Help me understand Impulse Response, Step Response, and Time Alignment

[DWPress]

Pics of my SB-1000 vs my 15" DIY sub from same measuring session dialing in 4-way XO:


SVS

DIY 15"

 

[bmc0]

FWIW, if you have good measurements (i.e. not corrupted by room modes) and a way to apply somewhat long FIR filters, you could correct the SVS's excess phase. Here's an example illustrating the time-domain response compared to a minimum phase model (based on @levimax's measurements)—
Magnitude:

Phase:

Group delay:

Impulse:

 

[levimax]

Pics of my SB-1000 vs my 15" DIY sub from same measuring session dialing in 4-way XO:


SVS
View attachment 496676

DIY 15"
View attachment 496675

While the step responses on the SVS "looks bad" I don't know how audible it really is. FR is more audible that time domain issues and if you look at the step response of a crossover it also "looks bad" but sounds fine. That is why I am trying to understand how filters affect the "cone movement" to help me understand what I am looking at. The way I am looking at it is that time domain issues are not as audible as they look on a graph but the technology exists to easily have both good FR and good time domain performance if you are willing to give up "small size". "Hoffman's Iron law" is apparently one of the laws of nature. .

FWIW, if you have good measurements (i.e. not corrupted by room modes) and a way to apply somewhat long FIR filters, you could correct the SVS's excess phase. Here's an example illustrating the time-domain response compared to a minimum phase model (based on @levimax's measurements)—
Magnitude:
View attachment 496677
Phase:
View attachment 496678
Group delay:
View attachment 496679
Impulse:
View attachment 496680

That looks better, what allpass filters are you using? I have a PC based system so Taps are not an issue.

 

[Keith_W]

I think you guys need to understand that part of the appearance of the step response is due to the minimum-phase nature of subwoofers itself. Here is a sim to show you.

I created a bandpass with 4th order minimum-phase Butterworth curves. The HPF for both is at 25Hz, and the LPF is at 50Hz (red) and 100Hz (green). This will be our subwoofer simulation.

Now examine the step response of both "subwoofers". Bear in mind that there is NO MANIPULATION of these curves, it is the pure minphase bandpass. The green curve looks nicer, does it not? It's narrower, and the ringing is slightly shorter. The rise time is sharper, the peak is higher. The red curve is time stretched and almost has the appearance of inverted polarity.

Since we are here, let's increase the gain of the green curve by 5dB. Wow, it looks even better! That red sub is going to the landfill!

If I were to show you both step responses without any context, you would say the green curve has the better looking step response. But - the only difference is that it has been crossed over higher.

In a real world scenario, you might look at a really wiggly long tail in your step response and you feel disappointed. That feeling of utter failure as an audiophile starts to creep in. Some doubt might creep into your mind - maybe it's time to end it all? Sell all the equipment and take up knitting? It is important to realize that the step response of your subwoofer is the total sum of filter ringing, driver ringing, and room ringing/decay. Sometimes you can reduce ringing by better filter design. Sometimes not. If you see an ugly looking step response, you need to ask yourself WHY.

 

[audiofooled]

While the step responses on the SVS "looks bad" I don't know how audible it really is. FR is more audible that time domain issues and if you look at the step response of a crossover it also "looks bad" but sounds fine. That is why I am trying to understand how filters affect the "cone movement" to help me understand what I am looking at.

I would hate to complicate it any further, but there's also this:

What is group delay?

As long as they're all showing up at the same time, should not make a difference, n'est pas? So long as there was not a specified arrival time, then yes.

www.audiosciencereview.com

What is group delay?

As long as they're all showing up at the same time, should not make a difference, n'est pas? So long as there was not a specified arrival time, then yes.

www.audiosciencereview.com

The last picture in the above post may be very informative but I suggest reading all of it for a better context.

 

[3ll3d00d]

Maybe I am being paranoid and not understanding but it seems like a terrible result to have ringing that strong and long

The design is to maximise low end in a small package and then cross to main speakers at a typical (say 60-100Hz). It's basically the design of every single svs sub ever (albeit perhaps not in a small package for some of them) and they are generally successful at achieving that goal. Comparing that to a DIY sub which is likely a raw driver in a box with either no dsp or minimal dsp (as opposed to custom dsp designed to maximise output and compress when overloaded without letting out the magic smoke) is going to produce different IRs.

