Difference between microphone transient response and frequency response (if any?)

Thread Starter

ebeowulf17

Joined Aug 12, 2014
2,953
So, I'm an regular reader and intermittent contributor over at the gearslutz audio engineering forums, and there are recurring debates over there about how to characterize transient response of high-end studio-grade microphones. There's no question that there's a dramatic difference, one that you can hear with your ears and see in a digitized waveform, in the transient response of certain mics that otherwise don't sound so dramatically different. It's especially obvious when recording drums and other forms of percussion, but also has a significant impact on instruments like acoustic guitar. The question is how, other than buying and trying a mic, you can know what to expect in terms of transient response.

One school of thought says that there's a simple mathematical transformation that allows you to take a frequency response plot and a phase response plot, and from the two of them you can derive everything you need to know about transient response. It's also claimed that the phase response of the microphone can be determined based solely on its frequency response. One (or maybe both) of those steps relies on the assumption that the microphone is a minimum phase device (or a time-invariant system.) Advocates of this thinking would say that any mic with especially good transient response has correspondingly higher response at high frequencies, possibly well beyond the range of human hearing in some cases.

The other school of thought says that microphones are imperfect transducers which do not qualify as minimum phase devices, and as such, no assumptions about transient response can be made solely on the basis of frequency response. They say that distortion, mechanical deformation, etc. break down the clean mathematical relationship between frequency response and transient response. There seems to be reasonable anecdotal evidence to support this, particularly in the case of ribbon microphones. Many studio-grade ribbon microphones exhibit very poor frequency response above 10-15kHz, but seem to have fantastic transient response, capturing a more powerful attack on plucked strings and percussion than their modest frequency plots would predict.

The people making the pro-math argument seem to know what they're talking about, and make fairly convincing arguments... but the other side's argument about non-linear, non-ideal behavior from real-world microphones also makes a lot of sense to me, and my personal experience thus far, not to mention the opinions of an awful lot of audio engineers whom I respect and trust, support the idea that transient response in microphones isn't defined exclusively by frequency response. I've tried to provide a reasonable summary of the salient points, but here are links to two of the larger debates for anyone who's interested in hearing both sides of this debate:

https://www.gearslutz.com/board/geekslutz-forum/1231275-there-measure-mic-transient-response.html

https://www.gearslutz.com/board/remote-possibilities-in-acoustic-music-and-location-recording/453859-ultra-thin-diaphram-fast-small-diaphrgam-condensers.html

I trust the engineers here more than most of the people over at gearslutz when it comes to math, science, and theoretical stuff, so I wanted to get your opinions. Keep in mind, I'm not questioning the validity of these mathematical transformations for testing electronics (preamps, transmitters, etc.) but just specifically for microphones. I'd love to hear what you all think.

Thanks!
 

Wolframore

Joined Jan 21, 2019
1,472
Hey I have a perspective on this that may lead you to make your own decision on this:

Transient response is really sensitivity to amplitude (or lack of) - the thing that ribbon mics make great drum mic's is that they're not as sensitive and can capture transients without pushing into overdrive. It has a natural compression effect which makes for great percussion recordings. Plus they're "warmer" because most of them have poor high freq response. A dynamic has similar traits... that's why we add XY cartoids/condensors on the cymbals to add back the highend.

Frequency response is just that... the ability for the transducer to be affected by different frequencies...

We are talking about two different characteristics and not necessarily co-dependent, especially when it comes to using in recordings. The colored mic's are sometimes very useful in capturing something. Otherwise we would all just use flat response Shure condenser mics and be done with it.

Same goes with your entire signal chain... I think of them as tools and paint rather than one better than another... certain ones are just more desirable and useful than others depending on what you're doing.

I've heard just about every mic out there and some of them are just beautiful...

Charlie
Chicago Engineering and Recording Society member
 

bogosort

Joined Sep 24, 2011
428
One school of thought says that there's a simple mathematical transformation that allows you to take a frequency response plot and a phase response plot, and from the two of them you can derive everything you need to know about transient response.
Typically this is done in reverse, though it works in both directions: we derive a system's frequency/phase response from its impulse response, where an impulse is a specific kind of transient. If the system is linear and time-invariant, the impulse response completely characterizes the system: once we know the impulse response, we know everything about the system. In math-speak, the Laplace transform of a system's impulse response is a complex-valued function called the system's transfer function. Taking the magnitude of the transfer function gives us the system's frequency response, while taking the argument gives us its phase response.

