response (including the transient behaviour). The frequency response is not determined by hanging around for the transients of applying an infinity of sinusoids to decay.
Actually it is. That is exactly what an FT does for you given the impulse. Or, for example, given the system transfer function, T(s) =A(s)/B(s) how do you find the frequency response? We let s = jw and calculate the amplitude and phase of T for each value of frequency of interest. The result is not a function of time, just of w. Thus, it is the steady state result.
It's really a chicken - egg argument.
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John, that was not the context of my objection to the terminology:
In all the posts there is the statement or implication that the system's frequency response is only useful for "steady state" behaviour. That is simply not true. The frequency response of a system gives all the information needed to determine its temporal behaviour, it is a complete characterisation of how the system will react to any input in time or frequency domains and we can skip back and forth between the domains via the FT. It is in no way a "limitation" nor does it require "steady stateiness" to be determined - there is an infinity of transient signals from which we can determine a system's frequency response, from impulses to sweeps, the only restriction is that the signal spectrum is not zero anywhere.
I agree. FR and impulse response for an LTI system are one in the same. Given one or the other the system response to any input is obtainable. I was just saying that how SS is determined really isn't the issue. That FR is a SS result is. I didn't interpret you comments in the correct context.
Hi John,
Isn't that like saying "the engine speed and throttle position gives all the information needed to determine fuel injection quantity on an engine"?
The analogy between speakers and engines may be useful; engines exhibit some dependence on transient inputs. For example, this is why acceleration enrichment is still required on fuel injected engines.
I would think that any system that exhibits energy storage and release cannot be completely understood from its "steady state" behaviour.
John and John
my guesstiimation of Elias line of arguments is that he is fully aware of what you outline - though he is emphasizing on the point that you need quite some "reading the tea leaves" experience to predict whats going on from IR and FR only.
As was made pretty clear - both IR and FR are just two sides of the same coin - *but* you always have to (in the sense of : easier you do so) look at both sides to get the rather complete pix and even then its way less (intuitively) clear how to "read" the IR/FR plots compared to a time-frequency plot.
So the difference in comparing "wavelet versus IR/FR" is more in the user friendliness and accessibility as well as in the scalability of wavelet versus IR/FR.
Sure the system is determined also by FR/IR but it simply is a PITA to distinguish reflections from normal resonance from looking at FR/IR only.
I guess this is why CMP behaviour hasn't made it into consciousness of audio geeks until now.
The second point in Elias line of arguments - as I see it - is valid as well.
The more "steadyness" we look for in measurement (making it a looooong process) the better accuracy - possibly not so much because we could not gain the exact same information by a very short pulse measurement - but simply from mere practical circumstances (background noise compromising accuracy).
Not in an "easy and intuitively" way IMO - but actually that it *can* be done for usual speaker systems is pretty much out of question.
I'm though still awaiting an outspoken answer if a CMP system is in any case determined by IR/FR just the same.
To me, a discontinuity in temporal behaviour - as happening in CMP systems - questions that (most obviously seen in the - theoretically - 100% correctability of deep nulls) - but as said - thats better be answered by the math magicians.
Michael
my guesstiimation of Elias line of arguments is that he is fully aware of what you outline - though he is emphasizing on the point that you need quite some "reading the tea leaves" experience to predict whats going on from IR and FR only.
As was made pretty clear - both IR and FR are just two sides of the same coin - *but* you always have to (in the sense of : easier you do so) look at both sides to get the rather complete pix and even then its way less (intuitively) clear how to "read" the IR/FR plots compared to a time-frequency plot.
So the difference in comparing "wavelet versus IR/FR" is more in the user friendliness and accessibility as well as in the scalability of wavelet versus IR/FR.
Sure the system is determined also by FR/IR but it simply is a PITA to distinguish reflections from normal resonance from looking at FR/IR only.
I guess this is why CMP behaviour hasn't made it into consciousness of audio geeks until now.
The second point in Elias line of arguments - as I see it - is valid as well.
The more "steadyness" we look for in measurement (making it a looooong process) the better accuracy - possibly not so much because we could not gain the exact same information by a very short pulse measurement - but simply from mere practical circumstances (background noise compromising accuracy).
Not in an "easy and intuitively" way IMO - but actually that it *can* be done for usual speaker systems is pretty much out of question.
I'm though still awaiting an outspoken answer if a CMP system is in any case determined by IR/FR just the same.
