Magnitude + Phase ... Bodeplots

Part1:  Bode plots.

Some 150 years ago a couple of brilliant guys like Fourier, Hertz and Helmholtz proved that every periodic signal (like music) can be seen as a combination of sine waves:

So, when we do understand the behaviour of sine waves in our system we do understand our system.
Two parameters of sine waves are quite obvious, amplitude for loudness and wavelength for pitch.
Taken in to account that both these are perceived in a logarithmic manner we often use magnitude in dB's and frequency in Hz for these two parameters.

Ok, so far everybody with only a wee bit of audio knowledge will know this and work with these two concepts on a daily base.

Now for a more tricky concept.
Phase:

Actually music consisting of only 1 sine wave at the time can be a bit boring ;)
So normally when we would decompose our music into sine waves we would see really a lot of them.
At this point we also have to include the way they are related in time, in the analytic description of our systems.
Phase expresses this in a manner of how these frequencies are related to each other, rather then absolutely, in time.
Certainly we could express this relation in much more intuitive units like seconds (or ms). (actually we sometimes do and call this groupdelay)
But 1ms of time shift on wave of 10kHz has a completely different impact (in terms of phase) than the same time shift on a wave of 100 Hz.
So expressing this time difference in phase degrees will work better in our case.


Now we all know these pictures and the concept of describing a system in terms of frequency response:

More highs, more lows and for the more literate we also have mids... ;)
By now we should know that this is only half the story.
If we do a frequency analyses like the above (also called a bode plot) we should  also include the phase behaviour.


So for correctly describing how system X responses to a input one should see this kind of pictures:

Don't think this will be a nice sounding loudspeaker system, but hey..

Time (delay) and phase are intertwined and there is really a lot of confusion out there.
Let's agree to some conventions:

• For now lets define phase difference as every time related difference between the sine components of our signals where one of them will be set to zero-time or 0' degrees shift. (usually the highest frequency).

• Furthermore: if a group of these sine components are delayed by a equal amount of time I will not regard this as a phase difference! (for example in a two way system in which both sources have a physical distance)

next part2: FFT analyses

Dual FFT analyzers..how to use SMAART

part 2: Getting to know dual FFT analyzers


Now we know that we need both magnitude and phase plots to describe the behaviour of our system under test.
So wouldn't it be nice to have some tool to measure these?
For some 20 years now our PC's are powerful enough to do the multiple, fast calculations we need to make use of mathematics developed by Jean Baptiste Fourier.
This clever guy proved that (within certain restrictions we will talk about later) you can also describe the behaviour of your system by looking at how it responses to a unity impulse (Dirac).
So we actually have two methods for analyses: one in the frequency domain (that's your bode plots), one in the time domain (that will be this impulse response).
And the mathematical system to go between these two domains is hence known as FastFourierTransform (FFT and iFFT for the reverse).

Enough of this math stuff which I also hardly understand, why dual FFT?

It's perfectly possible to do build a RTA (real time analyser, you know, these dancing light bars you find on every crappy audio device nowadays) with a single FFT.

I remember (back in the '70's!) staring at the dancing lights of that precious Klark Teknik DN60 while trying to do system tuning and thinking f... that, I can hear the EQ changes I apply before I see them on that thing.
Sorry friends no serious audio use for these toys!

Now for the dual FFT
Remember phase being a time related phenomena?
So if you want time information you need a reference point!
We will feed our dual FFT's machines with a reference signal as well as the measurement signal.
By mathematics (or magic if you are more inclined to the esoteric) the software will give you pictures like this:

This as a magnitude (above) and phase (lower) plot measured with Easera Systune.
But I am not biased, you may also prefer to use Smaart or Wavecapture or even Meyer SIM3 if you really want to fork out money. (I have been using all of these, I even used to own the very last working copy of MACFoH r.i.p. ;) )
There are also a zillion shareware utils out there ARTA or Holm being the more serious candidates. Linux people (like me) are not very lucky though..(2021 edit: got both Smaart and Systune to work under wine..)

Dual FFT performs a 'substrative' measurement, I will not go into the depth's of measurement techniques now but do realize that your measurement is relative to your reference signal.

Which can be anything!

So please stop the white noise terror: use your favourite music and everybody will be more happy.
Keep an eye on the coherence trace (the blue line in the picture above) as long as there is enough spectral content in your music source you're just fine.

Now how to set up a measurement?

I have seen people in (club) venues putting their measurement mic at some 20 meters away from a full blasting PA system (in stereo) pumping white noise into a reverberant room. What do you expect to measure?

Now surely, it is possible to exclude some reflections from your measurement by carefully trimming your measurement window.
But if you really want to make some scientifically viable measurements you will need to restrict the variables.
Think!
What's the wavelength of the lowest frequency you are investigating, how close is the closest reflecting wall/floor/ceiling, what part of the sound system am I investigating etc etc.