The math makes it all clear, but we can also think of it intuitively. Consider that an impulse is a very short burst of energy. Fourier analysis tells us that as the duration of the burst approaches zero, the bandwidth of the burst approaches infinity. A theoretically perfect impulse (Dirac delta) has constant energy at all frequencies, so, by feeding such a signal to a system, we are effectively "exercising" the system at all possible frequencies at the same time. In this way, the output tells us everything we need to know about how the system responds in frequency and phase.

It's also claimed that the phase response of the microphone can be determined based solely on its frequency response. One (or maybe both) of those steps relies on the assumption that the microphone is a minimum phase device (or a time-invariant system.) Advocates of this thinking would say that any mic with especially good transient response has correspondingly higher response at high frequencies, possibly well beyond the range of human hearing in some cases.
Frequency and phase are indeed related and can be represented by a single function of complex values. In other words, the aspects of a system that affect its frequency response are the same aspects that affect its phase response. I'm not sure how you're relating the condition of minimum-phase to this; as I see it, minimum-phase describes an invertible LTI system.

The other school of thought says that microphones are imperfect transducers which do not qualify as minimum phase devices, and as such, no assumptions about transient response can be made solely on the basis of frequency response. They say that distortion, mechanical deformation, etc. break down the clean mathematical relationship between frequency response and transient response. There seems to be reasonable anecdotal evidence to support this, particularly in the case of ribbon microphones. Many studio-grade ribbon microphones exhibit very poor frequency response above 10-15kHz, but seem to have fantastic transient response, capturing a more powerful attack on plucked strings and percussion than their modest frequency plots would predict.
From your mention of distortions and deformations, I believe you're referring to nonlinearities that would break the linearity requirement for a system to be characterized by its impulse response. I'd note that quality microphones are essentially linear devices when SPL is kept within spec. It's true that nothing in the physical world is perfectly linear, but we have to keep perspective on the significance of any nonlinearities. A simple resistor is not perfectly linear either, but for many applications there's no problem with treating it as if it were.

When it comes to microphone performance, frequency response is nothing more than a convenient way to quantify the general behavior of the mic. We can look at a plot and get a sense if the mic will be bright, or boomy, or even riddled with comb-filter problems. But I wouldn't buy a mic based solely on the plot of its frequency response. Ribbon mics are a great example: compared to quality small-diaphragm condensers, ribbons tend to have poor high-frequency response. We could say that, technically speaking, ribbons are not as good as SDCs at accurately capturing musical transients. But that says nothing about how ribbons sound to a human listener. For example, if I'm recording an aggressive violin part, I'm likely to choose a ribbon mic to help soften and smooth the harsh transients of the bow scraping against the strings.

The people making the pro-math argument seem to know what they're talking about, and make fairly convincing arguments... but the other side's argument about non-linear, non-ideal behavior from real-world microphones also makes a lot of sense to me, and my personal experience thus far, not to mention the opinions of an awful lot of audio engineers whom I respect and trust, support the idea that transient response in microphones isn't defined exclusively by frequency response.
You seem to be using the phrase transient response not in its technical meaning, but as a particular quality specific to sound perception. Sound perception is a whole 'nother (and equally interesting) ball of wax. Perhaps the confusion is due to people using different meanings for the phrase? Hopefully this post helped clear up how transient response (in the technical sense) relates to frequency response, and why it really doesn't matter when it comes to choosing a microphone for recording music.
 

Thread Starter

ebeowulf17

Joined Aug 12, 2014
2,953
From your mention of distortions and deformations, I believe you're referring to nonlinearities that would break the linearity requirement for a system to be characterized by its impulse response. I'd note that quality microphones are essentially linear devices when SPL is kept within spec. It's true that nothing in the physical world is perfectly linear, but we have to keep perspective on the significance of any nonlinearities. A simple resistor is not perfectly linear either, but for many applications there's no problem with treating it as if it were.
I'm not really 100% sure what I mean with references to distortions and such - just trying to understand both sides of the theoretical argument here. It sounds like you're saying that you do believe that differences between ideal and real microphone performance don't preclude transformations between frequency response data and impulse/transient/step response data.
I'm not sure how you're relating the condition of minimum-phase to this; as I see it, minimum-phase describes an invertible LTI system.
I'm not sure either! I tried reading a few descriptions of what minimum-phase means and relating it to this situation, but I didn't quite follow it. It was raised as a pre-requisite for the mathematical transformations, and then there was healthy debate or not mics qualified. Are you saying that minimum-phase and time-invariant-system descriptions are irrelevant? Can you reliably predict transient response from frequency response and vice versa without pre-supposing that the mic meets one or both of those criteria?
You seem to be using the phrase transient response not in its technical meaning, but as a particular quality specific to sound perception. Sound perception is a whole 'nother (and equally interesting) ball of wax. Perhaps the confusion is due to people using different meanings for the phrase?
Yes, you're absolutely right. Sorry about that. In audio recording circles the phrase "transient response" is widely used to describe a combination of factors (phase delay, slope, peak amplitude) in response to very sharp signals like plucked strings and percussion instruments. The collective effect of these things we refer to as transient response is that a mic with "better" or "faster" "transient response" is one which has less delay, less compression/clipping, etc. and more faithfully represents these sharp transients. I realize now that this isn't a very accurate use of the phrase, but it's a hard habit to break after 20-ish years of hearing it used this way constantly. Here are example images that may help provide context for you or other readers:
upload_2019-4-24_15-59-39.jpeg
You can see that the bottom trace in the image above reaches its peak sooner, has less overshoot/oscillation, and settles much faster. This would be described as "better transient response" in microphone discussions.