To me, a discontinuity in temporal behaviour - as happening in CMP systems - questions that (most obviously seen in the - theoretically - 100% correctability of deep nulls) - but as said - thats better be answered by the math magicians.
Michael
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Firstly, let me apologise to Elias as my posts probably come across as being directed personally at him. That is not my intent, what I am against is the common treatment of time and frequency domains as being somehow unconnected, which can creep into our thinking in many subtle and not so subtle ways and lead to erroneous conclusions. I'll illustrate using the post on sweeps, as it simultaneously contains much well considered truth and yet much fallacy.
- To measure a frequency response we should really use a frequency, and the more like a frequency (sine wave) the test signal is the better the result would be
- A sweep is a frequency signal and the shorter you make it the less like a frequency it becomes and so the less suited it is to measuring frequency response.
All signals are time signals, a sweep is no less a time signal than an impulse or a step. It is easy to start thinking about a sweep as if it lived in the frequency domain, we talk about its start and stop frequencies after all, but that is already the start of misunderstanding it. All time signals have spectra in the frequency domain, and those spectra are nearly always far richer than we imagine. A sweep that starts at 20Hz and ends at 200Hz has a spectrum that extends from DC to Nyquist, reducing the content outside the notional range of the sweep requires considerable care in tapering the onset and decay of the signal, and even then the levels outside the range we may be most interested in are only reduced, they are not zeroed. To address the DC comment, a single half cycle of a 1kHz sine wave has DC content. So does a single half cycle of a 1MHz wave or a 1GHz wave. An impulse has DC content! The slightest imbalance in a time signal yields DC content, constructing a sweep without DC is quite difficult (and of course unnecessary).
A system's frequency response is just a way of describing its behaviour (all its behaviour, not just "steady state"), that description is valid in the time and frequency domains. To determine it all we need to do is pass a signal through the system and divide the spectrum of the output by the spectrum of the input (that division is the reason the spectrum of the signal used must not be zero anywhere). The quality of the result is determined by how much energy the test signal contains across the whole spectrum -within the linear range of a system, a sweep that is twice as loud improves the accuracy of the derived frequency response by the same amount as a sweep that lasts twice as long. That frequency response can then be used to generate the time or frequency responses of the system to any stimulus by simply multiplying it by the spectrum of the stimulus. To see that as a time response, just inverse FT the product.
None of this is an argument against the utlility of different ways of presenting the information contained in a signal, whether they look purely at time, frequency or a combination of the two, but it is important not to treat frequency as being somehow independent of time or frequency responses as not containing full information about time domain behaviour.
In my view the basic misconceptions underpinning the above are:
- To measure a frequency response we should really use a frequency, and the more like a frequency (sine wave) the test signal is the better the result would be
- A sweep is a frequency signal and the shorter you make it the less like a frequency it becomes and so the less suited it is to measuring frequency response.
All signals are time signals, a sweep is no less a time signal than an impulse or a step. It is easy to start thinking about a sweep as if it lived in the frequency domain, we talk about its start and stop frequencies after all, but that is already the start of misunderstanding it. All time signals have spectra in the frequency domain, and those spectra are nearly always far richer than we imagine. A sweep that starts at 20Hz and ends at 200Hz has a spectrum that extends from DC to Nyquist, reducing the content outside the notional range of the sweep requires considerable care in tapering the onset and decay of the signal, and even then the levels outside the range we may be most interested in are only reduced, they are not zeroed. To address the DC comment, a single half cycle of a 1kHz sine wave has DC content. So does a single half cycle of a 1MHz wave or a 1GHz wave. An impulse has DC content! The slightest imbalance in a time signal yields DC content, constructing a sweep without DC is quite difficult (and of course unnecessary).
A system's frequency response is just a way of describing its behaviour (all its behaviour, not just "steady state"), that description is valid in the time and frequency domains. To determine it all we need to do is pass a signal through the system and divide the spectrum of the output by the spectrum of the input (that division is the reason the spectrum of the signal used must not be zero anywhere). The quality of the result is determined by how much energy the test signal contains across the whole spectrum -within the linear range of a system, a sweep that is twice as loud improves the accuracy of the derived frequency response by the same amount as a sweep that lasts twice as long. That frequency response can then be used to generate the time or frequency responses of the system to any stimulus by simply multiplying it by the spectrum of the stimulus. To see that as a time response, just inverse FT the product.
