# Transient response and stored energy



## npdang (Jul 29, 2005)

Transient response. One of the most misunderstood terms I've seen used in the world of audio. Some people will tell you it has to do with moving mass, motor strength, inductance, and an entire host of things. Forget what you've heard for a moment. Let's define "transient response" as how fast a speaker starts, and how fast it stops in reaction to a given input signal. You'll also hear this term referred to as "stored energy".

I can tell you right now that no single manufacturer spec or formula is going to give you an all encompassing, meaningful measure of the transient response of a speaker. Transient response is something that must be measured firsthand.

Let's first investigate the transient response of several tweeters at 2khz, using a shaped tone burst. Simply put, the speaker is excited with a special test signal (much like a quick ping) that is centered directly at 2khz, and the following graph is generated showing the initial rise time response and the decay times. A perfect transient response would show no delays during the rise of the signal, and no delay or bumps in the decay.










Looking at the response of all three tweeters, notice the initial rise time between 0 and 1 millisecond. They all rise rather quickly, and there's no issues here. 

However, we can see that after 1ms there is a drastic difference in the time it takes for the signal to decay between the three tweeters. That is what is commonly referred to as "overhang" or "stored energy". The signal has stopped playing, but the speaker has not.

As we can see, the black line Seas metal dome tweeter has the fastest decay time. That is not surprising, given that metal cones are often stiff and fast to respond.

The red line Max-fidelity tweeter lives up to its name with only slightly worse decay time than the Seas. 

Lastly, we can see the poor Morel dome in blue. It is clearly the worst performer of the three, taking quite a bit of time to come to a stop.

Let's examine another graph, this time a waterfall plot. Waterfall plots are a bit different way of looking at transient response. 










Looking at the above graph, we can see it looks much like a frequency response graph, except that there's a third dimension/axis that represents time. A waterfall plot shows you the frequency response over time. 

Now we should expect that for higher frequencies, the decay times would be very quick. For lower frequencies, the decay times should be progressively longer. For example, a 10khz sine wave is 10,000 cycles per second. That means it only takes 0.1 millisecond second to play 1 cycle of a 10khz sine wave. For a 1khz sine wave, it takes 1 millisecond, and for a 100hz sine wave it takes 10 milliseconds.

What's critical to look for are frequencies that decay very slowly along the time axis. We know that higher frequencies should decay quickly, and if they don't we have a problem.

If you look at the graph above, you can see at 10khz there is a clear problem. There's a very obvious ridge that doesn't disappear for nearly 3ms. The longer decay times below 2khz are fine however, because at lower frequencies we would it expect it to take longer for a signal to "finish playing". 

In this case, we wouldn't want to use this speaker anywhere near 10khz. 

So we can see here, that shape tone burst tests are great for examining the start and stop times of a speaker at a single frequency in great detail. The waterfall plots however, are great for examining a speaker's entire frequency response over time and finding "trouble spots"... frequencies that continue to play well after the original signal has stopped. 

Now, probably the most important thing is what does poor transient response sound like? A speaker with poor transient response generally sounds dull, and veiled. "Cloudy" would be the appropriate word. In most cases the differences are somewhat less dramatic, especially as you go below 200hz I would say they are almost inaudible. 

For example, at 20hz it takes almost 50 milliseconds to play 1 full cycle. So a 1 or 2 millisecond "overhang" is practically inaudible. At 1khz however, it only takes 1ms for a full cycle so that extra 1-2ms it takes to decay is actually quite significant.

One last important note... don't assume that sloppy, boomy bass, muddy midrange, or dull lifeless tweeters are the result of poor transient response. The effect of poor transient response is more like a haze, or dulling of the sound. The most extreme example I could think of is playing music inside a large gym with hard floors and walls... think of all the echoes and reflections that kind of wash out the sound. The music is actually continuing to play long after the original signal has stopped, due to all the echoes.

Sloppy bass on the otherhand, is often due to excessive distortion and improper tuning. The speaker is playing signals that weren't originally present, making it sound harsh, muddy, and well... distorted. Also, an imbalance in the frequency response can lead to any speaker sounding muddy, peaky, or slow. The sound is quite different than having poor transient response, if you know what to listen for.


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## 10K2HVN (Mar 8, 2005)

Awesome thread, NpDang!


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## Eastcoast (Mar 26, 2005)

*..*

This place is quickly becoming one of my first stops when I hop on line...

Thanks for the info...