Personally, having owned and tuned all of those sorts of subs over the years, am sceptical such differences matter much (whereas differences in extension/raw output level are plainly audible).

 

[levimax]

I would hate to complicate it any further, but there's also this:

What is group delay?

As long as they're all showing up at the same time, should not make a difference, n'est pas? So long as there was not a specified arrival time, then yes.

www.audiosciencereview.com

What is group delay?

As long as they're all showing up at the same time, should not make a difference, n'est pas? So long as there was not a specified arrival time, then yes.

www.audiosciencereview.com

The last picture in the above post may be very informative but I suggest reading all of it for a better context.

Thank you, I read through the article and need to do so several more times to grasp it better. A question that popped into my mind when reading this is about "steady state" vs "transient". It was my understanding that an impulse response shows the "transient" behavior of a system even though based on steady state measurements but after reading the article I think one of the points was that is not correct and an impulse response is just another view of a steady state measurement and not necessarily representative of the transient behavior of the system. Is that correct?

 

[levimax]

The design is to maximise low end in a small package and then cross to main speakers at a typical (say 60-100Hz). It's basically the design of every single svs sub ever (albeit perhaps not in a small package for some of them) and they are generally successful at achieving that goal. Comparing that to a DIY sub which is likely a raw driver in a box with either no dsp or minimal dsp (as opposed to custom dsp designed to maximise output and compress when overloaded without letting out the magic smoke) is going to produce different IRs.

Personally, having owned and tuned all of those sorts of subs over the years, am sceptical such differences matter much (whereas differences in extension/raw output level are plainly audible).

I agree SVS is looking at the fact that people are close to deaf at LF (80 dB threshold of hearing @ 20 Hz) so 100 ms GD and 40% THD @ 20 Hz are not really audible and they design their subs to be as small as possible (size is the biggest barrier to sales) that can meet these distortion thresholds with as high a SPL as possible when crossed at normal consumer points. As you say this is good engineering and they execute well.

Since this is a hobby is this approach really "Hi-Fi"? People seem to be concerned about SINAD over 100 for their electronics but subs with a SINAD of 8 are fine. It seems like a disconnect to me but if viewed strictly from an audibility perspective maybe SINAD of 8 is all you need for a sub?

 

[3ll3d00d]

Since this is a hobby is this approach really "Hi-Fi"? People seem to be concerned about SINAD over 100 for their electronics but subs with a SINAD of 8 are fine. It seems like a disconnect to me but if viewed strictly from an audibility perspective maybe SINAD of 8 is all you need for a sub?

How did you get to 8?

Regardless of the number clearly it's not an end game setup in absolute terms but you surely need significantly larger subs to achieve the same via native response and that is often physically impossible or impractical/unacceptable in a given room and/or prohibitively expensive so it may well be the highest fidelity you can achieve

 

[audiofooled]

I think one of the points was that is not correct and an impulse response is just another view of a steady state measurement and not necessarily representative of the transient behavior of the system. Is that correct?

It's also derived from steady state and sinusoidal input signals, caveat being that impulse response will be biased towards high frequencies. What we have in rooms is waves, so transients are a bit more complex but can still be calculated. Maybe this would help in distinguishing signals and waves:

Which way is up? (Which way does a loudspeaker driver move?)

I would beg to differ. A polarity flip can be readily identified in a ABX in a split second with many recordings when they contain kick-drums and/or thick bass-lines and when these signals are asymmetric around the zero line, which they often are. We don't need a study for that as it is one of...

www.audiosciencereview.com

 

[levimax]

How did you get to 8?

Regardless of the number clearly it's not an end game setup in absolute terms but you surely need significantly larger subs to achieve the same via native response and that is often physically impossible or impractical/unacceptable in a given room and/or prohibitively expensive so it may well be the highest fidelity you can achieve

8 is 40% THD at 20 Hz.... looking again at the Nuyes test the SVS SB3000 had 35% THD at 20 Hz at 95 dB so the be fair SINAD of 9 is more like it. I am not convinced the push to advertise the lowest extension and highest SPL with the smallest size is the optimum for Hi-Fi. At some point smoothly rolled off lows can sound better than highly distorted lows.

 

[bmc0]

That looks better, what allpass filters are you using? I have a PC based system so Taps are not an issue.