None of this is an argument against the utlility of different ways of presenting the information contained in a signal, whether they look purely at time, frequency or a combination of the two, but it is important not to treat frequency as being somehow independent of time or frequency responses as not containing full information about time domain behaviour.
Hello,
Is there any change to access these open baffle impulse responses? (I didn't read the whole thread
) It would be interesting to compare them to the one posted measured in the box.
- Elias
Is there any change to access these open baffle impulse responses? (I didn't read the whole thread
) It would be interesting to compare them to the one posted measured in the box.- Elias
Hi JohnPM,
In order to get over this issue you may like to review the properties of the Fourier transform, but you must know it already which makes it strange why someone would argue over such a basic fundamental definition. FT transforms time domain to frequency domain where time information is lost, thus frequency domain is steady state domain. FT does not care what you transform, anything in time domain signal will end up into steady state frequency domain signal even your time domain signal is not steady state. Frequency response is a frequency domain concept, thus frequency response is a steady state response.
This is what frequency response IS. One can use it freely to suit his purposes.
It's not clear to me if you agree or not about your quote below, but if you don't mind would you explain how to measure the frequency response? Yes, I emphasize to measure to differentiate it from to calculate, that is to determine frequency response without FT.
Anyway, I gave the answer earlier in my post which is also the definition of the frequency response, but it does not exclude you to give a different answer.
- Elias
In order to get over this issue you may like to review the properties of the Fourier transform, but you must know it already which makes it strange why someone would argue over such a basic fundamental definition. FT transforms time domain to frequency domain where time information is lost, thus frequency domain is steady state domain. FT does not care what you transform, anything in time domain signal will end up into steady state frequency domain signal even your time domain signal is not steady state. Frequency response is a frequency domain concept, thus frequency response is a steady state response.
This is what frequency response IS. One can use it freely to suit his purposes.
It's not clear to me if you agree or not about your quote below, but if you don't mind would you explain how to measure the frequency response? Yes, I emphasize to measure to differentiate it from to calculate, that is to determine frequency response without FT.
Anyway, I gave the answer earlier in my post which is also the definition of the frequency response, but it does not exclude you to give a different answer.
- Elias
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There are a few things to correct here, but let's start with the most important:
No information is lost when FT is performed
None at all. To understand why, it may help to consider what the FT is, but from a different perspective. There are many ways to represent a time signal. The most familiar are some equation in which time is a variable, or for our purposes a sequence of the amplitudes of the signal sampled at uniform time intervals. There are circumstances, however, where it would be useful to be able to represent that time signal in other ways that are more amenable to certain kinds of processing. One way to do this is to take some other set of time functions and see if we can find a way to create the original time signal through a combination of those functions. The functions are referred to in mathematical jargon as basis functions. In the case of FT (and here we are considering the Discrete Fourier Transfrom, since our signal is defined at the discrete time intervals at which we sampled it) we choose sine waves as the time functions. When we calculate the DFT of our signal we are finding the answer to the question "what are the amplitudes and phases of a set of sine waves that when added together will give the original time signal". (Strictly speaking, we calculate the real and imaginary parts of a complex number whose amplitude and phase are the answers we seek). We call that set of sine wave amplitudes and phases the frequency response. After we have done this, we can perform processing that alters the amplitudes and phases of those sine waves rather than altering the samples of the time signal directly, but we are still dealing with the original time signal, just represented in a different way. If we want to get back to the time series we just perform an inverse FT, which simply adds up the sine waves for us to get back to the time signal.
In the context of a system and its transfer function, the time signal we are representing is the impulse response. The system frequency response is just a different way of representing the impulse response, it still contains all the information the impulse response contains.
In the general case what we are trying to measure is the system's transfer function, either its impulse response or its frequency response (as we have seen above, they are same signal in two representations). If we determine one we have determined the other, since they are the same signal, but I'm sure you want more than methods of determining the impulse response, especially as you wish the FT to be banished
Fortunately, we can go directly from time signals to the real and imaginary parts of the frequency response amplitudes and phases. To do this we need a signal generator that will produce a sine and a cosine signal. We use our generator to produce two sweep signals, the sweep from the sine wave output goes to the system we want to measure. That sine sweep and the cosine sweep are both fed through time delays that match the time delay through the system we are measuring, in the acoustic case that would typically be the time it takes sound to travel from the speaker to the mic - we need to determine the delay in advance, but that is not difficult to do. The signal captured from the mic goes to two multipliers, in one of them it is multiplied by the sine signal, in the other it is multiplied by the cosine signal. Each multiplier has a low pass filter on its output. The output of the sine wave multiplier (after the low pass filter) is the real part of the frequency response, the output of the cosine multiplier is the imaginary part. Since we are controlling the generator we know the frequency the generator was at for each moment during our sweep, so we know what frequency the real and imaginary signals correspond to. So, in a matter of a few seconds of a sine sweep, we have captured the frequency response of the system. And the FT was left in the toolbox, unwanted.