OK, your just set up fine now.
Let's assume you are trying to tune a loudspeaker X-over (hell, this was this tutorial about..)
At this instance I am developing filters for this ancient cabinet (Axys T89)

This is a full range measurement (with Smaart for a change) from the 12' horn loaded part of this cabinet.

For the moment I have left the measurement compensation delay snapped to the maximum of the IR (here you can read more about this)

This is the same measurement (of course I adjusted the gain setting) a bit further away. See the difference?


So practice, practice, practice in measuring as long as is needed to acquire pictures which will help you in doing:

next part 3: X-over adjustment

Measuring and adjusting a X-over

part 3 : actual X-over setup


Now we did acquire some dual FFT measurement tool and we did get some practice in measuring.
So it's time to dive into actual system tuning.

First we have to establish some design goals:

I think there will be no doubt that, at first instance, we will go for a flat magnitude response.
Sure, certainly in live audio there is a big preference for sloped responses.
Be it a smiley curve or more seriously a sloped curve towards the sub end. (upwards that is ;)
But imho this is part of a sound engineers 'artistic' input in a same way as EQ-ing a system to his/her preference is.
Other people (with whom I don't always agree :) have called this toning (as opposed to tuning) a system.

A bit more complicated design goal will be to try to achieve an even controlled response off axis as well.
Why?
Not only will there be (paying?) people listening off axis; anywhere you listen to music, in enclosed spaces, there will be acoustics.
So also sound radiated off axis will (at a certain point) reach your (on axis being) ears.

One way to achieve this is by using  the well known Linkwitz-Riley X-over tuning.
Acoustically that is!
So if we would be applying a 24dB LR x-over in the electrical domain doesn't this necessarily give us an acoustic LR behaviour!

A simple trick to see what a desired 24 dB filter would look like in your measurement.
Just connect the output of your X-over to the measurement input, select a desired X-over trace and presto:

Here you have it, 1600 Hz /24 dB X-over LR style as a target trace.
Don't get confused by the out of band nonsense, no coherence there so no viable measurement data. (actually should have faded this, now where's that button in Smaart, hmm)

Now let see if we can take that measurement from that previous 12' horn loaded cabinet and see if we can give it a LR alignment.

Seems reasonable, hein?
Now look closely and you will notice in the bottom corner the soundweb setting:
2800Hz / 12 dB bessel!
Didn't expect that to work out for an acoustic filter of 1600Hz / 24dB?

We can now continu to see if the 1' driver horn will combine in the same way.
Remember I am measuring everything with music so at the same time I am listening if it sounds anywhere near ok.

Again if you try to read the small figures:
2300Hz / 12 dB Bessel!
From the days I still had to do these things by ear I have a real preference for Bessel filters, 25 years later this finally seems to make sense  ;)
(actually I am going to set up this cab with 18 db Bessels but these are not implemented in SW9088)

Final picture?

Microphone is now further away so this is why things start to get more bumpie again.
(for educational purposes unfortunately also in the X-region, no, no, this not a x-over artefact ;)


Are we finished now?
No, but this is a start. We can now start to compare this with different solutions.

Am I anywhere cheating here?
No, but I' am not telling you the whole story (see, this is my business, remember ;)
There is a lot more to be said about adjusting delay, applying EQ at the right places etc etc

You can read in between the lines though...

Valid dual FFT measurement

On several instances last week I have been talking to people how to acquire valid  dual FFT measurements.
It really stroke my how much miss understanding there still floats about, so here we go:
Crash course measurements in KBLsystems-style:

First::
Dual FFT means we can finally get rid of that annoying pink/white noise at terrorising levels.
Just use your favorite music!
As long as it contains enough spectral content it will be a perfect source and you can use both eye and ear in your system tuning.
Keep a keen eye on your coherence tracer to see if your program material works.

Second:
Surely you can use the 'delay finder' function to find the overall time difference between the reference signal and the measurement signal, but be aware that all software will make this snap to the maximum of the impulse response which can be
a) quite challenging when the IR is smeared over time e.a. in a filtered sub respons
b)  will yield the 'wrong' phase response

Let me elaborate the latter:

 

This represents a close mic measurement of mid speaker. Clearly one can see the low end magnitude roll-off and it's associated phase behavior of a small speaker in a closed cabinet.

In the following picture the above graph shows  the IR with the delay finder nicely snapped to the maximum of the IR:

We will now show you why this is wrong.

The high end roll-off in magnitude which is (partly) due to the inductance of the voice coil should also find its counterpart in phase behavior. But in above graph one is to believed that this would not be the case.

Next picture is a simulation I made:

 

The black ine is the imported measurement and the red line is a constructed model based on this measurement.

The thin red line show the associated (min.) phase behavior.

So the measured phase should look like this. In fact the thin black line all ready shows the 'correct' trend.

Now how do we acquire this?

Simple! 

Adjust the delay finder by hand to the ONSET of the impulse response:

Tada!

Valid measurement!

Now bust your heads on what I 'm actually doing here <grin> ;)