Here's another example showing that the peak response is generally both higher and earlier (faster?) for microphones with lighter diaphragms.


You get the idea!

Hopefully this post helped clear up how transient response (in the technical sense) relates to frequency response, and why it really doesn't matter when it comes to choosing a microphone for recording music.
I totally understand that "better" is totally subjective and depends on what you're trying to capture, so "better transient response" in the sense I'm familiar with isn't always what's wanted. I'm pretty comfortable with my creative and aesthetic choices when using mics. I'm not relying on graphs to decide which mic sounds best.

However, I am very interested in understanding a bit more of the theory behind things.

If the argument that frequency response completely predicts transient response is true, that would imply that any mic with "fast" transient response as shown in images above must necessarily have better high frequency response. Meanwhile, most recording engineers will tell you that ribbon mics have far "faster transient response" than dynamic mics, despite the fact that most of them have dramatic rolloff at high frequencies. Based on the frequency responses below, should I believe that the dynamic RE20 (with substantially more high frequency response) must have better transient response than the Beyer M130 ribbon?

EV RE-20:


Beyer M130:
 

Wolframore

Joined Jan 21, 2019
1,472
It’s funny you would think all mics with same or similar frequency response curves would sound the same but it’s definitely not the case. Sounds right that a diaphragm size would have a huge effect. I can see that it would be similar to playing the same note from a violin vs cello. Huge difference. Overtones are different as well as resonance and other timbral qualities.

I personally love those old Neumann mics. Until you hear them you would think all condensers are the same. Looking at the frequency curves you would think it shouldn’t be so different. We A/B equipment all the time through different studios. It’s amazing what some of them do. The best ones have some amount of character and color to them you wouldn’t see in those curves and can’t be explained by Nyquist theorems and such.

If I was recording violins or any classical I wouldn’t use anything to color it. We try to be absolutely pure with them. They need clean low noise signal chain with lots of dynamic range.
 

bogosort

Joined Sep 24, 2011
428
I'm not sure either! I tried reading a few descriptions of what minimum-phase means and relating it to this situation, but I didn't quite follow it. It was raised as a pre-requisite for the mathematical transformations, and then there was healthy debate or not mics qualified. Are you saying that minimum-phase and time-invariant-system descriptions are irrelevant? Can you reliably predict transient response from frequency response and vice versa without pre-supposing that the mic meets one or both of those criteria?
Minimum-phase is not a necessary condition for Fourier/Laplace analysis; we can just as easily perform the transformations on non-minimum-phase systems, e.g., mixed-phase or linear phase. In the audio context, you might be reading about the minimum group delay property of certain minimum-phase systems. Minimizing group delay can be important consideration for acoustic transducers, but it doesn't have anything to do with the analysis aspect.

As for time-invariance, that just means that a system's behavior is independent of time, i.e., its behavior won't change if we take our measurements today or tomorrow or next year. This isn't strictly true of any actual physical systems, as there are always time-dependent changes in real-world devices: for example, in the short scale, ambient temperatures and humidity levels fluctuate, which can affect behavior; in the long scale, components age and eventually stop working. But for all intents and purposes, we can treat every microphone as a time-invariant system, as you certainly expect your mic to sound the same whether you record your take at 12:00 pm or 12:15 pm.

In audio recording circles the phrase "transient response" is widely used to describe a combination of factors (phase delay, slope, peak amplitude) in response to very sharp signals like plucked strings and percussion instruments. The collective effect of these things we refer to as transient response is that a mic with "better" or "faster" "transient response" is one which has less delay, less compression/clipping, etc. and more faithfully represents these sharp transients.
Very sharp signals are impulse-like, which means they contain a lot of high-frequency energy. If a system could reproduce all those frequencies, its impulse response in the time domain would look exactly like the impulse at the input. The less the system can reproduce those frequencies, the more "smeared" the impulse will look at the output. The image you provided below is a nice example of that:



In everyday language, we'd say that the condenser mic is faster than the ribbon, which is faster than the dynamic. But this only means that the condenser has a larger bandwidth -- it can reproduce higher frequencies.