That process is called Time Delay Spectrometry, it is typically used with linear sweeps as that makes life a bit easier in terms of the effects of the low pass filters. It is the measuring principle used in the TEF system. TDS has some advantages (it is quite good at rejecting noise, distortion and room reflections) but also some disadvantages. Log swept sine is overall a better method of transfer function measurement, but of course that uses the dreaded FT
I hope that helps.
Hi,
I should have been more clear in my previous statement, but actually there IS information loss in the process of transforming the time domain to the frequency domain. What is lost is the information if the input signal was at the steady state or not before the FT was performed. There is no means in FT to express that information. FT allways outputs steady state freq response regardless of input, but when calculating IFT no information is available if the time domain signal should be at steady state or not. Usually one assumes it is, but generally it is unknown.
If one inputs arbitrary signal into the system and calculates FT at the output, the result will vary until the steady state is reached. In steady state input and output signals must also be steady state signals. This will take us back into measuring the frequency response and the importance of reaching the steady state before a freguency measurement achievement is declared. And of course since we are measuring at the steady state the frequency response must be a steady state measurement. LOL did I say it again Never mind
So, you are proposing using sweep. Are you sure it is steady state to be able to determine the frequency response accurately?
- Elias
I should have been more clear in my previous statement, but actually there IS information loss in the process of transforming the time domain to the frequency domain. What is lost is the information if the input signal was at the steady state or not before the FT was performed. There is no means in FT to express that information. FT allways outputs steady state freq response regardless of input, but when calculating IFT no information is available if the time domain signal should be at steady state or not. Usually one assumes it is, but generally it is unknown.
If one inputs arbitrary signal into the system and calculates FT at the output, the result will vary until the steady state is reached. In steady state input and output signals must also be steady state signals. This will take us back into measuring the frequency response and the importance of reaching the steady state before a freguency measurement achievement is declared. And of course since we are measuring at the steady state the frequency response must be a steady state measurement. LOL did I say it again
Never mindFortunately, we can go directly from time signals to the real and imaginary parts of the frequency response amplitudes and phases. To do this we need a signal generator that will produce a sine and a cosine signal. We use our generator to produce two sweep signals, the sweep from the sine wave output goes to the system we want to measure. That sine sweep and the cosine sweep are both fed through time delays that match the time delay through the system we are measuring, in the acoustic case that would typically be the time it takes sound to travel from the speaker to the mic - we need to determine the delay in advance, but that is not difficult to do. The signal captured from the mic goes to two multipliers, in one of them it is multiplied by the sine signal, in the other it is multiplied by the cosine signal. Each multiplier has a low pass filter on its output. The output of the sine wave multiplier (after the low pass filter) is the real part of the frequency response, the output of the cosine multiplier is the imaginary part. Since we are controlling the generator we know the frequency the generator was at for each moment during our sweep, so we know what frequency the real and imaginary signals correspond to. So, in a matter of a few seconds of a sine sweep, we have captured the frequency response of the system. And the FT was left in the toolbox, unwanted.
That process is called Time Delay Spectrometry, it is typically used with linear sweeps as that makes life a bit easier in terms of the effects of the low pass filters. It is the measuring principle used in the TEF system. TDS has some advantages (it is quite good at rejecting noise, distortion and room reflections) but also some disadvantages. Log swept sine is overall a better method of transfer function measurement, but of course that uses the dreaded FT
So, you are proposing using sweep. Are you sure it is steady state to be able to determine the frequency response accurately?