I generated those plots a while ago (for a rather tedious argument about minimum phase and absolute polarity) and didn't record the exact parameters. I remembered it being fairly simple though, so I derived parameters again: two 2nd-order allpass filters with a center frequency of 19Hz and a Q of 0.8 fits the excess phase pretty well. Applying the time-reversed allpass cascade to your measurements results in this—
Phase:

Impulse:

Step:

Gray dotted is the original measurement, magenta is the measurement with the FIR filter applied, and green is the calculated minimum phase version of the original measurement.
I've attached correction filters for 44.1kHz and 48kHz sample rates (filters for other rates can be generated by resampling, if needed). The filter latencies are 7524 and 8189 samples, respectively (~170.6ms), so hopefully you don't require low latency .

 

[gnarly]

It was my understanding that an impulse response shows the "transient" behavior of a system even though based on steady state measurements but after reading the article I think one of the points was that is not correct and an impulse response is just another view of a steady state measurement and not necessarily representative of the transient behavior of the system. Is that correct?

I think you've made an excellent realization.
I'm no bona-fide expert on Fourier analysis, but I understand it well enough to know it assumes/requires a steady state.
Whether we are looking at a transfer function or the impulse response, both are a snapshot in time, that assume a steady state.

Here's a really interesting experiment from the author F.Monteiro, of Crosslite+ an advanced measurement & processing simulation program.

The red and blue sine waves are the sides of a LR12 dB/oct crossover at 1kHz.
Normally, the two sides of a LR12 are polarity reversed, which gives a full cancellation at 1kHz. (for a 1kHz crossover)

In this example, the bold black squiggly line is the two sides 1kHz summation, while both of the sides are being amplitude modulated by a 100Hz wave.
The idea is that the 100Hz modulation is a simple form of introducing transient behavior into a summation measurement.

If the bold black like were straight flat across the example, modulation would have no effect. With the two sides showing full cancellation over the entire 100Hz period.
But note how the bold black only approaches being fully flat when the sines are cresting.

Any deviation from a flat zero summed response in the bold black line, is response bleeding past the full cancellation. Fourier analysis leads us to think such doesn't happen, again assuming steady state.
Hey, is music steady state or transient modulated?? ... i say hmmmm, as always I got some more thinking to do

 

[NTK]

I think you've made an excellent realization.
I'm no bona-fide expert on Fourier analysis, but I understand it well enough to know it assumes/requires a steady state.
Whether we are looking at a transfer function or the impulse response, both are a snapshot in time, that assume a steady state.

Here's a really interesting experiment from the author F.Monteiro, of Crosslite+ an advanced measurement & processing simulation program.
View attachment 496834


The red and blue sine waves are the sides of a LR12 dB/oct crossover at 1kHz.
Normally, the two sides of a LR12 are polarity reversed, which gives a full cancellation at 1kHz. (for a 1kHz crossover)

In this example, the bold black squiggly line is the two sides 1kHz summation, while both of the sides are being amplitude modulated by a 100Hz wave.
The idea is that the 100Hz modulation is a simple form of introducing transient behavior into a summation measurement.

If the bold black like were straight flat across the example, modulation would have no effect. With the two sides showing full cancellation over the entire 100Hz period.
But note how the bold black only approaches being fully flat when the sines are cresting.

Any deviation from a flat zero summed response in the bold black line, is response bleeding past the full cancellation. Fourier analysis leads us to think such doesn't happen, again assuming steady state.
Hey, is music steady state or transient modulated?? ... i say hmmmm, as always I got some more thinking to do

Below are the frequency responses of a LR2 (2nd order Linkwitz-Riley) high pass filter, a LR 2 low pass filter, both with 1 kHz cutoff, and the summation response of these 2 filters. There is a deep null at exactly 1 kHz, as per theory.

However, the spectrum of the amplitude modulated signal shows it has 2 frequency components, F_signal - F_AM, and F_signal + F_AM, or 900 Hz and 1100 Hz. These 2 frequency components therefore do not get perfectly nulled by the two summed filters. That's why there is residual signal after the filter. Fourier works just fine.
[ sin(f1 t)×sin(f2 t) = ( cos((f1−f2)t) - cos((f1+f2)t) ) / 2 ]

Attached is the Python Jupyter notebook to generate these plots.

[Edit] Updated the Jupyter notebook to add time domain simulations and replicated the plots in Gnarly's post #33. Also corrected where frequencies should be ω (angular frequency, in rad/s) and when frequencies should be f (ordinary frequency, in Hz).