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## thechris (Jun 26, 2005)

well, yes. but the acoustic enviroment, ears, and crossover will have something to say about a "best case" transient response. still i think the crossover would have a hard time competing with a 1mSec delay. at least the upper highpass. from an engineering perspective, it may be a moot point if the ear is unable to detect changes in delay less then 25mS or the environment adds a delay of 25mS.

how do shaped tone bursts work? it seems that a frequency-time plot would suffer the same minimum error as would be seen in the heiseberg uncertinty priniple. basically errors in F and T are multipled, or dF * dT >= K where K is a constant based opun the transform's kernel function. i'm not sure perfect transient response whould be measured as having an infinate rise time, but probably a slope related to 1/K.

not arguing that transient response doesn't exist -- i have easy to read time based plots that clearly show the concept.

what affects does this have on sound? can the ear detect the differences easily? does it have a negative impact on sound, or an oddly positive impact?

i think it would be interesting to look at a few test cases. get a recording of about 30s. then add a decay time to the music to see how much additional decay time is needed.


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## npdang (Jul 29, 2005)

@thechris 

Hmm maybe you should write this article


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## Halonix (Jul 1, 2005)

Can I ask a stupid question of the origional poster? What are you using to record that?


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## Ludemandan (Jul 13, 2005)

A dull haze is slowly lifting from my understanding of all things mobile audio. It's becoming clearer and more defined.


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## thechris (Jun 26, 2005)

you might define transient response as the opposite of steady state.

think of it this way. start your car. let the engine run at idle. the engine runs a fairly predictable course. from past events you can reliably predict future events. this is because there is a lack of new information.

but when you first start the car, or when you shift into gear, these are transients. they are non-repetitive and short lived.

there is a lot of math behind these things, and for linear systems you can show transients for a given input signal (which must have transients if you want the output to have transients). note that once you have a pattern of transient events, you could use a fourier transform to show how the system would react...

so i guess its all in the hands of the observer. from a math perspective, it you can patternize the input, it can be expressed in the fourier domain. once you have non-repeating or non-patterned responses you have to use the laplace transform which has obvious transient terms.

Laplace response might be: sin(t) + sin(t)*e^(-0.001t)
Fourier would just be: sin(t)

if you wait long enough, the second term effectively disappears, giving fourier.


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## 300Z (Mar 20, 2005)

Didnt Dan W. said that transient response is just frequency response or something along the lines? or am i wrong? I think it was over at ECA a while back...


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## Guest (Mar 7, 2006)

The *Fourier Series* is defined only for periodic signals, but the *Fourier Transform* is well suited for a variety of non-periodic ... including transient ... signals. Periodicity in one domain (time or freq) means discrete in the other (freq or time) ... so a _frequency series_ is only useful for signals that are _periodic in time_. However, as soon as we allow a continuous frequency domain representation (Fourier Transform), non-periodic time domain signals are also allowed.

Alternatively, it can be shown than any periodic time signal only has energy at discrete frequencies in the frequency domain. The amplitude and phase at those discrete frequencies is precisely the Fourier Series.

A good example is the time-domain "gate" or "boxcar" function. A single "gate" ... which is a very transient signal ... has a well defined Fourier Transform. It's a classic "sin(f)/f" type fucntion. If you periodically repeat the time domain gates, the new periodic signal has a Fourier Series ... the frequency content only exists at _discrete_ points in the frequency domain. This is true for _any_ periodic function.

So it's accurate to say that a Fourier Series is only valid for periodic, repetitive signals, but a Fourier Transform is valid for non-periodic ... including transient ... signals.

One has to be careful how "transient response" is defined. The classical EE definition is : The complete time domain response to a transient input. The transient input may be an impulse, step, etc. The impulse is simply the time-domain derivative of the step, and consequently the two responses (of linear, time invariant systems) are also related through a simple derivative.

If we define the transient response as the _impulse response_ of the system, then yes .... the Fourier Transform of the transient response will identically be the Frequency Response. In other words, the "Impulse Response" and "Frequency Response" of any linear, time invariant system are Fourier Transform pairs.


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## squeak9798 (Apr 20, 2005)

300Z said:


> Didnt Dan W. said that transient response is just frequency response or something along the lines? or am i wrong? I think it was over at ECA a while back...


http://forum.carstereos.org/showthread.php?t=56659&highlight=transient+frequency+inductance



DanWiggins said:


> 2. Transient response IS frequency response. The two are the same thing. Take the transient response, run it through the Fourier transform, and you get frequency response. Take frequency response, run it through the inverse Fourier transform, you get transient response. You cannot talk about one without implicitly talking about the other!
> 
> Now, what IS transient response - how fast something responds to a signal, correct? What is that - it's the time it takes. Look at the immediately above paragraph - time IS frequency response. In fact, if your driver responds in 1 millisecond, 1/1000 of a second - it must have a frequency response out to 1 kHz at a minimum (1 Hz = 1 cycle per second; 1 kHz = 1000 cycles per second, which means each cycle takes 1/1000th of a second).
> 
> ...