- Elias
You didn't actually read my post Elias, did you, or you simply failed to take in any of the content. Never mind, I'm sure the penny will drop with you one day. If your misconceptions have left any room for new information that doesn't fit your steady state world view and you are curious to know how transfer functions are measured here in the real world, this paper has a good overview of the various methods that requires only very basic knowledge. http://www.anselmgoertz.de/Page10383/Monkey_Forest_dt/Manual_dt/aes-swp-english.PDF
I for myself wouldn't take that for granted - otherwise a Dirac impulse measurement would completely fail to be a valid FR measurements setup - *even in theory* (meaning - to let mere practical problems like bad signal to noise behaviour out of consideration) - which is not the case to all of my knowledge
Those discussion about measurement and calculation of steady state plots tells me how difficult it is to create common ground even at long known facts when it comes to very details - good thing for me, as I can see it more relaxed that CMP concept and "FR depending on time we look at" doesn't get accepted in a single day.
Michael
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Mike, whilst it is good to see a sense of humour around here, selectively editing my posts to make it appear as if I said the opposite of what I actually said is misleading for others browsing the thread, as Michael's post demonstrates, and kind of dishonest. To be fair to others you should edit your post.
Michael, the comments quoted were preceded by In my view the basic misconceptions underpinning the above are: which rather changes the meaning.
Mike, your earlier engine mapping analogy isn't really applicable to the discussion as engines are not LTI systems. I used to design engine, gearbox and chassis management and data acquisition systems for motorsport applications as it happens, though that was 2 decades ago. One of the most interesting engine types to deal with is two-strokes, as a single misfire has a dramatic effect on scavenging efficiency and hence the crankcase mixture - the early two-stroke fuel injection systems tried a simple map-based approach which was disastrous, as misfires caused havoc. The solution there is to model the crankcase mixture, using crankshaft acceleration as an input to the model. If that doesn't count as off-topic, even in a thread of 7,000+ posts, I don't know what does
Hi JohnPM,
I already can see youre more of a practical type of a person, but I'm trying to have a theoretical discussion here without that much of the restrictions of the practical world.
- Elias
I already can see youre more of a practical type of a person, but I'm trying to have a theoretical discussion here without that much of the restrictions of the practical world.
- Elias
Hi Michael,
"Dirac impulse measurement" does not actually measure "frequency response" but it measures impulse response in time domain. To get frequency response one needs to process the measurement as it is known.
One can measure "frequency response" directly by steady state sinusoid signals, without the need to measure impulse response.
Maybe it's the loose usage of terms that's causing the confusion?
One have to differentiate between the actual physical measurement and the processing and presentation of the results.
- Elias
"Dirac impulse measurement" does not actually measure "frequency response" but it measures impulse response in time domain. To get frequency response one needs to process the measurement as it is known.
One can measure "frequency response" directly by steady state sinusoid signals, without the need to measure impulse response.
Maybe it's the loose usage of terms that's causing the confusion?
One have to differentiate between the actual physical measurement and the processing and presentation of the results.
- Elias
Let's consider where the approach has got so far. Most impulse responses in loudspeaker measurements decay below the noise floor of even sensitive instruments in less than a second. If we put an impulse into such a system, record 64k samples at 48kHz (1.365s) we have captured the entire impulse response. Another few ms for an FFT and we have a 64k point frequency response. Does it not strike you as contradictory that in less than 1.5s we have managed to acquire information that your steady state model suggests should take us, assuming we only allow 1s for steady state at each frequency and changing frequency takes no time, more than 18 hours? When the world does not correspond to your models, it is the models that are wrong, not the world. If your position is based on the introductory course notes you posted previously I think I can see where the misunderstanding has arisen, and I'm happy to try (again) to explain it - a mistake one person can make, many people can make - but there is no point if you are not prepared to listen.
Here you go Elias, enjoy.
The Tale of the Transfer Function
Let’s revisit the “sinusoidal steady state” course notes and stick with the same notation: lower case for time domain signals, upper case for frequency domain; a system of impulse response h(t), transfer function (Frequency Response) H(omega), driven by an input u(t) and producing an output y(t). h/H are unknown, but we would really like to know H.
The notes show that if we use a sinusoid of radian frequency omega as the input then, in the steady state after the transient effects of applying the sinusoid have died away, the output consists of a sinusoid at the same frequency as the input but modified in magnitude and phase according to the value of H at that frequency. If we measure the amplitude and phase of the output relative to the input we have figured out H at one frequency. Hooray! We are on the way to finding H. But there is a problem. There are an infinite number of frequencies. And getting to steady state may take a long time, depending on how long h(t) lasts. Even if h(t) is fairly short, at infinity times a short time it is still going to take us well past tea-time to fully characterise H. Elias, for one, isn’t happy.