If the argument that frequency response completely predicts transient response is true, that would imply that any mic with "fast" transient response as shown in images above must necessarily have better high frequency response. Meanwhile, most recording engineers will tell you that ribbon mics have far "faster transient response" than dynamic mics, despite the fact that most of them have dramatic rolloff at high frequencies. Based on the frequency responses below, should I believe that the dynamic RE20 (with substantially more high frequency response) must have better transient response than the Beyer M130 ribbon?
We don't have enough information about the measurement parameters used in the plots to say definitively, but if we assume that both plots are comparable, then indeed the RE20 is the faster mic. Note that the RE20, with its relatively flat frequency response, isn't representative of most dynamic microphones. Look at plots for the SM57, SM7B, MD-441, etc., and you'll see the more usual upper-mid bump and the steep high-frequency rolloff typical of most dynamics. In other words, very generally speaking, ribbons are indeed faster -- and have better high-frequency response -- than moving-coil dynamics. You could say they tend to have better transient response, in both meanings of the phrase.
 

Wolframore

Joined Jan 21, 2019
1,472
Typical Ribbon:


From Sound on Sound: Although they are generally capable of handling impressive SPLs (up to around 165dB without distortion on some of the models we tested), simply blowing into a ribbon, or exposing it to any other burst of wind (for example, by slamming a door, or putting the mic in front of a kick drum) can tear the ribbon (resulting in a recorded sound like the distorted, slightly quiet swearing from the engineer).

Further: The overall output level of ribbons is often extremely low compared with condenser (and even dynamic) mics, so they can be very demanding when it comes to the choice of mic preamp they are plugged into: they often need 50dB or more of gain

Typical Dynamic:


With the dynamics you can pretty much stuff them in kick drums without issue and captures the transients and bass nicely...

It's also the placement of the mics... another important point is that you need to know how these response curves were generated... some are 6" away using white noise ... others have different criteria. Not sure that all use the same all the time.
 

Thread Starter

ebeowulf17

Joined Aug 12, 2014
2,953
Minimum-phase is not a necessary condition for Fourier/Laplace analysis; we can just as easily perform the transformations on non-minimum-phase systems, e.g., mixed-phase or linear phase. In the audio context, you might be reading about the minimum group delay property of certain minimum-phase systems. Minimizing group delay can be important consideration for acoustic transducers, but it doesn't have anything to do with the analysis aspect.

As for time-invariance, that just means that a system's behavior is independent of time, i.e., its behavior won't change if we take our measurements today or tomorrow or next year. This isn't strictly true of any actual physical systems, as there are always time-dependent changes in real-world devices: for example, in the short scale, ambient temperatures and humidity levels fluctuate, which can affect behavior; in the long scale, components age and eventually stop working. But for all intents and purposes, we can treat every microphone as a time-invariant system, as you certainly expect your mic to sound the same whether you record your take at 12:00 pm or 12:15 pm.


Very sharp signals are impulse-like, which means they contain a lot of high-frequency energy. If a system could reproduce all those frequencies, its impulse response in the time domain would look exactly like the impulse at the input. The less the system can reproduce those frequencies, the more "smeared" the impulse will look at the output. The image you provided below is a nice example of that:



In everyday language, we'd say that the condenser mic is faster than the ribbon, which is faster than the dynamic. But this only means that the condenser has a larger bandwidth -- it can reproduce higher frequencies.


We don't have enough information about the measurement parameters used in the plots to say definitively, but if we assume that both plots are comparable, then indeed the RE20 is the faster mic. Note that the RE20, with its relatively flat frequency response, isn't representative of most dynamic microphones. Look at plots for the SM57, SM7B, MD-441, etc., and you'll see the more usual upper-mid bump and the steep high-frequency rolloff typical of most dynamics. In other words, very generally speaking, ribbons are indeed faster -- and have better high-frequency response -- than moving-coil dynamics. You could say they tend to have better transient response, in both meanings of the phrase.
Thanks for the detailed response!

I don't know why I didn't think of this sooner, but I'm going to try my own experiment with this. I've got an RE20 with its unusually good high frequency response (for a dynamic) and an AKG CK94 with a fair bit of HF rolloff for a condenser. Based on specs, one might expect the AKG to be "slower" than the RE20. I think I'm going to try my own transient test with those two, and maybe a couple other mics. I'll post waveforms and sound files if/when I do the experiment.
 