 

[bmc0]

However, the spectrum of the amplitude modulated signal shows it has 2 frequency components, F_signal - F_AM, and F_signal + F_AM, or 900 Hz and 1100 Hz. These 2 frequency components therefore do not get perfectly nulled by the two summed filters.

And the reason it looks like it cancels cyclically is because of the cyclical constructive/destructive interference ("beating") of the two components. The beat is at twice the frequency of the modulating signal, exactly as expected from the two frequency components (1100Hz-900Hz=200Hz).

 

[3ll3d00d]

8 is 40% THD at 20 Hz.... looking again at the Nuyes test the SVS SB3000 had 35% THD at 20 Hz at 95 dB so the be fair SINAD of 9 is more like it. I am not convinced the push to advertise the lowest extension and highest SPL with the smallest size is the optimum for Hi-Fi. At some point smoothly rolled off lows can sound better than highly distorted lows.

it's true but it's also true that a sub that plays 20Hz, even with a decent chunk of distortion, at a decent level will *feel* materially different to one that doesn't (emphasis on feel). Whether you're better off moving towards tactile devices (there's a rich vein of DIY devices here that go well into massive overkill territory similar to how DIY subs tend to go) in combination with an array of smaller subs is another Q.

 

[levimax]

it's true but it's also true that a sub that plays 20Hz, even with a decent chunk of distortion, at a decent level will *feel* materially different to one that doesn't (emphasis on feel). Whether you're better off moving towards tactile devices (there's a rich vein of DIY devices here that go well into massive overkill territory similar to how DIY subs tend to go) in combination with an array of smaller subs is another Q.

Hobbies are all about massive overkill. If you have a massive overkill sub you don't need "tactile devices" . I have had better luck with one very capable sub as compared to an array of smaller subs. If 110 dB @ 20 Hz was your objective you could use one "big" sub or 3 smaller subs (Assuming 95 dB @ 35 % distortion). The difference is the "big" sub would have THD <10% instead of the 3 smaller subs playing with 35% distortion. Due to Fletcher-Munson with 35% distortion you will hear the 40 Hz distortion as being louder than the 20 Hz fundamental which is not Hi-Fi and can be audible. While the small sub array in theory can help "smooth out" the LF response, with the difficulty of LF in room measurements, it is not clear to me exactly what is being measured and how audible this "smoothness" really is. One good sub with a little DSP to knock down the mode peaks will usually sound better to me than a array of less capable subs even if the "graphs" show the array to be "smoother". YMMV. If it wasn't for the difficulty of reproducing LF in small rooms there would not be much left for hobbiest to chase

 

[gnarly]

However, the spectrum of the amplitude modulated signal shows it has 2 frequency components, F_signal - F_AM, and F_signal + F_AM, or 900 Hz and 1100 Hz. These 2 frequency components therefore do not get perfectly nulled by the two summed filters. That's why there is residual signal after the filter. Fourier works just fine.

Thx for showing what exactly is going on with the modulation producing the sidebands.
And please know I did not mean to question the validity of Fourier analysis.

What I do question though, is how fully representative a slice in time measuring, ie steady state, fully characterizes how our loudspeaker respond to a continually changing stimulus.

I guess my question to you and bmc0 and others more advanced in the math, is how much amplitude modulation occurs with music as it changes from instant to instant?
Is it at all significant?

 

[gnarly]

And the reason it looks like it cancels cyclically

I think it actually does cancel cyclically. The fellow who presented the slide encouraged us to generate the test signals into a mixer, and measure for ourselves.
Haven't done it yet, but curiosity is rising...

 

[3ll3d00d]

If you have a massive overkill sub you don't need "tactile devices"

an appropriately designed tactile device can deliver massively more tactile impact than massive overkill subs

The difference is the "big" sub would have THD <10% instead of the 3 smaller subs playing with 35% distortion

that's just comparing a bigger sub to a smaller one, The right comparison here is having sufficient smaller subs (or a heterogeneous set of subs) so that they have equivalent output to 1 bigger one (which may in itself have benefits such as not needing 1 massive amp, sharing the load over larger motors in total thus having reduced thermal compression etc).

with 35% distortion you will hear the 40 Hz distortion as being louder than the 20 Hz fundamental which is not Hi-Fi and can be audible

this is certainly true (though, for film soundtracks and in the absence of beq, can actually be quite enjoyable in itself even if it's not accurate)