So where does the problem lie? The first place to look is probably the input signal, because a sinusoid has infinite duration – it just never stops. Let’s go to the other extreme, how about if we use an impulse? At zero duration, that is going to be pretty short. Looking more promising already. We know from the notes that
Y = U H
i.e. FT of output = FT of input times Transfer Function. And we also happen to know that the FT of our impulse has unit magnitude and zero phase at all frequencies, so if u is an impulse then
Y = H
Now that’s more like it! Since the impulse has zero duration, we only need to capture the output for the duration of the impulse response, h, and take the FT and we’re done. Not only have we found H in a very finite time, we have found it at all frequencies at once. The mathematician in us is very happy indeed, job done and still time to polish off that Reimann Hypothesis proof before tea.
The Applied Mathematician in us is not so happy, however. He knows infinite amplitude, zero duration signals are a bit hard to come by in the acoustics lab, and the speakers may not react well to getting infinity up them. The techs are not likely to be too happy if we give them this method for finding H. So if zero duration signals are not so great after all, how about trying something of finite duration? In the time domain the system output is
y = u * h
where “*” indicates convolution. If h has finite duration, the duration of y will be the duration of u plus the duration of h. So if we capture that much of y, we have all the information the system is ever going to put out when fed u. If we capture that, and use our friend the FT again, we can figure out H from
H = Y / U
It will take longer than using the impulse, since we need to wait for the additional duration of our input signal u, but we still find the value of H at all frequencies at once and we’ll still get to tea in good time. Now all we need to do is pick a suitable input signal, u, of a suitably short duration. We know that dividing by zero is a no-no, so our input signal needs to have a spectrum that is not zero anywhere, but beyond that we can use pretty much whatever we like as far as the Applied Mathematician is concerned.
The Engineer in us is not completely happy yet. He knows that our loudspeaker and the amplifier driving it have limits, and our mic preamp has some noise, and it is not unknown for the techs to start chatting while the measurements are being made. Whatever signal we use had better not overstress the amps and speakers and have enough energy in it that the measurement noise becomes negligible. The Applied Mathematician has given us some candidates though, including a few noise sequences and sweeps, linear and logarithmic. The engineer isn’t keen on the noise sequences – they will have a high crest factor of 10dB or more (ratio of peak to average) which means the energy in the signal is going to be limited by accommodating the peaks. The sweeps are more promising, since their crest factor is only 3dB. The linear sweep has the same energy in each frequency band, but the engineer knows from the RTA that there’s more noise at low frequencies than high. The log sweep, from low frequency to high, puts more of its energy at the low end, so for a given input signal duration it should give us the best measurement result. So that settles it. The engineer picks the log sweep, measures the loudspeaker and knocks off for the day, and everyone lives happily ever after. Except Elias, who still isn’t happy.
The Tale of the Transfer Function
Let’s revisit the “sinusoidal steady state” course notes and stick with the same notation: lower case for time domain signals, upper case for frequency domain; a system of impulse response h(t), transfer function (Frequency Response) H(omega), driven by an input u(t) and producing an output y(t). h/H are unknown, but we would really like to know H.
The notes show that if we use a sinusoid of radian frequency omega as the input then, in the steady state after the transient effects of applying the sinusoid have died away, the output consists of a sinusoid at the same frequency as the input but modified in magnitude and phase according to the value of H at that frequency. If we measure the amplitude and phase of the output relative to the input we have figured out H at one frequency. Hooray! We are on the way to finding H. But there is a problem. There are an infinite number of frequencies. And getting to steady state may take a long time, depending on how long h(t) lasts. Even if h(t) is fairly short, at infinity times a short time it is still going to take us well past tea-time to fully characterise H. Elias, for one, isn’t happy.
So where does the problem lie? The first place to look is probably the input signal, because a sinusoid has infinite duration – it just never stops. Let’s go to the other extreme, how about if we use an impulse? At zero duration, that is going to be pretty short. Looking more promising already. We know from the notes that
Y = U H
i.e. FT of output = FT of input times Transfer Function. And we also happen to know that the FT of our impulse has unit magnitude and zero phase at all frequencies, so if u is an impulse then
Y = H
Now that’s more like it! Since the impulse has zero duration, we only need to capture the output for the duration of the impulse response, h, and take the FT and we’re done. Not only have we found H in a very finite time, we have found it at all frequencies at once. The mathematician in us is very happy indeed, job done and still time to polish off that Reimann Hypothesis proof before tea.