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## Guest (Mar 7, 2006)

careful ... Dan tends to be a little "loose" with his definition of transient response 

As stated in my earlier post, strictly speaking the "transient response" of a linear, time invariant system is :

*TRANSIENT RESPONSE :
The complete time domain response of a system to a transient input. The transient input maybe a step function, an impulse function (related to the step through a simple first derivative), a bursted sinewave, or any other input with short, quick dynamics.*

If we restrict the transient input to be an _impulse_, then yes ... the Fourier Transform of the "transient response" is _identically_ the Frequency Response.

*The Impulse Response and Frequency Response of linear, time-invariant systems are Fourier Transform pairs.*

For further clarification, the Frequency Response so defined is a _complex_ function of frequency. Meaning, the Frequency Response contains _both_ a Magnitude Function versus Frequency, as well as a Phase Function vs. Frequency. Oftentimes, the Magnitude Response alone is erroneously equated to Frequency Response.

And yes, I certainly do recognize that the Magnitude Response and Phase Response are not completely independent  For example, if we know a system is _stable_, this condition restricts the "poles" of the transfer function to the left-half plane. If we know a system is _minimum phase_, this condition restricts the "zeros" of the transfer function to the left-half plane. And these two conditions ... stability, and minimum phase ... allow us to unambiguously determine the Phase Response from the Magnitude Response. But nonetheless, the strict definition of the complex Frequency Response contains BOTH the Magnitude and Phase functions.


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## acuvox (Jul 19, 2006)

Quote: *The Impulse Response and Frequency Response of linear, time-invariant systems are Fourier Transform pairs.*

For further clarification, the Frequency Response so defined is a _complex_ function of frequency. Meaning, the Frequency Response contains _both_ a Magnitude Function versus Frequency, as well as a Phase Function vs. Frequency. Oftentimes, the Magnitude Response alone is erroneously equated to Frequency Response. (end quote)

I have two points regarding transient response:

1) Since the point is to correlate measurements with music, the best test signal is a rectangular gated sine wave. Musical instruments that are plucked, struck or blown can achieve full output on the first half cycle, and many can stop almost as quickly. 

2) Bass systems store huge amounts of energy to achieve low end extension. This is normally expressed as group delay. It does correspond to frequency response, but more to the shape of the curve than the f3. It is commonly quoted that Qtc of .5 has the "best transient response", but rarely admitted that this is a frequency response ideal. 

Combining points 1) and 2), try listening to a four cycle rectangular gated sine wave at an impedance peak frequency of your woofer and then vary the frequency up and down. Group delay sounds off-key because it is a resonance just as nasty as cone breakup, shifting envelope transients into a single note at both the leading and trailing edges. 

The lower the impedance peak, the less energy is stored and the better the response to envelope transients.


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## Excelsior (Dec 8, 2005)

thechris said:


> what affects does this have on sound? can the ear detect the differences easily? does it have a negative impact on sound, or an oddly positive impact?



I'm not sure that this relieves transient response as an important factor in discussing overall distortions.

The entire idea of a stiffer cone (diamond for instance) is to relieve this "ringing" much quicker and push break up nodes into the far ultrasonics.

This can only be good for SQ... and yes you could hear it depending on the situation. Poor transient response is generally frequency dependent as waterfalls show. 

Can you say for sure that this is an AUDIBLE source of distortion? maybe, on some particular drivers. Is this the most important cause of distortion? probably not

Highly dampened cones (plastics and papers, among a million better currently unused materials) tend to store energy and have a "worse transient response", however they also tend to have much lower resonances when they occur. And as everyone should agree, a flatter FR is going to be more audible than a speaker with decently more stored energy

It's all application oriented


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## donkeypunch22 (Nov 5, 2008)

Thank you!


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## astrochex (Aug 7, 2009)

Very informative, thank-you.


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## amungal (Mar 29, 2010)

Thanks to everyone who takes the time to contribute so that others may be enlightened. Great stuff! Keep it coming...


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## rdlhifi (Sep 8, 2013)

another great read npdang! TY!


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