MrAl

Joined Jun 17, 2014
6,782
Hello there,

The key qualifier in the linear system is the relationship between the friction and the mass and "springyness" of the mechanism that moves and the frame it moves in. That means that if we knew these quantities for a given mechanism we could determine it's frequency and transient response. We could probably measure these in some way.

In the non linear system however, almost anything goes. We could probably design systems that behave counter to each other in relation to their frequency and transient responses. The key question then is are there any limitations inherent in microphones that restrict them to a certain class of non linear systems where that class is known to have a certain frequency and transient relationship.
But then what happens if someone designs a very dynamic microphone the very next day that has an entirely different behavior. The only thing stopping that from happening would be if there were some constraint on the allowable physical structure, which again is hard to pin down because who says this wont change tomorrow anyway, or might have even changed already but we did not get informed yet.

So in this quest we are assuming that the physical system can only change in certain ways always being constrained to conform to what we assumed in the analysis. I think this would be a mistake, therefore testing each system would be mandatory.
 
There isn;t a lot here: https://en.wikipedia.org/wiki/Slew-induced_distortion

But it;s worth looking at TIM distortion or in essence slew-rate induced distortion.
I have built the Leach Amp which is optimized for low TIM distortion, by having a 100 V/uS slew rate and an obscene frequency response before it's intentionally rolled off. The frequency response is from 0Hz to about 800 kHz, I have a 4bx signal processor that can alter that attack type sound and I connected it to one amp I was repairing and it sounded really sick.

I can;t really find what I'm looking for, but a square wave or your transient can be modeled as the infinate series of the sum of sine waves. When you look at the first 3-5 terms, you can see that the edges get defined better at higher "frequencies".

That's why, I think audio frequency response needs to be from about 0.5 Hz to 100 kHz up until the speaker. There, we can intentionally roll it off.
 

Wolframore

Joined Jan 21, 2019
1,472
A square wave is the sum of odd order harmonics. After ~20k it can affect the waveform but we won’t be able to hear it. I believe it’s difference tones and summations that create the implied fundamentals we hear or maybe I mean sense. Our ears and brain does some interesting fill in the blanks at times. Another concern for me is bit depth and dither. With current systems recording at 24bit 192k without data compression it’s pretty close to real life at least as far as we can hear.

Here’s that bit about square waves
https://www.allaboutcircuits.com/textbook/alternating-current/chpt-7/square-wave-signals/
 

MrAl

Joined Jun 17, 2014
6,782
Hi,

Yes in a linear circuit the harmonics will be related to the transient response, but that's in a linear circuit.
The problem is that the exponential part is considered to have died out in the frequency response curve.
I'll try to provide some more insight to this later but i have to do a bunch of calculations to show that.

Also keep in mind when these topics are discussed it is usually within the context of a second order system. Higher order systems are much harder to correlate and may even not be as well defined as in the second order system. We could look at some examples i guess.
 
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Thread Starter

ebeowulf17

Joined Aug 12, 2014
2,953
Hi,

Yes in a linear circuit the harmonics will be related to the transient response, but that's in a linear circuit.
The problem is that the exponential part is considered to have died out in the frequency response curve.
I'll try to provide some more insight to this later but i have to do a bunch of calculations to show that.

Also keep in mind when these topics are discussed it is usually within the context of a second order system. Higher order systems are much harder to correlate and may even not be as well defined as in the second order system. We could look at some examples i guess.
I freely admit that I'm not completely following what you've said so far, but I'm excited to see where it's headed. Most of the math involved in working this stuff out is way over my head, so I'm counting on people like you to help me grasp as much as possible in simpler, more conceptual terms (not that I'm saying anyone should hold back when it comes to scientific and mathematical evidence, just that simpler summaries are also appreciated when possible.)

Thanks in advance for any insights you can share!
 

MrAl

Joined Jun 17, 2014
6,782
I freely admit that I'm not completely following what you've said so far, but I'm excited to see where it's headed. Most of the math involved in working this stuff out is way over my head, so I'm counting on people like you to help me grasp as much as possible in simpler, more conceptual terms (not that I'm saying anyone should hold back when it comes to scientific and mathematical evidence, just that simpler summaries are also appreciated when possible.)

Thanks in advance for any insights you can share!
Hello again,

Here is one little addition. The rise time of a second order system is:
tr=(pi-atan(sqrt(1-d^2)/d))/(w0*sqrt(1-d^2))

where d is the damping factor. To get this value we have to know certain things about the mechanical system like the mass, friction, and springiyness.
 
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