The Applied Mathematician in us is not so happy, however. He knows infinite amplitude, zero duration signals are a bit hard to come by in the acoustics lab, and the speakers may not react well to getting infinity up them. The techs are not likely to be too happy if we give them this method for finding H. So if zero duration signals are not so great after all, how about trying something of finite duration? In the time domain the system output is
y = u * h
where “*” indicates convolution. If h has finite duration, the duration of y will be the duration of u plus the duration of h. So if we capture that much of y, we have all the information the system is ever going to put out when fed u. If we capture that, and use our friend the FT again, we can figure out H from
H = Y / U
It will take longer than using the impulse, since we need to wait for the additional duration of our input signal u, but we still find the value of H at all frequencies at once and we’ll still get to tea in good time. Now all we need to do is pick a suitable input signal, u, of a suitably short duration. We know that dividing by zero is a no-no, so our input signal needs to have a spectrum that is not zero anywhere, but beyond that we can use pretty much whatever we like as far as the Applied Mathematician is concerned.
The Engineer in us is not completely happy yet. He knows that our loudspeaker and the amplifier driving it have limits, and our mic preamp has some noise, and it is not unknown for the techs to start chatting while the measurements are being made. Whatever signal we use had better not overstress the amps and speakers and have enough energy in it that the measurement noise becomes negligible. The Applied Mathematician has given us some candidates though, including a few noise sequences and sweeps, linear and logarithmic. The engineer isn’t keen on the noise sequences – they will have a high crest factor of 10dB or more (ratio of peak to average) which means the energy in the signal is going to be limited by accommodating the peaks. The sweeps are more promising, since their crest factor is only 3dB. The linear sweep has the same energy in each frequency band, but the engineer knows from the RTA that there’s more noise at low frequencies than high. The log sweep, from low frequency to high, puts more of its energy at the low end, so for a given input signal duration it should give us the best measurement result. So that settles it. The engineer picks the log sweep, measures the loudspeaker and knocks off for the day, and everyone lives happily ever after. Except Elias, who still isn’t happy.
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John - I don't think I got anything wrong - even so I did not emphasis on your points.
What I emphasized on was to correct Mikes summarizing in laymen terms - cause I thought it does not hold *as stated*.
Elias - I think I got your point to clearly distinguish in a "theoretical discussion here without that much of the restrictions of the practical world" - that "measuring" a FR by Dirac impulse actually is "calculating" the FR.
But again - in laymen terms I corrected what I thought does not hold *as stated*.
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Count me in - so there are at least two.
I would not doubt that FR can be measured or better "captured" by short signals (for usual LTI systems) - as said for the Dirac impulse "measurement", but you didn't drop a single word about CMP systems for now.
What do *you* think is "the steady state" of a CMP system - considering as outlined - there are *several steady states* in a CMP system - and given, we define "steady state" as having no SPL change over some delta time.
IMO, CMP simply does not fit *that well* into IR/FR world - at least not "in a single one".
Impulse response IMO is not compromised - but FR actually is - which shades some light on the swiss knife tool of FT and its usability and limitations in audio.
Michael
What I emphasized on was to correct Mikes summarizing in laymen terms - cause I thought it does not hold *as stated*.
Elias - I think I got your point to clearly distinguish in a "theoretical discussion here without that much of the restrictions of the practical world" - that "measuring" a FR by Dirac impulse actually is "calculating" the FR.
But again - in laymen terms I corrected what I thought does not hold *as stated*.
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The engineer picks the log sweep, measures the loudspeaker and knocks off for the day, and everyone lives happily ever after. Except Elias, who still isn’t happy.
Count me in - so there are at least two.
I would not doubt that FR can be measured or better "captured" by short signals (for usual LTI systems) - as said for the Dirac impulse "measurement", but you didn't drop a single word about CMP systems for now.
What do *you* think is "the steady state" of a CMP system - considering as outlined - there are *several steady states* in a CMP system - and given, we define "steady state" as having no SPL change over some delta time.
IMO, CMP simply does not fit *that well* into IR/FR world - at least not "in a single one".
Impulse response IMO is not compromised - but FR actually is - which shades some light on the swiss knife tool of FT and its usability and limitations in audio.
Michael
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